Vibrations of Spinning Disks

A MODELLING AND VIBRATION ANALYSIS OF SPINNING DISKS

Natalie Baddour

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mechanical and Industrial Engineering University of TorontO

Copyright @ 2001 by Natalie Baddour

1+1

,calada

National Li'braiy

Bibliothèque nationale du Canada

Acquisitions and Bibliagraphic Senrices

Acquisitions et senricas bibliographiques

395welungm Street

395, nrs Wellington Ottawa QN K1A ON4 Canada

OlEawaON K l A W canada

The author has granted a nonexclpsive licence dowing the Natiod himy of Canada to reproduce, loan, disûiiute or sell copies of this thesis in microfonn, paper or electronic formats.

L'auteur a accordé une licence non exc1usive permettant à la Biblioth6que nationale du Canada de reproduire, prêter, distribuer ou vendre des copies de cette thèse sous la fonne de micmfiche/nlm, de reproduction sur papier ou sur format électronique.

The author retains ownership of the copyright in this thesis. Neither the thesis nor substantial extracts fiom it may be printed or otherwise reproduced &out the author's permission.

L'auteur conserve la propriété du droit d'auîeur qui protège cette thése.

Ni la thése ni des extraits substantiels de celle-ci ne doivent ê e imprimés ou autrement reproduits sans son autorisation.

Abstract A ModeLing and Vibration Analysis of Spinning Disks Natalie Baddour Doctor of Philosophy Graduate Department of Mechanical and Industrial Engineering University of Toronto 2001

This thesis considers a modelling and vibration analysis of thin spinning disks. Both in-plane and transverse vibrations are considered. Kirchhoff and von Karman theory of plates dong with Hamilton's principle are used to derive Iinear and nonlinear equations of motion for the vibrations of a spinning plate. In-plane and rotary inertias are naturally accounted for in this procedure. The resulting equations are similar to those previously derived in the literature but contain some additional terms. These linear and nonlinear equations are subsequently analyzed. New orthogonality properties of the in-plane modes are derived and are used to construct solutions to the linear, forced in-plane vibration problem. The linear transverse vibration equations are derived with new terms.

These new terms are explained physicaily and their effect on the frequencies of vibrations is analyzed. Nonlinearly coupled transverse and in-plane vibrations are considered while including the effect of in-plane inertia which has dways been neglected by ot her researchers. It is found that the inclusion of in-plane inertia introduces the possibility of intemal resonance between the in-plane and transverse modes of vibration.

Contents 1 Introduction 1.1

Background .

1

.................................

1.2 LiteratureReview .

t .3

1

..............................

4

.............................

-4

...........................

9

1.2.1

Linear Theory

1.2.2

Noniinear Sheory

Summary of Existing Models .

........................

10

1.3.1 The linear membrane mode1 . . . . . . . . . . . . . . . . . . . . .

20

1.3.2 The linear membrane with bending stiffness mode1 . . . . . . . .

11

1.3.3

The nonlinear mode1 . . . . . . . . . . . . . . . . . . . . . . . . .

1.3

1.3.4

Drawbacks of Existing Models and Proposed New hIodel . . . . .

14

.............................

16

1.4 Outline of the Thesis

2 Modelling

19

2.1 Strain Energy and Kinetic Energy . . . . . . . . . . . . . . . . . . . . . .

20

2.1.1

StrainEnergy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.1.2

KineticEnergy . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

2.2 EquationsofMotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

09

Variation of the Lagrangian . . . . . . . . . . . . . . . . . . . . .

30

2.2.1

2.3

2.4

2.2.2

Nonlineax Equations of Motion . . . . . . . . . . . . . . . . . . .

32

2.2.3

Linear Equations of Motion Derived Rom Linear Strain . . . . .

33

2.2.4

Linear Equations of Motion Derived From Nonlinear Strain . . . .

34

Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

2.3.1

Linear Transverse Vibrations . . . . . . . . . . . . . . . . . . . . .

38

2.3.2

Linear In-plane Vibrations . . . . . . . . . . . . . . . . . . . . . .

39

2.3.3

Nonlinear Boundary Conditions . . . . . . . . . . . . . . . . . . .

40

Discussion and Compazison Between the New and Existing Models . . .

42

2.4.1

Nonlinear Equat ions . . . . . . . . . . . . . . . . . . . . . . . . .

42

2.4.2

Lagragian vs Eulerian Variables . . . . . . . . . . . . . . . . . .

43

2.4.3

Decoupled In-plane and Transverse Plate Problems . . . . . . . .

-1-1

2-44 The Membrane Problem . . . . . . . . . . . . . . . . . . . . . . .

45

Presence of Additionai Terms . . . . . . . . . . . . . . . . . . . .

46

2.4.6

3 Linear Transverse Vibrations 3.1 Introduction

..................................

3.2 Equations of Motion

50 O;

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3 Frequency .4n alysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

3.3.1

Solid Plate with Free Boundary . . . . . . . . . . . . . . . . . . .

57

3.3.2

Frequency Equation . . . . . . . . . . . . . . . . . . . . . . . . .

59

3.3.3

Mode Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

3.3.4

Nodal Circles

.............................

3.4

The Frequency Equation for pl . . . . . . . . . . . . . . . . . . . . . . .

3.5

Numerical Simulations

3.6

............................

3.5.1

Frequencies of Vibration . . . . . . . . . . . . . . . . . . . . . . .

3.5.2

Observations from Simulations . . . . . . . . . . . . . . . . . . . .

Discussion

3.6.1

...................................

Validity of the Linear Strain Assumption . . . . . . . . . . . . . .

3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Linear In-plane Vibrations

87

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

$7

4.2

Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . .

88

4.3

Free Vibration Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

4.4

Orthogonality Properties . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

4.4.1

Derivation of Orthogonality Conditions . . . . . . . . . . . . . . .

93

4.4.2

Eigenfunction Expansions . . . . . . . . . . . . . . . . . . . . . . L O I

4.5

Forced Vibration Analysis . . . . . . . . . . . . . . . . . . . . . . . . . .

103

4.6

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

108

4.6.1

Free Vibration Response Under Initial Conditions . . . . . . . . . 108

4.6.2

Forced Vibration Response under Special Forcing Conditions . . . 109

4.7 Simulations

..................................

5 Nonlinear Coupled Vibrations

114

121

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2'22 Simplification of Solution to a 2 DOF system via Galerkin's Method . . . 122

.........................

5.2.1

Strategy of Solution

5.2.2

Galerkin Method

5.2.3

Physical Interpretation of the Coefficients . . . . . . . . . . . . . 139

122

. . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.2.4 Simpl@ing the Coefficients . . . . . . . . . . . . . . . . . . . . . 136 .4nalytical Solutions of the Simplified 2

5.3.L

DOF System . . . . . . . . . . .

144

Canonical Perturbation Solution . . . . . . . . . . . . . . . . . . . 1.53

5.3.2 Tirne-Averaged Harniltonian . . . . . . . . . . . . . . . . . . . . .

156

Exact Solution of the 1 DOF mode1 . . . . . . . . . . . . . . . . . . . . .

166

Energv Considerations of the 1 and 2 DOF models . . . . . . . . . . . . 168

............................

1 72

5.6.1 Resonance Case . . . . . . . . . . . . . . . . . . . . . . . . . . . .

173

,\T urnerical Simulations

5.6.2

Non-Resonance Case . . . . . . . . . . . . . . . . . . . . . . . . . 178

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

189

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

194

6 Summary and Conclusions 6.1 Summary

195

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

6.2 Conclusions

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

6.2.1

New Terms in the Linear Equations of Motion . . . . . . . . . . .

6.2.2

EEect of Rotary Inertia . . . . . . . . . . . . . . . . . . . . . . . . 198

6.2.3

Assumption of Linear Strains . . . . . . . . . . . . . . . . . . . . 199

198

6.2.4

Overd Effect of Spin on Linear Vibration Frequencies . . . . . . 200

6.2.5

Couphg Between In-plane and Transverse Modes . . . . . . . . . 201

6.2.6

Effect of Inclusion of In-plane Inertia . . .

6.3 Future Work. . .

...... . ......

202

..... ...... .. ..... ....... . . .. ..

204

A Plate Theory A.1 Kirchhoff Theory (Linear Theory)

206

........ . . ... . .. ... . . .

206

-4.2 Von Karman Theory (Nonlinear Theory) . . . . . . . . . . . . . . . . . . 207

B Derivation of Nonlinear Strain-Displacement Expressions

209

C Some Usefid Calculus Results

214

D Self-Adjointness of the Operator L,

216

E Orthogonality Properties of the Eigenfunctions of the LSM

219

F Properties of the Eigenfunctions of the Linear Spinning Membrane

224

Bibfiography

227

List of Tables 3.1 Solutions of the frequency equation br a stationary disk without rotary inertia, ka=O. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8-1

3.2 Solutions of the Frequency equation for a stationary disk with rotary inertia, ka=&

................................,..

5.3

3.3 Solutions of the Frequency equation for a rotating disk withoiit rotary inertia. ka=1.65.

...............................

83

3.4 Solutions of the fkequency equation for a rotating disk without rotary inertia, ka=3.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

3.5 Solutions of the frequency equation for a rotating disk with rotary inertia.

ka=3.0.

....................................

86

3.6 Solutions of the fiequency equation for a rotating disk without rotary inertia, ka=0.036 . . . . . . . - . . . . . . . . . . . . . . . . . . . . . . .

86

List of Figures 3.1 Graphs of Pa vs fi for Cases 1 and 2 . . . . . . . . . . . . . . . . . . . .

3.2

Graphs of Pa vs Pb for Cases 1 and 3 . . . . . . . . . . . . . . . . . . . .

3.3 Graphs of Pa vs Pb for Cases l and 4 . . . . . . . . . . . . . . . . . . . .

Pa vs Pb for Cases 2 and 4 . . . . . . . . . . . . . . . . . . . .

3.4

Graphs of

4.1

First Dimensionless Natural Frequencies po 1 and Backward Frequencies Po1

4.2

' , -

vs dimensionless angular velocity

a for u = 114. . . . . . . . .

Second Dimensionless Natural Frequencies p02 and Backward Frequencies PM

-

&

YS

dimensionless angular velocity a for v = 114.

. . . . . . . . LIS

4.3 Third Dimensionless Natural Frequencies po3 and Backward Frequencies Po3

4.4

-

vs dimensionless angular velocity o! for v = 114.

........

First Dimensionless Xat u r d Frequencies po 1 and Backward Frequencies Po1 -

4.5

'

' ,

vs dimensionless anpuiar velocity d for v = 112.

. . . . . . . .

Second Dimensionless Natural F'requencies po2 and Backward Frequencies po2 -

& vs dimensionless angular velocity a for v = 112.

........

4.6 Third Dimensiodes Naturd Frequencies po3 and Backward Frequencies po3

- & vs dimensiodess angular velocity a for v = 112. . . . . . . . .

The numerical solution of

T

for the 1 DOF and 2 DOF models. . . . . . .

The nunerical solution of c for the 1 DOF and 2 DOF models. . . . . . .

Comparison of r with numerical solution. k t order canonical perturbation solution and the-averaged canonical perturbation solution . . . . . .

Cornparison of c with numerical solution. first order canonical perturbation solution and tirne-averaged canonical perturbation solution . . . . . . . . .

Comparison of r with numerical solution. first and second order canonical perturbation solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Comparison of c with numerical solution . first and second order canonical perturbation solution .

.............................

The numerical solution of r for the 1 DOF and 2 DOF models. . . - . . .

The numerical solution of c for the 1 DOF and 2 DOF models . . . . . . .

Comparison of

T

with numerical solution. first order canonical perturba-

tion solution and time-averaged canonical perturbation solution . . . . . .

.

3.10 Comparison of c with numerical solution first order canonical perturbation

solution and tirne-averaged canonical perturbation solution . . . . . . . . . 180 5.11 Cornparison of r with numerical solution, first and second order canonical

perturbation solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

180

3.12 Compaxison of c with numerical solution. first and second order canonical perturbation solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

5.13 The numericd solution of r for the 1 DOF and 2 DOF models. . . . . . . 182 5.14 The numerical solution of c for the 1 DOF and 2 DOF models. . . . . . . 183

5.15 Comparison of

T

with numerical solution. first order canonical perturba-

tion solution and thne-averaged canonical perturbation solution. . . . . .

5.16 Cornparison of c with numerical solution. first order canonical perturbation solution and time-averaged canonical perturbation solution. . . . . . . . .

5-17Comparison of r with numencal solution. first and second order canonical perturbation solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

5.18 Comparison of c with numecical solution first and second order canonical

perturbation solution. . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.19 The numerical solution of T for the 1 DOF and 3 DOF models. . . . . . .

5.20 The numerical solution of c for the I DOF and 2 DOF models . . . . . . .

5.21 Compaxison of r with numericd solution. first order canonical perturba-

tion solution and time-averaged canonicai perturbation solution. . . . . .

5.22 Compa.rison of c with numerical solution.6rst order canonical perturbation

solution and time-averaged canonical perturbation solution . . . . . . . . .

5.23 The numerical solution of 7 for the 1 DOF and 2 DOF models . . . . . . .

5.24 The numerical solution of c for the 1 DOF and 2 DOF models. . . . . . .

5-25 Comparison of

T

with numerical solution. first order canonical perturba-

tion solution and time-averaged canonical perturbation solution . . . . . .

5.26 Cornparison of c with numerical solutionyfi& order canonical perturbation

solution and tirne-averaged canonical perturbation solution. . . . . . . . .

5.27 Cornparison of r with numencal solution. k t and second order canonical

penurbation solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.28 Cornparison of c with numerical solution, first and second order canonical perturbation solution. . . .

... ... .... ..... . .. . .. .. . ..

192

Nomenclature plate density Young's Modulus plate half-t hickness Poisson's ratio normal and shear stresses transverse displacement of middle surface displacement in r direction of middle surface displacement in 6 direction of middle surface bending rigidity angular velocity of plate strain tensor shear rnodulus displacernent in r direction of arbitrary point displacement in 没 direction of arbitrary point displacement in z direction of arbitrary point Kinetic Energy Potential Energy In-plane rnodeshape corresponding to u In-plane modeshape corresponding to

1:

Transverse modeshape correspondhg to w Ueq

in-plane equilibrium displacement of disk

xiv

natural frequency due to membrane portion natural frequency due to plate portion natural frequency plate radius natural frequency radius of nodal circle spin variable integer ; number of nodal diameters

Iinear rnatrix operator in-plane modeshapes time-dependent portion of in-plane vibrations. nonlinear 2 DOF niodrl time-dependent portion of transverse vibrations, nonlinear 2 DOF mode1 generalized momentum corresponding to c generalized momentum corresponding to r Xew coordinates to replace c, r,p,, p,

Coefficients of Lagrmgan in 2 DOF model Hamiltonian natural fiequency of c oscillator natural frequency of r oscillator coefficients of nonlinear terms in 2 DOF model

Chapter 1 Introduction 1.1

Background

Spiming disks can be found in many engineering applications. Commoo indusrrial appli-

cations include circular sawblades, turbine rotors. brake systems. fans. Rywheels. gtws. grinding wheeis, precision gyroscopes and cornputer storage devices. Spinning disks rnay experience severe vibrations which could lead to fatigue failure of the system. Thus. the

dynarnics of spinning disks has attracted much research interest over the years.

A disk is usually defined as a thin, Bat, circular plate. Hence. the analysis of spinning

disks involves the theory of thin plates. The first step in investigating the vibrations of spinning disks is to set up a suitable mathematical model of the systern. The aim ĂŽs to set up a model that captures the essential physin of the problem while remaining

tractable. ~I%sumptionsabout the system under investigation rnust be made at the stage of deveioping a mode1 for the system. One such assumption is to mode1 the spinning disk

as either a plate or a membrane that rotates. In this manner the vast literature on the theory of plates and membranes can be used.

Over the years, many different plate theories and models have been employed to rnodel various different effectç of interest in the andysis of stationary (non-rotat ing) plates.

However, in the realm of thin plates, two plate theories have established themselves

as basic tools in the literature. These are the Kirchhoff and von Karman models of a plate. These models use the same fundamental assumptions. The difference between them lies in the choice of strain-displacement relations. The Kirchhoff theory uses linear strain-displacement relations. This amounts to assurning infinitesimal displacements and results in a linear theory. The Von Karman theory diEers in that nonlinear transverse displacement terms are retained in the st rain-displacement equat ions. This removes the assumption of small transverse displacements, resulting in a nonlinear disk model.

Hence, to develop a mode1 for a spinning disk, the simplest approach is to choose the appropriate plate or membrane representation and then to incorporate the ~ffectof the rotation. It is thus no surprise that this is the approach that has been foilowed in the existing literature. However, the exact manner in which the effect of the spin is incorporated into the equations of motion and corresponding boundary conditions leads to different equations with different solutions and techniques required to obtain those

solutions.

Most of the investigations on spinning disks have focused on andysis using linear plate or membrane theory. .halysis of spinning disks using nonlinear theories has rarely been discussed in the Ăźterature. Linear theories can be adequately used to represent the dynamics of a thin plate only when the transverse deflections of the plate are much smaller than the thickness of the plate. For very thin plates, the transverse defiections can easily become larger than the thickness of the plate, rendering the linear plate theory

inadequate to correctly model the vibrations of the plate.

Nowadays, the trend in industry is towards thinner disks wit h lighter materials. Thus. the applicability of the current 端near analyas becomes limiced. In addition. the growing communications industry is increasingly relying on storage devices such as floppy disks

and CDROMS.Such storage devices are usudly thin and light and thus may have large deflections. The existing linear andysis cannot be properly used to investigate such systems. These aforementioned factors create a need for a theoretical analysis of the vibrations of spinning disks using nonlinear t heory.

Hamilton's Principle may be used to derive the equations of motion and the accompanying boundary conditions using linear and nonlinear plate theories. One of the advantages of using Hamilton's principle is that the correct boundary conditions are automatically obtained. This t heoretically rigorous approach results in differeot equat ions and boundary conditions than previously seen in the spinning disk literature.

In the linear case, these efforts axe well rewarded by the resulting equation being easier to solve in closed-fom. In fact, the solution of the resulting equation in the linear case involves the use of Bessel hnctions and Modified Bessel functions. While these special functions have long been used in the solution of stationary linear plate problems. they have not previously been used to solve spznnzng linear plate problems.

It is the goal of this work to develop linear and nonlinear models for a spinning disk employing both Kirchhoff and von Karman plate theories.

The 端near in-plane and

transverse vibrations as well as the nonlinear conpled vibrations will be anal-tedd.

CHAPTER1- INTRODUCTION

Literature Review

1.2

The review of the 端terature on the subject of spinning disks will proceed in this section as follows

Linear theory

- Free transverse vibrations - Forced transverse vibrations

- In-plane vibrations

1.2.1

Linear Theory

Free Transverse Vibrations Mary authors have investigated the vibrations of spinning disks using linear

theory.

The original papers are by Lamb and Southwell [l]and by Southwell [2]. Lamb and Southwell investigated the free transverse vibrations of a spinning solid disk iising linear theory. Three separate cases were considered in their a n a s i s .

In the k t case, the disk was so thin and the rotation so rapid that the flexural forces were taken to be negligible. The resulting partial differentid equation is thus second order. Separation of variables leads to a hypergeometric equation in the radial variable.

The series solution is known to be logaxithmically d i n i t e at the outer edge unless the series terminates. Forcing the series to terminate leads to the natural fiequencies of vibration. Similar to a stationary disk, a mode of vibration is characterised by nodal

circles and nodal diameters. The corresponding hequency depends on the number of nodal circles and diameters, the spin rate and the poisson ratio of the material.

In the second case, the influence of rotation was ignored. This reduced the problem to that of a stationary disk. The modes of vibration, nodal circles and natural frequencies for this problem are widely avaiiable in the literature.

F i n d y , in the third and most complicated case, the influence of flexural rigidity and the rotation-induced centrifugai force were both taken into account. The resul ting partial differentiai equation is fourth order. Separation of variables leads to a fourth-order ordinary differential equation in the radial variable. It is the solution of this equation that causes difficulties. In theory it can be solved with a series. however this approach quickly becomes unwieldy and deduction of general results becomes difficult. Lamb and Southwell deal Nith this difficulty in two ways. First. they illustrate how Raleigh's method can be applied to the problem. More importantly. they propose an approximation to calculate

the eigenvalues of a spinning disk by using the calculated membrane eigenvalues (their first case) and the stationary disk eigenvalues (their second case). This approximation is aiso a lower Iimit to the true eigenvalue. The subsequent article by Southwell continued the investigation for the case of a circular annulus with clamped inner radius and free outer radius.

As previously mentioned, the incorporation of both the bending stiffness of the disk and the effect of rotation leads to a fourth order PDE that is difficuIt to solve- As a resuit .

various researchers have applied different solution techniques to the linear anaiysis of the free transverse vibrations of spinning disks [3, 4, 5, 61. Barasch and Chen [3] reduce the

fouah order ODE in the radial variable to a set of four hst-order equations subject to

arbitrary initial conditions. They then numericdly integated the resulting system of equations and showed that the Lamb-SouthwelI approximate cdculation of the naturd Erequencies is just ified.

Mignolet, Eick and Harish [6] treated the normalized disk bending stiffness as a small parameter. This permitted them to use perturbation techniques to investigrte the frrr vibration characteristics. Natural frequencies and mode shapes were found from various perturbation formulations and then compared to the "exact" values obtained from power series solutions of the eigenvalue problem. They considered the cases of a full disk. an annulus and a very narrow annulus. They found that singular perturbation solutions match the exact values well for s m d l st誰f誰ness, hub radii and nodal circles. Regular perturbation solutions provided good accuracy for the large hub radii. This extensive paper is testament to the difficuity of solving the resulting fourth-order equation.

Eversman and Dodson [4] solved the probiem of a spinning disk with clamped inner edge by using a power series solution. Solutions of the resulting eigenvalue equation werP found numerically and agreed wit h Southwell's work.

Another popular approach in the literature is to modei the spinning disks as a pure membrane with no bending stifhess [Tt 8, 9, 101. This corresponds to the first case as analyzed by Lamb and Southwell. The advantage of this approach is that the resulting partial and ordinary differential equation is of second order and thus more tractable than the fourth order equation.

Forced Transverse Vibrations In addition to the study of free vibrations of spinning disks, considerable research effort has also been spent on the analysis of the forced vibrations where the disks were modeled using linear theory. Huang and Hsu [Il] considered the forced vibration due to a circular traveling line load. Yano and Kotera [12] investigated the vibrations of a rotating disk

with an elastic support at one point under the action of a static in-plane load. Benson and Bogy [13]analyzed the transverse deflection of a spinning disk due to spatially stationary loads. They first approached the problem from the context of membrane theory. However. this resulted in the eigenvaiue problem being singular. They showed that the effect of the bending stiffness of the disk must not be ignored in order to keep the eigenvalue problern from becorning singular. Thus, the effect of bending stiffness. no matter how small.

must dways be included in the analysis. With bending stiffness retained. they solvr the concentrated load problem by using a Fourier series in conjunction with a numerical solution for the radial modes. -4nalysisof a rotating disk wit h a coupled read/ write hrad.

as applicable to magnetic recording has also been performed [M.15. 161.

In-plane Vibrations

In addition to considering the transverse vibrations of a spinning disk. the problem of free planar vibrations has also been investigated using 端near theory [l7. 18. 191. In particulax, Bhuta and Jones [19]studied the problem of coupled symmetric radial and torsionai vibrations. Doby [20] investigated the in-plane vibration problem using a different formulation. Burdess, Wren and Fawcett [21]generalized the problem to the case of

asymmetric in-plane vibrations and discwed the properties of forwards and backwards

traveling chcumferential waves. In t hese investigations the disk was always assurned full.

Subsequently, Chen and Jhu [22] considered the in-plane vibrations of a spinning annular

disk.

Adams [23] showed that a d o m flexible elastic disk clamped at its inner radius and rotating at constant speed is unable to support an a r b i t r q spatially fixed transverse load at certain critical speeds. This investigation can be thought of as the reverse eigenvalue problem. Most invest igators approach the spinning disk problem by finding the natural frequencies of vibration given a particular &ed) speed of rotation. in t his manner. when applying a transverse Load to the disk. the frequencies to be avoided are known. Adams approached the same problern by affuming that it is desired to apply a transverse load of known ( k e d ) frequency to the disk. The speeds of rotation that satisfy the frequency equation were then solved for. These are the critical speeds of rotation that must bรง. avoided.

Honda, Matsuhisa and Sato [24] analyzed the vibration of a rotating disk subjected to base excitation. The steady-state response to harmonic excitation was formulatecl as an eigenfunction series where the eigenfunctions were given in the form of rotating

waves. Chen [25, 26, 271 performed stability analysis of a spinning disk under stationary concentrated and distributed edge loads both anaiytically and numerically. He also

analyzed the parametric resonance of the disk under the effect of space-tiued puisating edge loads as well as the effect of a space-Lxed fnction force [28,291. Similarb analysis of the stability of a spinning disk under rotating, arbitraily large damping forces aas performed by Huang and Mote [30].

1.2.2

Nonlinear Theory

A s m d body of work exists on the analysis ofspinning disks using nonlinear Von Karman plate theory. Nowinski [31] was apparently the Brst to tackle this problern . The equations were f o d a t e d to descnbe the nonlinear transverse vibrations of a spinning disk. The transverse deflection of the disk and a stress b c t i o n were set up as the principal unknown variables. The disk was assurned free and its deflection was represented wit h a two-term polynornial. The results specialized to the linear case were in close agreement with the classical results of Lamb and Southwell [Il. Renshaw and Mote [32] developed a perturbation solution to the equations set up by Nowinski. In addition. experiments were performed to measure the stability of rotating disks under transverse point loading.

The numerically computed stability of the perturbation solution agreed quantitatively

with the experimental measurements.

Nonlinear transverse vibrations in spinning disks have also been studied by Advani

and Bulkeley [33]. They obtained two exact solutions to the nonlinear equations governing the transverse motions of spinning membrane disks. The membrane \vas taken to be whole at its centre and free at its edge. They compared their results to that obtained from 端near theories. Since membrane theory was used, the applicability of the work is limited. Advani [34] also obtained an exact solution for nonlinear flexural. aspmetric waves in a spinning membrane. This solution was in the form of harmonic waves and corresponds to a membrane spinning with no nodal circles and two nodal diameters.

Again, the use of membrane theory limits the applicability of the result.

CRAPTER1. INTRODUCTION

1.3

Siimmmy of Existing Models

X few dinerent models have been used to analyze spinning disks. At first, a spinning

membrane model was used. Next, bending stiffness was incorporated into the spinning membrane model. Finaliy, a nonlinear mode1 based on von K m a n ' s plate equations was developed. Each of these models will be examined in this section and the drawbacks

of each model will be discussed.

1.3.1

The linear membrane model

The firรงt step towards developing the equation of motion for a spinning disk h a traditionally been to consider the disk as a membrane. That is, the bending stiffness of the disk is ignored. For the small transverse deflections of a membrane. it is aรงsumed that these defiections do not affect existing in-plane stresses. W e n the membrane deforms. the in-plane stresses have a component in the vertical direction and this is what provides the restoring force for the transverse deflections. The in-plane stresses are those t h a t develop in a Bat, undefomed membrane as a result of the rotation of the membrane.

To solve for the required in-plane stresses, the usual assumption employed in the literature has been to use the equilibrium stresses that correspond to a Freely spinning plate. If an elastic plate is rotated in its own plane and dlowed to corne to equilibrium. the

resulting stresses will be dong and perpendicular to the radius vector. For the calculation of the eqdibrium stresses of a rotating disk, it is typically assumed that the displacement of particles is in the radial direction only. Since an equilibriurn configuration is assumed. the in-ptane inertia of the diรงk is ignored.

The equation of motion governing the transverse vibrations of a Bat spinning membrane is denved by Simmonds [7]as

where w is the transverse displacement, p is the density of the plate material and . , a oee, or@are the normal and shearing stresses. The variables r, 0 and t denote the usual

radial, angular and temporal variables. Note that the normal and shearing stresses are in the plane of the defomed membrane and thus have a vertical component when the

membrane is deformed. The normal and shearing stresses are assumed to have been found from a prior cdculation.

The aforementioned approach is appropriate for the derivation of the equations of motion of a spinning membrane. It must be noted that inherent in t his derivat ion is the neglect of the in-plane vibrations since an equilibrium situation is assumed for the inplane stresses. Aso. the transverse vibration problem is ioevitably linkerl to the solution

of the in-plane problem since the latter problem must be solved first .

1.3.2

The linear membrane with bending stiffness mode1

Thus far, the equation of motion does not have a term representing the effect of the bending stifhess of the disk. To remedy this, a bending stiffness term, proportional to

V4w is added to the right-hand side of equation (1.1) as rnentioned in [Il:

Here, E is Young's rnodulus, u is Poisson's ratio, and h is the plate haKthickness. Note that the assumptions on the displacements that lead to the

V4w tenn and those used

to find the equilibrium stresses in the disk axe not quite the same since the equilibrium stresses assume a displacement in the radial direction ody.

The fourth-order partial dserential equation given by equatioo (1.2) has historically been difficult to solve exactly. Note that thus far, a membrane model and a membrane with bending stiffness model have been derived and employed. The spinning disk modelled purely as a spinning plate with no membrane effects is an approach that has not been previously considered in the literature.

Furthemore, the effect of in-plane inertia is completely ignored in previous models.

For a spinning disk, the centrifuga1 and Coriolis forces act in the plane of the undeformecl disk. Therefore. it would not seem reasonable to neglect the in-plane inertia of the disk for the spinning disk problem. When modelling stationary (non-rotating) plates using lioear (Kirchhoff) theory, the in-plane inertia is included. It then turns out that the inplane and transverse free vibration problems decouple and can be solved independentlu.

Thus. there seems to be little theoretical justification for neglecting the in-plane inertia of

a rotating plate. Furthermore, given the precedent set by the station-

plate problem.

it might be expected that when incorporating the effect of the in-plane inertia of the plate, the in-plane and transverse vibration problems should decouple for the spinning plate problem as weH.

1.3.3

The nonlinear mode1

The nonlinear spiMing disk equations were fvst derived by Nowinski using the von

Karman plate equations as a starting point [31]. The equations are

where D = ~ q ~ - " o is ) the bending rigidity of the plate. R is its angular velocity. w is the transverse deflection and 6 is a stress function.

In modelling the spinning disk using the nonlinear von Karman t heory. the in-plane inertia of the disk is also ignored. The justification given for this is that the in-plane inertia of the plate is usually ignored when modelling the stationary plate. It was shown

by Chu and Herrmann [35] chat the equations resulting from ignoring the in-plane inertia

of a stationary plate are the first-order perturbation approximations to the full problem.

In other words, for the statzonary problem, ignoring the in-plane inertia is a good a p proximation to first order. The advantage of ignoring the in-plane inertia is that a stress function may be used. This reduces the equations to be solved from three nonlinearly coupled partial differential equations to two, a considerable simplification.

The same perturbation a r p e n t used by Chu and Hermann [35] for the stationsdisk was attempted for the spinning disk. The results did not indicate that the in-plane

inertia could be ignored as for the stationary problem. For this reason, it appears that the in-plane inertia of the disk shouid be retained in modelling a spinning disk, for bot h linear and nonlinear formulations of the problem.

Furthermore, in both the nonlinear and linear theories of spinning disks. the rotary inertia of the disk has been ignored. Again, drawing on the known literature of thin stationary (non-rotat ing) plates. t his is a valid approximation for the lower Frequency vibrations of thin disks. For the high frequency vibrations of thin disks. the rotary inertia of the disk must be taken into account. However, it is also known that the effect of the spin is to raise the natural frequencies of vibration. Thus, it is possible that the spin raises the naturd Frequencies enough that t hey are in the region where rotary inertia rnust be taken into account. It thus remains to be proven whether the rotary inert ia of a thin spinnzng disk c m be neglected.

1.3.4 Drawbacks of Existing Models and Proposed New Mode1 hl1 the aforementioned models were derived using the Sewtonian approach. A drawback of using a Newtonian approach to derive the equation of motion is that care must be taken in formulating the proper boundary conditions. For spinning disk problems. the outer edge is typicaily assumed to be free. Mathematically. this is usudiy interpreted by requiring the bending moment and the KirchhoEshem to be zero at the free edge. Cse of Hamilton's principle to derive the equations of motion and the boundary conditions wilI con端rm this. Furthermore, ail &ing

models ignore both rotary inertia and in-plane

inertia of the disk. A spinning disk mode1 with pure bending as the restoring force cannot be found in the Iiterature.

It is propoçed to derive slightly different linear and nonlinear equations of motion of a spinning disk using the following assumptions : The disk is modelled purely as a classical Kirchhoff or von Karman plate.

Effects

of transverse shear are thus not included in the modelling. Details of the classical assumptions made by Kirchhoff to mode1 a thin plate can be found in Appendk h or in [4O].

Only thin disks d l be considered.

0

The in-plane inertia of the plate will not be ignored as no theoretical justification of this can be found.

0

The nomally-neglected effect of the rotary inertia of the disk rvhich arises naturally in this formulation wiU be retained.

0

Lagrangian instead of Eulerian coordinates will be used throughout.

Hamilton's principle wiil be used so that the cornplete boundary conditions may be obtained. The equations derived here will be shown to be slightly different from chose previously used in the Ăźterature in both the linear and nonlinear strain cases. In the linear strain case, it wi11 be shown that it is possible to solve this new equation analytically.

Note that for a thick plate, Kirchhoff's assumption no longer holds and thus the effect of transverse shear would have to be included in the modehng.

Note from the above assumptions that effects of shear are not going to be included in the modeiling assumptions. This is in keeping with the modelhg assumptions made

by Kirchhoff and von Karman. It should be noted that including the effects of rot-

inertia and in-plane inertia is adding a level of complexity above what is currently in the literature. Adding the effect of shear would be quite a substantial leap in complexity.

Without the effect of shear, if it is known what the rniddle surface of the plate is doing. then it is known what the rest of the plate is doing. In other words. neglecting shear serves to reduce a three-dimensional continuum mechanics problern into one of two dimensions.

Including rotary inertia and/or in-plane inertia does not change the fact that the problem is still one of two dimensions.

However, including the effect of shear woultl turn the

problem into one of three dimensions. considerably adding to the complexity of the problem. Thus, for this work it was decided to include the efTect of rotary inertia and in-plane inertia but to neglect the effect of shear.

1.4 Outline of the Thesis The current chapter serves to lay the foundation of the later work. The concept and uses of a spinning disk are introduced as well as models that have been previously considered in the literature. The assumptions upon which the rest of the work will be based are introduced. The thrust of this thesis is to derive new models for the vibrations of a spinning disk based on the aforementioned assumptions and more irnportantly to examine the implications and predictions of the nem and existing models. These models inclucie factors that have previously been neglected and the focus is to examine the coosequences

of their inclusion on the predictions for the vibrations.

Chapter two is dedicated to developing a new mode1 for spinning disks. In this chapter, the assumptions o u t h e d in the previous subsection are used to denve three

particular models for the transverse vibrations of a spinning disk. The first model is linear and is based on the assumption of linear (Kirchhoff) strains. The second model is also linear but is based on the assumption of nonlinear (von Karman) strains. The

third model is Mly nonlinear and results in t hree nonlinearly coupled partial differential equations based on the assumption of nonlinear (von Karman) strains. The corresponding boundary conditions to these equations of motion are also delived and discussed. The major ideas of Chapter two can also be found in this author's paper [36].

The focus of Chapter three is the analysis of the linear free transverse vibrations of the spinning disk based on the newiy developed models of the second chapter. Both iinear models arising Eiom the assumption of linear and nonlinear strains are considered.

The assumption of linear strains leads to a simple equation that greatly resembles that

of the vibrations of a stationary (non-rotating) disk. it is thus easy to solve this equation analytically. While this simplicity if a highly desirable feature, it is not clear whet her or not the assumption of linear strains is as applicable to rotating disks as it is to stat ionan disks. This is the prima.

question that this chapter seeks to address. To t his end. the

predictions on the frequencies of vibrations of both linear models are considered. The ideas in this chapter can aiso be found in this authoryspapers (37. 381.

Chapter four considers the linear in-plane vibrations of a spinning disk. The model considered in this chapter is not new and corresponds exactly with those currently used in the literature [19,21].The new contribution in this chapter lies primarily in the analysis of the orthogonality properties of the in-plane modes of vibration. These orthogonality properties are extremely useful in that they can be used to easily constnict a general solution to the forced vibration problem. Furthemore, they can also be used to make

simple predictions about the free vibration frequencies. While the predictions made about the fiee and forced vibration problern are not new, the rnanner at which they. are arriveci is new. Furthemore, knowledge of the properties of the in-plane modes of vibration sheds some light on the more complex problem nonlinear problem where the in-plane and transverse vibrations are not independent but are nonlinearly coupled. The ideas of Chapter four can also be found in this author's paper [39].

The primary focus of Chapter five is to determine the effect of the inclusion of in-plane inertia on the nonlinear vibration problem where the in-plane and transverse vibrations are nonlinearly coupted. It must be kept in mind that t his is the tocus of the analysis and not the actual solution of the full nonlinear PDEs. To this end, a simple approximation is proposed whereby only one mode of each of the in-plane and transverse vibrations are

kept. The problem where the in-plane inertia is ignored thus reduces to one nonlinear ordinary differential equation, a one degee-of-freedom (DOF)model. When the in-

plane inertia is kept, the corresponding description is two nonlinearly coupled ordinary differential equations and this is referred to as the two degee-of-freedom rnodel. The rest of the chapter thus focuses on comparing the predictions on the vibrations between the 1 DOF model and its 2 DOF counterpart. This cornpa.r誰son is done by both analytic

and numericd means.

The sixth chapter is the final chapter of the thesis. In this chapter. the conclusions of the preceding work are suxrunarized and presented.

Chapter 2

Modelling As mentioned in the first chapter, current linear and nonlinear models of spinning disks ignore certain aspects in their modelling asçumptions. In both the nonlinear and linear

theories, the rotary inertia of the disk was ignored. Drawing on the known literature of station-

(non-rotating) plates, this is a valid approximation For the Iower Frequcnry

vibrations of thin disks. Since it is also known that the effect of the spin is to raise the naturai frequencies of vibration, it rernains to be proven whether the rota-

inertia

of a thin spinning disk can be neglected. Current nonlinear models €or spinning tiisks ais0 ignore the presence of in-plane inertia. Furthemore. al1 the aforementioned models were derived using the Newtonian approach. A drawback of using a Newtonian approach to derive the equation of motion is that care must be taken in formulating the proper

boundary conditions. A spinning disk mode1 derived from first principles based on the plate theones of Kirchhoff and von Karman has not been found in the literature.

In this chapter, it is propoûed to derived a new mode1 for a spinning disk from first principles, based on the plate theories of Kirchhoff and von Karman. New linear and nonhear equations of motion of a spinning disk wiil be derived using the bllowing

assumptions : The disk is rnodelled purely as a classical Kirchhoff or von Karman plate.

0

Only thin disks will be considered.

Lagrangian inรงtead of Eulerian coordinates wiI1 be used throughout.

Hamilton's principle will be used so that the complete boundary conditions can be obtained. In-plane and rotary inertiaรง will atise naturaiIy as a coasequence of the above assumptions. It has generally been conventional to drop the in-plane and rotary inertias. however in this analysis they will both be retained. The equations derived here will be shown to be slightly Merent from those previously used in the literature in both the linear and noniinear strain cases.

2.1

Strain Energy and Kinetic Energy

In order to use Hamilton's principle, expressions for the kinetic energy. elastic strain energy and work done must be lormuiated. For continuous systems this is typically done

by considering a small element of volume and then integrating over the entire volume of the solid in question. In considering large displacernents, the shape of the entire volume

changes as a hnction of tirne. The solid Iooks diierent at different times. Hence. a r e

must be taken when integrating over the entire volume. The question arises as to whether the integration shodd be performed over the curent volume or over the initial volume.

Since the current volume is usually unknown, t his issue is best dealt wit h by referring

ail quantities to the initial volume and then performing the integration over the initial

volume of the solid. In other words, Lagrangian and not Eulerian coordinates must be

used.

Assuming that a strain energy exists is akin to making a fundamental assumption about the material and how it behaves. If the form for the strain energy of any material is known, then the stress-strain relationship for the material may be deduced. Conversel?. if the stress-strain relationship of the material is known then so is the Form of the strain energy

Let us fint consider the strain energy hnction. There is a nonzero contribution to the strain energy stored in a deformable body only if the body bends. stretches or otherwise deforms. If it rotates or translates like a ngid body without deforming then these displacements will not contribute to the strain energy of the body. In rneasuring the displacements of particles in the body, it is irnperative to measure them with respect to the translated and/or rotated rigid body motion. This is accomplished by k i n g a coordinate frame to the body. The translation and rotation of this body-fixed h m e describe the cigid body motions of the body. Displacements with respect to this frame then contribute to the strain energy stored in the body. Since the displacements are measured with respect to the undefonned (not unrotated) body the discussion for the derivation of the strain energy proceeds in the same rnanner as for non-rotating bodies.

To denve an expression for the strain energy in t e m s of displacements. expressions for the foiIowing are required:

An expression for the strain energy in terms of the stress and strain in the body.

0

An expression giving stress in t e m s of strain (stress-strain expression).

An expression giving the strain in terms of the displacements (strain-displacement

expression).

The above would yield energy expressions in terms of displacements of any point in the body. In thin plate theory, the displacements of an arbitrary point are further related to the displacements of the rniddle surface of the plate via Kirchhoff's hypothesis. The problem thus reduces to one of solving for the deflections of one particular surface only.

Since the ensuing deflections do not depend on any vertical component. this serves to turn a three-dimensional continuum mechanics problem into one of two dimensions. In summary, four relationships will be required to express the strain energy in ternis of tht. displacements of the rniddle surface as measured in the body-Lxed frame.

The strain energy per unit volume is denoted by I.Vo and is given by

where oi, and eij are the stress and strain tenson respective[. Small strains are assumed so that stress-strain relationship will be taken to be Hookean (linear). Note that t his implies that the strains are small, but does not imply that the displacements are small.

For a flat plate, a plane stress condition may be a s s k e d . This implies that (Tor

O,,

= a,= -

= 0-

The strain-displacement relation is required in order to express the strain e n e r p in terms of displacements. It is at this point in the development that nonlinearities may

be introduced. Since all quantities are to be referred to the undeformed body. it is

the Lag~angianform of the strain tensor that is requited. The full nonlinear straindisplacement relationships for large displacements are derived in polar coordinates in Appendix B. These relations are nonlinear and complicated. At this point, Kirchoff and Von Karman plate theories can be used to simplik these expressions. The basic assumptions of Kirchoff and Von Karman plate theories are summarized in Appendix A.

The von Karman plate theory can be shown [40]tu lead to the following nonlinear strain-displacement expressions

where u,. ue and ut are the displacements of the disk in the r. t9 and r directions. resptXc-

tively. For the linear Kirchhoff theory, the nonlinear terms involving u , are dropped from the above expressions, leading to the foliowing linear strain-displacement relat ionships

The last required expressions are those relating the displacements of an a r b i t r a - point in the plate to those of the middle surface of the plate. In thin plate theory. it is usually

assumed that the h e a r f3aments of the plate initidy perpendicdar to the middle surface

remain straight and perpendicular and do not contract or extend. Transverse shear effects are thus negiected.

This assumption leads to a relationship between the displacements

of an arbitraxy point uv, ue and u, and the displacements of the middle surface u. c and W. They

are given by

Finally, the strain energy of the entire plate can be obtained by integating over th^ entire volume of the plate. Note that the strain energy will be a function of u.

c

and

tr

and of the vertical coordinate z. Furthemore, u, *u and w are themselves functions of the in-plane coordinates (r,O ) and of time, t. Thus the strain energy is an explicit function of z. This dependence can be elirninated by explicitly carrying out the integration over the thickness of the plate from z = -h to

t

= h, where h is the distance between the

middle surface of the plate and the plate bounding surface.

This procedure 6nal.y yields the strain energy of the plate as an explicit function of u(r,8 , t ) , v(r,8, t) and w (r,O, t) only. The strain energy of the plate is t hus given by

where Ri and R2denote the b e r and outer rad端 of the disk. For nonhear von Karman

strains, the above expression is expanded as

where G is the shear modulus and X is a constant which are related to the Young's

moddus, E, and Poisson's ratio v of the materiai by

Linear Kirchhoff strains could also have been ernployed to derive the expression (2.10) for strain energy. Since the linear strains are a special case of the nonlinear strains. the strain energy expression corresponding to linear strains can be found from equation (2.10) by retaining only second-order terms and dropping al1 third and fourth-order terms.

2.1.2

Kinetic Energy

While the strain energy of a stationary and a rotating disk are the same, it is in formulating the kinetic energy expression that the difference between a rotating and stationary disk becomes apparent. Let us set up two coordinate systerns. S and B. Suppose that

S is an inertial frame of reference and that B is rigid-body-fixed to the disk. so that B rotates with the disk at a spin rate of R with respect to S. Let r, denote the undeformed location of a particle in the disk and let u denote the corresponding displacement vector. Hence, the location of a particle originally at r. is

-

given by r = ro + u at any given time. These vectors are chosen to be expressed in terms of unit vectors belonging to the B kame. Then, the velocity of any particle is given by dtr

where it must be remembered that since the unit vectors are ÂŁked in the rotating

frame, their time derivative rnust be bund as well.

Let e,.be unit vector in the r direction such that e, = cos(6)iB+sin(6)jB. Note char is and jBare unit vectors in the x and y directions respectively, in the body-fked frame.

B. Similady, let ea = - sin(8)iB + cos(8)jB be a unit vector in the 0 direct ion. point ing in the direction of increasing B. Furthemore, let e, be a unit vector pointing in the z

direction such that G,ea, e, form a right-handed coordinate system.

The angular velocity vector of the body-fked hame is given by w = Re,. Thus the

inertial the-derivatives of the body-fixed unit vectors are given by

Points within the body are represented by the polar coordinates (r,B. 2). The original position of a particle is given by r, = re,

+ re,.

The deformed position of the sarne

particle is given by

where 4. ue and u, are the displacements in the q,ee, e, directions of a particle. respectively. Each of these displacements d l be a function of time and the original position of the particle in question. The velocity of this particle is given by

Since the unit vectors are orthonormal, the square of the speed is given by

Once the velocity as measured by an inertial observer of any particle has been found. the kinetic energy of a mail element of volume can be expressed as $pdVv v. where p is the density of the material and dV is an element of volume. Thus, the total kinetic

energy of the body can be found by integrating over the entire undeforrned volume. Note

that since the velocity has been expressed as a function of the undeformed location of the particle, the integration is to be performed over the undeformed volume of the body. not over the unknown deformed volume.

The kinetic energy e.cpression is now expressed as a hinction of

%. 端e

and ul. the

displacements of an arbitrary point on the disk in the r , B and z directions respectively.

As for the strain energy, equation (2.8) relating the displacements of an arbitrary point on the disk to the displacement of the middle surface can be used. The explicit dependence of the kinetic energy on z can be eliminated by integrating over the thickness of the disk. from z = -h to z = +h.

As before, this procedure finaily yields the kinetic energy of the plate as an explicit function of u(r,O,t), z)(T,6, t ) and W ( T ,0, t ) only. The kinetic energy of the plate is thus given by

where

Here p is the density of the disk, f2 is its angu1a.r velocity, h is its half-thickness. and the displacements of the middle surface are given by u. u and outer radĂź of the disk axe given by

W.

Note that the inner and

RIand &, respectively. The expression for kinetic

energy remains the same regardless of whether iinear or nonlinear strains are employed since it does not depend on the strain. In other wordst the nonlinearity of this problem is confined to the strain energy.

2.2

Equations of Motion

Hamilton's principle can be concise- stated as

where L = T

- ÇVo is the

Lagrangian Function. T is the kinetic energy and CIO is the

strain energy of the system. -4s a variational principle. it States t hat the variation of the integral of the Lagrangian from t h e to to time tLvanishes provided that the variations

of the displacements vanish at to and tLand also on those parts of the boundary where the displacements are prescribed.

The equations of motion and corresponding boundary conditions are derived by applying Hamilton's Principle. The variation of the Lagrangian leading to the correct equations with corresponding boundary conditions is illustrated in the next subsection.

2.2.1

Variation of the Lagrangian

-4fter the Lagrangian function has been assembled kom the kinetic and strain energies. the last step towards hding the equation of motion is to perfom the first variation of the Lagrangian. Since it is the displacements of the middle surface, u, v and w that are the

generalized coordinates of the systern, the variation is perfomed with respect to these coordinates. When this is done, the result will be an expression that involves the time and space derivatives of the variations of the coordinates, bu, du and

&W.

Integration

by parts and the two dimensional analogue of integation by parts (see Appendix C) can then be used to obtain expressions purely in t e m s 6.u. dv. bw. These will involw integrals over the domain and path integrals over the boundary as well.

Since the generalized coordinates are independent. requiring the coefficient of the variation of each coordinate to vanish over the domain will yield the equations of motion.

By requiring the boundary integral terms to vanish as well. proper boundary conditions can thus be derived. This procedure will be illustrated for two representative terms.

As a h t example, consider the following variation illustrated

:

In the above, D refers to the domain of integration. Note that çwitching the order of space and t h e integration was employed along with integration by parts. However. the boundary term disappeared since in using Hamilton's principle, the variation of the coordinates is assumed to be zero at tirnes to and tl.

To extract the first variation of any coordinate from the variation of its time derivative. the same technique as illustrated above may be used. To extract the firçt variation of a coordinate frorn variations of its space derivatives. some of the results in hppendis

C may be used. This d l again be illustrated for a representative term. Consider the variation of Vw - Vw, which occurs in the expression for kinetic energy :

Xow consider the integration over the space variables only :

Note that one of the results of Appendix C has been used to extract the first variation of w â‚Źrom the space denvative of the variation. This yields an integrai over the space domain and a path integral over the boundary.

The procedures illustrated above were carried out for the entire expression for kinetic and strain energies. The approach is ciear but tedious. Setting to zero the coefficient of each of bu, bu, 6w in the integration over the domain yields the equations of mot ion.

Nonlinear Equations of Motion

2 2.2

With the help of MAPLE, the full nonlinear equations of motion can be found to be

(l+u) r -么U + - r -(-1++-u ) ? s u 2 m & + 2 30 2 drd6

rawh ---

+

2

r - l3r2

2

r-

dCut du. ( 1 - ~ ) ~--r2dw$w ~ + + ,-2-2 2 d6 Dr2 drd8 dr au ( i + ~~3a2v ) + (1 +u)r3- + -r dr &de drd没 39

Note that the hiil nonlinear representation of the spinning disk problem requires the

solution of three nonlinearly coupled partial differential equation, equations (2. X ) ,(2.25)

and (2.26). By way of contrast, the nonlinear mode1 used by other researchers consists of two coupled partial Werentid equations, equations (1.3) and (1.$) [31]. The difference arises as a result of the use of Lagrangian coordinates as well as the inclusion of in-plane

inertia, Coriolis and rotary inertia terms. These t e m s are neglected in the mode1 that leads to the nonlinear formulation given by (1.3) and (1.4).

2.2.3

Linear Equations of Motion Derived R o m Linear Strain

The equations of motion azising from the use of linear (Kirchhoff) strain-displacernent expressions are

Note that equations (2.27) and (2.28) for the in-plane vibrations are the same equations

as those derived by other authors [19,22, 211. However, equation (2.29) for the linear transverse vibrations is not the same as equation (1.1) or (1.2), which are the equations currently used for the iinear transverse vibrations of a spinning disk. Equation (0.29)

bears a strong resemblance to the equation for the h e a r transverse vibrations of a stationary disk. The equation for the linear transverse vibrations of a stationary disk is also derived on the assumption of linear (Kirchhoff) strains. Thus, it should not corne as a

surprise that the equation of h e a r transverse vibrations of a rotating disk based on the assumption of linear strains should be sirnilac to its stationary count erpart .

2.2.4

Linear Equations of Motion Derived From Nonlinear Strain

Let us now consider the linearisation of the nonlinear equations of motion. In other worcls. consider small displacements from equilibrium. The important observation to make is that. due to the rotation of the disk, the equilibriqm value of u may not be small at all.

From the rotation of the disk and symmetry considerations. the equilibrium values for the three displacements will be ueq= u(r), ve, = O,

W.,

= O, so that the equilibrium value

of u is a function of radius only. Now consider the equations obtained by neglecting al1 nonlinear t e m s in the nonlinear equations of motion with the exception of terms containing u or

(corresponding to the equilibrium value of u ) . The equation for the

equilibrium value of u,, is given by

The equations for the s m d displacements of u,v and w are given by

Furthemore. note t hat correspondiag to an in-plane purely radial displacement ti ( r ) . use of linesr stress-strain and iinesr strain-displacements relationships leads to the following stress-displacernent relationships

Using t hese relationships. equation (2.34) can be rewri t ten as

With the exception of the

v2wand v22端,this is the same equation as obtained by Lamb

and Southwell [1] for the transverse vibrations of a spinning disk. The presence of the

v26term is not unexpected ; it is simply the term due to the rotary inertia of the disk.

The physical meaning of the V*W term wiil be explained later.

CHAPTER 2. MODELLING

Boundary Conditions

2.3

-4fter using the 2D analogue of inteqation by parts to isolate the variation of u, rr and

W.

the remaining boundary term can be found. Rom this boundary term, suitable boundary

conditions to the problem may be written down directly. R e c d that u. v and

UTare

the

generalized coordinates for the problem here and the entire boundary term is required to vanish

. Since u, u and w are independent,

the only way the entire boundary term will

vanish is if the following four conditions hold on the boundary 1.

Su = O or the coefficient of Su vanishes

2. du = O or the coefficient of du vanishes

3. 6w = O or the coefficient of dw vanishes

4. 6 9 = O or the coefficient of 6% vanishes The boundary term obtained from applying Hamilton's principle and integration by parts is given below. Recall that r here must be evduated on the b o u n d a . Thus. for

a solid disk, r is the radius of the disk. For an annulus, a set of boundary conditions is

required at the inner and outer radĂź, so r will take two possible values.

+ -b2 r2de2

$-

-rai.

There are a few points that are worth rnentioning. First. the above boundary term

was

obtained from the variation of the Lagrangian obtained with the nonlinear (von Karman) strain-displacement relations. Note that the corresponding boundary conditions are also nonlinearly coupled. Had the linear (Kirchhoff) strain-displacement relations been iisecl. the corresponding boundary conditions would also have been linear and they can be

obtained from the above expression by neglecting all nonlinear tems.

It was previously noted that fomuiating the problem in this manner automatically accounts for the effect of rotary inertia in the equations of motion. The corresponding term in the boundary condition is

&.That is, the Mnation of some particular part

of the kinetic energy expression gives rise to the

and to the

V~IĂœ

t e m in the equation of motion

term in the boundary condition. Hence, if the effect of rot-

inertia is

ignored (or included) in the equation of motion, then the corresponding term must dso

be ignored (or included) in the boundary condition.

From equation (2.38), the boundary conditions for the special cases of linear in-plane

and transverse vibrations c m be derived.

2.3.1

Linear Transverse Vibrations

For the linear transverse vibration problem. the bound-

conditions at a free edge

become

Xote that

Hence, the first boundary condition (2.39) states that the moment at the free edge is zero, and this is in agreement with boundary conditions given in the literature.

The standard second aรงsumption at a fkee edge is that the Kirchoff shear. I.; or 'edge reaction' [41] be set to zero. For stationary plates (neglecting rotary inertia). the Kirchhoff shear is given by

where D = 2~ is the bending stinness of the disk. R e c d that h denotes the half-

thickness of the disk. This Iast boundary condition corresponds basically to summing the transverse forces at the edge of the disk and setting the result equal to zero.

This same expression is also usually applied to the rotating disk. Even when the term corresponding to the rotary inertia of the disk is dropped from the equation of motion and boundary condition, the boundary condition given by equation (2.40) and the standard boundary condition, equation (2.421,do not quite agree. The derived boundary condition (2.40) differs by the inclusion of a term proportional to presence of R2, this term would vanish for a station-

Due to the

disk. It must be observed that the

variation of some particular portion in the kinetic energy gives rise to the R'G'U~ term in the equations of motion and to the

p ~ 2term g in the boundaxy condition : therefore

explaining the presence of one should explain the presence of the other.

The presence of the centrifuga1 term in the balance of forces at the edge (and interior) of the disk should not corne as a surprise. Recall that this term was obtained naturally as part of the variation of the kinetic energy of the disk and not in an ad-hoc manner.

The physical significance of these new terms will be addressed in the discussion.

2.3.2

Linear In-plane Vibrations

The boundary terms for linear in-plane vibrations are given by

For a solid disk, equations (2.43) and (2.44) rnust be t m e on the outer radius. For an annulus. equations (2.43) and (2.44) must hold a t both the inner and the outer radius.

Note that

Thus, equation (2.43) irnplies that on the boundary either the displacement of the middle surface in the radial direction must be specified o r the integral of the stress in the radial direction over the side of the disk must vanish. Simiiarly, equation (2.44) States that on the boundary, either v must be specified. or the integral of the shear stress over the side

of the disk must vanish.

2.3.3

Nonlinear Boundary Conditions

The b o u n d q conditions at a free boundary obtained from applying Hami1ton.s pririciple

and integration by parts are given below. Recdl t hat r here must be evaluated on the

CHAPTER2. MODELLING hee boundary. Thus, for a solid disk, r below is the radius of the disk.

If the linear stress-strain relations are used along with the nonlinear (von Karman) strain-displacement relations, t hen the above boundary conditions can be interpreted as easily as the 1inea.r boundary conditions. Note that the nonlinearity in the boundary

condition is a direct consequence of the nonlinearity of the strain-displacement relations.

Equations (2.47) and (2.48) are actually identical to the respective st atements t hat

Note that this is the same as for the Iinear boundary conditions. In other words. both

the h e m and noniinear boundary conditions at a free edge corresponding to the u and u variables are merely the statements that the integral of the radial and shear stresses

over the side of the disk must vanish. In addition, equation (2.49) remains unchanged fkom its linear counterpart and thus states that the moment at the free edge must be

zero. The h a 1 nonlinear boundary condition, equation (2.50) appears to be a great deal more compücated than its linear cousin due to the presence of nonlinear terms on the right-hand side of the equation. However, upon further examination. the right-hand side of equation (2.50) can be rewritten as (2% x LHS of equation(2.47)

+

(1-4

au x LHS of

equation(2.48)). In other words, if equation (2.47) and equation (2.48) are both satisfied. then the nonlinear portion of equation (2.50) is automaticdy equal to zero. Hence. the nonlinear boundary condition, equation (2.50), is identical to its linear counterpart

2.4

Discussion and Cornparison Between the New and Existing Models

2.4.1

Nonlinear Equations

The nonlinear equations of motion for a spinning disk are given by equations (1.24).

(2.25) and (2.26). Note that there are three nonlinearly coupled equations. implying that they must be solved sirnultaneousIy. The spin rate R is usually taken to be constant

so that

= O. By way of contrast, the previous nonlinear formulation of the problem

results in t v o nodinearly coupled equations, nameiy equations (1.3) and (1A).

It îs possible to use a stress function to reduce the three new equations (2.24). (2.25)

and (2.26) to two where the variables to be solved for would be the transverse displacement and the newly-introduced stress function. However, thîs would necessitate the omission of the in-plane inertias,

% and $$,as well as the Coriolis terrns, 2 and $.

Indeed, this is the approach taken by Nowinski in deriving the standard equations (1.3)

and (1.4).

It must be recalled that the centrifuga1 force is not really an external force at all. but rather a consequence of the fact that the reference f m e is rotating and thus non-inertial.

The Coriolis force is due to the same effect. At present, there is no t heoretical justification for neglecting the in-plane inertia and Coriolis tems. In-piane inertia is t-ypically ignored for station-

plates, and this has been s h o w to be a good approximation for stationanj

plates [35]. However, the same calculation fails for the rotating plate precisely because

of the presence of the centrifuga1 and Coriolis forces. In other words. it is because the plate is rotating that the same argument fails. Thus, the validity of rotating disk niodels that reduce the number of nonlinear equations to be solved from three to two must be quest ioned.

2 -4.2

Lagrangian vs Eulerian Variables

In the derived nonlinear equations (2.24), (2.25), (2.26) and their Iinear counterparts (2.2?), (2.28) and (2.29), the centrifuga1 force appears as a term proportional to R2(ri u).

This is ceasonable given that ( r + u ) is the current radiai position of a given particle. .-\ particle originally at radius r has radius (r + SU) after deformation takes place. This is consistent with the Lagragian description of the system that has been employed. In this description, the boundaries of the disk or annulus are at the original (known) radii. On

the other hand, if an Eulerian description of the system is used, the centrihigd force will be proportional to R2r. Now the location of the boundaries is an unknown : if the disk

stretches, the new location of the boundaries is part of the unhowns of the problem.

Most authors use the Eulerian description n2r with the boundaries (incorrectly) located at their (original) Lagrangian location.

This is the approach taken by Nowinรงki [31],as a fkst step towards finding the nonlinear equations of motion (1.3) and (1A). Also, recall that in the traditional approach to

the linear transverse vibration problem, the in-plane stresses resulting from the rotation of the disk must first be found. .4gain, in this first step, the traditional approach has been to represent the centrifugai force as proportional to

R2r [1, 2. 3. 4. 61.

While it rnay seem that for small displacements there should not be much difference between r and (r + u), Bhuta and Jones [19]showed that the actual solutions obtained for the linear in-plane problem can be quite different.

2.4.3

Decoupled In-plane and Transverse Plate Problems

Consider the linear equations of motion (2.27). (2.28) and (2.29) corresponding to the assurnption of linear strain. Xote that, although the equations for the in-plane displacements are still coupled, they are not coupled to the third equation for the transverse displacement. As for the stationary plate. the in-plane and out-of-plane free vibrations

are independent of each other. R e d 1 that. in the traditional linear formulation of the problem as a membrane or a membrane with bending stiffness, the in-plane equilibrium stress problem must be solved before the transverse vibration problem can be addressed.

In this new formulation, the two problems are independent. which proves to be a considerable simplification for the solution of the linear transverse vibration problem.

This

independence is a consequence of the sssumption of linear strains. The linear equation

(2.34) derived from the nonlinear strain assumption does not feature this decoupling

between the in-plane and transverse problems.

2.4.4

The Membrane Problem

Let us once again consider equation (2.37) but now under the assumption that the plate

is so thin that the h2 terms are srnaIl in cornpaxison with the other terms and can be neglected. In other words, we are supposing that the disk is a membrane. In this case. equation (2.37) reduces

which is the equation for a spinning membrane as derived by Lamb and Southwell Il].

Furthemore. if the in-plane stresses a, and

are equal and constant. then equation

(2.37) is exactly the traditional equation for the transverse vibrations of a membrane

Similady dropping the h2 terms in the boundary term. equation (2.38) and using

the

same lineaxization procedure that led to equation (2.37) gives

for the boundary t e m . Equation (2.55) gives the possible boundary conditions as either

w is Exed on the boundary (dw = O) OR r3a,g = O at the boundary.

For the

second of these boundary conditions, if o, = O on the boundary then the boundary conditions reduces to cequiring that

$ is bounded on

the boundary

Both of these

boundary conditions are what wouid be expected for boundary conditions for a vibrating membrane.

The preceding was merely meant to point out the interesting observation that the linear equations of a membrane can be arrived at by starting with the assumption of

nonlznear strains and then linearizing the resulting equations.

Starting out with the

linear plate equations based on the assumption of linear strains does not lead to the linear equations of a vibrating membrane. This is not meant to serve as an analysis on the derivation of the equations of membranes â‚Źrom those of plates. The interested reader is referred to Niordson [421 where this derivation is performed by means of perturbation techniques based on the dendemess ratio.

2.4.5

Presence of Rotary Inertia

It should also be noted that the rotary inertia h a , automaticaIIy been taken into account in both the linear and nonlinear formulations of the problem. The term representing the effect of the rotary inertia of the disk is proportional to V'G. To ignore the effect of rotan. inertia, it suffices to drop these terms From the equation for transverse vibrations and their counterpart in the boundary conditions. Again, this is a new aspect in the modelling of spinning disks since the effect of rotary inertia has hitherto not been considered. Khile the effect of rotary inertia was shown to be minor for the low frequency vibrations of stationary plates, it remains to be seen whether the same holds tme for spinning plates.

2.4.6

Presence of Additional Terms

As previously observed, the equations of motion for the h e a r transverse vibrations arising fĂŽom the assumption of both linear and nonluiear strains featured a term proportional to

RZV2wwhile the corresponding boundary conditions also contained a term proportionai

to Cl2. The usual formulations of spinning disk problems do not contain these ternis : therefore their presence and physical significance must be carefdly examined.

Suppose that the in-plane displacements and rotary inertia are ignored. Shen the kinetic energy of the rotating plate becomes

The variation of Vw Vw gives rise tu the new terms in the equation of motion and in +

the boundaq condition. Where does this term corne from?

First, note that due to the presence of h3, this term will not anse in the membrane problern. Since Lamb and Southwell [II derived their famous equation of motion by adding bending stiffness to the spinning membrane equation, it is not surprising that their formulation does not include the term in question. Furthemore. it turns out that the term in question is a consequence of the use of Lagrangian coordinates. To see this, consider the velocity of any element of the spinning plate. The contribution to the velocity of the element due to the rotation of the plate is w x r, where r = r,

+ u. Now

coosider w x u. where we consider the contribution to u from the transverse displacernent only.

In other words. take

Since w = Rk, it follows that

CHAPTER2. MODELLING and (w x u) (w x u) = z2n2vw v w ,

(239)

which e?cplaios the presence of the term in question in the kinetic energy. It arises as a

consequence of the contribution to velocity due to the rotation of the disk and

the use of

Lagrangian coordinates. But it is known that the w x r is an in-plane term eventually

giving rise to the centrifuga1 force. Why does it crop up in the equation of transverse vibrations? The answer lies in closer examination of equation (2.58). This is indeed an in-plane term. However, it is lineax in z and thus gives rise to a bending moment. tn other words,

$>

w x u z dz gives a non-zero contribution. If the equations of motion

were to be derived in the Newtonian way (for example, see [41])the equations summing the moments are used to sirnplify the equations summing the in-plane and transverse forces. In this way, the bending moment due to the w x u term tvould eventually make its way to the equation expressing the balance of forces in the transverse direction.

In short, the presence of the new R2 terms in the equatioo of transverse vibrations and its corresponding boundaxy condition reflects the contribution of the bending moment due to the w x u term. it is only relevant for plates (as opposed to membranes). It is aiso a consequence of the use of Lagrangian coordinates.

In summary, the traditional assumptions of the plate theories of Kirchhoff and von Karman were utilized to derive h e m and nonhear equations of motion of a spinning plate.

The traditionally-ignored in-plane inertia of the spinning plate was taken into account.

as was the rotary inertia. Hamilton's principle was employed to derive the equations of motion, thus automatically yielding the cornpiete boundary conditions. Three sets of equations were derived; noalinear equations corresponding to nonlinear strains. linearized equations corresponding to nonlinear strains and linear equations based on the assumpt ion of linear strains. These equations bear similĂŠxities to (respectively ) the plate equations based on nonlinear strains? iinear equations of a spinning plate and linear equations of a stationary plate. The new terms that anse are due to in-plane inertias. Coriolis terms, rotary inertia and bending moment due to the rotation of the plate.

The resulting boundary conditions were also examined. The presence of additional terms in the equation of motion also btings with it corresponding terms in the bountlary conditions. The centrifuga1 term appears in the boundary conditions For the free botinclary. The presence of these terms in the equations of motion and b o u n d q condition has

been interpreted as arising from a bending moment due to the centrifuga1 force. When introducing (or neglecting) particular effects in the equations of motion. it is important to introduce (or neglect) their corresponding term in the boundary conditions.

Chapter 3 Linear Transverse Vibrations Introduction The spinning disk has traditiondly been rnodelled as either a spinning membrane. or as a spinning membrane with added bending stiffness [Il. A different approach to modelling the dynamics of spinning disks was considered in the previous chapter. The disk was modelled as a plate using the plate theories of Kirchhoff and von Karman. Rotary inertia was automatically included. Two possible forms were presented for the equation of

motion of linear transverse vibrations. One equation is the result of using the assumption of linear strains. Let us refer to this as the linear-strain-mode1 (LSM).The second possible equation is linear in the transverse vibrations but is based on the assumption of nonlinear (von Karman) strains. Let us refer to this as the nodinear-drain-model (YLSM).

The LSM bears resemblance to the linear equation of the transverse vibrations of a plate. This is to be expected since this latter equation is also based on the assumption of Linear strains. In other words, the LSM is essentidy the Linear (station-)

plate

equation with a new term that is due to the rotation of the plate. The 'ILSbI bears

resemblance to the (linear) equation of motion of a spinning plate as derived by Lamb and Southwell [l]. Just as the Lamb equation contains the stationary plate equation as a special case, the

NLSM contains the LSM model as a special case. The LSM model

is obviously a much simpler equation to solve than the NLSM. We also know that for stationary plates the assumption of linear strains is a good one. In this chapter. we seek to investigate whether the assumption of h e a r strains is e q u d y useful for rotating plates or if nonlinear strains must be used to obtain meaningf' results.

3.2

Equations of Motion

Consider the pcoblem of small transverse vibrations of a thin spinning circular plate. The equations of motion describing these vibrations is derived in the first part of the paper.

The equation of motion that results from using the assumption of linear (Kirchhoff) strains (LSM)is given by

Recall that h represents the hdf-thickness of the plate. The effect of the rotary inertia of

the plate has been included. The rotary inertia of the plate may be ignored by dropping the

az

2

w t e m . The above equation was derived with the use of Hamiltonk principle.

Note that equation (3.1) is basically the equation for the transverse vibrations of a stationary plate with additional terms to account for the rotary inertia and the bending moment due to the centrifugd force. The latter terrn is the R ~ V * Zterm.

U

The boundary conditions for the problem are

CHAPTER 3. LINEARTRANS~.~RSE VIBRATIONS 1. Either

is specified or

2. Either w is specified or

The equation of motion that results from the assumption of nonlinear strains (NLSN) is given by

where ru,q(,) must first be found from

-4shas been shown in the first part of the paper, equation ( 3 . 4 is the same as the Lamb and SouthweiI [II mode1 (TLM)with the exception of the terms due to the rotan. inertia of the disk and the bending moment due to the centrifuga1 force. The boundap conditions to be used with equation (3.4) are given in equations (3.2) and (3.3).

We first consider the kequencies of vibration as predicted by the NLSLI of equation (3.4). In this regard, we will follow the approach taken by Lamb and Southwell in

their historic paper [Il. The same formula as proposed by Lamb and Southwell [Il is employeĂ . Specifically, in their paper they write "If the restoring forces which control the vibrations of an elastic system can be separated into two or more groups which affect the potential energy independently, and if the gavest frequency of vibration be found on the assumption that each group acts independently (the inertia being unchangecl). then the surn of the squares of the frequencies thus found is l e s than the square of the greatest frequency which can occur when ail the groups act simultaneously".

In particular. suppose that rotary inertia is ignored, then as an approximation to finding the exact frequency, the following formula is proposed :

Here

pl

is the naturai frequencies of vibration of equation (3.4) when al1 h2 terrns arc

dropped. Remark that the resulting equation (when all h2 t e m s are dropped) is identical to the equation for a spinning membrane as introduced by Lamb and Southwell [Il and Simmonds [ĂŻ]. This was shown in the first part of this paper. These frequencies have been found by Lamb and SouthwelI in their historic paper [l].Similar to their approach. pl

is taken as the natural frequencies of equation (3.1) with the rotary inertia dropped.

Why has the effect of rotary inertia been disregarded? This has been done so that we may apply the aforernentioned result which requires the inertia of the two subsystems to be

the same. Hence, the primary difference between t his consideration and the work of Lamb and Southweil is the inclusion of the f2*vZw tenn that represents the bending moment

due to the centrifugai force. It was shown in the k t part of this paper that this term actually &ses from the kinetic energy of the system. In particular. it is a consequence of the vaxiation of the Vw Vw term which only depends on space derivatives. Hence. although Vw - Vw is technicdy part of the kinetic energy, from a purely mathematical point of view, it could be easily lumped in with the potential energy of the system (with an appropriate change in sign). This would permit the use of the aforementioned result

which requires the inertias of the two subsystems to be the same.

Hence, to summarize. we shall use equation (3.6) to analyze the frequency predictions of the NLSM. Our pl is identical to the pl of Lamb and Southwell [II. Our pz will be the natural frequency of equation (3.1) . In compaxison. for Lamb and Southwell. p2 were the natural frequencies of the stationary disk equation. In other words. to compare the naturai frequency predictions of the NLSM to the predictions of the traditional linear mode1 (TLM)of Lamb and Southwell, it suffices to compare the two values of p2 . Thus. the primaq difference between the NLSM and the Lamb and Southwell mode1 (TL.\[) is in the cdculation of p2. In the TLM, p2 are the frequencies of vibration of the stationary plate while in the NLSM,

are the frequencies of the rotating plate of the LSM. Based

on thĂŽs Logic. a large portion of the rest of the chapter will focus on the analysis of the

LSMt equation (3.1) and its cornparison to the stationary plate equation.

The complete solution and andysis of the LSM will now be undertaken. It is the naturd frequencies of vibration and the modes of vibration that are of interest here. To

CHAPTER3. L~NEAR T R A N S ~ RVIBRATIONS SE fmd the frequencies of vibration, assume that w takes the form

w (T, 6,t ) = e " ' ~(r,O).

(3.7)

This forrn of w is substituted into equation (3.1). The remaining equation to be solved for the spatial dependence of w takes the forrn

w here

Note that 6

of

(R2+ A*)

> O and c 5 O.

Furthermore, if the rotary inertia is ignored. ail occurrences

are replaced with

R2. Equation (3.8) is much simpler to solve ezactly

than

the equations resulting from the TLM or the NLSM. Furthermore, the inclusion of the rotary inertia is a new development in the vibration of spinning disks. However. as can be seen from equation (3.8). the inclusion of the rotary inertia does not present much of

an additional complication since the form of the equation remains unchangecl whether it

is included or not. That is, 6 is non-zero for a spinning plate regardless of the inclusion of the rotary inertia of the plate.

For stationary (non-spinning) plates, the omission of the rot-

inertia simplifies

the form of the equation considerably since now b = O. For low-Frequency vibrations. this is the approach traditionaily taken for stationary plate problems. We see that for

spinning plates, the prixnary traditional motivation to ignoring the rotary inertia is not present. Furthemore, from the analysis of stationary plates, it is known t hat for highhequency vibrations, the rotary inertia can no longer be ignored wit hou t introducing significant errors. From previous works, it is known that spinning a plate raises its natural frequencies of vibration. This fact also provides further incentive to preserve the rotaxy inertia t e m in the ensuing analysis.

If v2is considered as an operator, the form of equation (3.8) suggests a quadratic in

v2.This observation can be used to "factor" equation (3.8) in the same manner that one would factor a quadratic. Thus in place of solving equation (3.8). i.e. one fourth-orcler

PDE,two second-order PDEs must be solved instead

:

w here

For future reference, the foUowing relationships between a and ,O follow as a consequence of their definition :

The simplicity of this formulation becomes apparent in equations (3.11). First. the problem has been reduced to solving two second-order equations, which is a much simpler task than solving a fourth-order equation. Second, these second-order equations happen to be Helmholtz equations. In polar coordinates, the solution of these equations witl involve the use of Bessel functions and modiied Bessel functions.

Solutions of Helmholtz equations via separation of variables is well documented in the literature, Note that the function X will be the sum of the solutions to the two Helmholtz equations (3.11). Thus, X is given by

[E,cos(n6) + F,, sin(nB)],

where A,,

(3.15)

Bn,Cn,D,,En,F, are constants, J, and In are Bessel ftinctions and motlified

Bessel function of the 6rst kind, respectively, while 1; and Kn are Bessel functions and modi6ed Bessel functions of the second kind, respectively.

3.3.1

Solid Plate with Fkee Boundary

Substitution of w = eLAt.Y(r, 6 ) into the boundary conditions as given by equations (3.2)

and (3.3) leads to the boundary conditions for a fiee boundary in terms of

as

CHAPTER 3. LINEART R A N S ~ R VIBRATIONS SE D e h e X, and XB such that they sat*

In other words,

From the definition of the Laplacian in polar coordinates, we may write

Sote that

$ = -n2.Y.

This follows from the separation of variables of the Helmholtz

equation in polar coordinates. Furthermore. n must be an integer for the solution to

be single-valued. From the above definition of .Y, and .Yd, it follows that V 'S

=

v'.Y,+ v2&= p2X0 - a2&. The above facts can be cornbined to simplify boundary condit ion (3.16) to the following expression

Boundary condition (3.17) can be similady simplified to

3.3.2

Frequency Equation

Consider the free transverse vibrations of a spinning solid plate with a free outer radius.

The Bessel and modified Bessel huictions of the second kind, Yn(w) and K,(,Br) are not bounded at r = O. They are thus discarded from the general solution. That is.

B, = D, = O for dl n. Furthemore, R J r ) = A, J,(cw) and Re ( r ) = CnIn ( 3 r ) .where n = 0,1,2,. ... These expressions are subsequently substituted into equations (3.21) and

(3.22) and evaluated at the outer radius of the plate. a. This yields two equations for the constants -4, and C,:

[ ( 1 - u)~'J,,(P,)- (P: + (1 - v)n2)P&(P.)]

-4,

[(I - U ) ~ * I ~ + ( P(P: ~ ) - ( 1 - y)n2) pbI;(pb)]

+ = 0.

( 3.24)

where Pa= aa, Pb = Ba and a is the radius of the plate. 'uote that the definitions of cr

and B have been used to simpl@ equations (3.21) and (3.22).

Equations (3.23) and (3.24) represent two homogeneous equations in terms of the two unknowns A, and Cn. For nontrivial solutions the determinant is required to ianish.

This yields the fiequency equation of the system :

Note that the new mode1 has yielded a frequency equation in closed form for the spinning

disk problem. -4 closed-form expression for the spinning disk problem with bending stiffness taken into account was hitherto unavailable in the literature. While this is

a considerable simplification, we must still investigate the validity of the linear strain assumption which leads to this equation. Also note that the above frequency ecpation has the exact same form as that for a stationary circular plate (also based on the linear

strah assumption). In fact, it can be shown that the frequency equation for a stationary circular plate can be obtained as a special case from equation (3.25). as would be expectecl.

3.3.3

Mode Shapes

The mode shapes for a solid disk c m be found from equation (3.13) . For reference. thesc are given here by

where

Naturdy,

Paand Pb in the above expressions are those s a t m g the frequency equation.

CHAPTER 3. LINEARTRANSVEXSE VIBRATIONS

3.3.4

Nodal Circles

Now consider b d i n g the radius of the nodal circles, r* , associated wit h a part icular mode of vibration. For the radius of a particular nodal circle, it follows that

R(T*)= &Jn(ar')

+ CnIn(P7*)= O.

(3.28)

From the definition of Paand Pb and using the frequency equation. the above equation

c m be rewritten as

Thus, for particular values of Paand Pb which correspond to a particular mode of vibration, equation (3.29) can be solved for

$ or r'.

Equation (3.29) can further be rewritten to match the expression derived by Colwell and Hardy (431 for stationary plates :

FoiIowing the argument given in [43], the modified Bessel hnction In is like a hyperbolic hnction in that it starts a t small values and increases rapidly to infinity. For values of r'

not near the edge,

f < 1, it follows that In(Pb)> In(Pbr*/a)and IL(Pb)> In(P b r 8 / a ) .

Thus, the right hand side of equation (3.30) becornes vanishingly small for small r'. In other words, for high kequencies and for nodal circles not close to the edge. the nodal

CHAPTER3. LINEARTRANSVERSE VIBRATIONS circles may be approximated by

That is, the values of the nodal circles rnay be approxirnated from the zeros of the Bessel function. Note that the effects of rotary inertia and of the spin do not enter into the argument. It must be emphasized once again that the above approximation is for high frequencies and For smail r'.

3.4

The Frequency Equation for p:!

The frequency equation (3.25) represents an equation in terms of two variables. Pu ancl

Pb. Of course, these two variables are not independent, but are related through equation (3.13). The actual frequency, X rnay consequently be found from equation (3.14). In fact, equation (3.25) is the frequency equation for four possible circular plate cases ancl it is the relationship between Paand Pb that determines precisely which case is being considered. These four cases are Non-spinning plate. rotary inertia not taken into account.

O

Non-spinning plate, rotary inertia taken into account.

Sp첫ining plate, rotary inertia not taken into account.

Spinning plate, rotary inertia taken into account.

In its present forrn, equation (3.25) depends on the parameters n and v? as weil as the two va,riables Pa and Pb. Note that thus far the spin rate andfor the plate thickness-to-

diameter ratio do not appear directly in equation (3.25). It tums out that these factors

will appear when the appropriate relationship between Paand Pb is forrnulated. thus entering the frequency equation indirectly through this second equation. The aforementioned four cases are for a splluiing or non-spinning (stationary) plate with or wit hout

rotary inertia taken into account.

Define the variable k such that

Note that k represents the effect of the spin. For a stationary (non-spinning) plate. R = O and thus k = O as well. Also define the bending stiffness of the plate as

where h is the half-thickness of the plate. Recall that the following relationships hold between

a!

and ,Oas a consequence of their definition

Each case will now be considered separately.

1. Stationary plate, no rotary inertia In this case. R = O and rotary inertia is

neglected. Thus, b = O and equation (3.34) yields

2. Stationary plate, rotary inertia included. Here

is nonzero. In fact, b =

p 1 u2 A* + .

R

= O as before, but now b

However, equation (3.35) can be solved for X

in terms of P. and Pb. This procedure yields the required relationship between Pa and Pb

a

Note that the ratio o l plate-thiches~to-diameter, now enters the frequency equation through this relationship. Since rotary inertia is being taken into account. this

seems a reasonable result.

3. Spinning plate, no rotary inertia. In t his case. b is nonzero. Howver. h = 'k since rotary inertia is being neglected. Thus the relationship between Pa and Pb is given by

Note that in this case, the vaziable k has entered into the frequency equation. As previouslp mentioned, k represents the effect of the plate spin and thus it is to be expected that it appears in the caiculation of natural frequencies.

4. Spinning plate, rotary inertia included. Based on the results of the previous cases, it wouid be expected at this point that both the thicknesto-diameter ratio of the plate and the effect of rotation appear in the relationship between Pa and

4. This is indeed the case. Equation (3.35) c m be solved for X and the resulting

expression substituted into equation (3.34). This procedure yields the following relat ionship

Pb =

Thus to find the natural frequencies of vibration of a circular plate for one of the above four cases, it suffices to solve equation (3.35) in conjunction wit h one of equat ions (3.36), (3.37), (3.38) or (3.39).

Once the appropnate values of Pa and Pb have been found from the frequency equation, the frequency itself may be found from (3.14)

Again, note that this is merely a generalized version of the classical relationship found in many texts for stationary circular plates. Since the classical case corresponds

to

the

no-spin, no rotaxy inertia case, it may be obtained by substitut ing Pa= Pb into the above equation.

The frequency equation (3.25) can be rennitten in a sirnpler Form as

CHAPTER3. LINEAR TRANSVERSE VIBRATIONS where

iin = ( 1 - u)P.P~(P:+ Pb).

As previously mentioned. this equation must be solved in conjunction with one of equations (3.36), (3.37), (3.38) or (3.39). For numerical simulations. the easiest a p proach is to substitute one of equatioos (3.36): (3.377, (3.38) or (3.39) into the frequency equation. This will yield an equation in Pa only, which can then be solved numerically.

For each value of n, there will be an infinite number of solutions.

However, there is another way of viewing the above problem which lends itself to insightful observations. Instead of substituting one of equations (3.36). (3.37). (3.38)

or (3.39) into the frequency equation, it is more interesting to consider the problem of solving the frequency equation and one of equations (3.36), (3.37). (3.38) or (3.39) simultaneously. In t his case, the fiequency equation expresses a relat ionship between

Pa and Pb and is thus a curve in the Pa - Pb plane. Similarly, equations (3.36). (3.37).

(3.38) and (3.39) are relationships between P . and Pb and are thus also curves in the

Pa - Pb plane. Hence the d u e s of Pa and Pb that satisfy 60th the Frequency equation and one of equations (3.36), (3.37), (3.38) or (3.39) are intersections of the two curves in the Pa - Pb plane. By sketching these curves, numencal simulations are not required to obtain insights into the effect of certain parameters on the natural frequencies. Xote that

kom their definition, interest is confined to the case where both Pa and Pb are positive.

That is, only the h t quadrant of the Pa- Pb plane will be considered.

Case 1: Stationary Plate, No Rotary Inertia In this case, we have Pa= Pb. In the P . - Pb plane, this is a straight iine wit h unit dope passing through the origin. This is the classic problem whose solution is well documented.

. of the frequency equation depend only on the value of o and can be The solutions P found in the literature [41].A s a first approximation, Kirchhoff hm given the value

where s is the number of nodal circles and n is the number of nodal diameters [43. 41).

This value can be used as an initial guess when using a numerical method to solve the frequency equat ion.

Case 2 : Stationary Plate, Rotary Inertia Included This case is Uustrated in Figure 3.1, with case 1 inciuded for cornparison. As can be seen from the figure, the P. - Pbcurve for case 2 resembles a parabola opening up onto the P . axis. For smdl values of Pa,it matches the Pa = Pb straight line of case 1 exactly.

However, as

Paincreases, the cuve approaches the horizontal asymptote of Pb = TdZa.

From this, it c m be concluded that at Low fkequencies, rotary inertia may be safely

ignored, since there is virtudy no Merence between the Pa - f i curves for the two different cases. However, at hi& fiequencies, there is a marked difference between the

Pb vs Pa for Cases I and 2 I

-

F

1

Case 2, Ha=O.Ol - Case 2, Ha=0.04

Figure 3.1: Graphs of Pa vs Pb for Cases 1 and 'Z

two cases and rotary inertia should be taken into account. The thinner the plate. the s m d e r the thickness-to-diameter ratio

and thus the hvo curves become different at

higher and higher frequencies. Furthemore, it rnay be seen that compared to case 1. the vaiues of Pj, that sati*

both the frequency equation and the Pa - Pb relation are much

smaller. From equation (3.40), it can be seen then that the net effect of the inclusion of the rotary inertia is to lozuer the natural frequencies of vibration.

By gaphing the P .

- Pb c m s for case 1 and case 2 in this rnanner. the frequency at

which rotary inertia must be included, can be deduced. This occurs wherever the P .

- Pb

curves for the two cases start to diverge. The fact that the rotary inertia can be ignored at low frequencies for thin plates and that this becomes l e s valid as the thickness of the plate or the frequency of vibration increase is well known. While these conclusions are not new, the ease Nith which they were obtained is emphasized.

Case 3 : Rotating Plate, N o Rotary Inertia This case is illustrated in Figure 3.2. From the hrm of the equation or from the figure. it may be seen that the Pa - Pb relation for case 3 is a hyperbola. The straight line

P. - Pb corresponding to case 1 is an asymptote to the curve, and furthemore. Pa 3 ka.

From this, we may deduce that the rotation affects mostly the low frequencies where the two curves are different. At high frequencies, the naturd frequencies of a rotating plate

and those of a stationary plate will be very similar, although they would not necessarily correspond to the same mode of vibration. Thus, different modes rnay correspond to

the same fiequencies. This is an interesthg result since if the (high) natural fiequencies

Figure 3.2: Graphs of

P. vs Pb for Cases 1 and 3

are of interest without regard to which mode they belong to, those of a station-

plate

may be used instead. This fact is usehl since these values are readily available in the Ăźterature and are easier to measure experimentdly.

Furthemore, the effect of the rotation of the plate is readily apparent.

In the low

frequency range, the values of Pathat satise both the frequency equation and the Pa-Pb relation are higher than in the first case, while the values of Pb rnay be comparable. This follows since

Pacannot be less than ka. From equation (3.40), it

can be deduceci that

the effect of the rotation generally is to raise the naturd frequency of vibration. The frequency of vibration can also be similarly raised by increasing D, the stiffness of the plate. In other words. the rotation has the general effect of stiflening the plate. This stiffening has been previously obsenred in the field of rotating structures.

The frequency equation may once again be rearranged

Xow consider the low frequencies for the case where the spin rate. R. is very large. This

implies that Pais large and Pa > Pb. From equation (3.43). if P . dominates. t hen the right hand side is very smaii.

Ln other words, for Pa Âť Pb, equation (3.43) becornes

Since Pa is assumed to be large, the asymptotic expression For J, may be used

Thus, we have nn

&(Pa)= cos (P. - - E ) = O. 2 4

(3.46)

The zeros of the cosine function are the odd multiples of f. Furthemore, recall that

Pa must be at least as large as ka and that Pa is assumed to be large. Thus the above argument yields as a fiat approximation

where n is the number of nodal diameters, n = 0,1,2,... and m = 0.1.2.. ... While n is the number of nodal diameters as in the stationary plate case, m does not necessarily

correspond to the number of nodal circles as it would for the stationary plate case. Note how this approximation reduces to Kirchhoff's first approximation when k = O. as it should. Furthemore, it implies the raising of the naturd frequencies due to the effect of the spin, so it captures the aforementioned stinening effect of the rotation. This is a cmde approximation but it may be used as an initial guess in numerical solutions.

Pb vs Pa for Cases 1 and 4 1

I

I

Case 4, Ka=50, Ha=O.Ol

Figure 3.3: Graphs of P. vs Pb for Cases 1 and 4

Pb vs Pa for Cases 2 and 4

Figure 3.4: Graphs of Pa vs Pb for Cases 2 and 4

Case 4 : Rotating Plate, Rotary Inertia Included As would be expected, the individual features of case 2 and case 3 are now both in play.

The Pa - Pb c w e s in Figure 3.3 compare case 1 and 3 where the equivalent curves are shown; case 1 without the effect of rotation or rotary inertia and case 4 including both.

The P .

- Pb curves seen in Figure 3.4 compare the equivalent curves with or without the

effect of rotation only since both curves include the effect of rotary inertia. -4s previously observed, for plates with larger thickness-tediameter ratios at high frequencies. the effect of rotary inertia should not be disregarded. Note that when the spin rate

R

is high. the

'low' frequencies of vibration are actually high compared to those of a stationary plate.

In other words. the effect of rotary inertia should be included even at low frequencies of vibration of a plate spinning at a high spin-rate. This is true even if the plate is thin.

Again, to reiterate, the effect of rotary inertia may be safely ignored at low frequencies of vibration of a thin stationary plate. but not for a thin plate that is rotating very rapidly. Note that this conclusion could not have corne out of the analysis of a stationary plate. This serves to point out that some of the approximations that are well-accepted for stationary plates may not be applied to spinning plates as they may not necessarily

be as valid for the spinning problem.

3.5

Numerical Simulations

Numerical simulations were carried out to solve the fkequency equation for various cases.

For aII cases, the values of E = 2 10" N/m, v = 0.25 , p = 7.8 - 103 kg/rn3, plate radius a = 0.6 moand plate half-thickness h = 0.01 rn were used. These values correspond

exactly to those used by Lamb and Southwell [II in their pioneering paper.

3.5.1

Frequencies of Vibration

The k t case simulated was for Q = O, that is, no spin. In this case P. = Pb and the solutions to the frequency equation should be those for a stationary plate. These solutions are widely available in the literature, thus providing a way of veri@ng the accuracy of

the developed equations and code. The resulting values of Paare given in Table 3.1.

Nat, a stationary plate (Q = O) with rotary inertia was considered. The resulting solutions Pa of the frequency equation are given in Table 3.2. Note that taking the effect of rotary inertia into account results in srnaller values of P . In addition. the lower frequencies do not differ much from their values without rotary inertia taken into account.

It is at high frequencies that the presence of rotary inertia becomes apparent. In other words, the effect of rotary inertia on the frequencies of vibration of a stationary disk can be ignored For the low frequencies. However, at high frequencies of vibration. it should

be taken into account. .\gain, this is a well known result and serves to verify the validity of the developed equations.

Now consider higher d u e s of the dimensionless spin parameter ka. wit hout the effect of rotary inertia. It appears that certain modes of vibration of the stationary plate get

. corresponding to ka = 1.65 are given in cut off. For example, the resulting values of P Table 3.3. For ka = 1.65, the n = 2, s = O mode is cut off. That is, our mode1 predicts that this mode should not be observed at all in a spinning plate. The n = 2? s = O mode wodd have been the mode with the lowest frequency of vibration had it not been cut

off. Now that this mode is cut off, the mode with the lowest fiequency of vibration is

the n = O, s = 1 mode. However, the n = O, s = 1 mode corresponds to Pa = 2.5912 for ka = 1.65, compared with

Pa= 2.9816 for ka = O. In other words, the effect of the spin

is to lover the fiequency of vibration of a partieular mode even though the frequencies of vibration are overall now higher due to the absence of particular modes.

Since Pacannot be l e s than ka, we c m now see why certain modes get cut off resulting in overall higher vibration frequencies. In essence, the effect of the spin is to lower the frequency of vibration of a particular mode until it becomes disdlowed as Pa must alrvays be greater than or equd to ka. This points to the existence of critical speeds of rotation.

At some particular value of the spin rate

R, some modes

may cease to erist. Again.

note that despite the cutting off of certain modes and the lowering of the frequencies of neighbouring modes, the higher modes of vibration (those not near the cutoff point) are

hardly affected by the effect of spin. This confirms the predictions made by investigating the shapes of the P .

- Pb curves.

For values of P . rnuch Iarger than ka. the P, - Phcurws

for a spinning plate asymptotically approach those of a stationaxy plate. The prediction that the modes and frequencies of vibration of a stationary plate can be used in place of those of a spinning plate for the higher modes has thus been verified niimerically.

The same trend is observed for higher values of dimensionless spin parameter ka. For higher values of ka such as ka = 3, even more modes are cutoff. The kequencies of modes adjacent to the cutoff modes are Iower compared to the corresponding d u e s for

a stationaxy pIate, while those of modes far from the cutoff modes are once again hardly different (although still lower) from the corresponding values for a stationary plate. The r e d t i n g values of Pac m be found in Table 3.4.

Now consider the effect of both rotary inerfia and the spin of the disk. The solutions

to the frequency equation Pa for ka = 3 with the effect of the rotary inertia included are given in Table 3.5. From the table, it is easy to deduce that the presence of rotinertia affects a spinning plate the same way as it affects a stationary plate. In other words, rotary inertia tends to lower the frequencies of vibration and mostly affects the higher frequencies of vibration.

The next case simulated was one for which

R = 100q

without the effect of rotary

inertia taken into account. This is the same value used by Lamb and Southwell [l]and corresponds to ka = 0.036. The resulting values of Pa can be found in Table 3.6. Note that the smaliest non-zero value of Pa that is a solution to the frequency equation for a stationary plate is Pa = 2.3476. Since Pa,Pb and ka are dimensionless quantities. the small value of ka for this case leads us to expect that the spin will not have much effect

on the resulting solutions to the fiequency equation. This is indeed the case. as can be seen â&#x201A;Źrom Table 3.6. OnIy the smaller values of Pa are affected. and even so only

at the

fourth decimal place. The higher frequencies do not appear to be affected at al1 by the spin of the plate. The value of ka is not large enough to cut off any of the stationary plate modes. In other words, for the spin rate used by Lamb and SoathwelI. the uaiues of pz predzcted by the LSM and the

3.5.2

TLM are nearly identical.

Obsenmtions from Simulations

Some Stationary PIate Modes Get Cut Off .Mer performing rnany more simulations with various values of the spin parameter, a generd trend starts to become apparent. It is the relationship between the value of dimensiodess spin parameter ka and dimensionless naturd frequencies P .

of the stationary plate that determines the results for the rotating plate. If the hensionless frequency

Pa of a particular stationary plate mode is less t han

ka.

then that mode will be absent for the rotating plate. If the stationary plate value

of Pa is Larger than ka but close in value to ka. that mode may or may not exkt for the rotating plate. Recall that the spin will lower the frequencies of a particular mode and thus it is not clear if a particular Erequency will be lowered to less than

ka or not. Finally, if the stationary plate value of Pais much greater t han ka. t hen the frequency and mode shape of that particular mode are hardly affected by the

rotation of the plate.

Stationary Plate Modes Can be Used For Higher Modes For values of

Pagreater than but close to ka, a certain amount of care must be

taken. In fact, it is these modes that will have the lowest frequencies of vibration and are thus of the greatest concem. These modes will also be the rnost different

hom their stationary plate values. For values of Panot too close to ka. the stat i o n plate ~ values of

Pamay be used as starting points in the numerical solution

of the rotating plate. In fact, the stationary plate values rnay aiso be used as an approximation to the rotating plate values for PaÂť ka since it is known that they

do not differ much.

Rotary hertia The question of whether or not the rotary inertia must be inchded can also be

addressed in a similar fashion. R o m the stationary plate problem, it is known that for thin plates rotary inertia w i l l tend to decrease the values of Pathat satis- the fiequency equation. Furthermore, it is known that the effect of r o t q inertia only

. Thus, for a given plate, it is usudly known how large afFects the higher values of P Pa has to be before the rotary inertia of the plate plays a role in the calcuiation of the naturd frequencies. So, let us suppose that for Pa> P,' the rotary inertia rnust be included to correctly calculate the fiequency. This criticd value of

P. is still

applicable to the rotating disk. Furthermore, if ka > Pl,then the effect of rota^ inertia rnust be taken into account even at the lowest frequencies of the rotating plate. Wote that this will correspond to high angulsr speeds of the disk and the assumption of small in-plane displacements inherent in the mode! may cease to be valid.

3.6

Discussion

Now that the frequency predictions of the LSM have been thoroughly analyzed. we may return to compare the LSM, NLSM, TLM and discussing the validity of the linear strain assumption. It is now obvious that the linear strain assumption for rotating plates is

NOT a good assumption. First. we note that for the spin rate taken by Lamb and SouthweU, m for the stationaxy plate and the rotating plate of the LSM are identical. If the

LSM mode1 alone is used to predict frequency, we would have p =

the TLM of Lamb and Southweil, p =

d m . 端niess pl

However. for

is insignificant. the LSLI will

underpredict the ftequency. Furthermore, we note that the predictions of the NLSM LI11

match those of the TLM since the effect of p, is included in this rnodel.

Based on the foregoing analysis of the

LSM,the frequencies predicted by the NLSSI

will be similar to but slightly s m d e r than the predictions of the TLM. For the anguiar velocity used by Lamb and Southweli, the two models are in complete agreement. At higher angular velocities, the bending moment due to the centrifugal force will become more apparent and the NLSM and TLM will begin to differ in the predictions of the fiequency of the lower modes of vibration. Since it was obsewed â&#x201A;Źrom the simulations that the stationary plate frequencies can be used for the higher frequencies of the LSh I. at high frequencies the YLSM and TLM will once again agree. The ease and simplicityof-use of the LSM cornes at a price ; it is evidently not valid for angular velocities large enough to render the assumption of lineat strains inadequate. The LSM can be used to accurately refiect the dynamics of a rotating plate only for angular velocities small

enough to make pl negligible.

3.6.1

Validity of the Linear Strain Assumption

One of the conclusions that can be made here is that the assumption of Iinear strains. while successful for stationary plates is not valid for rotating plates.

FVhy is t his so?

The rotating plate problem is best thought of as two problems : one of in-plane stretch and then one of vibrations.

When the plate is rotated, it fmt stretches in its plane

and achieves an equilibrium between the centrifugal force resdting fiom the rotation of the disk and the elastic restoring force.

On top of this in-plane stretch. the in-plane

and transverse vibrations are superimposed.

For srna11 amplitudes, the in-plane and

transverse vibrations do not affect the in-plane stretch and do not affect each other.

CHAPTER3. LINEARTRANSVERSE VIBRATIONS

82

However, for larger amplitudes, the stretching and vibration problems couple and cannot be thought of as independent.

Linearizing at the strain-displacement expression is

prernature linearization. Recall that we may think of the in-plane equilibriurn stretch of the disk as being known and &ed (and thus not subject to variation when using Hamilton's principle) . Thus when quadratic transverse t e m s in the strain-displacement expression are retained, there are terms in the resulting strain energy expression that are products of the known equiiibrium stretch and quadratic unknown transverse terms.

Recall that linear transverse terms in the strain-displacernent expressions will lead to quadratic transverse terms in the strain energy expression and linear terms in the equation of motion. From the point of view of variation, the products of known stretch terms and quadratic transverse terms will lead to linear transverse terms in the equation of motion since they are quadratic in the unknown transverse terms. However. had the Kirchhoff strain expression been used, these terms would never even have appeared in the expression for strain energy.

Although this anaiysis was done for a spinning disk. the same would hold true For

any disk with an underlying fked in-plane stretch such as a pre-stressed disk. In other words, if the plate has an underlying stretch imposed by rotation or other rneans. then Linearization should not be performed at the strain-displacement level but rather at the strain energy level.

Linea,rizing at the strain energy level involves retaining quadratic

t e m s in the unknowns since these Iead to h e a r tems in the equations of motion after the variation has been performed.

The equilibriurn stretch affects the Erequencies of transverse vibration.

For srnail

amplitudes, the transverse vibrations are not large enough to change the underlfing

stretch and thus the stretch and vibration problems are uncoupled and can be considered independently.

However, for large transverse amplitudes of vibration, the transverse

displacements axe large enough to change the underlying stretch of the disk and this in turn affects the frequencies of vibration. The nonlinear coupling between modes d l be considered in a later chapter.

In summary, two Iinear models of spinning disks were considered, one arising from the use of linear strains and the other from the use of von Karman strains. The linear equation of motion for small transverse vibrations of a spinning disk modelled as a spinning plate with linear strains has been solved and analyzed. The natural frequencies of vibration were obtained. In contrast with the traditional mode1 of a spinning disk. the frequency equation was found in closed form. This frequency equation is a relationship in two variables. The four cases of stationary or rotating plate with or without rotary inertia have the same frequency equation. The particular case being considered depends on the relation between the two variables in the frequency equation. Thus. finding the natural frequencies of vibration amounts to solving two equations in two &ables.

It tums

out that the solution of the stationary or rotating plate problems are quite similar and the weH-known resuits for the stationary plate can be used as a starting point for the rotating disk problem. This new mode1 predicts that certain modes corresponding to the station-

plate will be cut off if the plate is rotated at particdar angular speeds.

The net effect of having some of the Iower modes cut off is the raising of the lowest

CHAPTER 3- LMEAR TRANSVERSE VIBRATIONS

84

Table 3.1: Solutions of the frequency equation for a stationary disk without rotary inertia.

Pa for a stationa y disk, ka = O

frequencies of vibration. However, cornparison with existing work makes it evident that t his equat ion cannot accurately represent the spinning disk at angular velocit ies t hat are

not extremely small. For angular velocities that are typical for spinning disks. the second

equation based on nonlinear strains must be used. This second equation is closer in form to the traditional model yet contains a L w terms neglected in the traditional model.

It was shown that this second equation is in agreement with the traditional model for the angular velocities used. At higher anguiar velocities. the bending moment due to the centrifuga1 force should be taken into consideration and thus the NLSM and TLll may begin to differ appreciably in their predictions of the lower modes .The most subtle

conclusion is that inspite of its efficacy for stationary plates, the assurnption of linear strains quickly becomes inadequate in the realm of spinning bodies.

CHAPTER 3. LINEARTRANSVERSE VIBRATIONS

85

Table 3.2: Solutions of the frequency equation for a stationary disk with rotary inertia. ka=o.

I

Pafor a stationary disk with RI, ka = O

Table 3.3: Solutions of the frequency equation for a rotating disk without rotary inertia.

Pafor a rotating disk. ka = 1.65. nl s = O l

s=11

s=21

s=31

Table 3.4: Solutions of the frequency equation for a rotating disk without rotaq inertia. ka=3.OI

n

Pa for a rotating disk, ka = 3.0.

II

Table 3.5: Solutions of the frequency equation for a rotating disk with rotary inertia. ka=3.0.

Table 3.6: Solutions of the frequency equation for a rotating disk wit hout rotary inert ia. ka=0.036

P . for a rotating disk, ka = 0.036.

Chapter 4 Linear In-plane Vibrations 4.1 Introduction In a certain sense, it is possible to view the first two nonlinear equations of motion for the spinning disk as linear in-plane forced equations.

That is. the nonlinear terms in

the transverse displacernent c m be thought of as forcing terms to the linear in-plane equations. Thus, considering the linear in-plane problem sheds some Iight on the more general nonlinear coupled problem.

Burdess. Wren and Fawcett [2l]solve the general non-symmetric in-plane vibration problem. The free motion mode shapes that are denved in their paper can be used to construct a general solution to the forced vibration problem. Following the example of Wickert and Mote [Ml, recasting the problem in state space d o w s for orthogondity properties of the aforernentioned mode shapes to be easily derived. Once complete orthogonality properties have been denved, construction of solutions becomes a simple matter. This chapter investigates the orthogonality properties of the Eree motion mode

shapes and demonstrates how these properties cm be put to use in constmcting general

solutions to the forced in-plane vibration problem.

4.2

Statement of the problem

Consider a typical point P in a disk which is rotating about its polar avis with constant angular velocity Q. The position of P is d e h e d by polar coordinates ( r ,0) which are measured with respect to axes fixed in the disk. Assume that the motion of a particle in the disk oniy occurs in the plane of the disk and is given by u = (u,, uejT. where u, and

ud

are the radial and tangentid displacements respectively. For srnall displacements. linear stress-strain and linear strain-displacement relationships can be assumed. Furt hermore. for a thin disk plane stress conditions can be assumed. The preceding assumptions were shown in Chapter 2 to lead to the following equations of motion :

where R is the spin rate and L is the rnatrix operator

The operator Ln is derived from L by setting

& = in,where i = a.

In the above.

E is Young's modulus, u is Poisson's ratio and p is the density of the disk. Furthemore.

D,u and F are given by

where (Fr,Fe)T are the radial and circumferentid body forces applied at a point in the disk.

The boundary conditions for a disk with a Free boundary are given by or, = ors = O at the boundary of the disk, r = a.

Using linear stressstrain and strain-displacement

relationships, these equations can be written in tems of u, and

as

Note that for the remainder of what follows, the Foliowing inner product will be used :

where the

* denotes the cornplex conjugate.

4.3

Free Vibration Analysis

The free motion mode shapes are required for the construction of the general solution to the forced vibration problern. When F = O, then general solution to equation (4.1)can be f o n d through use of Helmholtz's theorem.

This development is outlined in [21] and

d l be included here for completeness. To solve equation (4.1),u may be expressed in

terms of the Functions 4 and ,tir such that

If these equations are substituted into equation (4.1),some algebra yields the following two equations for iIr and @ :

where

The general solution of the above equations for a solid plate can be written in terms of Bessel h c t i o n s of the first End as

and

in the above. n in an integer and is the circurnferential node number. p is an unknown frequency and 0, (j= 1,2) are the positive roots of

and

The factors A, and B, are constants to be found from the boundary conditions.

Xote

that equations (4.19) and (4.20) are solutions onIy if g?- f R2. When p2 = R2. then

p, = 0 and the Bessel functions Jn(Plr) are undehed.

CHAPTER 4. LINEARIN-PLANSV~RATIONS The displacements 2 ~ ,and ue are thus given by

To derive particular values for

p and u the boundary conditions must be satis-

么ed. Substituting equation (4.24) into the boundary conditions (4.10) and (4.11) yields equations for -4, and B, given by

where the elements of the matrix Q are given by

To obtain nontrivial d u e s of

A, and B,, the fkequency p must be chosen so that det Q@,R) = 0.

(4.30)

For each nt the frequency equation (1.30) has an infinite number of solutions. responding to each root

pnk,

Cor-

the factor Bnr c m be found and it defines a mode shape

(Ud,iVd).These mode shapes are important in that they will be used to construct solutions to the general forced vibration problern.

Xote that the solutions of the free motion problem are of the form u = [

, IT lLp

where bot h u and u are real functions. Thus, for the rernainder of t his paper. attention will be confined to functions of this form.

4.4 4.4.1

Ort hogonality Properties Derivation of Orthogonality Conditions

In this subsection: the orthogonality properties of the eigenfunctioos will be derived. To this end, let us now write the equations of motion in state-space. Let

Then equation (4.1) c m be rewritten as

This can further be rewritten as

This becomes

w here

CHAPTER4. LINEARIN-PLANE VIBRATIONS Let

be a vector in state-space. Again, note the special form of the constituent vectors pl

and w, are al1 real functions. It then follows that

=

1' O

where the ~e~adjointness of

[(w;IT~ n F h+ a2(w;lTPI + ( w ; ) ~

rdr

Ln and the special form of pl, p2, WI and wt

have been

employed. The self-adjointness of the operator Ln is demonstrated in Appendix D. In particular, we used the fact that

For the above to be tme, recall that we require the pj and wj to satisfv the boundap conditions and equation (D.8) so that the Ln is a self-adjoint operator. Using the same

argument,

rdr

~ ; ) ~p (2 w ~ ) ~ ~ rdr R D ~ ~

rdr

(4.4 1)

Hence A, and

B, are self-adjoint opetators on the statespace of Functions that satisfy

the boundary conditions. equation (D.8)and are of the form

Yow suppose that

in state-space so that w = -zpw. For the free vibration problem then =L,w - B,w = O becomes -zpA,w

- B,w

= O.

This can be expanded as

If we put O2 = -ipOl then equation (4.43) is satisfied and equation (4.44) reduces to

which is exactly the same as equation (4.1) with F = O. u = B ~ ( T ) ~ - ' Pand ~ the operator

L replaced with Ln.

This indicateรง that the function

01

is exactly the eigenfunction

of the in-plane problem written in normal space (that is, not in state-space). @ . *=

[

4.k

-2~nkm.k

1'

Let

be the state-space eigenfunction corresponding to the kt h

eigenvdue of the nth operator. That is.

Consider the following :

where we have used the seKadjointness of

An. Sirnilarly,

CHAPTER4. LINEAR IN-PLANE VISRATIONS Subtracting equation (4.47) ÂŁrom equation (4.48) yields

T ~ Ufor S j # k and distinct pnj,pnk we obtain

Furt hermore. for nonzero eigenvalues, we also obtain

fore, for j # k

This implies for j # k

CHAPTER4. LINEARIN-PLANE VIBRATIONS where

Similady, O = (a,,

&ank)can be evpanded to read

where

Rom the self-adjointness of Ln,it follows that

= ilkj. Furthemore. if the @M are

( unJ iL'k3 ) .

1

in the space of functioos of the t o m

then by syrnrnctry

If j and k are interchanged in equation (4.56) and the symmetry relationships (4.58) and (4.59) are also employed, then we can also mite

CHAPTER 4. LINEARIN-PLANE VIBRATIONS Adding equation (4.56)and equation (4.60)gives

Hence for j # k and nonzero eigenvdues, it lollows that Djr = O. Equatioa (4.60) then implies that

Hence for distinct eigenvalues, it follows that

This last relationship in conjunction with equation (4.53) implies that For nonzero eigenvdues and j # k that

ujk

= O. In summary, for nonzero, distinct eigenvalues and j # k.

the following relations are true : .Ajk = O. a j = ~ 0.4, = 0.

It is aiso useful to consider how to normalize the eigenfunctions. Clearly. we c m normalize iljj,ajj or ,Ojj,although they are not independent quantities and care must

be taken in the normalization. Let us consider how these quantities are related. From equation (4.1) the kt h mode-shape must satisfy

Premultiplying this equation by #ikr and integrating fiom r = O to r = a yields

CHAPTER4. LINEARIN-PLANEVIBELATIONS

102

Interchanging j and k and using the known symmetry relationships, we can write

Subtracting the two above equations yields

Thus, this yields the following orthogondity relationship

Yow if equations (4.65) and (4.66) are added, it may be shown using equation (4.67) that

Equations (4.68) and (4.69) provide the necessary relations between rl,,, 3,) and a,,. For the case of a stationary disk,

R = O and simpler relationships can be used. In this case.

equation (4.65) becomes

Interchanging j and k and using the fact that Ajk = Akj,aj&= a k j . then we c m also mite

Subtracting the preceding two equations yields the simpler orthogonality conditions

4.4.2

Eigenfunction Expansions

Given the above orthogonality relationships, arbitrary functions cm be expanded in tems of the eigenfunctions. Consider expanding an arbitrary function G ( r) as

To find the coefficients h k , prernultiply both sides of the above equation by @, integrate from r = O to r = a.

The 1st line follows since ajk = O for j # k. Hence, the coefficient Gj is given by

r

and

4.5

Forced Vibration Analysis

We now consider the problem of forced in-plane vibrations. The forcing function F cm

The response of the disk to this forcing function c m be found by finding the response of the disk to a typical harmonic Fn(r,t)e'n8 and then using the principle of superposition.

In other words.

where u,&"'

is the response of the disk to F,(r, t)etnB. By considering the 6 dependence

of the response to be einBand summing over n,the differential operator L can be replaced with

Ln, and u(r,8, t) can be replaced with &(r, t). Thus, the general problem becornes

one that depends on r and t only.

Now consider the solution of this 'simpMedr forced in-plane problem for u, using the free motion mode shapes and using the laplace transform. Recall that the laplace transform of a n a r b i t r q function q ( t ) is given by

It is ais0 useful to note that

The PDE in question is given by

with initial conditions

Let & = L(&(r, t ) ;p ) = %(r, p) be the laplace transform of &(r, t). Similarly. let Ê,' = L(F,(r. t ) ; p ) = ~ , ( r . ~be) the Laplace transform of F,(r. t ) . Taking the laplace transform of equation (4.82) yields

Ln% = p2û, - p h (T, O ) - ii,(T, 0)-R*& 1+2QpDû, - 2QDu,,(r,O) - -F, P

Now suppose that an expansion for û can be found in tems of the eigenfunctions. That is, suppose that

Note that to find the solution to the equation (4.82), it sufEces to find the coefficients

in the expansion. In the above equation, the &@)

are the coefficients of the expansion

of &, the laplace transform of h,not those of u, itself. ClearIy the k k ( p ) are the laplace transforms of the ~ ~ ( which t ) , are the coefficients in the expansion of u, itself.

The use of the laplace transform only affects the time domain. The space domain is unaffected by the laplace transform and thus the eigenfunctions can still be used for the space domain.

Since the forcing huiction F. is known, its expansion in terms of the eigenfunctions can be explicitly found:

where

Sirnilarly, because the initial conditions are known. expansions for those Functions can also be found :

where the constants g,k and

Lkc m be found fkom

Substituting equations (4.86), (4.87) (4.89),and (4.90)into equation (4.85) gives

Premultiplying both sides of the preceding equation by

c&

r and integrating frorn r = O

to r = a, then using the orthogonality property of the eigenfunctions and rearranging the resulting equation for &, yields

Observe that E,,, is the sum of two terrns.

The first term is the response of the system

to the initial conditions. The second term is the response of the system to the forcing.

As is well known, the response of a Linear PDE can be considered as the sum of the responses of two problems : one with initial conditions and no forcing and the other with zero initial conditions and with a nonzero forcing function. This is in fact reflected in equation (4.93). Let us write

S=

<: +cj,where cf:

right hand side of equation (4.93) and side of equation (4.93). Thus,

<y

represents the first term on the

6 represents the second term on the right hand

is the response of the system to the initial conditions

and Cj is the response of the system to forcing.

Note that if

fnj

= O then

cj= O.

That is, the forcing only excites the çame modes

that are present in the forcing huiction. If the k j are known, then ĂŽ, is known. To

find u, itself, the inverse transform of equation (4.93) is required. To End the inverse transform of equation (4.93),the following t hree facts are required:

Convolution Theorem

Inverse Laplace Transform

Shift Property

With these three facts, the inverse transform of &, cm easily be found. coefficients k

j

Once the

are known' then b,and thus the solution to the generai forced vibration

problem is known.

CHAPTER4. LINEARIN-PLANEVIBRATIONS

4.6

Applications

The above formulation can be applied to several problems to illustrate the solution met hod.

4.6.1

F'ree Vibration Response Under Initial Conditions

To solve the general free vibration problem with general initial conditions. it suffices to find the inverse Iaplace transform of expression for

Cn).

Csing equations (4.68) and (4.69). the

4:can be written as

(4.91)

Note that the free vibration problem for u, results in a harmonic term of frequency p,, .

as would be expected. Ln addition, a term of fiequency pnJ from its definition, a,, must be a positive quantity.

& is d s o present.

Therefore,

p,,

1 -is a11

Clearly. a lower

Erequency than pn3. If p,, = O, then the displacement corresponding to t hat frequency becomes fixed in the disk.

Let us suppose that pnj = O at R = Ra. From equation

(4.68),the value of Ro can be found to be Ro =.'

Similarly, if pnl - -i-= O. then the

2 a ~ ~

033

displacement correspondhg to that fiequency is sĂ&#x17D;milarly Lxed in the disk. Define R I to be the spin rate when pnj

is found that SIi =

-A = O. a33

Again, using equation (4.68) to solve for Ri. it

= -ao.

'In essence, the mathematics shows what is intuitively S J J

obvious : any phenornena that happens at a certain spin rate R should dso occur at -R.

In other words, fiom symmetry considerations it should not matter if the disk is spiming clockwise or counterclockwise.

Consider the special case for the stationary disk. where R = O. Here. instead of using equations (4.68) and (4.69) to sirnplifjr the expression for E,:. equations (4.72) and (4.73) are used instead.

This yields

The inverse laplace transform can easily be found to be

It is interesting to compare

4:for the stationaq disk case and for the rotating disk case.

For a stationary disk, u, contains harmonic terms of frequency pnJ only. However. For

the rotating disk u, results in harmonic terms of frequency pnj and terms of frequency Pnj - G i

4.6.2

Forced Vibration Response under Special Forcing Condi-

tions Using the convolution theorem and the expression for Cjtit can be s h o w that

CHAPTER 4. LINEARIN-PLANE VIBRATIONS where

Note that g(t,u) = e - i ~ n j t-

=-(p.,

-L)i(t-U) J

is actudly the green's function for the prob-

lem.

Force of Constant Magnitude -4s a particular example, suppose that the forcing function is of constant magnitude as would be the case if there were a srnail asymmetry in the mass distribution of the disk.

In this case. ,.f are constants and it can easily be shown from equation (4.100) t hat

O I ~

1j

j

+

j

ajj

1 - Pnjajj

Clearly two resonance conditions are possible. One resonance condition is when p,, = 0.

As previously discussed above. from equation (4.68) it can be shown t hat such a condition occurs at a value of R which can be found to be

' 0.

4=1 .

Similady. if p,, 24, a11

=

then another resonance condition is established. Again, using equation (4.68) to solve for the corresponding value of RI, it is found that RI = -1 - -4. This is also physical24,

intuitive ; if resonance is established a t some particular value of spin rate: then it would f o h w that resonance would also occur if the disk were spun at the same rate in the opposite direction.

Harmonic Excitation Rotating with the Disk As another simple euample, consider the case of harmonic excitation that rotates wit b the disk at a frequency W . In this case,

It then follows that

where

Clearly, resonance conditions occur at w = f

pnj

(

=;,)-

and at w = f pnj - -

Let

us suppose that Q = C& when w = f pnj. From equation (4.68), it can be shown that

=

'f2wQis. 33j1

Substituthg the other resonance condition w = & ( p n j

-

J-)into

equation (4.68) and solving for the corresponding value of R = QP, it can be shom that

= -4, as wodd be expected.

As before, the solution predicts that such physical

phenomena is independent of the direction of rotation.

Once again, let us consider as a special case the stationary disk for which R = 0.

Using equations (4.72) and (4.73) to simplify the expression for

G,we obtain

The convolution t heorem yields

If the excitation is harmonic S U C ~that f n j ( u ) = s,sin(wu) + bnj cos(wu). t hen

The resonance conditions for this case are w = kpnj. Furthemore. the only frequencies present in the response to the simple harmonic excitation are clearly pnj and

&.

That is.

the response contains the natural hequency of that mode and the excitation frequency.

In cornparison, the frequency content of the response for the spinning disk is p , , . ~ and pl

- &.

Recall that for the initiai value problem, the frequencies p,,

are present in the response.

. and p,,

-

, 1

Hence, the harmonic forcing only introduces the forcing

frequency into the response, as would be expected.

Harmonic Excitation Fixed in Space As mothier simple example, consider the case where the body force acting at a point on the disk is fked in space. In this case, the forcing h c t i o n can be written as

where 8 = O + Rt. Thus to consider the response to a typicd harrnonic. consider the response to fn (r.t)einnt

. This case proceeds as in the previous example except t hat norv

we must write

Once again, it can easily be shown From equation (4.100) that

where

Here, resonaace occurs at

pnj

=

-nR iz w and at

pnJ - 2 = QJI

-(no f J).

hgain.

equation (4.68) can be used to find the values of $2 at which these resonances occur. At pn, = -nR k w it can be shown that

Similady, at pwajj = 1 - a j j(ni2I u ) ,the corresponding value of R can be found to be

once again demonstrating the independence of the phenornena to the direction of rotation.

4.7

Simulations

Simulations were performed to calculate the naturai fkequencies of the in-plane vibrations for the case of n = 0, the symmetrical case. For ease of computation. the approach

of Bhuta and Jones [19]was used.

In the foIlowing, the actual fiequencies of vibra-

tion can be found fiom the dimensionless fiequencies p~ by mdtiplying the dimenionless

frequencies by the spin rate R. Note that here we are using pi, to denote the dimensiodess natural frequency when previously it was used to denote the actual frequency.

This should not cause any confusion if one remembers how the dimenionless and actual frequencies are related to one another. The dimensionless rotation rate ii is defined as

where recall that a is the radius of the so端d disk. Similarly, P is defined by

For completeness, the frequency equation as given in Bhuta and Jones [19] is @en

where

by

The in-plane eigenfunctions can then be found from

The ratio of Ci to

C2can be found kom ClklrnI J2(kl)+ C2k2m2 J 2 ( b )= 0 .

To find Ci itself, the normalization condition that relates a,, to 3;,

is used

2Rpn,~jj+ 2R4,, = t.

In this manner. the natural frequency of vibration of a particular mode. poJ ccan be bund.

Referring to the section on the free vibration response. the ot her dimensionless nat ural irequency (the 'backward' frequency) can be found Erom

(2)

Pnj

1 - Pnj - -Rajj

The results for the first, second and third natural frequencies as function of dirnensionless angular velocity 6 are given in figures (4.4) to (4.3). Bhuta and Jones also solve for poJ in their paper (191 and the results obtained in these numerical simulations exactly match

their resdts.

It can be seen that the natural frequencies p~ tend to decrease with

increasing angular velocity.

The other (backward) frequency tends to have opposite

sign (hence the backward designation) and its magnitude dso tends to decrease with

uicreasing frequency.

These results are dso in-line with those obtained by Burdess.

Wren and Fawcett [21].

Figure 4.1: First Dimensionless Natural Frequencies pal and Backward Frequencies pal vs dimensionless anguiar velocity a for u = 114.

Qtl

Detailed orthogonality properties of the in-plane eigenfunctions are derived by cecasting the original problem in state-space.

端sing these orthogona端ty properties. a general

analytic solution to the in-plane forced vibration problem c m be constructed.

The

details of thk construction were given. This general solution can be easily appIied to solve particular problems. A generd solution to the free vibration problem with general

-15l

O

l

0.5

I

t

1

1.5

a

2

1

2.5

DimensionlessAngufat Velocity

Figure 4.2: Second Dimensionless Naturai Frequencies po2 and Backward Frequencies -J - vs dimensionles angular velocity 6 for u = 114. a?:! initial conditions is presented for bot h spinning and stationary disks. While For stationary disks each mode-shape is associated with only one frequency, b r the spinning disk each

mode-shape has two possible frequencies associated wit h it. Responses to simple forcing functions are also considered. Constant and harmonic excitations rotating wvit h the disk

and stationary in space are considered. The aforementioned methodology permits the generd solution to these problems to be easiiy derived. Resonance conditions associated with these excitations are considered. Although it is physically intuitive. it is shown analytically that the resonance conditions are independent of the direction of rotation of the disk.

Third lnplane Natural Fmuencies vs Angufar Vdocity for nu=1/4 -l

-frequency

-25

I

O

0.5

I

1

1 15 Oimensionless Angular Velocity

1

2

2.5

Figure 4.3: Third Dimensioniess Naturai Frequencies Po3 and Backward Frequencies p03 vs dimensionless angular velocity o! for u = 114.

033 First lnplane Naturai Frequem'es vs Anguiar Veldt'y for nu=l/Z IOl

I

l

1

1

-frequency

t

05

Y

1

I

I

1.5

2

l 25

Dirnensionless Angular Velocity

Figure 4.4: First Dirnensionless Naturai Frequencies p01 and Backward Frequencies pot vs dimensionless angdar velocity fi for v = 1/2. at1

Secand lnphne Natufal Frequencies vs Angular Velocity for n e 1 M

Figure 4.5: Second Dimensionless Natural Frequencies po2 and Backward Frequencies - vs dimensionless angular velocity a for v = 112.

p02

Third Inplane Natural Frequenciesvs Angular Velocity for nu=112

-frequency

- - backwardfrequency

-20 1 O

1

t

1

I

1

O5

1

1.5

2

2.5

Dimensionless Angular Veiocity

Figure 4.6: Third Dimensionless Natural Frequencies po3 and Backward Frequencies po3 vs dimensionless angular velocity Ă ! for u = 112.

433

Chapter 5 Nonlinear CoupIed Vibrations Introduction Recall that in chapter 2. equations For the nonlinearly coupled vibrations of a spiri-

ning disk were derived using Hamilton's principle and the von Karman expressions for nonlinear strain.

In this chapter, an approximate solution to these nonlinear coupled

vibrations is considered and anal-yzed. The main focus of this chapter is to inuestigate the effect of including the in-plane inertia on the resulting dynamics of the problem. This is done by representing the solution as one mode for each of the transverse and in-plane

vibrations and assurning a form for the space dependent part of the solution. The timedependent parts of the problem is then solved for. Since the in-plane inertia is incitided. the unknoms (degrees of freedom) are then the time dependence of the transverse vibrations and the time dependence of the in-plane vibrations. Through this method. three

nonlinear partial differential equations become condensed into two nonlinear ordinan; dinerential equations, for the t h e dependence of the transverse and in-plane vibrations.

The two degees of fkeedom are the tirnedependent portions of the in-plane and trans

verse vibrations which are related through the two coupled clinerential equations. This two degree of freedom model c m be used to determine the effect of the inclusion of inplane inertia on the dynamics of the problem. The thrust of this chapter is to derive the simplified two degree of freedom model, to consider analytical and numerical solutions and to compare these solutions in order to gain some insight into the dynamics of the nonlinear vibrations. The determination of the effect of the inclusion of the in-plane inertia on the dynamics of the problern is also an important goal in this chapter.

5.2

Simplification of Solution to a 2 DOF system via Galerkin's Method

5.2.1

Strategy of Solution

To consider the effect of the inclusion of in-plane inertia. we choose to proceed with Galerkin's procedure to produce a simplified problem. The result will be a

two

degree

of freedom mode1 - one for each of the time dependences of the in-plane and transverse vibrations. When the in-plane inertia is included, the time dependence of the in-plane vibrations is an independent quantity that must be taken into account. Thus. the result is two coupled nonlinear orĂ hary differential equations for the time dependence of the transverse and in-plane vibrations. To this purpose, let us choose the displacements as

~ ( 6, r ,t) = u&)

+ U(r) cos(m8) c ( t )

u(r,0, t ) = -V(r) sin(rn0)c(t) w (r, 0, t) = W (r)cos(n8) ~ ( ,t )

where m and n are integers and u,,, U,V and W are lmown and will be discussed moment d y . The only unknowns here are the time dependences, c(t) and ~ ( t )The . u,, portion

is the in-plane equiiibrium displa~ementdue to the spin of the disk. The implication here is that there is a symmetrical tirneindependent in-plane displacement that occun in the plane of the disk as a result of its rotation (that is, due to the centrifugai force).

The form for u and u follows from equation (4.24),which gives

Taking the real part of equation (4.24) leads to the forms for u and u indicated in equations (5.1) and (5.2). Now. U and where Umk and

vmkare the

ç'

will be taken to be LLkand

lkk respectively.

(m, k)th in-plane mode shapes found by solving the

Iinear in-plane vibration problem.

kee

Other choices for U and V can also be made and

we are not Iimited to choosing them the be the in-plane mode shapes.

However. the

advantages of taking U and V to be the in-plane mode shapes are that ( i ) we have already proven orthogonality properties for the in-plane modeshapes and more importantly (ii) the in-plane mode shapes s a t i e the boundary conditions rr, = are = O at the boundary of the disk. Similady, W will be taken to be Ă&#x2021;Vnk,where bVnk is the (n. k)th mode shape of the h e a r strain model of the spinning disk. That is,

where

Pak, and ekare the (n,k)th solutions of the frequency equation for the Linear

strain model of the spinning disk as given in equation (3.25). R e d that the appropriate

relationship between

PO

and

must be specified.

Other choices for the space-

dependent portion of the solution can easily be made and substituted into the above expressions.

Once a choice for the space dependent part of the solution has been made. it still remains to use this choice to derive the equations of motion for the temporal part of the solution. One option is to substitute the assumed forms of the solution into the equations of motion and then apply the method of Galerkin. A completely equivalent calculation is to substitute the assumed form of the solution into the expressions for kinetic and potential energies. Since the spatial form of the solution is known. the integation can be performed explicitly over space. The kinetic and potential energies are thus reduced ) in equations (5.1). (5.2) and (3.3). that is ive have a 2 to functions of c ( t ) and ~ ( tgiven

DOF system. Lagrange's equations (which is equivalent to performing the variation OF the Lagrangian on the continuous system) then yield the two equations of motion that are sought. The advantage of actually calculating the potential and kinetic energies of the system. as opposed to directly using Galerkin's method is that conserved quantities

are more readily apparent. Such conservation laws can be used to simpli- the equations of motion and could aid in the fmding of an analytic solution. Furthemore. the machinery of Hamilton's equations and canonical perturbation theory is also available in this formulation. For these reasons, the simplified 2

DOF ordinary differential equations of

motion are calculated by f%t constnicting the kinetic and potential energies, integrating over space and then using Lagrange's equations.

CHAPTER5. NONLINEAR COUPLEDVIBRATIONS

5.2.2

Galerkin Method

Recall that the linear stressstrain expressions, assuming a condition of plane stress are

given by

and that the von-Karman nonlinear strains are given by

The potential and kinetic energies can then be found From

where the integration over z has already been incorporated into the expression for the

khetic energy and KEI and KE3 are given by

Once the integrations over z. B and r are performed, the resulting expressions for the kinetic and potential energies become simple functions of c and r and resemble the expressions of a two degree of freedom system. Indeed, by taking only one modeshape for each of the in-plane and out of plane vibrations. we have restricted the system to having only two degrees of freedom. The expression for the kinet ic and potential energies in terms of the two degrees of freedorn, c and r are given by

The coefficients in equations (5.16) and (5.17) c m be expressed as

where

Here oz,oii are the equilibrium stresses in the rotating disk resulting from the equilib-

rium displacement u,,

.

Q = phn

Qio = O

lu

+ 1")rdr

([i2

for rn # 2n

hE7r Qio = 2(1- u2)

(3.29)

$, {Y [nzT

W

&V (2d;

PV

)i

-

&V

&V

+ v ; i ; (d;- 2n2E)] r

The above calculations indicate that Ql0 = O for m #

272.

(5.30)

This point is worth some

consideration since it considerably affects the resdting dynamics if QIo is indeed zero.

In this case, Qio = O foIIows as a result of our choosing trigonometric hnctions for the 0 component of the t n d soiution to the problem and the fact that the sine and

cosine functions possess orthogonality properties. A slight ly different choice for the 8 component of the solution would not necessarily lead to Qla = 0.

Note that for a stationary disk, where

R = O, it would follow that Q4= O unless the

disk is prestressed and that Qs = Qs = Qg = 0. -411 other expressions are the same for a stationary disk, although the actual mode shapes used in these expressions will be different.

Once expressions for the kinetic and potential energies are found, the Lagrangian

can be constructed and the equations of motion for the simplified system can be found.

Given the above expressions For kinetic and potential energies. the Lagrangian For the '2

DOF system can be written as

Since L has no time dependence, then

i g + èg - L is a constant for the system.

Let

us denote this constant by hl so that

That is, the quantity M is conserved, where iCI is given by

= QT?

f

Qat2

+ ((Q4 - Qg) + (QI- Q5)2 + Qg4 + (Q3- QG)13 + Q ~ ~ C ?(5.34) .

Note that the quantity .VI is not the total energy of the system as given by PE

+ KE.

However, suppose we d e h e the effective kinetic and potential energies of the system as

then Ad is indeed the sum of PEeff and K Pf f.

The equations of motion for the system can be found from the Lagrangian and by using Lagrange's equations :

For the variable c this gives

C +(QI- Qs) c+&a

Qio 2Qs

T2

=O.

Similady, Lagrange's equation for T is given by

Now suppose that new variables are introduced such that

CHAPTER 5. NONLINEAR COVPLED VIBRATTIONS Then the Lagrangian Eom equation (5.32) can be written as

where

Thus the equations of motion for the system becorne

In future discussions, f and r* WUbe replaced with c and r for ease of notation.

It is important to note that if Qio = O then kt = O and the two equations of motion (5.47) and (5.48) are uncoupled

That is, for QIo = 0, the -stem

behaves like two

independent oscillators, where the in-plane oscillator is Iinear and the transverse one is a

nonlinear duffing oscillator. What are the physical implications of this? Recall that to arrive at equations (5.47)and (5.48), two particulair modes were selected at the outset.

If kI = 0, the implication is that the vibration of these particuiar modes are not coupled.

Since there are an i n h i t e number of in-plane and transverse modes of vibration, this

would not necessdy imply that none of the in-plane and transverse modes of vibrations are coupled. However, it does not follow that al1 the transverse modes are coupled to a l l the in-plane modes and t his is the physical implication of

5.2.3

kl = 0.

Physical Interpretation of the Coefficients

It is clear from the preceding section that the expressions for the

Qi

are complicated.

However, upon closer inspection of these expressions and the expressions for the effective kinetic and potential energies given by equations (5.35) and (5.36),some physically meaningful interpretations may be obtained. Consider equations (5.35) and (5.36). restated here for reference :

It is obvious that these expressions resemble those for two coupled harrnonic oscillators.

Here, Qi and Qsare the effective masses of these oscillators. Examining the expressions for Q7 reveals a few things. First, Q7is a result of the transverse vibrations. thus the fact that is it the 'effective m a s yfor the T oscillator makes sense. Second. no rnatter the forrn taken by Ă&#x2021;V, Q7 will always be positive, which is consistent with its interpretation as a mass t e m . Third, Qi has a h3 component and h component. Ă&#x2021;Ve c m think of these as a 'plate' term (h3) and a 'membrane' (h) term. This is relevant since we could

in theory consider h to be so çmaii that h3 is negligible. For our mode1 to be physically

meaningfd, we still need it to have an effective m a s , even if we are ignoring the plate terms and we see that this is indeed the case.

Examining Qsreveals similar insights. Narnely? Qgis always positive and is a result of the in-plane vibrations. This term stays the sarne regardless of whether h is small or not

- that is, if we were modelling a membrane, this term would still be present.

These

observations are consistent with Qgbeing the effective m a s for the c oscillator as seen Erom equation (5.35).

Yext, let us examine the effective stiffnesses of the osciilators. Clearly. from equations

(5.36) and (5.16), it is obvious that QIis part of the stiffness of the c oscillator. From equation (5.18), we can see that QIitself is purely a result of in-plane terms and this is consistent with it playing the role of part of the stiffness of the c oscillator.

But

from equation (5.36): w e see that the 'effective stiffness' of the c oscillator is actiially

QI- Q5. Yow from equation (LI?), we see that Q5is actually a hnetic energy terrn.

Furthemore, from the defining equation for Q5,equation (5.24), we see that Q5is a spin

(R2) term and is a result of the in-plane tems. If the disk were not spinning. this term would not be present. rUthough this term is actuaily a kinetic energy term. is behaves

like a potential energy term and has the effect of reducing the effective stiffness of the c oscillator. This is consistent with previously made observations that the effect of spin is to lower the linear in-plane frequencies of a spinning disk [19,211. oscUator, uncoupled to the

T

For a linear c

o s d a t o r , its squared hequency would be given by the

ratio of its effective stitfness over its effective m a s . Thus, reducing its effective st iffness reduces the fiequency of vibration of the c oscillator.

The same comments may be made regarding Q3and QG, except that some additional

comments need to be made regardhg Q3.Here, Q3 is the actual potential energy term.

as seen fiom equation (5.16). Qsis the actud kinetic energy term as seen from equation

(KU), but it is a positive spin (a2)term and serves to reduce the effective stifhess of the T

oscillator. From equation (5.25), it may be shown that

Q6 is actually directly related

to the bending moment due to the rotation of the disk term that was derived in Chapter two. Thus, we can now see why this bending moment due to rotation term had the effect

of lovering the frequencies of vibration for the linear strain model. Again. although it is a kinetic energy term, it serves to reduce the effective stiffness of the oscillator and thus reduces the linear vibration frequencies of the T oscillator. Now. the QJterm bean hrther investigation.

It should be clear that increasing Q3 will serve to increase the

linear naturai frequencies of vibration of the r oscillator.

Now. examining equation

(5.20) reveals that Q3itself is made of two parts, one part being a 'plate' (h3) term and the other being a 'membrane' (h) term. From equation ( X I ) , it can be seen that the plate portion of Q3is the result of the bending stiffness of the disk. This contribution is purely dependent on the W terms and would still be present if the plate were not spinning

(R = O) or if the plate had no in-piane equilibrium displacement equation (5.22), it can be seen that the membrane portion of

(zu.,

= O).

From

QIis actually dependent

on the transverse deflection ( W ) and on the in-plane equiiibriurn stresses in the disk (O~,O;:).

For a spinning disk the equilibrium stresses are the result of the spin of

the disk, however, it is possible to pre-stress a disk and these equilibriurn stresses do not necessarily arise as a consequence of the rotation.

In fact, for a regular stretched

membrane, the 'plate' portion of Q3wouid not be present and the only contribution to the effective stifhess wodd be from the membrane stresses. However? the most important

observation to make here is that the membrane portion of Q3resulting from the spin of the disk serves to increase Q and thus increases the effective stiffness of the T oscillator.

This in tuni increases the Linear frequencies of vibration of the r oscillator. This is in agreement with observations made in previous chapters and in previous works [II. Thus. we see that the spin of the disk aEects the frequencies of vibration in several ways. Fint. the in-plane equilibrium stresses add to the effective stifhess of the disk. t hus increasing the Linear hequencies of vibration. However, the spin term that gives rise to the bending moment due to the rotation of the disk serves to decrease the effective stiffness of the

disk, lowering the frequencies. -4lthough this term is kinetic energy term. it can be considered to effectively be a potential energy term. From previous chapters. ive know that the effect of the in-plane stresses (raising the frequencies) is more pronounced than the effect of the bending moment term (lowering them) and the net effect of rotation is to raise the overall vibration frequencies.

The rest of the coefficients provide less insight? but are worthy of comment. First.

Q2is clearly the coefficient of the nonlinear T term. From equation (5.19). it can be seen that it is purely dependent on transverse t e m s and that they are OF fourth order. these

observations being consistent with the fact that Q2is the coefficient of r4 in the effective potential energy. Q4and Qsare also not very interesting. They are purely dependent on the in-plane equilibrium displacement u,.

From equation (5.36), it may be seen that

they d icontribute to the overall potential energy of the system, but wi-ill not change

anything in the equations of motion. In other words, by choosing our reference potential energy Ievel dserently, these tems can be made to drop out of the potential eneru expressions. Qlo serves to couple the two oscilIators. From equation (5.31), we can see

that it depends on both the in-plane and transverse terms, which is consistent with it being the coefficient of

d in the potential energy term.

Furthemore, under certain

conditions, it is possible for Qlo to be zero. What does this mean? If Qlo = O. this says that the two particular modes that we have chosen to represent our 2 DOF system are not coupled and will vibrate independently. In other words, not al1 in-plane modes are necessarily coupled to al1 transverse modes, which seems reasonable. modes are chosen and they do not vibrate independently, then

5.2.4

However. if two

Qio will not be zero.

SimpliQing the Coefficients

The following calculations axe tedious, but they show rigorously that exactly what we would expect them to be

J,

and s, are

- namely the frequencies of vibration of the

linear in-plane and transverse vibrations.

Let us first consider QIto see whether or not this expression can be simplified. Recall t hat

QIis given by

The following facts are required to simplify the expression for QI:

rdr = ru-1; dr -

l d u d2u [(;d; + =)U rdr

(

dV

V+ r

r

rdr =

la(s)2

idr - La2-dV dr

(-Lrr -

m:)

rdr

Using these expressions. QIreduces to the sum of a boundary term and an integral term

w here

boundafil - O by virtue of the boundary conditions a , = are = O at r = a. If Note that Q,

= LImk,V = VJ, is substituted into Q,integral , denoting the (m, k)t h modeshape. then

where recall t hat

Now, using the orthogona端ty properties of the in-plane modes from the previous chapter.

for a spinning disk and that

CHAPTER5- NONLINEAR COUPLEDVIBRATIONS for a stationaxy disk. Thus for a spinning disk, wa c m be written as

and for a stationary disk, wt cm be written as

In other words, w, is precisely the frequency of vibration of the in-plane mode that was

chosen. This is a naturally intuitive result, which the caIcuIations justify.

Let us now consider how to calculate w,.

Recd that w, is defined by

That is, w: can be considered to have two components ; a 'plate' part where al1 terms are proportional to h3 and a 'membrane' part which is proportional to h. ClearIy the ~ ' - ~ h thenmembrane d part is g誰ven by plate part is given by ~3Q7

$.

Let us Bnt consider the plate portion of w,. Fi& consider f:

[(%)2

1

n2W* +7 rdr,

which occm in both Q6and Q7.This can simplified using integration by parts to give

Now consider the h3 component of Q3,namely ~f :

=

la

V ~ W V:W rdr + 2(1-

U)

&V

bV

dLW

l&V

L ~ { ; ( X - ~ ) (x~ - Z2 ; ) )

where we used the fact that

Let us first consider the boundary condition at r = O. If we take LC' functions of the linear strain mode1 of the spinning disk, then

to be

the eigen-

rd.

It follows by using the properties of Bessel functions that

since Jn+I(O) = I,+l (O) = O for n 2 O. Thus at r = 0, the boundary condition becomes

The boundary condition at r = O is zero since the above expresion is zero for n n = 1, and since for n

> 2,

= O or

% = 0.

Thus the boundary condition at r = O disappears and we are left with the boundaq condition at r = a. which can be simplifieci using the two boundary conditions (E.4) and

where Ank is the (n,k)th frequency of vibration of the linear strain mode1 of the spinning

disk. Thus for numerator of the plate component of w,, we obtain

CHAFTER5. NONLINEAR COUPLEDVIBRATIONS where

Gkj=

Hkj =

Jrawnkv:wnj

rdr

lawnkv:

Wnjrdr

Using equation (E.10),which is basicdly the equation of motion restated, we can simpli-

the above expression hrther to

From equation ( 5 . 7 4 , we can m i t e

Hence, it easily follows that the plate portion of w: is given by

That is, the plate portion of wf

is the square of the fkequency of the linear strain

model of the spinning plate. Considering the fact that the Wnr were chosen to be the modeshapes of this model, this is an intuitive result.

Xow consider the membrane portion w:.

-Q3=

Q?

"

J:

g {oz (F)* + oep rdr [($)' + $91rdr + ph* :ĂŽ CV2 rdr

rh

eq n2W2

} - 7 ,

'

Now, for W = Wnkywhere the Wnkare the eigenf'unctions of the h e a r strain model of the spinning plate, the above expression does not permit as nice an interpretation

aรง

the

plate component of W: did. However, for the sake of illustration, suppose that in equation

(5.91) we take

= W,L, the eigenfunctions of the iinear spinning membrane.

First.

the term proportional to h3 in the denominator of equation (5.91) would not be present for a membrane. Thus, the denominator becomes phn

5; kV2 rdr. Furthermore. if the

eigenfuoctions of the spinning membrane are normalized, then using equations (F.3) and

(F.4), equation (5.91) reduces to

that is, the square of the natural frequency of the (n, k)th mode of the spinning membrane. Again, this is a naturai result. However. in this developrnent. it is the eigenfunctions of the linear strain mode1 of the spinning disk that are being used and thus the rn membrane component of urf will not give exactly (A,+)

2

but rather some approximation

to it.

Yote how this development further confirms the previously used result that to a p proximate the natural frequencies of vibration of the linear spinning disk model derived from nonlinear strains, it d i c e s to take the square root of the sum of the squares of the two component problems ;the spinning membrane and the linear strain model of the spinning disk. That is,

where LSM refers to the linear strain model and NLSM refers to the nonlinear strain model. the eigenfunctions of the As another note, suppose that we had taken W = WiILSMo nonlinear strain model. Note that these eigenfunctions are not actually available to us. but it is instmctive to see what would happen is we codd use them in this calculation.

A quick review of the above developments would indicate that the result to be expected is u: = ( A ~ , L ~ " .)

5.3

Analytical Solutions of the Simplified 2 DOF System

The coupled equations of motion derived in the above sections are given by

Clearly, w, and w, are the natural frequencies of the h e u r in-plane and transverse modes that were chosen. The expressions for kl and k3 are complicated. but it suffices to note that they will depend on the amplitudes of the chosen modes.

For certain choices of

modes, it is possible that kl = O, which indicates that the in-plane and transverse modes in question are uncoupled and will vibrate independently. However. for other choices of

modes, kl w i l l not be zero and the in-plane and transverse vibrations will be coupled.

Equations (5.94) and (5.96) d r i l be referred to as the two degrees of fieedom (2 DOF) model since the equations of motion are w e n t i d y those of a nonlinear two degree of

Ereedom systern. Thus, current work considers equations (5.94) and (5.95) to be the fundamental equations describing the dynamics of the system.

Equations (5.94) and (5.95)can also be anived at â&#x201A;Źrom the point of view of Hamilton's equations. That is. equations (5.94)and (5.95)are equivdent to

where p, and p, axe the conjugate momenta to c and

T.

and the Hamiltonian is given by

Equations (5.96)represent a general Hamiltonian system. Since the equations of motion of the 2 DOF mode1 c m be derived frorn a Harniltonian. they are a Hamiltonian system.

The general objective here is to transform the given Hamiltonian systern into a simpler Hamiltonian syçtem and to use the result to constmct an approximate solution.

Generally, a transformation of variables fiom generaiized coordinates q, p to q*. pr can be of the forrn

with correspondhg transformed diaerential equations

Suppose that equations (5.99) have the same symmetry as the original equations. That is, suppose that there exists a function H*(q*, p*, t) such that

Then the transformation (5.98) is known as a canonical transformation [G]. That is, canonical transformations transform Hamiltooian differential equations into another system of differential equations of the same txpe.

One possible approach to solving the equations of motion is to End a canonical t ransformation such that the transĂŽormed Hamiltonian H* is a constant function. that is a function chat does not depend on tirne or any of the generdized coordinates or momenta.

Then equations (5.100) reduce to the simple form of q" = f = O and can be easilj- integrated.

In redity, finding such a transformation can be just as difncult as solving the

original equations of motion. However, it is possible to use a canonical transformation that effectively transforrns a simple Hamiltonian to aid in the construction of an a p p r o k a t e solution for a problem with a more complicated Hadtonian. That is. take a transformation that essentially solves a simple related problem and apply it to the more cornplicated problem. Even though such a transformation will not solve the more

complicated problem, it will help in the construction of an approximate solution. This is the approach that will be undertaken here.

Consider the Hamiltonian of the 2DOF mode1 :

This Hamiltonian can be expressed as

where

The motivation for doing this is that H(O) can be recognized as the Hamiltonian of two uncoupled linear harmonie oscillators. H(')brings in coupling by way of the third order k1 term. The fourth order term in H(I) does not couple the two oscillators. Rather.

it

has the effect of turning the linear r oscillator b t o a nonlinear duffing oscillator. The k l term is the only term that couples the two oscillators.

Thus the

kit

step in our approximate solution will be to End a canonical transfor-

mation that transforms H(') into a constant. This can easily be done by soltĂŽng the Hamilton-Jacobi equation associated with H(O) [45]. Now t his canonical transfomat ion

is equally MĂźd no rnatter wEch Hamiltonian is employed. It just has the particular

advantage of simpliijting this particular Hamiltonian H(O), but it can be used with the

entire original Hamiltonian H. After thus transforrning the Hamiltonian, approximate solutions for the ensuing Hamiltonian equations d l be sought

As previously mentioned, the first step in the procedure is to seek the complete solution S(c,r,p,, p,, t) of the Hamilton-Jacobi equation associated wit h H(O)(c. r. p,. p,) .

Xamely, solve

That is, solve

To achieve this, write

where h = al +a2 is a constant. If equation (5.107) is substituted into equation (5.106). then equation (5.106) can be satisfied if

The preceding equations can be integrated direct- to y誰eld

The required transformation can now be found from

Sirniladl we obtain

T=-

J2a;sin w T ( t - 4 ) .

WT

Furthermore. the generdized rnomenta can also be transfonned

Note t hat equations (5.113) to (5.1 16) are essentially a canonical transformation from

the old variables c: T , p,, p, to the new variables al,a*, 3,. It cm be verified that if (5.1 13) to (5.116) are substituted into H(O),the r e d t is the constant al + a2. This is no

.

co茂ncidence : the transformation was specifically constructed to do so. In fact equations (5.1 13) to (5.116) are precisely the solution to the uncoupled two oscillator problem where al,ai,PL, ,没2 are to be taken as constants of integration. The transformed Hamilt onian

for the uncoupled two oscillator problem is given by H(*) + g, which is precisek zero

(by construction). With the t r d o r m e d Hamiltonian being zero, Hamilton's equat ions

assume a particdarly simple, easily-integrable form. Namely, they give the result that al,q,p,, 6 must be constants to s a t i e the new Hamilton's equations.

Now, while this canonical transformation transfoms

H(O) into zero, it does not do

so for the entire Hamiltonian H = H(O) + H(').The transformed Hamiltonian becornes

H* = H(O) +

么t

+ H(=)= H(~)(LZ~, a*,Pl, p2,t).

Ushg these results, the 'perturbiog

Hamiltonian' H(')c m be expressed as

a2

~('1 =&c

wr4

(-8 k~J2a;w l sin (w,t - wc 3 , )

Note fkom this definition of H(l) that there is a difference in its form depending on whether or not w, = b,. AIthongh it rnay not be obvious yet, if w, = 2w,, then the system will display intemal resonance and

H(')will take on a different fom.

From the above, using the new variables a L,a*?,Ol and &, Hamilton's equations can

aH(') 1

& = -= - a 2 k i 2 - ( - 2

w,

3

Wr

cos (-w,t

+ w,&)

+ 2k3a2dCsin (-2w,t + 2 w J 2 ) ) ,

dH(l' = ia2k~ 1

J~= --

ack

[

fi w%c*

(3.l18)

(2sin (-uCt+ wc,8,)

- sin (-wct + wcPi- 2w,t + 2w&) + sin (w,t - wc& - 2wTt + 2wT,ü2)).

1) p2 = --6 ~ (- ~-(-2klJ2a;wF

802

.iW&

+4 k I & f i u 3

sin (-wct

sin (-w,t

+wC& - 2w,t + 2w&)

+ w,&) + 2 k i d K w 3 sin (w,t - u,@,- 2wrt + 2 ~ ~ 3-, )

3k3a2w, - k3a2wccos(-4w,t

+ 4 4 , ) + 4k3azw,COS (-2wrt + 2

Now, equations (5.117) to (5.120) are exact this point.

(5.119)

- no

~ ~ 4 ) ) . (5.120)

approxîmations have been made at

We7ve used a valid transformation to change the variables and obtained

the equations of motion for the tran莽formed variables.

If these equations could be

solved, they would provide the solution to the original problem.

In practice though.

these equations aze no easier to solve than the original equations of motion. However. equations (5.117) to (5.120) offer a good starting point For approximate solutions and

make certain features of the solutions easy to spot.

In this work, we will consider two approximate solutions to equations (5.117) to

(5.120). These are (i)a canonical perturbation approach and (ii) replacing H(l) with its t ime average.

For the first technique, the approximations are made at the equations (5.117) to (5.120)

.

In this approach. a l ,a2,pl,

on the right hand side of equations (5.117) to

(5.120) are replaced by their initial constant values. Equations (3.117) to (5.120) are then integrated to yield the first order canonical perturbation solution . For a second order canonical perturbation solution, al,az,,没,, & on the right hand side of equat ions

(5.117) to (5.120) are replaced with the first order perturbation solution found in the previous step. This procedure can be continued indefinitely, although the expressions can get unwieldy after only a few iterationsFor the second technique, the approximations are made at a slightly earlier stage.

namely at

H(')itself- Here the perturbing Hamiltonian c m be further separated and

replaced with its temporal mean. Hamilton's equations for 8 1. ci2> $-jl,j2are then found using t his time-averaged Hamiltonian instead of the original Hamiltonian. The benefit of this approach is that for our problem the ensuing HamiItonYsequations can be solved

exactly.

It should be noted that this approach is also referred to in the literature

as canonicai perturbation.

However, since we're

&O

considering the t e m 'canonical

perturbation' in this work to refer to a slightly dinerent approach, the term 'time-averaged Hamiltonian' will be used to refer to this approach.

Each of these approximate solutions will be considered in more detail.

5.3.1

Canonical Perturbation Solution

RecaIl that the exact solution to the original problem (5.94) and (5.95) could hypotheticaliy be found by Gnding the exact solutions of equations (5.117) to (5.120) and using the

canonical transformation equations to transform back to the original variables.

How-

ever, as an exact solution to these equations is probably difficult if not impossible. an approximate solution to equations (5.117) to (5.120) will be constructed. To obtain the first order canonical perturbation solution to equations (5.117) to (5.120) . a,.a~>.

Ji. 4

are replaced on the right hand side of equations (5.117) to (5.120) by their initial values

4,4,A,P;.

If the initial conditions for (5.94) and (5.95) are given by

then the correspondhg initial values for ai,a2,Bi, P2 can be worked out from the trans-

formation equations (5.113 ) to (5.116) to be

Once the al,(22

PI,,O2are replaced in the right hand side of equations (5.117) to (5.120)

with their initiai values al, <rgt b!, & then equations (5.117) to (5.120) can be easily integrated to give the first order canonicd perturbation solutions for cr 1.

3 1. 4.

Clearly, a quick g l a c e at some of the values in the denominators reveals that the above expressions for ai:a:, piT& are not valid if the resonance condition of

In that case, the limit of the above expressions as w,

+2wT is taken.

dc =

2 4 , holds.

This fields the

first order canonical perturbation solution for the intemal resonance case

&R

=-

:

+

(4klco + k3ri) (4klco 3 k 3 4 sin& t) cos(w',t) sin(u,t) cos3(w, t) b: &3,

(5.136)

8u:

Thus this procedure gives a first order solution for both the interna1 resonance case and the non-interna1 resonance case. To h d the second order solution, cr 1, a l , dl.4 on the right hand side of equations (5.117) to (5.120) are replaced with their Brst order values a f .ah. O:,

6.Once again. the right hand sides of equations (5.117) to (S. 120) can be 2 integrated to yield the second order perturbation solution a:, a;, ,BI, a.

The resulting expressions for Q I , ci2, &, ,b2 are actudy difficult to integrate analytically and are best 2

integated numericdly. Thus, the canonical perturbation approsch easily provides a first order approximate solution. While higher order approximations are possible with this method, it tends to get rather unwieldy.

Recall t hat ai,a*,B,, P, do not provide the solution to the original problem directly.

These are obtained Erom the transformation equations, namely

c = -sin wc(t - Pi) uc

O O R e d that a!,a!,O,, ,4 are constants, thus the resulting solut ion for c and r for the

two uncoupled oscillators are simply periodic trigonometric functions, namely the linear solution for simple h m o n i c oscillators that are so familiar. For the coupled problem. a:,4,pi, & are actudly periodic hnctions themselves. Thus the resulting solutions for c and T are now periodic trigonometric functions 6 t h periodically changing amplitudes.

5.3.2

Time-Averaged Hamiltonian

Consider again the expression for H(I) h m which equations (5.117) to (5.120) are derived. The temporal mean of H(')cm eaรงily be calculated. Note that the temporal mean of J, = 2w,

H(')will have two different values depending on whether or not then sin((uc - 2 4 t + (w,&

- 2w,&))

J, = 'lw.,.

If

is not a function of time. and will not

average out to zero. Thus we obtain

(1)

&,IR

- 3 k 3 4 kla2 \/2a;; 8 n ( 2 4 P 2 - P l ) ) - -2 w f &4,

TOconstmct an approximate solution to our problern, HA:)

and

HL:!,

are used to derive

the Hamilton equations of motion instead of the exact value of H(').There are thus two possible sets of equations of motion depending on whether or not the condition for

internai resonance is met.

However, the advantage of this formulation is that in both

cases the Hamilton equations of motion can be solved exactly.

Case 1 : No Interna1 Resonance

For the case when there is no internal resonance, that is where w, # 2w,, then the Hamilton equations of motion are given by

Integrating t hese four equations yields

Thus the solution to the original problem becomes

rn

c = -sin (u,(t - p;)) WC

Note that for this case, since a:,a!,fi:,

& are d constants, the solution for c and

T

have constant amplitudes. Furthemore, note that the frequency of r depends on k3

so that we have the familiar result of fiequency dependmg on amplitude. Substitut ing O from equations (5.125) to (5.128) for a:,a!,Pl, &O yields

c = cg COS (w,t)

f 5.131)

Note that the solution for c in this case is exactly what it would be if the oscillators were uncoupled - it is not afTected by the coupling.

Case 2 : Interna1 Resonance

For the case when there is interna1 resonance, that is where uc= 'Lwrt then the Hamilton equations of motion are given by

Since

HA?

ddoes not contain the tirne t explicitly, it must be a conserved q u a n t i . that

HA:)

= gl. Adding equations

+ a2 = g*,

where g* is a constant of

is, a constant. Let us denote this constant as g,, so that (5.153) and (5.154) yields ck1

+

= O. or al

integration. Both constants gl and a can be found ÂŁrom the initial values of al,mt 8,

CHAPTER5. NONLINEAR CO~TLED VBRAT~ONS and

a. This yields

Solving equation (5.140) for the sine term, using the fact that ai = g*

- a2. and

using

the tngonometric identity sin2(x) = 1 - cos2(x), it c m be shown that a2 must satisfy

Note that equation (5.159) is an equation purely in terms of

a2.

That is. this ODE

can be integrated for a*. Integating this ODE will be discussed presently. Once a? is known, the other variables can be found by integation from

Thus, the c m of the problem is to solve equation (5.159) for a*.

Recall that a2 is the amplitude r. Before even solving for q,it is wonhwhile to remark that in the non-resonance case s was a constant and in this (resonance) case. it will clearly not be a constant. This is the most important observation that can be

made here since even though we cm solve for s exactly, the calcdations are not very intuitive.

CHAPTER5. NONLINEAR COUPLED VIBRATIONS

160

It can be verified that (2a2- LI:) is a factor of the quartic in equation (5.159). Thus. to solve equation (5.159), consider a change of variables by Letting

Then equation (5.159) can be written as

Note that this assumes that the roots to the above cubic are real. Should two of the roots be cornplex. a different procedure must be considered and will be discussed after

this one.

The solution of equation (5.165) is given by

CHAPTER5. NONLINEAR COUILEDVIBRATIONS where

and sn denotes the Jacobian elliptic function with parameter k. Yote t hat sn is a generalization of the trigonometric sine function. In fact, sn reduces to the sine function for

k = O. The jacobian elliptic function can be defined fiom dx

C/(l - x2)(1 - k2x2)

= s d (x),

where sn-l(x) denotes the inverse Jacobian elliptic function. That is. x = s n ( u . k ) . B y differentiating, it can be shown that

Note the similarity with equation (5.165). In fact, equation (5.171) and equation (5.165)

are related by a simple change of variable of the form x = d

m .

It still remains to find to. This can be found from the initiai conditions. Recall that

az(t=O) =~i:/2 i f r O= 1. This Ă?mpiies that y -t w at t = O .

Mplies that -klto = iKr where

Thus sn(-Mto.k) + x.

and

+ k2 = 1. Note that F denotes the complete elliptic integral of the first kind

with parameter ll. It follows that

1

= -ns(ilIt, k).

k

Thus fuiaily we have that

For the more general initial condition (where ro # l)?it follows that aĂ&#x17D;(t= 0) = i j ~ i / ? .

Thus y ( t = 0) = T{-i.

This yields

Hence the solution for y becomes

Y = (312 - y 1 ) s n 2 ( ~ (-t t o ) , k) + Y1

The above solution is only valid if y~,y2,and y:, are real. For irnaginary roots the solution

proceeds as fdows. The equation for a2 is written as

[g] (%)*

= (a*-

2)

(a2

- X I )[(ch- xd2 + XE]

7

where xl,xd and xs are red and

(S. 183)

Let Zi = (a2-

$) (

~2 xl) and

Z2= (a2-

+ 23.

Xow consider Zi

- hZ2 :

Clearly, ZI - XZ2 is a quadratic in a*. This quadratic is a perfect square if we choose

h such that the discriminant of the quadratic is zero. Thus, let XI and X2 denote the roots of the discriminant :

Equation (5.185) is itself a quadratic in X and as such has two possible solutions. Further. it is not a quadratic if x g = O, as this case would reduce to the previously discussed case

of red yi*. Solving for & yields

The expression under the square root sign is dways positive so Xi and X2 are both real quantities.

Take X2 to be the quantity associated with the positive sign in equation

(5.186) while AL is associated with the negative sign.

Since the coefficient of X2 in

equation (5.185) is negative while the constant term is positive. it is easily seen that XI < O. Solving the e-pression for A2 in equation (5.186) for w j , it can easily be shown

that for W: to be a real quantity that A2 2 1. It can be shown that X2 = 1 results in

Zl and Z2 being proportional, clearly an irnpossibility since one has real roots while the other has irnaginary roots. Thus, it follows that X2 > 1.

It follows that

where

(S. 189)

(.?.NO)

Solving for ZI and Z2 gives

Now consider the change of variables y =

2-Then aher some algebra, equation (5.182)

CHAPTER5. NONLINEAR COUPLED VIBRATIONS can be written as

where

In its present forrn, equation (5.193) cm be solved to give

this can be solved for q to give

-

Y* define .yo as the value It rernains to find to. uSing the initial condition q (t = 0) = 1,

of y at t = O such that

It then follows that

5.4

Exact Solution of the 1 DOF model

Recall that the nonlinear model of a spinning disk used in the literature neglects the presence of in-plane inertia. Thus, the model derived in this t hesis, equations (5.94) and (3.95) cm be reduced to the model used in the literature by ignoring the in-plane inertia

in equation (5.94). Thus, equation (5.94) becomes

This equation can be solved br c and the result substituted into equation (5.95) to yield the one degee of freedorn model (1DOF) :

Xote that this is the equation of motion of a nonlinear one degree of freedom system

and wiil be referred to as the one degee of freedorn (1DOF) model. Previous work has ignored the effect of in-plane inertia, and thus equation (5.203) is the equation describing the dynamics of the system with that underlying assumption.

The Hamiltonian associated with equation (5.203) can be written as

R e d that p, =

2 and that

the Hamiltonian must be conserved since it does not

explicitly depend on time, that is

is a constant.

Using those two facts. the

equation (5.203) can be integrated to give T as a huiction of time.

If the initial conditions are given by

then the solution is given by

where

In equation (X!O?), cn is the Jacobian elliptic functicli that is a generalization of the cosine function. The function cn(wcL)t, k(')) is a periodic function with period

where K(')(k('))= F ( k ( ' ) , ~ / 2is) the complete ekptic integral of the first kind. Xote that the above solution is only valid for red k('). That is, for k3u: - 2k: > O. The solution to the 1 DOF mode1 is thus a penodic function which reduces to the cosine

function when k@) = O. Furthemore, the frequency of this hinction is given by

dl)

which depends on w,, as we wouid expect, as well as ro,kl,k3 and w,. Here, k L and k3 may depend on the amplitudes of vibration and thus we have the familiar result of the frequency of vibration being dependent on the amplitude.

5.5

Energy Considerations of the 1 and 2 DOF models

In this section, stability of the 1 and 2 DOF models is considered by examining the potential energy expressions for each model. Recall that equations (5.94) and (S.95) can also be arrived at from the point of view of Hamilton's equations where the Hamiltonian is given by

Note that the above Hamiltonian is not an explicit function of tirne and is thus a conserved quantity.

That is. the total energy of the system is conserved and the systern is a

conservative one.

For the lDOF rnodel, the in-plane inertia is neglected and c =

-yis substituted

into the Hamiltonian to give the lDOF Hamiltonian

-4s before, this Harniltonian does not explicitly depend on time and is thus a conserved quantity.

The Hamiltonah H c m be considered as the sum of potential and kinetic energies.

H = T + V where

The potential energy function V is a function of c and r and can be considered to be a three dimensional surface. Local maxima and minima of this surface can be found using simple techniques of calculus. To find an extrema of a surfacet first consider

Now consider solutions to

any others?

= O,

= O. Cleaxly, c = T = O is a solution. but are there

Solving equation (5.215) for c and substituting the result into equation

(5.216) yields

Thus if 2k:

- k3u: < O,

clearly there is no red r which satisfies the preceding equation

and the trivial solution is the only solution to

=

$ = O. If 2k:

- k3u: > O. then

another solution to

=

= O is

Let us now consider whether the critical points are maxima or minima.

To this end.

we need to consider

The value of D at the previously-derived critical points needs to be considered. It is simple to show that

This implies that c = T = O is a local minimum, as would be expected. Sirnilarb

This implies that the second equilibrium point is a local maximum, if it exists. That is.

if 2e - k3u: > O. Thus this second equilibrium point is an unstable equilibrium point.

if it exists.

The value of the potential energy Km= at this local m e u r n can be calculated to

However, whether or not this local maximum is ever achieved depends on the energy initially in the system. Recall that

=T

+ V = E,n,t,.l? since energy is conserved.

Thus the potentid energy must always be less than the initial energy, C'

< Et,,t2al. It is

possible that the parameters of the system axe such that the potential energy function

has a Iocal maximum but it is never achieved because bi,,

> E,n2t,al. However. if

the system has a local maximum and sufficient initial energy then it will achieve this

maximum. The situation here can be compared to that of a simple pendulum.

The

pendulum has two equilibrium points, when it is hanging straight down and when it is balanced upside down. Clearly, one equilibrium point is stable and the other is unstable.

Whether the unstable point of equilibrium is ever reached depends on whether or not the pendulum has enough energy in the system initially.

This observation can be used to predict instability in the numerical solution of the system of equations (5.94) and (5.95). Instability in the system occurs if

2k:

- k3&:

>O

and if &nztzd > h maxThe preceding analysis can be repeated for the lDOF model. Recail that the lDOF model can be obtained fkom the 2DOF model by ignoring p, (which doesn7t aifect the

-:<

potential energy function) and by substituting c = -L into the 2DOF potential energy hinction. However, recdl that c =

y is exactly the solution to

the rest of the analysis is identical for the one for the

= O. and t hus

2DOF model. In other words.

-k1w2 +

c = 2k, - k 3 ~ , 2' T =

function, provided that 2k:

is d s o a local maximum for the lDOF potential energy

- k 3 4 > 0, as before. The maximum value of I.' is also the

same, and instability will occur if Ehitid

.>,K

Thus, the 1 DOF and 2 DOF models

yield the same stability criteria. However, it is important to realize that although the criteria for stability is the same, it is possible to h d an example of a case where the 1 DOF model will be stable while the 2 DOF model wilI be unstable. How is this possible?

To see this, it is important to realize that whether or not stability occurs depends on the initial conditions ; if Einitioi> 6

and 2k:

- k3wf > O. then instability will occur.

However, for the 2 DOF model, the value of Einiad depends on the initial values of c and T

of

and their initial time derivatives. However, in the 1 DOF rnodel. c is not independent T

but is related to it via c =

,y.Thus. for the 1 DOF model, on-

r and its tirne

derivative can be specified as part of Ein*tial.Thus, for a given initial value of r that does not yield instability with the 1 DOF model. it is possible to find initial values of c that will make Einitii larger than the maximum value of I.' and cause instability in the system.

With an extra degree of fieedom, the 2 DOF model will display a larger

number of possible behaviours than the 1 DOF model.

5.6

Numerical Simulations

In this section, numerical simulations of the lDOF and 2 DOF models are considered. along with the canonicd perturbation solution and Time-Averaged Harniltonian solution.

.AU numerical results were obtained by using MATLAB. For numericd soiutions to the nonlinearly coupled equations, fourth and fifth order Runge-Kutta formulas were

employed via built-in MATLAB hinctions.

Resonance Case

5.6.1

For the first case, the foilowing values for the constants are taken : kl = 2, k3 = 2, u, = 4, w, =

240

= 0.2, ro = 0.2. This is clearly a case where w, = 'Zw,. thus interna1

resonance is expected.

Consider first a compazison of the numerical solution of the 1 DOF and 2 DOF models.

This is shown for r and c in Figures (5.1) and (5.2) respectively. Yow. recall that the approximation c = -

3 is used in order to obtain the lDOF model. Thus, for the lDOF

mode1 it is to be expected that there is a large discrepancy between c ( t ) as obtained from the solution of the 1 DOF model and that obtained from the 2 DOF model. This is in fact shown explicitly in Figure (5.2). What is interesting and not predicted by the 1 DOF model is the intemal resonance. From Figure (5.1) it can be seen that while the

amplitude of r ( t )remains constant for the 1 DOF model. the 2 DOF rnodel displays a periodically varying amplitude, in keeping with the phenomena of internai resonance.

It can be seen from Figures (5.1) and (5.2) that energy is transferred back and forth from the

T

oscillator to the c oscillator. Indeed, each achieves its maximum amplitude

variation when the other achieves its minimum amplitude variation.

Now that we have a better understanding of how the exact solution of the 2 DOF model behaves, let us consider how the approximate solutions fare in capturing the periodic variation of amplitude. In Figures (5.3) and ( 5 4 , the exact numerical solution for r and c are compared to the approximate solutions generated by the fkst order canoni-

cal perturbation t heory and the time-averaged canonical perturbation t heory- The 6rst

order perturbation solution clearly does not capture the amplitude variation of r or c.

However, it can be seem that the solution produced by the the-averaged perturbation solution does track the actual amplitude vaxiation of the solution. In fact. in this case the tirne-averaged perturbation solution is good approximation to the actual solution.

Next, the =act solution is cornpared to the first and second order canonical perturbation solutions in Figures (5.5) and (5.6). As before, the first order canonical perturbation solution does not have the correct amplitude variation. However. it can be seen from the figures that the second order canonical perturbation solution is a much better approximation to the actual solution since it begins to capture the variation of amplitude of the solution. However in this particular case, it is the time-averaged canonical perturbation solution that produced the approximate solution that most closely matched the actual solution.

As a second euample, consider the case kl = 4, k3 = 6, w, = 2, w, = 1.co = 0.1. ro = 0.1. This is dso a case where w, = 2w,, thus internal resonance is expected.

The graphs of r and c For the exact sohtions of the 1 DOF and 2 DOF model are shown in Figures (5.7) and (5.8) respectively. The graphs of r and

c

for the exact

solution of the 2DOF model, the first order canonical perturbation solution and the tirne-averaged Hamiltonian perturbation solution are shown in Figures (5.9) and (5.10) respectively.

-&O,

the graphs of r and c for the exact solution of the 2DOF model. the

fkst and second order canonicai perturbation solutions are shown in Figures (5.11) and (5.12) respectively

Tau for 1 and 2 DOF modeĂŽs

Figure 5.1: The numerical solution of T for the 1 DOF and 2 DOF rnodels.

c for t and 2 DOF models

Figure 5.2: The numerical solution of c for the 1 DOF and 2 DOF modeis.

tau for exact, Canonical Perturbationand Time-Averaged Hamiltonian Solutior 0.5

4.5;

+

I

L

I

10

5

,

v

b

15

1

20

I

25 Time

1

30

I

1

35

I

40

1

45

1

50

Figure 5.3: Cornparison of r with numerical solution, first order canonical perturbation solution and time-averaged canonical perturbation solution. c for exact, Canonical Perturbation and Tirne-Averaged Hamiltonian Solutiont 0.25

4.2

t

L

5

r

1

10

r

1

I

1

1

tS

20

25 Time

r

.

30

1

1

1

1

1

35

40

45

1

50

Figure 5.4: Comparison of c with numencd solution, 6rst order canonicd perturbation solution and time-averaged canonicd perturbation solution.

Tau for Exact. First and Second Oder Canonid Perturbationsolutions 0.5

O

1

I

5

l

i'0

I

i5

20

l

25 Time

1

&

O;

&

45

O?!

Figure 5.5: Comparison of r with numerical solution, first and second order canonical perturbation solution. c for Exact, First and Second Order Canonical Perturbation solutions 0.25

1

1

r

1

8

1

1

1

t

1

l

1

5

1

10

I

15

I

20

Figure 5.6: Comparison of c with n&cal perturbation solution.

t

25 Time

I

I

I

1

1

30

35

40

45

50

solution, first and second order canonicai

Figure 5.7: The numericd solution of

T

for the 1 DOF and 2 DOF models.

Many of the comments made regarding the previous case can also be made here. Note how the 1 DOF model does not predict the variation of the amplitude. In addition. the

first order canonical perturbation solution does not correctly refiect this same change in the amplitude of the solution. Wh端e the second order canonicd perturbation solution fares better at representing the solution of the 2 DOF model than the first order canonical perturbation solution. for this case, the tirne-averaged Hamiltonian canonical perturbation solution is the best approximate solution to the 2 DOF model.

5.6.2

Non-Resonance Case

As another example, consider the case ki = 2, k3 = 2, w, = 2, w, = 4, co = 0.1, ro = 0.1.

This is not a case where w,= 2w,, thus no resonance is expected.

c for 1 and 2 DOF modefs O.15(

b

b

1

Figure 3.5: The numerical solution of c for the 1 DOF and 2 DOF models.

tau for exact, CanoniW Perturbation and Time-Averaged Hamiltonian Solutior

Figure 5.9: Cornparison of r with numerical solution, ÂŁirst order canonical perturbation solution and the-averaged canonical perturbation solution.

c for exact. Canonfcal PerturbaĂŽionand Tfme-Averaged HamiftonianSolutiom 0.1 5

1

1

1

1

i

i

1

r

t

Figure 5.10: Cornparison of c with numerical solution. first order canonical perturbation soiution and t ime-averaged canonical perturbation solution.

Tau for Exact, Arst and Second Order Canonical Perturbation solutions 0.2St

.

.

,

l

l

1

r

r

,

Figure 5.11: Cornparison of r with numerical solution, fkst and second order canonicd perturbation solution.

c for Exact, Rrst and Second Order Canonicat Perturbationsolutions 0.1 5-

-0.15' O

I

5

I

10

1

15

I

20

1

25 Time

1

.

30

b

;

I

I

I

1

1

35

40

45

50

Figure 5.12: Cornparison of c with numerical solution, first and second order canonical perturbation solution.

The graphs of r and c for the exact solutions of the 1 DOF and 2 DOF model are shown in figures (5.13) and (5.14) respectively. The graphs of

T

and c for the exact

solution of the 2DOF model, the first order canonical perturbation solution and the time-averaged Hamiltonian perturbation solution are shown in figures (3.13) and (5.16) respectively. -41~0,the graphs of r and c for the exact solution of the 2DOF model. the

first and second order canonicd perturbation solutions are shown in figures (5.17) and

(5.18) respectively.

.4s expected, the solutions for c for the exact 1 DOF and 2 DOF models differ sigdicantly, but this is expected as a result of the assumption required to amve at the 1

DOF model. Other than that discrepancy, d other solutions match perfectIy. That is. the tirne-averaged Hamiltonian canonical perturbation, 6rst and second order canonid

Tau for 1 and 2 OOF rnodels 0.1 Sr

+

m

e

8

+

1

1

1

1

Figure 5.13: The numencal solution of r for the 1 DOF and 2 DOF models. perturbation solutions are al1 in pecfect agreement.

Consider a similar example with kI = 2. k3 = 2. J, = 2. d, = 4. co = 0.4. ro = 0.4.

This is the same as the previous case, with a slight change in the initial conditions. The graphs of r and c for the exact solutions of the 1 DOF and 2 DOF model are shown in figures (5.19) and (5.20) respectively. The graphs of r and c for the exact solution of

the 2DOF model, the first order canonicd perturbation solution and the time-averaged Harniltonian perturbation solution are shown in figures (5.21) and (5.22) respectiveThe second order canonicd perturbation solutions are not shown since the first order canonical perturbation solution produces results that are in good agreement with the exact solution-

-0.1 51 O

2

*

1

I

I

4

6

8

10 Time

t

12

l

I

14

16

I

18

20

Figure 5.14: The numerical solution of c for the 1 DOF and 2 DOF models. tau for exact, Canonical Perturbationand Time-Averaged Hamiltonian Solutior 0.15

r

1

1

1

c

r

-Exact

s

t

Canonical Perturbation

- - Tirne-Av Hamiltonian

k

I

1

I

I

2

4

6

8

10

Time

, 12

I

14

1

16

1

18

1 20

Figure 5.15: Cornparison of r with numerical solution, Fust order canonical perturbation solution and time-averaged canonical perturbation solution.

c for exact, Canonical Perturbation and Time-Averaged HamiBonian Solutiom 0.1 5

I

8

I

8

I

r

I

-Exact

Canonical Perturbation

- - Time-Av Hamittonian

0.1

0.05

Q

1

-

0-

-0.05

-0.1

-0.15

*

O

I

I

2

4

6

8

L

L

I

1

1

10 T誰me

12

14

16

18

20

Figure 5.16: Comparison of c 6 t h numerical solution, first order canonical perturbation solution and t ime-averaged canonical perturbation solution.

Tau for Exact. Rrst and Second Order Canonical Perturbationsolutions 0.15-

1

1

1

1

1

-Exact First Order Canonical

- - Second Order Canonical

Figure 5.17: Comparison of r with numerical solution, f b t and second order canonicai perturbation solution.

Figure 5.18: Cornparison of c with numerical solution, first and second order canonical perturbation solution.

Once again, there is good agreement between the approxhate solutions and the exact solution. Note also how the frequency of the solutions depends on the initial conditions.

The only change in this case from the previous case is in the initial conditions. However. note that in this case the time-averaged canonicd perturbation solution is the worse of the approximate solutions. Although it correctly captures the frequency of the solut ion. it slzghtly underpredicts the minimum of the solution for c. However, the fint order canonicd perturbation tracks the =act solution and does not show this same underprediction.

Consider a simiiar example with kl = 2, k3 = 4,wc = l , w , = 3, co = 1 . q = 1.

There is no intemal resonance predicted in this case. However, a quick check of the

Tau for 1 and 2 DOF rnodeis

4.5

IDOF 2

@

4

6

8

@

IO

p

12

fl

14

,

16

a

18

20

Time

Figure 5.19: The numerical solution of r for the 1

DOF and 2 DOF models.

c for 1 and 2 DOF models

Figure 3.20: The numericd solution of c for the 1 DOF and 2 DOF models.

tau for exact, Canonical Perturbation and Tirne-Averaged Hamiitonian Solutior r

I

-OS1 0

,

2

r

t

4

I

I

r

1

*

L

1

6

8

10 Time

L

t

12

I

I

14

1

1

1

t6

18

20

Figure 5.21: Comparison of r wit h numerical solution. first order canonical perturbation solution and time-averaged canonical perturbation solut ion. c for exact, Canonical Perturbation and Time-Averaged Hamiitonian Solution:

-OS1 O

2

t

4

I

I

1

6

8

10

Time

t

12

I

14

1

1

1

16

18

20

Figure 5.22: Comparison of c with numerical solution, first order canonicai perturbation solution and the-averaged canonical perturbation solution.

parameters will reveaithat w: <

9and that the initial energy in the systern is greater

than the maximum of the potential energy function. Thus, our stability malysis predicts instability in the system.

The graphs of r and c for the exact solutions of the 1 DOF and 2 DOF rnodel are shown in figures (5.23) and (5.24) respectively. The graphs of

T

and

c

for the exact

solution of the 2DOF model. the first order canonical perturbation solution and the t ime-averaged Hamiltonian perturbation solution are shown in figures (52 5 ) and (5-26) respectively. Also, the graphs of r and c for the exact solution of the 2DOF model. the first and second order canonical perturbation solutions are shown in figures (5.27) and

(5.28) respect ively.

-4quick glance at the figures reveals that the 2 DOF model does indeed display the expected instability. Indeed, the solutions for r and c both blow up in finite tirne for the 2 DOF model. However. it should be noted that neither the 1 DOF model. nor any of the approximate solutions display the same behaviour. The 1 DOF model solutions and the approximate solutions remain bounded.

On the other hand. if the same case is considered with only a change in initial conditions to co = 0.1, ro = 0.1, the only thing this changes is the initial energy in the system so that the initial energy is now less then the maximum value of the potential energy With the initial energy being less than the maximum value of the potential energy. there

is no instsbility in the system. Only periodic solutions are observed with the 1 DOF and 2

DOF models producing the same solution for T . The gaphs of the solutions resem-

ble those of the previous case and are thus not included. -4s for the previous case. the

Tau for 1 and 2 DOF models

1

I

-506

,

5

10

I

15

,

20

,

25

30

Time

Figure 5.23: The numerical solution of r for the 1 DOF and 2 DOF models. approximate solutions and the exact solutions agree quite well.

5.7

Conclusions

By employing various analytical and numerical tools, a better picture of the similarities and differences between the 1 DOF and 2 DOF models begins to emerge.

First. the

1 DOF mode1 predicts a periodic solution for the time dependence of the transverse

vibrations, 7. -4lthough the hequency of vibration depends on various parameters of the solution, the amplitude of the solution is constant. For this model, the time dependence of the in-plane vibrations, c is 端nearly related to r since the in-plane inertia is dropped.

Although this will not accurately represent the dynamics of the in-plane vibrations? it

,

c for 1 and 2 DOF models

10' I

F

I

V

I

O

-2 -1

5-3

-

-4

-

-5

-

-6

-

A

-7, '

I

1

t

5

O

20

15

10

25

30

Time

Figure 5.24: The numerical solution of c for the 1 DOF and 2 DOF models.

Wu for exact. Cananical Perturbation and Time-Averanad Hamiltonian Solutior

I 200

Canonical Perturbation

-

- - Time-Av Hamiltonian

1

1

150-

g 100-

1

50 -

t I 1

0

-50' O

.

-

2

5

0 -

--

--

I

L

tO

15 Time

.

---.-C

----ac-----

1

L

1

20

25

30

Figure 5.25: Cornparison of T with numerical solution, tirst order canonical perturbation solution and t ime-averaged canonical perturbation solution.

c for exact, Canonical Pemirbatfon and Vme-Av. Hamiltonian

4

1

K

i

L

1

I

O

-.

Canonicai Perturbation

- - Time-Av Hamiltonian

Figure 5.26: Comparison of c with numerical solution, first order canonical perturbation solution and tirne-averaged canonicd perturbation solution.

Tau for Exact, FĂ?rst and Second Order CanonicaJ Perturbation solutions 250-

s

200

-

150

-

- - Second Order Cananical

too-

50 -

O

i

-3

I

1

5

t

1O

1

15

TĂŻme

I

20

1

25

30

Figure 5.27: Comparison of T 6 t h numericd solution, k t and second order canonicd perturbation solution.

1-

c for Exact, Fnst and Second Order Canonkat Perturbation r

i

I

1

O

Rrst Order Canonical

- - Second Order Canonical

Figure 5.28: Comparison of c with numerical solution, first and second order canonical perturbation solut ion. simplifies the overall problem enough to permit a closed-form solution to the transverse problem.

If kl = 0, for the 1 DOF model we get the seemingly nonsensical result

that c = O. This is actually a reasonable statement which follows from the underlying assumption.

Namely, by ignoring the in-plane inertia, we are essentiallp stating that

the in-plane problem is a static problem. Further assuming that kl = O states that there is no coupling between the in-plane and transverse vibrations. Hence, no cotipling to a

static problem leads to the prediction that the static problem remains unchanged.

In

other words, c = 0.

It becomes quickly apparent with the 2 DOF model that the solutions will exhibit

a broader range of possible behaviour than for the 1 DOF model. This model is best viewed as a nonlinear duffing oscilIator for the transverse vibrations and a linear harmonic

oscillator for the in-plane vibrations.

These are nonlinearly coupled.

For kI = 0'

there is no coupling between the two oscillators. Thus the two oscillators c m oscillate independently.

Thus, the first ciifference between the 1 DOF and 2 DOF oscillator

becomes apparent : depending on the modes in question, it is possible t hat t hese modes are coupled. It is also possible that the vibrations of the chosen modes are not coupled.

For the case where the modes are not coupled, the results for the vibrations of the transverse oscillator are very simiiar to the results of the 1 DOF transverse oscillator. In both cases, they are independent duffing oscillators.

For the case of coupled oscillators, the possibility of intemal resonance between the oscillators arises.

This represents a marked difference from the 1 DOF model where

there is no possibility of internal resonance since there is only one degree of freedorn.

In the case of internal resonance, there is a sort of 'energy sharing' between the two oscillators.

The amplitudes of vibrations of both are not constant but rather V a r y

periodically. This is reminiscent of beats between two linear harmonic oscillators. For the linear harmonic oscillator. the beat fiequency is easily found, whereas for the nonlinearly coupled oscillators it is not. For the non-resonant case numerical simulations showed that the predictions of the 1 DOF and 2 DOF models may a g e e or may V a r y in their

predictions of the amplitude of the solution.

For the 2 DOF model, energy anaiysis and numerieal simulations predict the possibility of instability in the system. From the numerical simulations, it can be seen that sometimes the 1 DOF model predicts a periodic solution while the 2 DOF model predicts instability.

In conclusion, by including the effect of in-plane inertia the dynamics of the system

become richer. The system dynamics may match those predicted by the model where in-plane inertia is neglected.

However, new possibilities such as intenial resonance or

non-unifomity of the amplitude of the solution arise. These are not predicted by the sirnpler model where in-plane inertia is neglected.

5.8

Summary

In this chapter, the three nonlinear coupled partial differential equations of motion were considerably sirnpiified by considering only one mode for the in-plane and transverse vibrations. If in-plane inertia is ignored the result is a simple 1 DOF nonlinear model.

If in-plane inertia is included the result is a nonlinear 2 DOF rnodel. these developments were given in the chapter.

The details of

.2nalytical expressions and numerical

simulations were derived for the solutions of each model.

Through the use of these

analpical and numericd tools, it becomes apparent that the inclusion of in-plane inertia gives rise to new possibilities in the dynamics of the system t hat are not predicted if the in-plane inert ia is ignored.

Chapter 6 Summary and Conclusions This thesis studied the vibrations of a spinning disk by employing a new modeling approach. A new model was derived from h s t principles and its success and implications were investigated.

6.1 Summary Chapter 2 considered modelling of the spinning disk. In this chapter, a new model was derived from first principles. This model subsequently led to three sub-models for the

transverse vibrations.

The models assume chat the disk is a thin plate and use the

plate theories of Kirchhoff and von Karman. The first model is linear and is based on the assumption of linear (Kirchhoff) strains.

The second model is also linear but is

based on the assumption of nonlinear (von Karman) strains. The third model is fully nonlinear and results ui three nonlinealy coupled partial differential equations based

on the assumption of nonlinear (von Karman) strains. The correspondhg boundary

conditions to these equations of motion are also derĂŽved through the use of Hamiltonk

principle and are discussed. A linear model for the in-plane vibrations of the spinning diçk with

its correspondhg boundary conditions is aiso derived.

The third chapter of the thesis considers the solution of the linear models based on linear and nonlinear strains.

These models that were derived in the second chapter.

The equation for a spinning plate based on the assumption of linear strain is simple to solve.

Given its simplicity, it would have been nice if it were also successful in

correctly predicting the vibration frequencies of the spinning disk. Ăźnfortunately. the investigation in chapter three reveals that this is not the case. The assumption of linear strains is not as successful for spinning plates as it is for non-rotating plates. The second model derived was a linear rnodel that resulted from the linearization of the nonlinear model based on nonlineax von Karman strains. In this version, the in-plane equilibrium problern must be solved first. This approach yields the same equation as derived by Iamb and Southwell [l]for the transverse vibrations with the addition of the two terms due to rotary inertia and bending moment resulting from the rotation. It turns out that the predictions for the frequencies match those predicted by the work of Lamb and Southwell [Il. Thus, this rnodel is to be considered successful for the spinning disk.

Before considering the third (nonlinear) mode1 for the transverse vibrations of a spinning disk, the in-plane vibrations of the spinning disk model were considered in the fourth

chapter.

New- the mode1 derived for the in-plane vibrations is not new - it matches ex-

actly models derived by other authors [19,211. However, in this chapter. considerable attention is given to deriving new properties of the in-plane modes of vibration. These orthogonality properties serve to shed some Iight on the in-plane vibrations themselves as well as their subsequent couphg to the transverse vibrations in the nonlinear model.

The orthogonslity properties may be used to constmct a general solution to the forced in-plane vibration problem. Responses to simple forcing b c t i o n s are also considered.

Resonance conditions associated with constant and h m o n i c excitations are considered.

Mthough it is physically intuitive, it is shown analytically that the resonance conditions are independent of the direction of rotation of the disk.

The penultimate chapter considers the nonlineady coupled equations of motion that result from the use of nonlinear strains.

The major consideration here is the effect of

the terms due to the in-plane inertia of the disk. For the analysis of non-rotating plates. it iรง generally accepted that the in-plane inertia of the disk may be ignored. This was assumed to be the case for spinning plates [31] without much analytical justification.

It is quite advantageous to do so since it perrnits the use of a stress function and thus reduces the number of equations to be simultaneously solved from three to two. The thrust of the analysis here

tvas

to investigate the effect of retaining the in-plane inertia

terms on the ensuing calculations. To this end, an ertrernely simplified analysis of the three noniinearly coupled equations with in-plane inertia was performed. This consisted of considering the coupling between one transverse mode and one in-plane mode only. In essence, this amounts to reducing the problem to one with only two degrees of freedom from one that originally had an infinite number of degrees of freedom.

6.2 6.2.1

Conclusions New Terms in the Linear Equations of Motion

The linear strain model for the spiunhg disk yields equations and boundary conditions that are quite similar to the classicd corresponding (Kirchhoff) linear strain modei for a non-rotating disk. There are two terms present in the model of the spinning disk that are not present in the classical model of the non-rotating disk. One term is the rotary

inertia t e m . This term cornes as no surprise and its existence and importance is well known in the redm of non-rotating disks.

It is present in both the equatioo for the

transverse vibrations and the corresponding boundary condition. The second new term is a term resulting from the bending moment due to the rotation of the plate.

It is a

result of the spin of the plate and the use of Lagrangian coordinates. Again. this term is present in bot h the equation of motion and the boundary conditions. In addition. t hese

two t e m s are also present in the linear model of a spinning disk based on the assumption of nonlinex strains.

6.2.2

Effect of Rotary Inertia

The presence of the rotary inertia term and its implications on the resulting frequencies

of vibration were also considered.

Now, for non-rotating plates it is well known that

rotary inertia can be ignored for the low frequencies of a thin plate.

Rotary inertia

must be taken into account for the calcdation of the high frequencies of thin plates. For spinning plates, this is also true. However, it was ais0 found that rotary inertia for the 1ow frequencies of a plate that is spinning very rapidy should be taken into account. -4s

the spin of the plate increases, the role of the rotary inertia assumes greater importance at lower freguencies. This is another example of how assumptions that are quite successful for non-rotating plates are not as succeรงsful for spinning plates.

6.2.3

Assumption of Linear Strains

The equation of motion for a spinning disk based on the assumption of linear strains mas derived and solved.

R e c d that the assumption of linear strains is quite effective

for stationary (non-rotating) plates and has been extensively used. It was shown in t his

work that the assumption of linear strains is not successful for spinning disks. It underpredicts the frequencies of vibration of the spinning disk.

However, the linear mode1

of a spinning disk based on the assumption of nonlinear strains did correctly predict the oatural frequencies of vibraiton.

The interesting observation is that to correctly

predict the frequencies of vibration with a linear model, we had to start out with the assumption of nonlinear strains and then linearize the resulting equat ions. Equivalent1 . we codd have linearized the expression for strain energy after establishing that the inplane equilibrium displacement is a known quantity.

However, linearizing the strain-

displacement expressions to give linear (Kirchhoff) strains is Linearizing premat urely this does not field accurate calculations of the frequencies of vibration.

It

should be

noted that although this observation was made for the case of a spinning disk. it should hold tme for any plate that has some sort of in-plane stretch, such as a pre-st ressed disk.

CHAPTER 6. SUMMARY A N D CONCLUSIONS

6.2.4

200

O v e r d Effect of Spin on Linear Vibration Frequencies

If we confine ourselves to linear in-plane and transverse vibrations, how does the spin of the disk affect its frequencies of vibration? We have seen that the spin results in in-plane fiequencies that are lower that those of a stationary disk. Similarly, the term in the transverse vibration equation that results frorn the bending moment due to the centrifuga1 force tends to lower the frequencies as compared to those of a stationary disk. The in-plane stretch terms in the transverse vibration equation tend to raise the frequencies. While the rotary inertia term is not due to the spin of the clisk. it is also instructive to consider in this discussion. Rotary inertia tends to locver the frequencies of vibration.

1s there a discernible trend to how these terrns affect the frequencies of

vibration derived with or without their effect?

The answer is yes.

It may be seen

that in general, if the effect in question ends up adding positive quantities to the overall strain energy, then the frequencies of vibration will be raised. This is the case for the equilibrium stretch due to spin. On the other band? if the effect in question adds positive quantities to the overall kinetic energy of the disk, then the effect is to lower the natural frequencies of vibration.

For instance, rotary inertia tends to lower the Ă&#x2021;requencies of

vibration as compared to a mode1 where rotary inertia is ignored. This is since rotary inertia essentiaily results in a larger kinetic energy term by virtue of its inclusion. When

the spin of the disk is taken into account, for the in-plane vibration these t e m s contribute to the kinetic energy and thus serve to Lower the overd frequencies.

To see why increaSing strain or kinetic energies results in higher or lower frequencies of vibration, it is instructive to consider a different characterization of the natural fie-

CHAPTER6. SUMMARY AND CONCLUSIONS quencies.

'201

It is possible to characterize hding the naturd fiequencies of vibration as

a variational problem. Roughly speaking, rninimizing with respect to a certain class of functions the ratio of strain energy to kinetic energy tviU yield the natural frequencies.

This is the backbone of Raleigh's method and of Ritz's method. In a similar vein. for a simple harmonie oscillator, the frequency of vibration is w2 =

5, where k is the stiffness

and m is the mass of the oscillator. However, we can also think of k as a sort of meaçure of strain energy that the oscillator is capable of storing and m as a rneasure of the kinetic energy that the oscillator may produce. Thinking of k and m in those terms allows us to see that once again the natural frequency is related to a ratio of strain energy over kinetic energy. Clearly, increasing the strain energy that a system can store will serve to increase the natural frequencies. Similady, increasing the kinetic energy that a systern

can produce will decrease the natural fiequencies. This is certainly true of the simple harmonic oscillator where a n increase in k increases w and an iacrease in m decreases W.

While for cornplex multi-dimensional systems this is a relatively crude rule of thumb.

it is still useful to have a feel for how including certain effects tvill affect the nat ural frequencies. To determine this, a qui& approach is to determine how the inclusion of this efect will affect the strain or kinetic energy of the system. This in tum should shed some light as to how the natural frequencies will be affected.

6.2.5

Coupling Between In-plane and Transverse Modes

In the nonlinear problem, the nonlinear transverse tenns in the in-plane equations can be considered to be forcing t e m . Since any function cm be eqanded in terms of the in-plane modes, in particular these nonĂźnear transverse 'forcing' terms can be expanded

CHAPTER6. SUMMARY AND CONCLUSIONS

202

as weU. In generd, d in-plane modes wiU be required for this expansion. Furthemore,

if a particular in-plane mode is present in the forcing hinction, it wĂźl excite the vibration of that mode in the forced vibration response.

Thus, we have the interesting result

that one transverse mode of vibration may excite al1 the in-plane modes of vibration.

Hence, although a particula. in-plane mode may not be initially vibrating, the nonlinear coupiing between the in-plane and transverse modes rnay eventually excite it. This is different From the situation with linear vibrations since generally only modes that are initially vibrating will continue to do so. While for stationary disks each mode-shape is associated with only one Frequency, for the spinning disk each mode-shape has two possible frequencies associated with it.

6.2.6

Effect of Inclusion of In-plane Inertia

A similax analysis to that done in Chapter five

was perforrned by Nowinski for the

spinning disk but ignoring the effect of in-plane inertia (311. It is Xowinski's analysis that foms the basis for cornparison.

In his andysis, since the in-plane inertia

tws

neglected, the equivdent mode1 only has one degree of freedom. Thus. in its most basic form, retaining the in-plane inertia adds one additional degree of freedom to the problem for each in-plane mode. With the most basic two degree of freedom model. the possibility of intemal resonance between the modes arĂŽses.

This is clearly not a possibility with

the one degree of freedom model where the in-plane inertia is neglected.

Even when

there is no intemal resonance, the two degree of Freedorn model sometimes predicts an amplitude that varies with time.

W e under some circumstances. the one and two

degree of fieedorn models agree in their predictions, the two degree of freedom model

CHAPTER6. SUMMARY AND CONCLUSIONS

203

displays some behaviours that axe not predicted by the one degree of freedorn model.

This is the resutt of considering just one mode for each of the in-plane and transverse vibrations. -4s was discussed for the in-plane vibrations, in general each transverse mode may excite al1 the in-plane modes. Thus the possibility of interna1 resonance between an

in-plane mode and a transverse mode increases.

Finally, the conclusions of this thesis may be concisely stated as

0

Deriving the spinning disk model from fust principles leads to new terms in the equations of motion. In particular, a new term results from the bending moment due to the rotation of the disk.

Rotary inertia of the disk may be ignored except for high frequencies or for high

spin rates.

The assumption of linear strains for spinning disks does not lead to reasonable results.

Spin terms in the kinetic energy lower the vibration frequencies rvhile spin terms in the potent ial energy raise the vibration frequencies.

Resonance phenomena are independent of the direction of rotation of the disk.

Inclusion of in-plane inertia shows that certain behaviours are possible that are not predicted by modek that neglect in-plane inertia. Specincally, there is a possibility of intenid resonance between some of the in-plane and transverse modes of vibration.

6.3

Future Work

Aithough this thesis has answered many questions, it has &O opened up the possibility of further questions being asked and further research being performed. One observation was that (von Karman) strains were required to correctly capture the dynamics of a spinning

disk, even for the linear model. Furthemore, it is necessary to use von Karman nonlinear strains for any thin plate where there is an in-plane stretch

- not just

one that results

lrom spin. Now. von K m a n strains are nonlinear but are not fully nonlinear. They are only nonlinear in t e m s of the transverse displacement. It would thus be instructive to pecform the same modelling and derivation of equations of motion starting with the basis of fully nonlinear strains. To make this calculation simpler. the fully nonlinear strains do not have to be modelled for a spinning plate, but rather any pre-stressed plate. If this pre-stress is taken as constant, it would considerably simplify the ensuing equat ions. but it would still serve to determine how the added complexity of taking the Full nonlinear strains affects the predict ions about the dynamics.

Another effect chat has yet to be modelled for spinning plates is the effect of shear deformations.

As mentioned in the first chapter, this would considerably complicate

the dynamics since the problem would now be one of three dimensions instead of two.

Knowledge of the middle surface of the plate wouid not be sufficient to determine the dynamin of the rest of the plate and thus the problem is considerably more cornplex.

One of the conclusions of this thesis is that rotary inertia must be taken into account for high frequencies or for hi&

spin rates. It would be extremely u s e N to determine

some sort of numericd criteria for when rotary inertia must be taken into account. For

CHAPTER6. SUMMARY AND CONCLUSIONS instance, given a particular spin rate, plate thicknesรง-to-radius ratio and a tolerance. at

which mode must we start taking rotary inertia into account?

A sirnilar statement c m be made regarding the in-plane inertia. This thesis has showo that there may be dinerences in the dynamics if in-plane inertia is included - in particular in the case of internai resonance.

There were also differences in stability predictions

whether or not in-plane inertia is included.

In other words, if the in-plane inertia

is neglected, it is possible to get inaccurate results when doing numerical simulations.

Again, what is required are some sort of criteria that would indicate exactly when in-

plane inertia must be included and when it can be safely neglected. This would be of value to those solving the equations numerically. As we know, inclusion of the effect of in-plane inertia for the nonlinear problem involves solving three nonlinear PDEs instead of two when it is neglected. If these equations are to be solved numericdly, it would be useful to know precisely when each mode1 should be used.

It would also be instructive to consider the vibrations in a thick spinning plate. For a thick plate, it would dways be necessaqi to include the effect of rotary inertia. Furthemore, for a thick plate, Kirchhoff's assumption no longer holds and it would become necessary to include the effect of shear deformation.

in this case, Kirchhoff and von

Karman pIate theories would no longer be valid and a more sophisticated plate theowould have to be employed.

Appendix A

Plate Theory -4 plate is a body bounded by two surfaces of small curvature. The distance between

these two surfaces is c d e d the thickness and is assumed to be small in cornparison to the dimensions of the surface. That is, the plate is assumed to be thin. The middle surface is the surface equidistant to the two bounding surfaces. If the thickness of the plate is

constant, the plate is said to be of unibrm thickness. The plate is said to be Bat if the middle surface is a plane in the undefomed configuration. The plates in this work NiIl be Bat, homogeneous and of uniform thickness.

A. 1 Kirchhoff Theory (Linear Theory) One of the principle features of straining a plate that alIows major simplifications to be

made is that the stresses acting on the surfaces pardel to the middle surface are maorders of magnitude smailer than the maximum bending or stretching stresses in the body. For a very thin plate, if the stresses on the extemal surfaces are smail. they are

s m d on any surface paralIeI to the middle surface. The two basic assumptions of the

Kirchhoff theory are that the stresses acting on any surface p a r d e l to the midĂ le surface are negligible and that the strains vary linearly throughout the plate t hickness. These

two basic assumptions can be shown to be equivalent to Kirchhoff's hypothesis which states that every straight line in the plate that was origindy perpendicular to the plate middle surface remains straight and perpendicular to the middle surface after the strain.

In the theory of small defle ctions of plates. the following assumptions are made.

0

The plate is thin, with the thickness of the plate being much smaller than the typical plate dimension.

0

The displacement of any particle in the plate is infinitesimal. That is. the defiections

are considered to be much smaller than the thickness of the plate.

The dope is everywhere small.

Linear strain-displacement relations are used.

iU1 drain components are small so that Hooke's law holds. That is, linear stressstrain relations are used. a

Kirchoff's hypot hesis holds.

A.2

Von Karman Theory (Nonlinear Theory)

When the deflections of the plate are comparable with its thickness, the results of the Kirchhoff theory become inaccurate and a different theory must be emptoyed. A well known theory to account for the large deflection of plates was developed by Von Kar

man. This theory employs the folIowing amimptions :

0

The plate is thin, with the thickness of the plate being much smaller than the typical plate dimension.

The displacement of any particle in the plate may be of the same order as the thickness of the plate, but must be small in cornparison with the plate dimensions.

0

The slope is eveqnvhere smdl.

O

The tangentid displacements are infinitesimal. Xonlinear strain-displacement relations are used, but the only nonlinear terrns retained axe those that depend on the derivatives of the middle surface transverse displacement ( i.e. the slope) . Ml ot h er nonlinear t e m s are neglected.

Al1 strain components are small so that Hooke's law holds.

0

Kirchoff's hypothesis holds.

Appendix B Derivat ion of Nonlinear Strain-Displacement Expressions Here. expressions for the nonlinear lagrangian strain-displacement relations in polar

CO-

ordinates will be derived.

The strain tensor is a description of the change in distance between two points in the body. Here, points in the body will be labelled according to their position before deformation. That is, Lagrangian coordinates shall be used. Consider ttvo points -4 and

B in the body in its undeformed state. The vector from -4 to B is given by dr.

Norv

suppose that the particle at A is displaced by a vector bf u, while B is displaced u + du.

The vector from the new position of -4to the new position of B is given by dr' = dr + du.

Thus

However, the partial derivatives can easily be cdcdated fkom the definitions of the vectors as

*

de =

(2

-w)

%+

(% +&)

au,

Q

+-ez dB

Thus, the explicit expression for dr' cari now be found

+

[reg+

(as.

--

Ue)4+($+~~)ee+$e:]dO

Thus the squares of the old and new distances between .4 and B are given by

APPENDIX B. DERIVATION OF XOXLINEAR STRAIN-DISPLACEMENT EXPRESSIONS 211

Thus the Merence in the squares of these distances is given by

dr' dr'

- dr - dr = [gr - gr - 11 dr2 + 2g,

ger d r d + ~ zg, - gzdr dz

+ [ge -ge - 1]r2dd2+ge . & r d ~ d t+ [g, - g, - 11 d t 2 .

(B.5)

The strain tensor is defined in so as to describe this change in the length of the line elements in the following manner

For completeness, the explicit expressions are given belouv.

'lote that ur,ue and u, are the displacements of the disk in the r , 0 and

2

directions

respectively.

The Von Karman plate theory assumptions dso imply that instead of

the

full non-

linear strain-displacement relations, the expressions for the required relations can be simplified considerably by ignoring ail nonlinear tems ezcept those that depend on the derivatives of the transverse displacement :

Kirchhoff's hypothesis can be shown to imply that the displacements u,, ue and u, can be written as a function of the displacements of the middle surface u, v and w as follows

Appendix C Sorne Useful Calculus Result s Consider a domain D of the x - y plane bounded by a simple closed curve C that consists of a finite nurnber of smooth arcs. The following line integals are carried out along C such that an observer walking around C in the direction of integration always has D on the left.

Assume that P ( x , y) and Q ( x ,y) are everywhere continuous in D and piecervise continuous dong C. Assume that D may be subdivided into a finite number of subdom端ins in each of which the first paxtial derimitves of P and Q are continuous.

Green's theorem in the plane states that

By writing P = vG7 Q = V F , the two-dimensional analogue of integration by parts is

By k t i n g q = @, G =

2, F = 3 in the above, the following useful resuit is obtained

Appendix D

Self-Adjointness of the Operator Ln Lemma 1 Ln is a self adjoint operator in the space of functions that satisfy the boundary conditions (2.39) and (2.40) and thnt o n of the f o m

[u

iv

] T wliere u and u o r e reol

fimctions. Here i = G.

Proof.

Only functions of the form

uj

=[

,

considered. Consider

( ~ 2 LUI) r

=

la[

u2 -iv2

IL.

ut T ~ T

ivl

iu, ]* where

.uj

and u, are real will be

It may also be verified that the integral (Le. non boundary) portion of equation (D.3)

r

is equivaieot to (ul,Lnu2)= f ' [

.,-iui ]Ln 1

a2

i

1

rd'.

Hence it follows that

provided that the b o u n d q term in equation (D.3) disappears.

Thus the operatot is

self-adjoint provided that

Now suppose that each of ul = [

.,iv,

ITand ul = [

a q conditions, equations (2.39) and (2.40). true

It then follows that equation (D.5)reduces to

.,

iV2

IT

satisfy the bound-

That is. for j = 1,% the E'olloiving are

APPENDKD. SELF-ADJOINTNLSS OF THE OPERATORLn

218

Hence, provided that uland uz satisfy the boundary conditions and equation (D.8)is

satisfied, the operator Ln is self-adjoint. m

Remark 1 The solutions of the free in-plane vibration problem are of the form u =

[ u iv

IT

where 60th u and u are real functions. It is for this reason that our attention

is conjined to functions of this fonn.

Appendix E

Ort hogonality Properties of the Eigenfunctions of the LSM Let w(r,0, t ) = ~ ~ . ~ ~ ( r ) e ' ~ be*the ' e '(n, " ~k)th Eigenfunction of the spinning disk mode1

using linear strains, with n being an integer. That is, the Wnkmust satisfy

where

d2 +--+1d v*= 창r2

rai

d2

ae2

v : =a2~ + l;a~ -n2.

The corresponding boundary conditions for a solid disk with a h-ee boundaxy are

APPENDME. ORTHOGONALITY PROPEKPIES OF THE EIGENFUNCTIONS OF THE LSM 220

Let

Conjecture 2 The ezgenfunctions corresponding to the iznear strain mode1 of a spinning disk with rotary inertia àncluded satish the folloving relationships

If rotary inertzo is neglected f~umthe mode1 then the eigenfvnctions are orthogonal and can be normolized to yield the following relationships

(E. 11)

PTOO~.Multiplying both sides of equatàon (E.1)by Wnjr and i n t e p t z n g fmm r = O r = a yields

to

APPENDKE. ORTROGONALITY P R O P E ~ I OF E STHE EIGENFUNCTIONS OF THE LSM 221 Now wing integmtion by parts, it con easily be shoum that for any two arbitray Jimctions

f and g that t

rit

Using this result, zt cun further be s h o w that

llsing equations (E.14)and (E.15) to simplih equation (E.13) gives

Using equution (E.5) to simplib this fvrther yields

Now equation (E.4) con be rem*tten as

v;bvnk+ n2(1r2-

(1- u ) a v n k i-a

APPENDIX E. OE?THOGONALITY PROPERTIES OF THE EIGENFUNCTIONS O F THE LSM 222 whàch clearly simplifies equatàon (E.17) to

In other words, it has been s h o w that

and that

Cieudy Ijk= Ikj, hence equutzons (E.20) and (E.21) together yield

In terchanging j and k in equation (E.20) gives

whzch c m then be subtmcted Rom equation (E.21) to give

If rotary inertia is neglected, then the x

~ t e~ m would G ~be rnzsszng ~ from equations

(E.20) to (E.23) und thus equation (E.24) would instead become

APPENDKE. O ~ H O G O N A L IPROPEF~TLES TY OF THE EIGENFUNCTIONS OF THE LSM 223 Thup for

j

# k, the eigenvalues

Ikj= O

and Lj are diffemt and we obtazn

k #j

NO Rotary Inertia

Furthemore, tue can novr -te h2

-R2Gjk 3

[Ij*- "

+ 3p(lEh2 - u2)Ifjk= gk

1

R ~ t Inertia q No Rotary Inertia,

which are the statements we were tyĂ ng to prove.

(E.29)

Appendix F Properties of the Eigenfunctions of

the Linear Spinning Membrane The equation of motion for a spinning membrane is given by

d2w

P w = ;&

+a,,Fa Z , e9

(O$)

2

where oz and O;: are the equilibrium stresses in the disk induces by its rotation.

Let

W(T.

0, t ) = W , ~ ( r ) e ' ~be z ~the e ~( n~, k) ~ th eigenfunction of the spinning mem-

brane model, with n being an integer. The superscript rn is to serve as a reminder that these eigenvalues and eigenfunctions refer to those of the spinning membrane. and should not be confused with those of the spinning plate. The tVnk must satisb

Conjecture 3 The ezgenfunctions corresponding to the h e u r spinning membrane rnodel

APPENDM F. PROPEKPIES OF TEE EIGENFUNCTIONS OF THE LINEAR SPINNING MEMBRANE 225 are orthogonal. They con be nonnalized to yield

Furthemore, the follomhg relationship holds

Proof. Multiplying both sades of equation (F.2) by W,fr and integrating from r = O to r = a yields

hovever recall that the boundwy condition for a spinning membrane zs 02 = O at r = a . thvs the above ezpmssion simplifies to

InterchangĂ ng j and k yields

Su btiactzng the preceding two equotions gives

T h w for j # k , the eigenfunctions are orthogonal. If the eigenfvnctzons are nonnaizzed.

APPENDIX F. PROPERTIES OF THE EIGENFUNCTIONS OF THE LINEAR SPINNING MEMBRANE 226 we have the folioving remit

Thus for j = k, equatzon (F.8) becomes

(F.11)

Bibliography [1] H Lamb and R.V. Southwell. The vibrations of a spinning disk. Proceedzngs of the

Royal Society

O/

London, Series A, 99272-280. 1921.

[2] R.V. Southwell. Free transverse vibrations of a uniform circdar disk clamped at its centre. Proceedzngs of the Royal Society of London, Series A. 101:133-153. 1922.

[3] S. Barasch and Y Chen. On the vibration of a rotating disk. Journal

01Applied

Mechanics, 39:ll&-llU. 1972.

[4] W. Evenman and R.O. Dodson. Free vibration o f a centraliy clamped spinning disk.

AIAA Journal, 7(10):201没-2012, 1969.

(51 K.A. Cole and R.C. Benson. -4 fast eigenfunction approach for computing spinning

disk defiections. Journal of Applzed Mechanics, 35:453-457, 1988.

[6] M.P. Mignolet, C.D. Eick, and M.V. Harish. Free vibration of flexible rotating disks.

Journal of Sound and Vibration, 196(5):537-577, 1996. [i'] J.G.SMmonds. The transverse vibrations of a flat spinning membrane. Journal of

Aeronautzcal Science, 29:1618, 1962.

BIBLIOGRAPHY

228

[8] W. Evemnan. Tkanmerse vibrations of a cIamped spinning membrane. AIAA Jour-

nal, 6:L395-1397, 1968.

[9] J.G. Simmonds. kxisymmetnc, transverse vibrations of a spiMing membrane clamped at its centre. AIAA Journal, 1 (5):l224-1225, 1963.

[IO] MW. Johnson. On the dynamics of shdow elastic membranes. Proceedzngs of the Symposium of Thin Elastic Shells, pages 281-300, 1960.

[Il] S.C. Huang and B.S. Hsu. Vibrations of a spinning annular plate with multi-circular

line guides. Journal of Sound and Vzbration? 164(3):535-547. 1993.

[12] S. Yano and T. Kotera. Instability of a rotating disk with a support under the action

of a stat ic in-plane load. Archive of Applied Mechanics, 61(6-7):373-38 1. 1991.

[13]R.C. Benson and D.B. Bogy. Deflection of a very flexible spinning disk due to a stationary transverse load. Journal of ApplĂ ed Mechanics. 45(3):636-642. 1978.

[14]H.J.Greenberg. Flexible disk-readlwrite interface. IEEE Trans. Magnetics. 14:336338, 1978.

[l5] G.G. Adams. .Analysis of the flexible disk/head interface. Journal of Lubncation

Technology, 102:8690, 1980.

[16] G.G. Addams. Procedures for the study of the flexible disk to head interface. IBM

Journal Res. Deu., 24512-517, 1980.

(171 K. Yamada. Vibration of a turbine disk in its plane. Proceedzngs of the Second Japanese National Gongress on Applied Mechonies, pages 343-347, 1952.

BIBLIOGRAPHY

229

[18]R-Gramme1and C.B. Bieaano. Engineering Dynamzcs. D.Von Nostrand Company, Inc., Princeton, New Jersey, 1954.

[19]

P.G. Bhuta and J.P Jones. Symmetric planar vibrations of a rotating disk. The Journal of the Acoustical Society of America, 35(7):982-989, 1963.

[')O]

R. Doby. On the elastic stability of corioiis-coupled oscillations of a rotating disk.

Joumal of the Franklin Instihrte, 288(3):203-212, 1969.

[21]J.S. Burdess, T.Wren, and J.N. Fawcett. Plane stress vibrations in rotating disks.

Proceedings of the Institution of Mechanical Engineers. 201(Cl):37-44. 1987.

[22j J.S. Chen and J.L. Jhu. On the in-plane vibration and stability of a spinning annular disk. Journal 01Sound und Vibrations, 195(4):585-593. 1996.

[23] G.G. Adams. Critical speeds for a flexible spinning disk. International Journal of Mechanical Science, 29 (8):XXPSI, 1987.

[24] Y. Honda, H.Matsuhisa, and S. Sato. Vibration analysis of a rotating disk subjected

to base excitation.

Tmnsactions of the Japon Society of Mechanical Engrneers.

60(579):3749-3754, 1994.

[25] J.S. Chen. Stability andysis of a spinning elastic disk under a stationary concentrated edge load. Journal of Applied Mechanics, 61(4):788-792, 1994.

[26] J.S. Chen. Stabiiity analysis of a spinning elastic disk under station-

edge loads. Journal of Apphed Mechanics, 63(2):439-444, 1996.

distributed

BIBLIOGRAPHY

230

[27] J.S. Chen and D.B. Bogy. Effects of load parameters on the natural fkequencies and

stability of a flexible spinning disk with a stationary load systern. Journal of Applied

Mechanzcs, 59(2):230-235, 1992.

[28]J.S. Chen. Pararnetric resonance of a spinning disk under space-6xed pdsating edge Ioads. Journal of Applied Mechanzcs, 64(1):139-143, 1997.

[29]J.S. Chen. The effects of a space Lxed friction Force on the in-plane stress and stability of transverse vibrations of a spinning disk. Journal of Applied Mechanics.

60:646-648, 1993.

[30]F.Y. Huang and C.D. Jr. Mote. Mathematical analysis of stability of a spinning disk under rotating, arbitrarily large damping forces. Journal of Vibration and Acougtics.

118(4);657-662.1996.

[31] J.L. Nowinski. ?ionlinear transverse vibrations of a spinning disk. Journal of Applied

Mechanics, 31:72-78, 1964.

[32]A.A. Renshaw and C.D.Jr. Mote. A perturbation solution for the flexible rotating disk : Nonlinear equilibrium and stabiiity under transverse loading. Journal of

Sound und Vibmtzon, l83(2):309-326,1995.

[33]S.H.Advani and P.Z. Bulkeley. Nonlinear transverse vibrations and waves in spinning membrane disks. International Journal of Nonlineur Mechanzcs, 4:123-127.

1969.

[34]S.H.Advani. Stationary waves in a thin spinning disk. lnternatzonal Journal of Mechanicol Sciences, 9:307-313, 1967-

BIBLIOGRAPHY

231

[35] H.N.Chu and G. Herrrnann. Muence of large amplitudes on free flermrd vibrations of rectangdar elastic plates. Journal of Applied Mechanzcs, 23:532-540, 1956.

[36] N. Baddour and J.W. Zu. A revisit of spinning disk models, part i : Derivation of

equations of motion. Accepted for publication in Applied Mathematical Modelling.

2000.

[37]

Y.Baddour and J.W. Zu. Solution of a new mode1 For a spinning disk. 17th Canadian Congress of A pplzed Mechanics, 19%.

(381 Y. Baddour and J.W. Zu. A revisit of spinning disk models. part ii : Linear transverse vibrations. Accepted for publication in Applied Mathematical 1Vodelling. 2000.

[39]N. Baddour and J.W. Zu. General solution of the forced in-plane vibration problem for a spinning disk. Transactions of the CSbIET24(1B):307-321. 2000. [do] Y.C. Fung. Foundatzons of Solzd Mechanzcs. Prentice-Hall, Inc., Englewood Cliffs.

New Jersey, 1965.

(411 A. Leissa. Vibmtions of Plates. NASA SP-160,Washington, D.C.. 1969.

[42] F. Niordson. . . . asymptotic t heory for circular cylindricd shells. Int emational Journal of Solzds and Structures, 27(13):181?-1839,?OOO. [43]

R.C. ColwelI and H.C.

Hardy. The frequencies and nodal systems of circular plates.

Phzlasophicul MagaRne, Series 7,24(165):1041-1055, 1937.

[44] J.A.. Wickert and C.D. Jr. Mote. Classicd vibration analysis of axially moving

57:738-744, 1990. continua. The Journal of Applied Mechanzc~~

[45] H. Goldstein. CIussical Mechanics. Addison-Wesley, Reading, Massachusetts, 1980.

[46]R. Weinstock. Calculus of Variations with Applications to Physics and Engineering.

Dover, New York, 1974.

[47]L .C.Andrews. Elemen tary Partial Diffemntaai Equations. -4cademic Press. New York, 1986.

[48] A.E.H. Love. A T~eatiseon the Muthematical Theory of Elasticity. Camebridge

University Press, Camebridge, 1952.

[49]Lord Raleigh. Theory of Sound. Macmillan and Company, London. 1937.

[JO] A.P.French. Vibrations and Wavea W.W.Norton and Company. 'iew York. 1971.

[51] J.E. Lagnese and J.-L. Lions. Modelling, Analysis and Control of Thin Plates.

Springer-VerIag, Berlin. 1989.

[52] R.M. Chalasani. Ride performance potential of active suspension systems part 1:

Simplified analysis based on quarter-car model. In Symposium o n Simulation and

Control of Ground Vehicles and Transportatzon Systems, pages 187-204. . N D volume 80, XSME, 1986.

(531 S.A. Tobias. Free undarnped nonlinear vibrations of imperfect circular disks. Pro-

ceedzngs Of the Instztute of Mechanical EngĂ neers, l?l:69 1-701, 1957.

[34]J.L. Nowinski. Stability of noniinear thennoelastic waves in membrane-like spinning

disks. Journal of T h e m a l Sciences, 4:l-11, 1981.

BIBLIOGRAPHY

[5q

233

C.D. Jr. Mote. Free vibration of initially stressed circdar disks. Transactions of the ASME, pages 258-264, 1965.

[56]S.A. Tobias and R.N. Arnold. The influence of dynamical imperfections on the vibration of rotating disks. Proceedzngs of the Institute of Mechanicd Enpneers.

171:669-690, 1957.

[57] S.A. Tobias. Free undamped nonlinear vibrations OF imperfect circular disks. Proceedings of the Institute of Mechonical Engineers. l7l:69l-?lti. 1937.

[58]C.D. Jr. Mote and L.T. Nieh. Control of circular disk stability with membrane stress. Ezperirnental Mechanics, 11:490-498. 1971.

(391 K. Ono and

T. Maeno. Theoreticai and experimentd investigation on

dynamic

characteristics of a 3.5 inch flexible disk due to a point contact head. Tiiboiogy and Mechanics of Magnetic Storage Systems. 3: 144- 13.1. 1987.

[60]P. Likin. F. Barbera, and V. Baddeley. Mathematical modeling of spinning elastic bodies for modal analysis. A IAA Journal, 11(9):l%l-1258. 1973.

[61]D.J. Segalman and C.R. Dohrmann. A method For calculating the dynamics of rotating flexible structures, parts 1 and 2. Journal of V2brution and .4coustics.

118:313-322, 1996.

[62]

J.C.Simo and L. Vu-Quoc. On the dynamics of flexible beams under large overall motions-the plane case, parts 1 and 2. Journal of Apptzed Mechanics, 53(4):849-863.

1986.

[63]J-C. Simo and L. Vu-Quoc. The role of nonĂźnear theories in transient dynamic sndysis of flexible structures. Journal of Sound and VĂ bmtzon, 1 19:48?-j080 1987.

[64]T.R.Kane, R.R. Ryan, and A.K. Banerjee. Dyuamics of a cantilever beam attached to a moving load. AIAA Journal of Guidance, Control and Dynamics. 10:139-131.

1987.