A Revisit of Spinning Disk Models, Part I: Derivation of Equations of Motion

Applied Mathematical Modelling 25 (2001) 541±559

www.elsevier.nl/locate/apm

A revisit of spinning disk models.

Part I: derivation of equations of motion N. Baddour, J.W. Zu

*

Department of Mechanical Engineering and Industrial Engineering, University of Toronto, 5 King's College Road, Toronto, Ontario, Canada M5S 3G8 Received 1 November 1999; received in revised form 30 August 2000; accepted 5 October 2000

Abstract Previous models of spinning disks have focused on modelling the disk as a spinning membrane. The e ect of bending sti ness was then incorporated by adding the appropriate term to the previously derived spinning membrane equation.

A pure spinning plate model does not exist in the literature. Furthermore, in both existing linear and nonlinear models of spinning disks, the in-plane inertia and rotary inertia of the disk have been ignored. This paper revisits the derivation of the equations of motion of a spinning plate. The derivation focuses on the use of Hamilton's principle with linear Kirchho and nonlinear von Karman strain expressions. In-plane and rotary inertias of the plate are automatically taken into account. The use of Hamilton's principle guarantees the correct derivation of the corresponding boundary conditions. The resulting equations and boundary conditions are discussed. Ó 2001 Elsevier Science Inc. All rights reserved.

Keywords: Spinning disk; Hamiltonian; Modelling; Derivation of equations

1. Introduction Spinning disks can be found in many engineering applications. Common industrial applications include circular sawblades, turbine rotors, brake systems, fans, ¯ywheels, gears, grinding wheels, precision gyroscopes and computer storage devices. Spinning disks may experience severe vibrations which could lead to fatigue failure of the system. Thus, the dynamics of spinning disks has attracted much research interest over the years.

The ®rst step in investigating the vibrations of spinning disks is to setup a suitable mathematical model that captures the essential physics of the problem while remaining tractable. Assumptions about the system under investigation must be made at the stage of developing a model for the system. It is obvious that di erent assumptions will lead to di erent equations of motion and thus di erent solutions and techniques are required to obtain these solutions.

In this paper, existing models used to analyze spinning disks will be thoroughly reviewed.

Subsequently, new linear and nonlinear equations of motion with corresponding boundary conditions will be derived via Hamilton's principle. *

Corresponding author. Tel.: +1-416-978-0961; fax: +1-416-978-7753.

E-mail addresses: baddour@mie.utoronto.ca (N. Baddour), zu@mie.utoronto.ca (J.W. Zu).

0307-904X/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved.

PII: S 0 3 0 7 - 9 0 4 X ( 0 0 ) 0 0 0 6 5 - 2

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2. Review of existing models A few di erent models have been used to analyze spinning disks. At ®rst, a spinning membrane model was used. Next, bending sti ness was incorporated into the spinning membrane model.

Finally, a nonlinear model based on von Karman's plate equations was developed. Each of these models will be examined in this section and the drawbacks of each model will be discussed.

2.1. The linear membrane model The ®rst step towards developing the equation of motion for a spinning disk has traditionally been to consider the disk as a membrane. That is, the bending sti ness of the disk is ignored. For the small transverse de¯ections of a membrane, it is assumed that these de¯ections do not a ect existing in-plane stresses. When 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 de¯ections.

The in-plane stresses are those that develop in a ¯at, undeformed 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 allowed to come to equilibrium, the resulting stresses will be along and perpendicular to the radius vector. For the calculation of the equilibrium stresses of a rotating disk, it is typically assumed that the displacement of particles is in the radial direction only.

Since an equilibrium con®guration is assumed, the in-plane inertia of the disk is ignored.

The equation of motion governing the transverse vibrations of a ¯at spinning membrane is derived by Simmonds [1] as

o2 w 1 o ow ow 1 o ow rhh ow rrrr rrh rrh

; 1 q 2 ot r or or oh r oh or r oh where w is the transverse displacement, q the density of the plate material and rrr , rhh , rrh are the normal and shearing stresses. The variables r; h and t denote the usual radial, angular and temporal variables. Note that the normal and shearing stresses are in the plane of the deformed 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 calculation.

The aforementioned approach is appropriate for the derivation of the equations of motion of a spinning membrane. It must be noted that inherent in this derivation is the neglect of the in-plane vibrations since an equilibrium situation is assumed for the in-plane stresses. Also, the transverse vibration problem is inevitably linked to the solution of the in-plane problem since the latter problem must be solved ®rst.

2.2. The linear membrane with bending sti ness model Thus far, the equation of motion does not have a term representing the e ect of the bending sti ness of the disk. To remedy this, a bending sti ness term, proportional to r4 w is added to the right-hand side of Eq. (1) as mentioned in [2]

o2 w 1 o ow ow 1 o ow rhh ow Eh2 q 2 2 rrrr rrh

rrh

r4 w: ot r or or oh r oh or r oh 3 1 m2 Here, E is the Young's modulus, m the Poisson's ratio, and h is the plate half-thickness. Note that the assumptions on the displacements that lead to the r4 w term and those used to ®nd the

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543

equilibrium stresses in the disk are not quite the same since the equilibrium stresses assume a displacement in the radial direction only.

The fourth-order partial di erential equation given by Eq. (2) has historically been di cult to solve exactly. Note that thus far, a membrane model and a membrane with bending sti ness model have been derived and employed. The spinning disk modelled purely as a spinning plate with no membrane e ects is an approach that has not been previously considered in the literature.

Furthermore, the e ect of in-plane inertia is completely ignored in the previous models. For a spinning disk, the centrifugal and Coriolis forces act in the plane of the undeformed 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 linear (Kirchho ) theory, the in-plane inertia is included. It then turns out that the in-plane and transverse free vibration problems decouple and can be solved independently. Thus, there seems to be little theoretical justi®cation for neglecting the in-plane inertia of a rotating plate. Furthermore, given the precedent set by the stationary plate problem, it might be expected that when incorporating the e ect of the in-plane inertia of the plate, the in-plane and transverse vibration problems should decouple for the spinning plate problem as well.

2.3. The nonlinear model The nonlinear spinning disk equations were ®rst derived by Nowinski [3] using the von Karman plate equations as a starting point. The equations are D 4 o2 w 1 ow r w q 2 qX2 r2 r2 w qX2 r h ot 2 or

2 2 o w 1 o/ 1 o / 1 ow 1 o2 w o2 / 2

or r or r2 oh2 r or r2 oh2 or2 1 4 r/ E

2 1 m X2 q

1 ow 1 o2 w o2 w

r or r2 oh2 or2

2 o2 w ow 1

r3 oroh oh r2

o2 w oroh

o 2 or

2

1 ow r oh

1 4 r

ow oh

o or

1 o/ ; r oh

3

2 ;

4

where D Eh3 = 12 1 m2 is the bending rigidity of the plate, X is its angular velocity, w the transverse de¯ection and / is a stress function.

In modelling the spinning disk using the nonlinear von Karman theory, the in-plane inertia of the disk is also ignored. The justi®cation 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 [4] that the equations resulting from ignoring the in-plane inertia of a stationary plate are the ®rst-order perturbation approximations to the full problem. In other words, for the stationary problem, ignoring the in-plane inertia is a good approximation to ®rst 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 di erential equations to two, a considerable simpli®cation.

The same perturbation argument used by Chu and Hermann [4] for the stationary disk 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 should be retained in modelling a spinning disk, for both linear and nonlinear formulations of the problem.

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Furthermore, in both the nonlinear and linear theories of spinning disks, the rotary inertia of the disk was ignored. Again, drawing on the known literature of stationary (non-rotating) plates, this 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. For thick disks, it is well known that neither the rotary inertia nor the transverse shear of the disk can be ignored. Since it is also known that the e ect of the spin is to raise the natural frequencies of vibration, it remains to be proven whether the rotary inertia of a thin spinning disk can be neglected.

2.4. Drawbacks of existing models and proposed approach All 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. For spinning disk problems, the outer edge is typically assumed to be free. Mathematically, this is usually interpreted by requiring the bending moment and the Kirchho shear to be zero at the free edge. Use of Hamilton's principle to derive the equations of motion and the boundary conditions will con®rm this. Furthermore, all existing models ignore both rotary inertia and in-plane inertia of the disk. A spinning disk model with pure bending as the restoring force cannot be found in the literature.

It is proposed to derive new linear and nonlinear equations of motion of a spinning disk using the following assumptions : · The disk is modelled purely as a classical Kirchho or von Karman plate. Thus, bending rigidity of the plate is the only restoring force. · Only thin disks will be considered. · The in-plane inertia of the plate will not be ignored as no theoretical justi®cation of this can be found. · Hamilton's principle will be used so that the correct boundary conditions are automatically obtained. · The e ect of the rotary inertia of the disk will automatically be included in this new formulation of the problem.

The equations derived here will be shown to be slightly di erent from those previously used in the literature in both the linear and nonlinear strain cases. In the linear strain case, it will be shown that it is possible to solve this new equation analytically.

3. 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 formulated. 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 displacements, the shape of the entire volume changes as a function of time. The solid looks di erent at di erent times. Hence, care must be taken when integrating over the entire volume. The question arises as to whether the integration should be performed over the current volume or over the initial volume. Since the current volume is usually unknown, this issue is best dealt with by referring all 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.

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545

3.1. Strain energy Let us ÂŽrst consider the strain energy function. In measuring the displacements of particles in the body, it is imperative to measure them with respect to the translated and/or rotated rigid body motion. This is accomplished by ÂŽxing a coordinate frame to the body. The translation and rotation of this body-ÂŽxed frame describe the rigid 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 undeformed (not unrotated) body, the discussion for the derivation of the strain energy proceeds in the same manner as for non-rotating bodies.

The strain energy per unit volume is denoted by W0 and is given by 1 W0 ˆ rij ij ; 2

…5†

where rij and ij are the stress and strain tensors, respectively. Small strains are assumed so that stressÂąstrain relationship will be taken to be Hookean (linear). Note that this implies that the strains are small, but does not imply that the displacements are small. For a ÂŻat plate, a plane stress condition may be assumed. This implies that rzz ˆ rrz ˆ rhz ˆ 0.

The strainÂądisplacement relation is required in order to express the strain energy 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 Lagrangian form of the strain tensor that is required. The von Karman plate theory can be shown [5] to lead to the following nonlinear strainÂądisplacement expressions:

2 our 1 ouz ‡ ; …6† rr ˆ or 2 or ur 1 ouh 1 ‡ 2 hh ˆ ‡ r r oh 2r

1 our rh ˆ 2r oh

ouz oh

2 ;

ouh ouz ouz ‡ uh ‡ r ; or or oh

…7†

…8†

where ur , uh and uz are the displacements of the disk in the r, h and z directions, respectively. For the linear Kirchho€ theory, the nonlinear terms involving uz are dropped from the above expressions, leading to the following linear strainÂądisplacement relationships: rr ˆ

our ; or

ur 1 ouh ‡ ; r r oh

1 our ouh rh ˆ uh ‡ r : 2r oh or

hh ˆ

…9† …10† …11†

The last required expressions are those relating the displacements of an arbitrary point in the plate to those of the middle surface of the plate. In thin plate theory, it is usually assumed that the linear Žlaments of the plate initially perpendicular to the middle surface remain straight and perpendicular and do not contract or extend. Transverse shear e€ects are thus neglected. This

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assumption leads to a relationship between the displacements of an arbitrary point ur , uh and uz and the displacements of the middle surface u, v and w. They are given by ow z ow ; uh v r; h : 12 or r oh Finally, the strain energy of the entire plate can be obtained by integrating over the entire volume of the plate. Note that the strain energy will be a function of u, v and w and of the vertical coordinate z. Furthermore, u, v and w are themselves functions of the in-plane coordinates r; h and of time, t. Thus the strain energy is an explicit function of z. This dependence can be eliminated by explicitly carrying out the integration over the thickness of the plate from z h to z h, where h is the distance between the middle surface of the plate and the plate bounding surface.

This procedure ®nally yields the strain energy of the plate as an explicit function of u r; h; t , v r; h; t and w r; h; t only. The strain energy of the plate is thus given by Z 2p Z R2 Z 2p Z R2

13 Wp r dr dh hW1 h3 W3 r dr dh; W0 uz w r; h ;

0

ur u r; h

R1

z

0

R1

where R1 and R2 denote the inner and outer radii of the disk. For nonlinear von Karman strains, the above expression is expanded as " 2 4 2 G ov ow u2 4 k G ov k 2G k G 4 k G 2 W1 k 2G or or r2 oh r 2

4 k 2G ou v ou k G ow

2 k 2G 2

2 r oh r oh r4 oh 2 2 2

2 k G ow ow ou ow v ov

4 k G 2 k 2G r2 oh or or or r or

2

2 2 k 2G ou ov 4 k G ov ow u ow 4 k G 3

r oh or r3 oh oh r oh

2 k 2G ou ow ow 2 k 2G ov ow ow

2 r oh oh or r or oh or

2 2

u ov 2k ov ow u ou u ow 4k ou ov 2k 4k

8 k G 2

r oh r oh or r or r or r or oh #

2 v ow ow ou v2 ; 14 k 2G 4 k G 2 k 2G 2 r or oh or r

2 4G k G ow

3 k 2G r2 or 2

2

2 4Gk ow ow o2 w 4G k G o2 w 4G o2 w

2 r 3 k 2G r2 oh2 or or2 3 k 2G r4 oh2 3r oroh 2 2 2 8G ow ow 4G k G o w ;

3 3r oh oroh 3 k 2G or2

8G k G W3 3 k 2G r3

ow or

o2 w oh2

4G 4 3r

ow oh

2

15

where G is the shear modulus and k is a constant which are related to the Young's modulus, E, and Poisson's ratio m of the material by

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Em ; 1 m 1 2m E G : 2 1 m k

547

16 17

Linear Kirchho strains could also have been employed to derive the expression (14) 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 Eq. (14) by retaining only secondorder terms and dropping all third and fourth-order terms.

3.2. 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 di erence between a rotating and stationary disk becomes apparent.

Let us setup two coordinate systems, S and B. Suppose that S is an inertial frame of reference and that B is rigid-body-®xed to the disk, so that B rotates with the disk at a spin rate of X with respect to S. Let r0 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 r0 is given by r r0 u at any given time. These vectors are chosen to be expressed in terms of unit vectors belonging to the B frame. Then, the velocity of any particle is given by dr=dt, where it must be remembered that since the unit vectors are ®xed in the rotating frame, their time derivative must be found as well.

Let er be unit vector in the r-direction such that er cos h iB sin h jB . Note that iB and jB are unit vectors in the x- and y-directions, respectively, in the body-®xed frame, B. Similarly, let eh sin h iB cos h jB be a unit vector in the h-direction, pointing in the direction of increasing h. Furthermore, let ez be a unit vector pointing in the z-direction such that er , eh , ez form a right-handed coordinate system.

The angular velocity vector of the body-®xed frame is given by x Xez . Points within the body are represented by the polar coordinates r; h; z . The original position of a particle is given by r0 rer zez . The deformed position of the same particle is given by r r0 u r ur er uh eh z uz ez ;

18

where ur , uh and uz are the displacements in the er -, eh -, ez -directions of a particle, respectively.

Each of these displacements will be a function of time and the original position of the particle in question. The velocity of this particle is given by i h dr 19 v u_ r Xuh er u_ h X r ur eh u_ z ez : dt Since the unit vectors are orthonormal, the square of the speed is given by i2 h v v u_ r Xuh 2 u_ h X r ur u_ 2z :

20

Once the velocity as measured by an inertial observer of any particle has been found, the kinetic energy of a small element of volume can be expressed as 1=2 q dV v v, where q 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 undeformed volume. Note that since the velocity has been

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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 expression is now expressed as a function of ur , uh and uz , the displacements of an arbitrary point on the disk in the r-, h- and z-directions, respectively. As for the strain energy, Eq. (12) 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 ®nally yields the kinetic energy of the plate as an explicit function of u r; h; t , v r; h; t and w r; h; t only. The kinetic energy of the plate is thus given by Z 2p Z R2 Z 2p Z R2

KEp r dr dh hKE1 h3 KE3 r dr dh; 21 KE 0

R1

0

R1

where

ov KE1 qX2 v2 u2 r2 2ru 2qX u ot " #

2 2 2 ou ov ow

;

q ot ot ot

v

ou ov r ot ot

" 2

2 # 2 2 2 # ow q o2 w ow 2 ow 2 r 2 r oh or 3r ohot orot

2qX o2 w ow o2 w ow :

3r ohot or orot oh

qX2 KE3 2 3r

22

"

23

Here q is the density of the disk, X is its angular velocity, h is its half-thickness, and the displacements of the middle surface are given by u, v and w. Note that the inner and outer radii of the disk are given by R1 and R2 , respectively. The expression for kinetic energy remains the same regardless of whether linear or nonlinear strains are employed since it does not depend on the strain. In other words, the nonlinearity of this problem is con®ned to the strain energy.

4. Equations of motion Hamilton's principle can be concisely stated as Z t1 L dt 0; d t0

24

where L T U is the Lagrangian function, T the kinetic energy and U is the strain energy of the system. As a variational principle, it states that the variation of the integral of the Lagrangian from time t0 to time t1 vanishes provided that the variations of the displacements vanish at t0 and t1 and 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 Appendix A.

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549

4.1. Nonlinear equations of motion The full nonlinear equations of motion are

q 1 m2 o2 u oX ov 2 v X r u 2X E ot2 ot ot

1 m o2 u 1 m o2 v o2 u

2r2 oh2 2r oroh or2 " 2 1 1 m ow 3 m ov

r 2 or 2r oh

q 1

ow oh

2

u ou

r or

#

ou X v 2X ot

25

2

1 m ow o2 w 1 m o2 v

2r or oroh 2 or2

1 1 m ow ow 1 m ov 3 m ou

r 2r oh or 2 or 2r oh

1

m v 2 r

1 m o2 u 1 o2 v 1 ow o2 w 1 m ow o2 w

; 2 2 3 2r oroh r oh r oh oh2 2r oh or2

m2 o2 w h2 X2 2 rw

E ot2 3

h2 4 1 ow 3 ow

r w 4 r oh 2 oh 3

q 1

1 m ow o2 w ow o2 w 1 m ow o2 w

; 2r2 or oh2 or or2 2r2 oh oroh

m2 o2 v oX r u 2 E ot ot

1 m 2r2

h2 o2 2 rw 3 ot2

26

o2 w o2 v

r oh2 oh2

r ow ow 1 m ou

r 2 oh or 2 oh

2 1 m 2 o2 u 1 m r2 ow o2 w 1 m 3 o2 v 2 o w ow

r r

rv

r 2 2 oroh 2 oroh or 2 or 2 oh or2 " 1 ow 2 o2 w ow ou 1 m 3 o2 v 1 m 2 o2 u 1 m r3 4 r r

r r or oroh oh or 2 oroh 2 oh2 #

2 2 r3 ow r2 ow o2 w 3r4 ow o2 w 1 m 2 ov 4o u r

r

or2 2 oh 2 or 2 or oh2 2 or or2

1 o2 w ov o2 w ou m ov 2 ou u 4 2 mr r u 2

r oh or oh or or r oh

1 m o2 w ou ov v r :

2 r oroh oh or

1

m 2

ov r or

2

27

Note that the full nonlinear representation of the spinning disk problem requires the solution of three nonlinearly coupled partial di erential equations (25)±(27). By way of contrast, the non-

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linear model used by other researchers consists of two coupled partial di erential equations (3) and (4) [3]. The di erence 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 terms are neglected in the model that leads to the nonlinear formulation given by (3) and (4).

4.2. Linear equations of motion derived from linear strain The equations of motion arising from the use of linear (Kirchho ) strain±displacement expressions are

m 2 o2 u E ot2

ov X r u 2X ot

2 2 2 1 m o u 1 m o v o u 1 ou

2r2 oh2 2r oroh or2 r or

q 1

oX v ot

2

m 2 o2 v oX r u 2 E ot ot

3

m ov 2r oh

ou X v 2X ot

2 2 1 m o u 1 m o v 1 1 m ov 3 m ou

2r oroh 2 or2 r 2 or 2r oh

q 1

q 1

m2 o2 w h2 X2 2

rw E ot2 3

u ; r

28

2

h2 o2 2 rw 3 ot2

h2 4 r w: 3

1

m v 1 o2 v 2 2; 2 r r oh

29 30

Note that Eqs. (28) and (29) for the in-plane vibrations are the same equations as those derived by other authors [6±8]. However, Eq. (30) for the linear transverse vibrations is not the same as Eq. (1) or (2), which are the equations currently used for the linear transverse vibrations of a spinning disk. Eq. (30) bears a strong resemblance to the equation for the linear 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 (Kirchho ) strains. Thus, it should not come as a surprise that the equation of linear transverse vibrations of a rotating disk based on the assumption of linear strains should be similar to its stationary counterpart.

4.3. Linear equations of motion derived from nonlinear strain Let us now consider the linearization of the nonlinear equations of motion. In other words, consider small displacements from equilibrium. The important observation to make is that, due to the rotation of the disk, the equilibrium 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 , veq 0, weq 0, so that the equilibrium value of u is a function of radius only. Now consider the equations obtained by neglecting all nonlinear terms in the nonlinear equations of motion with the exception of terms containing u or ou=or (corresponding to the equilibrium value of u). The equation for the equilibrium value of ueq is given by d2 ueq 1 dueq

r dr dr2

ueq q 1 m2 2

X r ueq 0: E r2

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The equations for the small displacements of u; v and w are given by

q 1 m2 o2 u oX ov 2 m X u 2X E ot2 ot ot

1 m o2 u 1 m o2 v o2 u 1 ou 3 m ov u

;

2r2 oh2 2r oroh or2 r or 2r oh r

m 2 o2 v oX r u 2 E ot ot

ou X v 2X ot

2 2 1 m o u 1 m o v 1 1 m ov 3 m ou

2r oroh 2 or2 r 2 or 2r oh

q 1

551

32

2

1

m v 1 o2 v 2 2; 2 r r oh

33

m2 o2 w h2 X2 2 h2 o2 2

rw rw E ot2 3 3 ot2

h2 4 o2 w dueq ueq ow 1 m dueq d2 ueq 1 o2 w dueq ueq 2 2 m

r w 2 m

: or or r r oh 3 dr r dr dr2 dr r 34

q 1

Furthermore, note that corresponding to an in-plane purely radial displacement u r , use of linear stress±strain and linear strain±displacements relationships leads to the following stress±displacement relationships:

E du u m ; 35 rrr 1 m2 dr r

E du u 36 rhh m : 1 m2 dr r Using these relationships, Eq. (34) can be rewritten as 2 o w h2 X2 2 q

rw ot2 3

h2 o2 2 rw 3 ot2

Eh2 1 o ow rhh o2 w 4 w

: rr

r rr r or or 3 1 m2 r2 oh2

37

, this is the same equation as obtained by Lamb and With the exception of the r2 w and r2 w term is not Southwell [2] for the transverse vibrations of a spinning disk. The presence of the r2 w unexpected; it is simply the term due to the rotary inertia of the disk. The physical meaning of the r2 w term will be explained later.

5. Boundary conditions The boundary term obtained from applying Hamilton's principle and integration by parts is given below. Recall that r here must be evaluated on the boundary. 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 radii, so r will take two possible values

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"

2 2 # u ou 2m ov ow m ow

dh 3E r du 2m 2 2 r or r oh or r oh 0

Z 2p Z 2p 1 ou ov 1 ow ow ow 3 3 2 3E 1 m r dv d v dh 2Eh r r oh or r oh or or 0 0 2

Z 2p o w m o2 w m ow o3 w 2 ow 2 2 3

r dw X

dh dh 2 1 m qh or2 r2 oh2 r or or orot2 0

Z 2p o 2 1 m o2 ow w 2 3 r dw r w dh 2Eh or r2 oh2 or r 0 " 2

3 Z 2p 1 m ow ov 1 ow ow ou ow ow 3

r dw 2 3E

r oh or r2 or oh or or or 0

# 1 m ow ou 2m ow ov 2 v u dh: r oh oh r or oh Z

2p

3

38

There are a few points that are worth mentioning. 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 (Kirchho ) strain±displacement relations been used, the corresponding boundary conditions would also have been linear and they can be obtained from the above expression by neglecting all nonlinear terms.

It was previously noted that formulating the problem in this manner automatically accounts for the e ect of rotary inertia in the equations of motion. The corresponding term in the boundary condition is o3 w= orot2 . That is, the variation of some particular part of the kinetic energy ex term in the equation of motion and to the o3 w= orot2 term in the pression gives rise to the r2 w boundary condition. Hence, if the e ect of rotary inertia is ignored (or included) in the equation of motion, then the corresponding term must also be ignored (or included) in the boundary condition.

From Eq. (38), the boundary conditions for the special cases of linear in-plane and transverse vibrations can be derived.

5.1. Linear transverse vibrations For the linear transverse vibration problem, the boundary conditions at a free edge become

o2 w m ow 1 o2 w 0; 39

or2 r or r oh2 o 2 1 m o2 r w or r2 oh2 Note that Z h rrr z dz h

ow or

w r

1

m2 q o3 w E orot2

2

2Eh3 o w m ow 1 o2 w

:

3 1 m2 or2 r or r oh2

ow X : or 2

40

41

Hence, the ®rst boundary condition (39) states that the moment at the free edge is zero, and this is in agreement with boundary conditions given in the literature.

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The standard second assumption at a free edge is that the Kirchho shear, Vr or `edge reaction' [9] be set to zero. For stationary plates (neglecting rotary inertia), the Kirchho shear is given by 2

or w 1 m o2 ow w

0; 42 Vr D or r2 oh2 or r where D 2h3 = 3 1 m2 is the bending sti ness of the disk. Recall that h denotes the halfthickness of the disk. This last 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 Eq. (40) and the standard boundary condition (42 ), do not quite agree. The derived boundary condition (40) di ers by the inclusion of a term proportional to qX2 ow=or . Due to the presence of X2 , this term would vanish for a stationary disk.

It must be observed that the variation of some particular portion in the kinetic energy gives rise to the X2 r2 w term in the equations of motion and to the qX2 ow=or term in the boundary condition; therefore explaining the presence of one should explain the presence of the other.

The presence of the centrifugal term in the balance of forces at the edge (and interior) of the disk should not come 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 signi®cance of these new terms will be addressed in the discussion.

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

ou m ov

u du 0; or r oh

1 ou v ov

dv 0: r oh r or

43 44

For a solid disk, Eqs. (43) and (44) must be true on the outer radius. For an annulus, Eqs. (43) and (44) must hold at both the inner and the outer radius.

Note that

Z h 2Eh ou m ov u rrr dz

; 45 m 1 m or r oh h

Z h 1 ou v ov rrh dz h

: 46 r oh r or h Thus, Eq. (43) implies that on the boundary either the displacement of the middle surface in the radial direction must be speci®ed or the integral of the stress in the radial direction over the side of the disk must vanish. Similarly, Eq. (44) states that on the boundary, either v must be speci®ed, or the integral of the shear stress over the side of the disk must vanish.

5.3. Nonlinear boundary conditions The boundary conditions at a free boundary obtained from applying Hamilton's principle and integration by parts are given below. Recall that r here must be evaluated on the free boundary.

Thus, for a solid disk, r below is the radius of the disk

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2 2 u ou m ov 1 ow m ow

m 2 0; r or r oh 2 or 2r oh

1 ou ov 1 ow ow v 0; r oh or r oh or o2 w m o2 w m ow 0;

or2 r2 oh2 r or

o 2 1 m o2 ow w 1 m2 q 2 ow o3 w r w

X or r2 oh2 or r E or orot2 "

2 3

3 1 m ow ov 1 ow ow ou ow ow 1 m ow ou

2

2 2h r oh or r2 or oh or or or r2 oh oh #

2m ow ov

u : r or oh

47 48 49

v 50

If the linear stress±strain relations are used along with the nonlinear (von Karman) strain±displacement relations, then the above boundary conditions can be interpreted as easily as the linear boundary conditions. Note that the nonlinearity in the boundary condition is a direct consequence of the nonlinearity of the strain±displacement relations. Eqs. (47) and (48) are actually identical to the respective statements that Z h rrr dz 0; 51 h Z h rrh dz 0: 52 h

Note that this is the same as for the linear boundary conditions. In other words, both the linear and nonlinear boundary conditions at a free edge corresponding to the u and v variables are merely the statements that the integral of the radial and shear stresses over the side of the disk must vanish. In addition, Eq. (49) remains unchanged from its linear counterpart and thus states that the moment at the free edge must be zero. The ®nal nonlinear boundary condition (50) appears to be a great deal more complicated 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 Eq. (50) can be rewritten as f2 ow=or LHS of Eq. (47) + 1 m =r ow=oh LHS of Eq. (48)g. In other words, if Eqs. (47) and (48) are both satis®ed, then the nonlinear portion of Eq. (50) is automatically equal to zero. Hence, the nonlinear boundary condition, Eq. (50), is identical to its linear counterpart

o 2 1 m o2 ow w 1 m2 q 2 ow o3 w 0: 53 r w

X or r2 oh2 or r E or orot2 6. Discussion and comparison between the derived and existing models 6.1. Three nonlinear equations vs two The nonlinear equations of motion for a spinning disk are given by Eqs. (25)±(27). Note that there are three nonlinearly coupled equations, implying that they must be solved simultaneously.

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The spin rate X is usually taken to be constant so that oX=ot 0. By way of contrast, the previous nonlinear formulation of the problem results in two nonlinearly coupled equations, namely Eqs. (3) and (4).

It is possible to use a stress function to reduce the three new Eqs. (25)±(27) to two where the variables to be solved for would be the transverse displacement and the newly introduced stress function. However, this would necessitate the omission of the in-plane inertias, o2 u=ot2 and o2 v=ot2 , as well as the Coriolis terms, ou=ot and ov=ot. Indeed, this is the approach taken by Nowinski in deriving the standard Eqs. (3) and (4).

It must be recalled that the centrifugal force is not really an external force at all, but rather a consequence of the fact that the reference frame is rotating and thus non-inertial. The Coriolis force is due to the same e ect. At present, there is no theoretical justi®cation for neglecting the inplane inertia and Coriolis terms. In-plane inertia is typically ignored for stationary plates, and this has been shown to be a good approximation for stationary plates [4]. However, the same calculation fails for the rotating plate precisely because of the presence of the centrifugal and Coriolis forces. In other words, it is because the plate is rotating that the same argument fails. Thus, the validity of rotating disk models that reduce the number of nonlinear equations to be solved from three to two must be questioned.

6.2. Lagrangian vs Eulerian variables In the derived nonlinear Eqs. (25)±(27) and their linear counterparts (28)±(30), the centrifugal force appears as a term proportional to X2 r u . This is reasonable given that r u is the current radial position of a given particle. A particle originally at radius r has radius r u after deformation takes place. This is consistent with the Lagrangian 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 centrifugal force will be proportional to X2 r. Now the location of the boundaries is an unknown; if the disk stretches, the new location of the boundaries is part of the unknowns of the problem. Most authors use the Eulerian description X2 r with the boundaries (incorrectly) located at their (original) Lagrangian location.

This is the approach taken by Nowinski [3], as a ®rst step towards ®nding the nonlinear equations of motion (3) and (4). 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 ®rst be found. Again, in this ®rst step, the traditional approach has been to represent the centrifugal force as proportional to X2 r [2,10±13].

While it may seem that for small displacements there should not be much di erence between r and r u , Bhuta and Jones [6] showed that the actual solutions obtained for the linear in-plane problem can be quite di erent.

6.3. Decoupled in-plane and transverse plate problems Consider the linear equations of motion (28)±(30). Note that, although the equations for the inplane 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. Recall that, in the traditional linear formulation of the problem as a membrane or a membrane with bending sti ness, 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 simpli®cation for the solution of the

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linear transverse vibration problem. This independence is a consequence of the assumption of linear strains. The linear equation (34) derived from the nonlinear strain assumption does not feature this decoupling between the in-plane and transverse problems.

6.4. The membrane problem Let us once again consider Eq. (37) but now under the assumption that the plate is so thin that the h2 terms are small in comparison with the other terms and can be neglected. In other words, we are supposing that the disk is a membrane. In this case, Eq. (37) reduces

o2 w 1 o ow rhh o2 w rrrr ; 54 q 2 2 ot r or or r oh2 which is the equation for a spinning membrane as derived by Lamb and Southwell [2]. Furthermore, if the in-plane stresses rrr and rhh are equal and constant, then Eq. (37) is exactly the traditional equation for the transverse vibrations of a membrane. Similarly dropping the h2 terms in the boundary term, Eq. (38) and using the same linearization procedure that led to Eq. (37) gives

ow

3 0 55 r dwrrr or boundary for the boundary term. Eq. (55) gives the possible boundary conditions as either w is ®xed on the boundary dw 0 or r3 rrr ow=or 0 at the boundary. For the second of these boundary conditions, if rrr 0 on the boundary then the boundary conditions reduces to requiring that ow=or is bounded on the boundary. Both of these boundary conditions are what would 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 nonlinear strains and then linearising 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 from those of plates. The interested reader is referred to Niordson [14] where this derivation is performed by means of perturbation techniques based on the slenderness ratio.

6.5. Presence of rotary inertia It should also be noted that the rotary inertia has automatically been taken into account in both the linear and nonlinear formulations of the problem. The term representing the e ect of the . To ignore the e ect of rotary inertia, it su ces to rotary inertia of the disk is proportional to r2 w 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 e ect of rotary inertia has hitherto not been considered. While the e ect 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 true for spinning plates.

6.6. Presence of new terms As previously observed, the equations of motion for the linear transverse vibrations arising from the assumption of both linear and nonlinear strains featured a term proportional to X2 r2 w

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while the corresponding boundary conditions also contained a term proportional to X2 . The usual formulations of spinning disk problems do not contain these terms; therefore their presence and physical signi®cance must be carefully examined.

Suppose that the in-plane displacements and rotary inertia are ignored. Then the kinetic energy of the rotating plate becomes # Z 2p Z R2 " 2 ow qh3 X2 rw rw r dr dh: 56

qh KE ot 3 0 R1 The variation of rw rw gives rise to the new terms in the equation of motion and in the boundary condition. Where does this term come from?

First, note that due to the presence of h3 , this term will not arise in the membrane problem.

Since Lamb and Southwell [2] derived their famous equation of motion by adding bending sti ness to the spinning membrane equation, it is not surprising that their formulation does not include the term in question. Furthermore, 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 x r, where r r0 u. Now consider x u, where we consider the contribution to u from the transverse displacement only. In other words, take

ow z ow u z ; ;w : 57 or r oh Since x Xk, it follows that x u

zx rw

58

and x u x u z2 X2 rw rw;

59

which explains 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 x r is an in-plane term eventually giving rise to the centrifugal force. Why does it crop up in the equation of transverse vibrations? The answer lies in closer examination of Eq. (58). This is indeed anRin-plane term. However, it is linear in z and thus h gives rise to a bending moment. In other words, h x uz dz gives a non-zero contribution. If the equations of motion were to be derived in the Newtonian way (for example, see [9]) the equations summing the moments are used to simplify the equations summing the in-plane and transverse forces. In this way, the bending moment due to the x u term would eventually make its way to the equation expressing the balance of forces in the transverse direction.

In short, the presence of the new X2 terms in the equation of transverse vibrations and its corresponding boundary condition re¯ects the contribution of the bending moment due to the x u term. It is only relevant for plates (as opposed to membranes). It is also a consequence of the use of Lagrangian coordinates.

7. Summary and conclusions In summary, the traditional assumptions of the plate theories of Kirchho and von Karman were utilized to derive linear and nonlinear 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

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inertia. Hamilton's principle was employed to derive the equations of motion, thus automatically yielding the correct boundary conditions. Three sets of equations were derived; nonlinear equations corresponding to nonlinear strains, linearized equations corresponding to nonlinear strains and linear equations based on the assumption of linear strains. These equations bear similarities to the plate equations based on nonlinear strains, linear equations of a spinning plate and linear equations of a stationary plate, respectively. The new terms that arise are due to in-plane inertia, Coriolis terms, rotary inertia and bending moment due to the rotation of the plate.

The resulting boundary conditions have also been examined. The presence of new terms in the equation of motion also brings with it corresponding terms in the boundary conditions. The centrifugal term appears in the boundary conditions at the free boundary. The presence of these terms in the equations of motion and boundary condition has been interpreted as arising from a bending moment due to the centrifugal force. When introducing (or neglecting) particular e ects in the equations of motion, it is important to introduce (or neglect) their corresponding term in the boundary conditions.

The second part of the paper will focus on the analysis of the two equations for the linear transverse vibration problem along with the appropriate corresponding boundary conditions. The two linear equations to be considered are the one arising from the assumption of linear strains and the other arising from the linearization of the equations derived with nonlinear strains. In particular, it is important to determine whether the assumption of linear strains for a rotating plate will be as successful as for a stationary plate.

Acknowledgements This research was ®nancially supported by the National Science and Engineering Research Council of Canada.

Appendix A. Variation of the Lagrangian After the Lagrangian function has been assembled from the kinetic and strain energies, the last step towards ®nding the equation of motion is to perform the ®rst variation of the Lagrangian.

Since it is the displacements of the middle surface, u, v and w that are the generalized coordinates of the system, the variation is performed 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, du, dv and dw. Integration by parts can then be used to obtain expressions in terms of du, dv, dw. These will involve integrals over the domain and line integrals over the boundary as well.

Since the generalized coordinates are independent, requiring the coe cient 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 ®rst example, consider the following variation:

Z t1 Z Z Z t1 Z Z Z t1 Z Z 1 2 1 2 dw dA dt: d w_ dA dt w_ dA dt w A:1 d 2 t0 D 2 t0 D t0 D In the above, D refers to the domain of integration. Note that switching the order of space and time integration was employed along with integration by parts. However, the boundary term

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disappeared since when using Hamilton's principle, the variation of the coordinates is assumed to be zero at times t0 and t1 .

To extract the ®rst variation of any coordinate from the variation of its time derivative, the same technique as illustrated above may be used. For example, consider the variation of rw rw, which occurs in the expression for the kinetic energy Z t1 Z Z Z t1 Z Z rw rw dA dt d rw rw dA dt: A:2 d t0

D

t0

D

Now consider the integration over the space variables only Z Z Z Z Z Z Z ow 2 d rw rw dA 2rw r dw dA dwr w dA dw ds: on D D D C

A:3

This yields an integral over the space domain and a line integral over the boundary.

The procedures illustrated above were carried out for the entire expressions of the kinetic and strain energies. The approach is clear but tedious. Setting to zero the coe cient of du, dv, dw in the integration over the domain yields the equations of motion.

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H. Lamb, R.V. Southwell, The vibrations of a spinning disk, Proc. Roy. Soc. Lond., Ser. A 99 (1921) 272±280.

J.L. Nowinski, Nonlinear transverse vibrations of a spinning disk, J. Appl. Mech. 31 (1964) 72±78.

H.N. Chu, G. Herrmann, In¯uence of large amplitudes on free ¯exural vibrations of rectangular elastic plates, J. Appl. Mech. 23 (1956) 532±540.

Y.C. Fung, Foundations of Solid Mechanics, Prentice-Hall, Englewood Cli s, NJ, 1965.

P.G. Bhuta, J.P. Jones, Symmetric planar vibrations of a rotating disk, J. Acoust. Soc. Am. 35 (7) (1963) 982±989.

J.S. Chen, J.L. Jhu, On the in-plane vibration and stability of a spinning annular disk, J. Sound Vib. 195 (4) (1996) 585±593.

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R.V. Southwell, Free transverse vibrations of a uniform circular disk clamped at its centre, Proc. Roy. Soc. Lond., Ser. A 101 (1922) 133±153.

S. Barasch, Y. Chen, On the vibration of a rotating disk, J. Appl. Mech. 39 (1972) 1143±1144.

W. Eversman, R.O. Dodson, Free vibration of a centrally clamped spinning disk, AIAA J. 7 (10) (1969) 2010± 2012.

M.P. Mignolet, C.D. Eick, M.V. Harish, Free vibration of ¯exible rotating disks, J. Sound Vib. 196 (5) (1996) 537± 577.

F. Niordson, An asymptotic theory for circular cylindrical shells, Int. J. Solids Struct. 27 (13) (2000) 1817±1839.