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Full Paper Proc. of Int. Conf. on Advances in Robotic, Mechanical Engineering and Design 2012

Dynamic Stability of a System Including Three Shafts M. Soltan Rezaee1, A.A. Jafari2, M.R. Ghazavi3 1,3

Tarbiat Modares Univ., Tehran, Iran soltanrezaee@gmail.com ghazavim@modares.ac.ir 2 K. N. Toosi Univ. of Tech., Tehran, Iran ajafari@kntu.ac.ir

Abstract— A system including three torsionally elastic shafts which they have been linked through two universal joints with different rotation axises is analyzed. The system stability has been studied by means of a three degree-of-freedom model in a spatial coordinate (three dimensional). Equations of motion for the system were derived. The differential equations consist of a set of Mathieu–Hill equations. Their stability is analyzed via a monodromy matrix method. Finally dynamic stability regions have been shown on system parameters such as rotational velocity, misalignment angle’s of shaft axis, shaft’s stiffness and rigidity. The results are presented in the form of stability maps constructed on different parameters and have been discussed. Index Terms— Shaft System, Dynamic Stability, Parametric Resonance, Universal Joint.

a case like Porter and Gregory. Chang [6] investigated both the linear and non-linear one-degree-of-freedom models, to obtain higher approximations to the instability zones of the models. Zeman [7] investigated a case consisting of n shafts connected via universal joints [8]. He derived the linearized equations of motion and gave approximations for the primary parametric resonance of a 2-dof system via Mettler ’s parameter expansion approach. Asokanthan and Hwang [9] used the method of averaging and maked closed-form dynamic stability zones related to combination resonance. Then in another work [10], the same case is investigated through Lyapunov exponent calculations. Finally, Asokanthan and Meehan [11] revisited a 2-dof nonlinear model. They demonstrated via numerical simulations that the shaft system may display chaotic behavior under specific conditions. DeSmidt and et al [12] explored the effects of damping, misalignment, load inertia and load-torque on the stability of a segmented shaft connected with U-Joint couplings operating at sub and supercritical speeds. A non-dimensional, linear, periodically time-varying model was developed and numerical Floquet theory was used to explore the effects of different system parameters. Recently, Bulut and Parlar [13] considered a 2dof model. Their system consisting of two torsionally elastic shafts interconnected through a Hooke’s joint. They linearized the equations of motion and stability of the solutions was analyzed by means of a monodromy matrix method [14]. Also Mazzei [15] studied a shaft system which motion was restricted to one plane. For the case studied, he has shown that it is possible to drive through the instability if the dwell time is not more than forty times the period of the parametric excitation of system. The aim of this work is to study the stability analysis of shaft system interconnected through universal joints via a monodromy matrix method. In the previous works, almost all the authors have focused on systems with one joint. These systems have two degree of freedom. While in many power transmission systems, it is necessary to use two universal joints to transfer the rotational motion.Therefore, a three DOF model including three torsionally elastic dimensionless massless rotating shafts,each carrying an inertia disk at one end, and connected via two universal joints is investigated. The linearized equations of motion of the system are obtained, and instability maps are presented on different system parameters to investigate the effect of different parameters on the instability.

I. INTRODUCTION Shaft system is one of the types of power transmission systems. It has many applications due to high speed and low weight capability. This system is composed of several shafts connected together and the rotational motion of the driver shaft is transmitted to the driven shafts. Based on the application, the shafts can be misaligned. To connect misaligned shafts, there are many ways. A universal joint is a positive, mechanical connection between rotating shafts, which are usually not parallel, but intersecting. They are used to transmit motion, power, or both. The simplest and most common type is called the Cardan joint or Hooke joint. It consists of a pair of hinges located close together, oriented at 90° to each other, connected by a cross shaft. Universal joints have many advantages including high torsional stiffness, possibility of large angular displacements, low price and easy repair. It also has several disadvantages. It requires lubrication to reduce wear, velocity fluctuation increases with operating angle, shafts must lie in precisely the same plane and transforming a constant input speed to a periodically fluctuating one. This system is parametrically excited and has dynamic instability conditions. Determining instability zones are important. A single-dof model was considered by Porter [1] for stability investigation. He used a linearized model to predict the instability zones of a map obtained by using the Floquet theory [2]. This analysis was later extended by Porter and Gregory [3] to a non-linear model and used the first approximation of the Krylov and Bogoliubov [4] to the original non-linear equation of motion for the prediction of the amplitude of the oscillations. Eidinov et al. [5] considered © 2012 AMAE DOI: 02.ARMED.2012.2. 505

Full Paper Proc. of Int. Conf. on Advances in Robotic, Mechanical Engineering and Design 2012 II. EQUATIONS OF MOTION The shaft system under study, including three shafts which is driven through two universal joints as illustrated in Fig. 1 Each shaft has one-degrees-of-freedom, so the system is a three-dof one. The shafts are uniform along the length and including a torsionally flexible dimensionless massless rotating bar consisting of a torsional spring ki, connected to a disk of rotary inertia J, at their right end. The input angular velocity Ω0, is kept constant, due to an attached flywheel. It is considered that the energy dissipation associated with the system is proportional to the viscous damping ci, exerted on the shaft. This rotational system having an angular misalignment of βi, is intended to be small. Their flexural vibrations are assumed to be entirely prevented via support bearings. (The bearings are ideal long, therefore no lateral flutter is possible.) The twist vibrations of the shaft system can be displayed to be governed by (see Appendix A)

J11  c11   01c22  k11   01k1 2  0 J 2 [2   011  01 ( 0  1 )] c22  02 c33  k2 2   02 k3 3  0 J 3 [3   02 (2  ( 011  01 ( 0  1 )))  (   (   ))]  c   k   0 02

3 3

Figure 1. Schematic of a shaft system in three dimension (driver shaft and second driven shaft can be in different planes)

Introducing the dimensionless system parameters

(1)

   0t ,   (2)

1  (3)

out1 cos 1  in1 (  1 ) 1  sin 2 1 sin 2 1

J k J2 k ,  2  3 , 1  2 , 2  3 (8) J1 J2 k1 k2

and then putting in the equations of motion, becomes

3 3

where θ i is twist coordinates, overdots show differentiation with respect to time, and ηi is the positiondependent velocity transfer ratio of a universal joint as becomes [8]

1 

0  1 1  1                2   E   2    F   2    01 (9)                  3  3  3   01 02 

(4)

where E and F are 3×3 matrices with the following elements:

E11 

2 

 out 2 cos  2 2   in 2 ( 2 ) 1  sin 2  2 sin 2  2

(5)

1   0t  1

(6)

E12  

 2  out 1t   2

(7)

E13  0

21 01   

is the overall angular position of the primary segment of the universal joint. The nonlinear Equations (1-3) can be linearized by expanding them into McLaurin series of their parts  ,  ,  ,  ,  ,  ,  ,  ,  and ignor e all the

    E 21  01

nonlinear terms under the considerations of small fluctuation amplitudes and frequencies.

E22 

0 c c1 c ,  , 1  2 ,  2  3 c1 c2 k 1 J1 k 1J 1

2 01   

21  2 1 01       1 

Full Paper Proc. of Int. Conf. on Advances in Robotic, Mechanical Engineering and Design 2012 III. ANALYSIS

2  E23   2  02    1

In this study the zones of instability are investigated via a numerical method based on the so-called monodromy matrix. The monodromy matrix method is very computationally intensive, but has some advantages over other method like perturbation techniques and Hill’s infinite determinant. For example, the method is a simple reliable one that does not have any approximations and can capture the whole instabilities within a parameter area (to within numerical exact). This technique is briefly described below. A state–space display of homogeneous system may be expressed as

  02    E 31  01 2      1 02   E 32  02  1 E33 

2 2 

1 2   02    1  1 

{ y }  [G ( , 1 ,  2 , 1 ,  2 ,  1 ,  2 , 1 ,  2 , )].{ y}

(12) It is a first order form of the equations of motion, where

1 2  F12   12 01    F11 

{ y}  1 ,  2 ,  3 , 1,  2 ,  3

(13)

and the matrix [G] is a 6×6 π -periodic matrix defined as

 0 F

G   

F13  0 1 F21  2 01   

I   E 

(14)

where 0 is a 3×3 zero matrix and I is a 3×3 unit matrix. The basic matrix is denoted by

 A( )   { y1 ( )},{ y2 ( )},...,{ yn ( )} F22 

1  2 1     2  01     1 

where

{ y1 ( )},{ y2 ( )},...,{ yn ( )}

1 0  A(0)    I      0

  02    F31  01     F32  2 02  1

2  1 2 1        02 2   1 2 

0 1  0

... ...  ...

0 0    1

(17)

Based on Floquet theory, a basic solutions’ matrix of system (9) can be defined as

A( )  D ( )e P

primes show differentiation with respect to τ, and

 01     01      cos 1 1  sin 1 sin  2

(18)

where D(τ) is a π-periodic matrix. There are a constant matrix B, called monodromy matrix. P is also a constant matrix which is associated with B, by P=(1/π)logB. If the basic matrix is normalized so that A(0)=I, then B=A(π).The eigenvalues σi; i=1,2,3,4,5,6 of the monodromy matrix.The system is stable if and only if mod(σi)d”1 for all i. In this case, the matrix B is obtained by numerically integrating (12) and its eigenvalues are computed to investigate dynamic stability of the system. Runge Kutta technique with step size adaptive control (RK45) is used for numerical integrating and the value of error control tolerance is taken 10-10.

(10)

 02     02      cos  2 1  sin 2  2 sin 2  (11) Equation (9) is for the torsional motion of the three-dof system. It consist of three shafts which connected throw universal joint and has periodically varying coefficients. There are a collection of linear differential equations with πperiodic ratios (or a collection of Mathieu–Hill equations). It is known that closed-form solutions to these equations are not available, but, insight into the dynamic stability behaviour may be obtained by using of the linearized homogeneous part of the equation. A stability analysis is performed by considering the homogeneous system. © 2012 AMAE DOI: 02.ARMED.2012.2. 505

(16)

are any n linearly independent solutions of the system. Integrating equation (12) n times from 0 to τ with the n initial conditions

    F23   22 02  1

F33 

(15)

Full Paper Proc. of Int. Conf. on Advances in Robotic, Mechanical Engineering and Design 2012 Also, the effect of the damping ratio α2 are investigated and the maps are illustrated in Fig. 4 and Fig. 5. An inspection of these figures reveals that: The value of α2 does not have a prominent effect on the width of the resonance areas in the high velocity ratio but has a more prominent effect in the low velocity ratio (Fig. 5). Therefore, increasing α2 (the damping ratio of the second driven shaft) lead to the the narrower instability areas especially in the case of low velocity ratio (Fig. 4).

IV. NUMERICAL CASES In this part numerical cases are investigated. The results are expressed in the form of stability maps constructed on Ω–α 2 and Ω–β 2 to reveal the effects of three system parameters on the system stability. These parameters are the shaft angular velocity, the source of energy loss of the system and misalignment angle. In all of these cases, the parameters γ1, γ2, µ1, µ2, α1 and ζ are set to γ1 =1, γ2=10, µ1=1, µ2=1, α1=1, ζ=0.005 and then stability is checked point by point. The grid period is taken 0.01 or less to catch very narrow instable zones. First, the effect of misalignment angle β2 is investigated for a system with α2 =1 and two distinct values of β1 (β1=0.2 and β1=0.4), and results are represented in the form of stability maps constructed on the Ω–β system parameter plane in Figs. 2 and Figs. 3,where shaded points show unstable zones. After investigation of these maps, notes that; The parameter â (misalignment angle) has a significant effect on the instability of the shaft system. As the angle (both β1 and β2) increases from 0 (where no parametric excitation exists) to π/2 radian (where the joint cannot move) the unstable zones become wider (In practical applications, the angle β does not usually increase π/4 because of design limitations). Also, the points of some areas are not obviously distinguish; it is known that if the system is assumed undamped, these points would be seen

Figure 3. Stability analysis: effect of the misalignment angle β 2 (α2 =1, β 1=0.4).

Figure 4.Stability charts: effect of the parameter α 2 (β 2 =0.4, β 1 =0.2).

Figure 2. Stability analysis: effect of the misalignment angle β 2 (α2 =1, β 1 =0.2).

Full Paper Proc. of Int. Conf. on Advances in Robotic, Mechanical Engineering and Design 2012 first joint.The first driven shaft is under the effect of the response torque of the output segment of the first joint. The equations be written as

c22  k2 2  M 1*  0

(A.2)

 ]  c   k   M J 2 [2   out 1 2 2 2 2 2

(A.3)

where θ2 is the rotational co-ordinate of J2 with respect to left end of the first driven shaft and Ωout1 is the rotation velocity of the output part of the first joint. The relation between the input and the output of the first universal joint be written as

M 1  01 M 1* ;01   out1 in1

(A.4)

The second driven shaft is under the effect of the response torque of the output segment of the second joint. The equations be written as

c33  k33  M 2*  0

(A.5)

 ]  c   k  J 3 [3   out 2 3 3 3 3

(A.6)

where θ3 is the rotational co-ordinate of J3 with respect to left end of the second driven shaft and Ωout2 is the rotation velocity of the output part of the second joint. The relation between the input and the output of the second universal joint be written as

M 2  02 M 2* ;02  out 2 in 2

(A.7)

Therefore, substitute for M 1 from (A.2) into (A.4) and for M*2 from (A.5) into (A.7) and then for M1 from (A.4) into (A.1) and M2 from (A.7) into (A.3) to generate, together with (A.6). Finally, equations (A.8-A.10) are the equations of motion of the discussed shaft system.

Figure 5. Stability analysis: effect of the misalignment angle β 2 (α2 =10, β 1 =0.2).

CONCLUSION

J11  c11   01c22  k11   01k1 2  0

The dynamic stability of a three-degree-of-freedom system including three shafts linked by two universal joints has been studied by means of a monodromy matrix technique. Instability areas have been shown graphically in several system parameter planes.The effects of the misalignment angle, the velocity ratio and damping ratio are investigated. Increasing damping ratio cause the narrower instability regions especially in the case of low velocity ratio and increasing velocity ratio result in more instability.

 ]  c    c   k    k   0 J 2 [2   out 1 2 2 02 3 3 2 2 02 3 3 (A.9)

 ]  c   k   0 J 3 [3   out 2 3 3 3 3

[1] B. Porter, “a Theoretical Analysis of the Torsional Oscillation of a System Incorporating a Hooke’s Joint”, J. Mech. Eng. Sci., Vol. 3, No. 4, pp. 324 –329, 1961. [2] G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques, Ann. École Norm. Sup. 12, 47–88, 1883. [3] B. Porter, R.W. Gregory, “Non-linear Tor sional Oscillation of a System Incorporating a Hooke’s Joint’, J. Mech. Eng. Sci., Vol. 5, No 2, pp. 191 –200, 1963. [4] N. Kryloff and N. Bogoliuboff, “Introduction to Non-linear Mechanics” Annals of Mathematics Studies , Vol. 11. Princeton: Princeton University Press, 1947. [5] M.S. Eidinov, V.A. Nyrko, R.M. Eidinov, V.S. Gashukov, “Torsional Vibrations of a System with Hooke’s Joint”, Ural Polytechnic Institute, Sverdlo vsk , Translated from, Prikladnaya Mekhanika, Vol. 12, No. 3, pp. 98 –106, 1976.

The equations of motion of the shaft system be achieved by a combination way. Therefore, the system is studied as three distinct parts (the driver, the first driven and the second driven shaft). The equation of motion of twist vibrations for the driver shaft be written as (A.1)

where overdots show differentiation with respect to time, θ1 is the rotational co-ordinate of J1 with respect to the flywheel and M is the response torque of the input segment of the universal joint. Ωin is the rotation velocity of the right end of the driver shaft and also of the input segment of the © 2012 AMAE DOI: 02.ARMED.2012.2. 505

(A.10)

REFERENCE

APPENDIX A EQUATION OF MOTION OF TORSIONAL VIBRATIONS

J11  c11  k11  M 1

(A.8)

Full Paper Proc. of Int. Conf. on Advances in Robotic, Mechanical Engineering and Design 2012 [6] S.I. Chang, “Torsional Instabilities And Non-linear Oscillation of a System Incorporating a Hooke’s Joint”, J. Sound Vibrat. Vol. 229, No. 4, pp. 993 – 1002, 2000. [7] V. Zeman, “Dynamik der Drehsysteme mit Kardagelenken”, Mech. Mach. Theor., Vol. 13, pp. 107– 118, 1978. [8] H.C. Seherr-Thoss, F. Schmelz, E. Aucktor, Universal Joints and Driveshafts: Analysis, Design, Applications, second ed., Springer-Verlag, 2006, pp. 5-9. [9] S.F. Asokanthan, M.C. Hwang, “Torsional Instabilities in a System Incorporating a Hooke’s Joint”, Trans. ASME, Vol. 118, pp. 368 –374, 1996. [10] S.F. Asokanthan, X.H. Wang, “Characterization of Torsional Instabilities in a Hooke’s Joint Driven System via Maximal Lyapunov Exponents”, J. Sound Vibrat., Vol. 194, No. 1, pp. 83–91, 1996.

[11] S.F. Asokanthan, P.A. Meehan, “Non-linear Vibration ofTorsional System Driven by a Hooke’s Joint”, J. Sound Vibrat., Vol. 233, No. 2, pp. 297–310, 2000. [12] H.A. DeSmidt, et al, “Stability of a segmented supercritical driveline with non-constant velocity couplings subjected to misalignment and torque” J Sound Vibrat., Vol. 277, pp. 895– 918, 2004. [13] G. Bulut, Z. Parlar, “Dynamic Stability of a Shaft System Connected Through a Hooke’s Joint” J. Mechanism and Machine Theory, Vol. 46, pp. 1689– 1695, 2011. [14] L. Meirovitch, Methods of Analytical Dynamics, New York, McGraw-Hill, 1970, pp. 263-292. [15] A.J. Mazzei Jr, “Passage through resonance in a universal joint driveline system”, J. Vibration and Control, Vol. 17, No. 5, pp. 667-677, 2011.