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Journal of Sound and Vibration

Journal of Sound and Vibration 333 (2014) 3691–3701

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E-m1 Pe

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Dynamic stability analysis of torsional vibrations of a shaftsystem connected by a Hooke's joint through a continuoussystem model

Gökhan Bulut 1

Faculty of Mechanical Engineering, Istanbul Technical University, Gümüşsuyu 34437, Istanbul, Turkey

a r t i c l e i n f o

Article history:Received 14 August 2012Received in revised form10 March 2014Accepted 14 March 2014

Handling Editor: W. Lacarbonara

element scheme. The stability of the solutions of the linearized equations of motion,

Available online 24 April 2014

x.doi.org/10.1016/j.jsv.2014.03.0220X/& 2014 Elsevier Ltd. All rights reserved.

ail addresses: gokbul@yahoo.com, bulutgo@irmanent address: Faculty of Engineering, G

a b s t r a c t

Stability of parametrically excited torsional vibrations of a shaft system composed of twotorsionally elastic shafts interconnected through a Hooke's joint is studied. The shafts areconsidered to be continuous (distributed-parameter) systems and an approximatediscrete model for the torsional vibrations of the shaft system is derived via a finite

consisting of a set of Mathieu–Hill type equations, is examined by means of amonodromy matrix method and the results are presented in the form of a Strutt–Incediagram visualizing the effects of the system parameters on the stability of the shaftsystem.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Shaft systems including a Hooke's joint (also known as a Cardan joint, universal joint or U joint) have a wide range ofapplications in industry and vehicles to transmit rotary motion, and accurately predicting the instability characteristics ofsuch systems is becoming progressively significant due to the increasing tendency to design more slender and lightermachine elements, generally combined with a high rotation speed, in modern machinery. As is well known, a Hooke's jointtransforms a constant input speed to a periodically varying output one, and under certain conditions, this disturbingbehavior may cause loss of stability of torsional vibrations of a shaft system interconnected through Hooke's joints. Such ashaft system which corresponds to a parametrically excited one possesses a number of specific resonance conditions, andthe problem of determining the occurrence conditions of these resonances is referred to as dynamic stability analysis.

Since the first work by Porter [1] in 1961 on the stability of a shaft system incorporating a Hooke's joint, a number ofstudies dealing with the same problem have been performed through linear as well as nonlinear models. Porter [1]considered a single-degree-of-freedom (single-dof) linearized model of a shaft system consisting of two massless shafts andused Floquet theory [2] to obtain the parametric resonance zones for various values of the stiffness ratio of the shafts. Porterand Gregory [3] reconsidered the same problem through a nonlinear model and used the first approximation of the Krylov–Bogoliubov method to investigate the system behavior, and found that the system would settle into one of several possiblelimit cycles. Zahradka [4] considered a 1-dof nonlinear model of a shaft system composed of three massless shaftsconnected by two Hooke's joints and used a perturbation technique, the approximate Van der Pol method, to investigate the

tu.edu.tr, gokhan.bulut@gedik.edu.tr.edik University, Yakacık 34876, Kartal, Istanbul, Turkey.

G. Bulut / Journal of Sound and Vibration 333 (2014) 3691–37013692

parametric resonance zones and to obtain the domains of attraction for different resonant solutions. Eidinov et al. [5]considered a 1-dof nonlinear problem very similar to that of [3] through a similar perturbation method.

Zeman [6] considered a multi-shaft system consisting of n shafts interconnected via Hooke's joints, derived thelinearized matrix equations of motion of the resulting n-dof system and gave approximations for the first-order instabilityzones of a 2-dof system via Mettler's small parameter expansion method. Kotera [7] reconsidered the same problem asZeman, developed an infinite determinant method based on Floquet theory, used 12th-order determinants in thecalculations, and obtained several higher-order instability zones among which was the principal combination resonancezone of the summation type. Asokanthan and Hwang [8] considered again a 2-dof linear model, and used a method ofaveraging to obtain the two primary parametric resonance zones and the main summation-type combination resonancezone. They also showed, in the framework of their computational approach, that difference-type combination resonancescannot occur in the considered system. In another study, Asokanthan and Wang [9] considered the same problem throughLyapunov exponent calculations. Chang [10] revisited the nonlinear single-dof model given in [3] to obtain higherapproximations to the parametric resonance zones of the linearized model as well as the basins of attraction of thenonlinear one. Asokanthan and Meehan [11] considered a 2-dof nonlinear model and concluded, by using numericalsimulations, that the systemmay exhibit chaotic behavior under certain circumstances. Bulut and Parlar [12] considered a 2-dof linear model of a shaft system composed of two massless shafts connected by a Hooke's joint, used a monodromy matrixmethod [13] and presented several stability charts visualizing the effect of the system parameters on the stability of thesystem.

In all of the above studies, the shaft systems are considered as 1- or 2-dof lumped-mass models consisting of torsionallyelastic, massless shafts. However, it is clear that an accurate and thorough prediction of the instability characteristics ofthese shaft systems cannot be expected to be achieved using such approximate low-dimensional lumped-mass (lumped-parameter) models.

Unlike the studies mentioned above, the current study presents a more realistic model for the dynamic stability analysisof torsional vibrations of a shaft system connected via a Hooke's joint. To this end, the shafts are considered as distributed-parameter systems, an approximate discrete model for the torsional vibrations of the shaft system is derived by means of afinite element scheme, and a set of ordinary differential equations with periodic coefficients (or Mathieu–Hill equations) isobtained. Then, the dynamic stability of this parametrically excited system is performed via a monodromy matrix method,and the results visualizing the effects of the system parameters on the stability of the system are given in the form of aStrutt-Ince diagram or, in other words, in the form of stability charts constructed on parameter planes consisting of aparticular pair of system parameters.

2. Mathematical model

Consider a shaft system, as shown in Fig. 1, composed of two torsionally elastic, continuous and circular cross-sectionshafts connected via a Hooke's joint with misalignment angle β. Let each shaft have mass density ρi, polar area moment ofinertia Ii, shear modulus Gi, and length ℓi. Assume that the left end of the driver shaft is rotating at a constant speed Ω0, dueto an attached flywheel (that can be regarded as an inertial reference), for instance, and also assume that the shaft system isdriving a mechanism with a constant inertia, so that the right end of the driven shaft is connected to an inertia disk with amass moment of inertia Jd.

Now, let the discretized model of the distributed parameter shaft system be obtained using the finite element methodthrough a synthetical approach and the Rayleigh (or proportional) damping model be used for modeling the structuraldamping. Thus, consider the shafts separately, divide the driver and the driven shafts into m and n equal-length elements,respectively, with three nodes (two of which are located at the ends and the third one is located in the middle), use aquadratic finite element scheme, take into account the fact that the shafts are under the effects of the input and the outputreaction torques, T and �T n, respectively, exerted by Hooke's joint, and obtain the linear equations of motion of the

0Ω

β

Ωin Driver shaft 1111 ,,, GIρ

Driven shaft 2222 ,,, GIρ

Flywheel

Ωout

θ0 θ1 ... θi ... θ2m

0

j

2n

1

Inertia disk, Jd

Fig. 1. Schematic view of the considered shaft system.

G. Bulut / Journal of Sound and Vibration 333 (2014) 3691–3701 3693

torsional vibrations of the driver shaft in absolute coordinates and of the driven shaft in relative coordinates as

ρ1I1ℓ1 UM1 U €θþ a1 Uρ1I1ℓ1 UM1þb1 UG1I1ℓ1

UK1

� �U _θþG1I1

ℓ1UK1 Uθ¼ T Ue2m; (1)

ρ2I2ℓ2 UM2þ Jd Uf2nþ1 U ðfT1þfT2nþ1Þh i

U €φþ a2 Uρ2I2ℓ2 UM2þb2 UG2I2ℓ2

UK2

� �U _φþG2I2

ℓ2UK2 Uφ¼ �Tn Uf1; (2)

respectively, where overdots denote differentiation with respect to time t, and ai and bi are the Rayleigh dampingcoefficients. The vectors and the matrices in Eqs. (1) and (2) are defined as follows: ej and fj are the j-th 2m and (2nþ1)-dimensional unit vectors, respectively, in which all elements are zero except the j-th one that is equal to 1, such as e2¼{0 10… 0}T for instance. θ¼{θ1 θ2… θ2m}T is the 2m-dimensional vector of nodal torsional coordinates of the driver shaft, whoseelements are defined with respect to the flywheel (note that θ0¼0 due to the fact that the left end of the driver shaft is fixedto the flywheel). φ¼{φ0 φ1 φ2… φ2n}T is the (2nþ1)-dimensional vector of nodal torsional coordinates of the driven shaft,whose elements φi; i¼1,2,…,2n are relative with respect to the nodal coordinate, φ0, of the left end of the shaft (note that theabsolute coordinates can be simply converted to the relative ones using the expressions φ0 ¼ φ0, φi ¼ φ0þφi; i¼1,2,…,2n,where the overline denotes the absolute coordinates). ρ1I1ℓ1 UM1 and G1I1=ℓ1 UK1 are the 2m�2m global consistent massand stiffness matrices of the driver shaft, respectively, and are derived by considering the fact that the left end of the drivershaft is fixed and the right end is free. ρ2I2ℓ2 UM2 and G2I2=ℓ2 UK2 are the (2nþ1)� (2nþ1) global consistent mass andstiffness matrices of the driven shaft, respectively, and are derived by taking into account the fact that both ends of thedriven shaft are free. The structures of the matrices M1, K1, M2, and K2 are given in Appendix A.

The equations of motion for torsional vibrations of each shaft given in Eqs. (1) and (2) can now be non-dimensionalizedas

M1 Uθ″þ2ζΩðν1 UM1þK1ÞUθ0 þ

1Ω2 UK1 Uθ¼

1Ω2T Ue2m; (3)

M2þλUf2nþ1 U fT1þfT2nþ1

� �h iUφ″þ2ζ

Ω

αμ

γν2 UM2þK2ð ÞUφ0 þ 1

Ω2

μ

γUK2 Uφ¼ � 1

Ω2

1γTnUf1 (4)

by introducing the following non-dimensional parameters:

τ¼Ω0t; Ω¼ Ω0ffiffiffiffiffiffiffiffiffiffiffik1=J1

p ; ζ¼ cK1

2ffiffiffiffiffiffiffiffiffik1J1

p ; α¼ b2b1; γ ¼ J2

J1;

λ¼ JdJ2; μ¼ k2

k1; ν1 ¼

cM1

cK1

; ν2 ¼cM2

cK2

; T ¼ Tk1; T

n ¼ Tn

k1; (5)

where

J1 ¼ ρ1I1ℓ1; k1 ¼G1I1ℓ1

; cM1 ¼ a1J1; cK1 ¼ b1k1;

J2 ¼ ρ2I2ℓ2; k2 ¼G2I2ℓ2

; cM2 ¼ a2J2; cK2 ¼ b2k2; (6)

and where ζ is the dimensionless damping ratio, λ the dimensionless disk moment of inertia, and γ and μ are thedimensionless inertia and stiffness ratio, respectively, of the shafts. After non-dimensionalization, the primes in Eqs. (3) and(4) denote differentiation with respect to the dimensionless time τ.

The position-dependent velocity ratio η that defines a nonlinear relationship between the input (Ωin) and the output(Ωout) rotation speeds of the Hooke's joint is given by [14] as

η¼ Ωoutð ¼ _φ0ÞΩinð ¼Ω0þ _θ2mÞ

¼ cos β

1� sin 2β sin 2ðΩ0tþθ2mÞ; (7)

where Ωin and Ωout are the rotation speeds of the right end of the driver shaft and of the left end of the driven shaft,respectively. By using Eq. (7), the relationship between the input (T) and the output (T

n) torques of the Hooke's joint can be

shown to be

T ¼ ηTn; (8)

and the following non-dimensional kinematic expressions can be written:

φ00 ¼ ηð1þθ02mÞ; (9)

ϕ″0 ¼ η0ð1þθ02mÞþηθ″2m: (10)

Now, the equations of motion of the whole shaft system can be obtained by synthesizing Eqs. (3) and (4) usingEqs. (8)–(10). As expected, the combined equations of motion are nonlinear owing to the nonlinear expressions (8)–(10), andneed to be linearized by using a power-series expansion. To this end, substitute for φ0

0 and φ″0 from Eqs. (9) and (10) into

Eq. (4), assemble Eqs. (3) and (4) by eliminating the reaction torques T and Tnusing Eq. (8), expand the nonlinear terms into

G. Bulut / Journal of Sound and Vibration 333 (2014) 3691–37013694

MacLaurin series of their arguments θ; φ; θ0; φ0; θ″; φ″, and retain the linear terms under the assumption of small torsionaldisplacements together with the assumption of small torsional frequencies for the torsional vibrations of the shaft system toobtain

MUΨ″þ 2ζΩðCMþCK ÞþCn

� �UΨ0 þ 1

Ω2 UKþ2ζΩ

UKnþKnn

� �UΨ¼ �2ζ

ΩUp�q; (11)

where Ψ¼{θ1 θ2… θ2m φ1 φ2… φ2n}T is the 2(mþn)-dimensional nodal coordinate vector of the whole shaft system. Thestructures of the matrices and of the vectors in Eq. (11) are given in Appendix B, and these matrices and vectors have thefollowing forms for m¼1 and n¼1:

M¼ 130U

16 2 0 02 4þ5γη20 2γη0 �γη00 20η0 16 20 ð5þ30λÞη0 2 4þ30λ

266664

377775; CM ¼ 1

30U

16ν1 2ν1 0 02ν1 4ν1þ5αμν2η20 2αμν2η0 �αμν2η00 20αμ

γ ν2η0 16αμγ ν2 2αμ

γ ν2

0 5αμγ ν2η0 2αμ

γ ν2 4αμγ ν2

266664

377775;

CK ¼ 13U

16 �8 0 0�8 7 �8αμη0 αμη00 0 16αμ

γ �8αμγ

0 0 �8αμγ 7αμ

γ

266664

377775; Cn ¼ 1

30U

0 0 0 00 5η00ð1þγη0Þ 0 00 40η00 0 00 2ð5þ30λÞη00 0 0

266664

377775; K¼ 1

3U

16 �8 0 0�8 7 �8μη0 μη00 0 16μ

γ �8μγ

0 0 �8μγ 7μ

γ

266664

377775;

Kn ¼ 130

U

0 0 0 00 10αμν2η0η00 0 00 20αμ

γ ν2η00 0 0

0 5αμγ ν2η

00 0 0

266664

377775; Knn ¼ 1

30U

0 0 0 00 5γðη020 þη0η

″0Þ 0 0

0 20η″0 0 00 ð5þ30λÞη″0 0 0

266664

377775; p¼ 1

30U

05αμν2η2020αμ

γ ν2η0

5αμγ ν2η0

8>>>><>>>>:

9>>>>=>>>>;; q¼ 1

30U

05γη0η0020η00

ð5þ30λÞη00

8>>>><>>>>:

9>>>>=>>>>;;

(12)

where

η0ðτÞ ¼ η0ðτþπÞ ¼ cos β

1� sin 2β sin 2τ: (13)

3. Stability analysis

Eq. (11) is the governing equation of torsional vibrations of the considered shaft system, and consists of a set of non-homogeneous linear ordinary differential equations with π-periodic coefficients (or a set of the Mathieu–Hill equations). Theloss of stability of the solutions of Eq. (11) occurs under specific resonance conditions and the problem of determining theseconditions is referred to as dynamic stability analysis which requires the homogeneous part of Eq. (11) to be considered:

MUΨ″þ 2ζΩðCKþCMÞþCn

� �UΨ0 þ 1

Ω2Kþ2ζΩKnþKnn

� �UΨ¼ 0: (14)

Stability of the solutions of Eq. (14) will be examined via the monodromy matrix method. This method is summarizedbelow.

A state-space representation of (14) may be given as

u0 ¼Hðα; β; γ; λ; μ; ν1; ν2; Ω; ζ; τÞUu; (15)

where u¼ fΨT Ψ0TgT and H is a 4(mþn)�4(mþn)-dimensional π-periodic matrix defined as

Hðα; β; γ; λ; μ; ν1; ν2; Ω; ζ; τÞ ¼ 0 I�M�1K �M�1C

� �; (16)

where 0 and I are 2(mþn)�2(mþn)-dimensional zero and unit matrix, respectively, and where

C¼ 2ζΩðCK þCMÞþCn

� �; K¼ 1

Ω2Kþ2ζΩKnþKnn

� �: (17)

According to the Floquet theory, a fundamental solutions' matrix for (15) can be expressed as

ΦðτÞ ¼Q ðτÞeRτ ; (18)

where Q(τ) is a π-periodic matrix and R is a constant matrix which is related to another constant matrix S, referred to as amonodromy matrix, by R¼(1/π)log S. If the fundamental matrix is normalized so that Φ(0)¼I, then S¼Φ(π). Theeigenvalues sk; k¼1,2,…,4 (mþn) of the monodromy matrix, referred to as Floquet multipliers, govern the stability ofthe system, and the system is stable if and only if mod(sk)r1 for all k (in particular, equality corresponds to the system

G. Bulut / Journal of Sound and Vibration 333 (2014) 3691–3701 3695

being marginally stable). In this study, the matrix S is obtained by numerically integrating Eq. (15) using the Runge–Kuttamethod with step size adaptive control (RK45), and its eigenvalues are calculated to assess the stability of the system.

The monodromy matrix method is a reliable numerical method to determine the instability conditions of aparametrically excited system. The method is used as a scatter plot method to obtain the stability charts on parameterplanes consisting of selected pairs of system parameters. Unlike the other methods, such as perturbation techniques and themethods based on Hill's infinite determinant, it allows one to obtain all instability zones (harmonic and sub-harmonicparametric resonance zones as well as combination resonances) with no embedded approximation besides the numericalapproximation. It must be noted that the main disadvantage of the method is that it requires much more time as thedimension of the problem increases and as the grid period decreases. In the current study, the dimension (and hence thecomputation time of the numerical integration) of the problem stated by Eq. (15) depends on the element numbersm and n.

4. Numerical applications

This section is devoted to numerical examples which illustrate the effects of the system parameters on dynamic stabilityof the system. The results of the numerical examples will be presented in the form of the stability charts constructed on Ω–β,Ω–λ, Ω–γ and Ω–μ parameter planes. In all numerical examples, the remaining system parameters α, ν1 and ν2 are set to α¼1,ν1¼1 and ν2¼1, and the parameter planes are checked point by point through a grid-like procedure. The grid period is takenas 10�2 or less to describe very narrow instability zones.

Before presenting the stability analysis results, it is useful to present a brief discussion on the convergence of the finiteelement model (FEM) and on the expected features of the stability charts. For this purpose, an undamped finite elementmodel of an aligned-shaft system is obtained by setting β¼0, ζ¼0, and Ω¼1 in Eq. (11), and then the dimensionless naturalfrequencies ωi of this system, calculated for different values of the system parameters, and of the element numbers m and nare compared in Table 1 with those calculated through the exact frequency equation

a sin ðaωÞðμ sin ωþλω cos ωÞ� cos ðaωÞð cos ω�λγω sin ωÞ ¼ 0; a¼ffiffiffiγ

μ

r: (19)

Before discussing Table 1, it should be noted that the convergence of the first few natural frequencies is mostly sufficientto get a satisfactory stability analysis in slender shaft systems operating at rotation speeds corresponding to those above thefundamental torsional frequency. A perusal of Table 1 reveals that m¼1 and n¼1 (4-dof system) provide, as expected, anadequate prediction only for the fundamental frequency. This means that the instability zones related to only the first modecan be obtained accurately. On the other hand, m¼2 and n¼2 (8-dof system) provide a satisfactory prediction for the firstthree natural frequencies, so that a more accurate stability analysis which exhibits the effects of higher modes can beperformed. Thus, it can be said that the element numbers should be taken as at least n¼2 and m¼2 for accurate predictionof the instability characteristics of the considered system.

Furthermore, it is well known that the locations of the instability zones on the stability charts depend on the naturalfrequencies of the system, given in Table 1. Accordingly, if the parametric excitation disappears on the Ω-axis of theconsidered parameter planes, and if no damping exists, the k-th order harmonic and sub-harmonic parametric resonance

Table 1Comparison of natural frequencies calculated through FEM and the exact frequency equation (β¼0).

ω1 ω2 ω3 ω4 ω5 ω6

γ¼1, μ¼1, λ¼1m¼1, n¼1 0.538468 1.834725 3.322471 5.762434 – –

m¼2, n¼2 0.538439 1.822646 3.304203 4.904827 6.409012 8.834298m¼3, n¼3 0.538437 1.821968 3.292301 4.834604 6.434804 8.113089Exact 0.538437 1.821799 3.289167 4.814780 6.361149 7.916806

γ¼1, μ¼1, λ¼10m¼1, n¼1 0.216421 1.608513 3.178735 5.682034 – –

m¼2, n¼2 0.216420 1.602412 3.169536 4.801986 6.332816 8.784161m¼3, n¼3 0.216420 1.602057 3.159970 4.740877 6.360091 8.044953Exact 0.216420 1.601968 3.157423 4.722974 6.291132 7.860342

γ¼10, μ¼1, λ¼10m¼1, n¼1 0.068691 0.642758 1.749992 3.470894 – –

m¼2, n¼2 0.068691 0.639773 1.526745 2.512363 3.370194 4.561258m¼3, n¼3 0.068691 0.639545 1.508854 2.433353 3.076210 3.992004Exact 0.068691 0.639486 1.503570 2.385521 3.061884 3.644755

γ¼1, μ¼0.1, λ¼10m¼1, n¼1 0.093622 0.891722 1.638788 4.595700 – –

m¼2, n¼2 0.093621 0.889053 1.546831 2.162337 3.548225 4.733973m¼3, n¼3 0.093621 0.887874 1.536813 2.164932 3.016813 4.321571Exact 0.093621 0.887551 1.533676 2.137456 2.998146 3.886657

G. Bulut / Journal of Sound and Vibration 333 (2014) 3691–37013696

regions corresponding to the i-th vibration mode will emanate from the points

ΩHik ¼

ωi

2k; ΩS

ik ¼ωi

2k�1; k¼ 1;2;3;…; (20)

respectively, on the Ω-axis, and the k-th order sum or difference-type combination resonance regions corresponding to i-thand j-th vibration modes will emanate from the points

ΩC7ijk ¼ωj7ωi

2k; k¼ 1;2;3;…; j4 i: (21)

In the first three numerical examples, the effect of misalignment angle β on the stability of the considered shaft system isexamined for different values of finite element numbers m and n, and of the damping ratio ζ, and of the disk moment ofinertia λ. The results are presented in the form of stability charts constructed on the Ω–β parameter plane. It must be notedthat the operating range of the considered system is 0rβoπ=2, where β¼ 0 corresponds to no parametric excitation, andwhere β¼ π=2 corresponds to the joint losing its movability. But, in practice, the misalignment angle β does not generallyexceed π/4 rad due to design limitations.

First, a shaft system with μ¼1, γ¼1, λ¼1 and ζ¼0.0001 is considered for different values of the element numbers m andn (m¼n¼1 and m¼n¼2) and the stability charts are presented in Fig. 2a and b, where shaded zones represent unstableparameter regions. In order to give an idea of what type of instability zones exist in the considered parameter range, some ofthe unstable zones are labeled according to the emanating points calculated through Eqs. (20) and (21). On the other hand,the emanating points of some regions are not clearly distinctive; it is known that if the shaft system is considered as anundamped one, these points would be observed clearly. A comparison of Fig. 2a and b indicates that as the element numbersincrease, new higher-order instability zones related to higher modes begin to appear, most of which are located in thehigher range of β (where is not practically important), while the others appear, in the form of sum-type combinationresonances, mostly in the middle range of β. Furthermore, the instability zones related to higher modes shift toward lowervalues of Ω as a result of increasing the convergence of the natural frequencies of the systemwith an increase of the elementnumbers, and some of the existing zones (e.g., ΩCþ

142; ΩCþ343) become smaller with an increase of the convergence of the

analysis. Eventually, it can be said that n¼m¼1 is adequate only for the parametric resonance zones of the fundamentalmode, while n¼m¼2 provides a satisfactory prediction for both the parametric and the combination resonance zonesrelated to higher modes in the considered velocity range 0rΩr2. Thus, in all other numerical examples in this section, theelement numbers will be set to n¼2 and m¼2.

One notes that although the element numbers need to be increased for a better convergence of the stability analysisresults, it must be kept in mind that the computer time required for the stability analysis increases very quickly with theelement numbers. Therefore, the element numbers should be determined by taking into account the velocity rangeconsidered.

Furthermore, a detailed inspection of Fig. 2 reveals the following: (i) The misalignment angle β has a very significanteffect on the system stability. As the value of β increases, the width of the instability zones gradually increases, and newinstability zones begin to appear. (ii) Certain harmonic and sub-harmonic parametric resonance zones of the first mode (ΩH

11,ΩS

12, ΩH12, Ω

S13 and the other higher-order ones) exist sequentially in the velocity range at the left side of the region ΩS

11 thatcorresponds to the fundamental torsional frequency of the system, and this implies that sub-fundamental resonance speeds

Ω

β+ΩC

133+ΩC

154+ΩC

143 +ΩC132

S32Ω

+ΩC153

+ΩC121

+ΩC142

+ΩC343

+ΩC163

−ΩC152

H31Ω +ΩC

242+ΩC

152

+ΩC135

+ΩC123

−ΩC142

S11Ω

H11Ω

S12Ω

H12Ω

+ΩC132

+ΩC144

S13Ω S

32Ω H42Ω

+ΩC142

H31Ω

S21Ω

β

+Ω C121

+ΩC243

+ΩC343 +ΩC

131

+ΩC122

H32Ω

H21Ω

H22Ω

H11Ω

S11Ω S

21Ω

Fig. 2. Stability charts; effect of misalignment angle β (α¼1, γ¼1, μ¼1, ν1¼1, ν2¼1, λ¼1, ζ¼0.0001); (a) for m¼n¼1 and (b) for m¼n¼2.

G. Bulut / Journal of Sound and Vibration 333 (2014) 3691–3701 3697

may be unsafe in the considered shaft system. (iii) All types of unstable regions exist and difference-type combinationresonance zones (ΩC�

142, ΩC�152), unlike the other types of unstable ones, appear at very high values of β.

Second, in order to exhibit the effect of damping on the stability of the system, the first example is reconsidered forζ¼0.001 and ζ¼0.01, and the results are presented in Fig. 3a and b. This figure, along with Fig. 2b, illustrates the following:(i) As the damping ratio ζ increases, higher-order instability zones (especially narrow ones) begin to disappear (as seen inFig. 3b, most of the higher-order instability zones are eliminated). (ii) The instability zones gradually decrease and their tipsbecome round with an increase of the damping ratio. It can also be expected that instability zones shift slightly toward thelower range of Ω as a result of decreasing the vibration frequencies with an increase of the damping ratio. Eventually, it canbe said that an increase in damping ratio has a considerable stabilizing effect on the parametrically excited torsionalvibrations of the considered shaft system.

Furthermore, it should be noted that if material damping of the lighter and slender shaft systems operating at highrotation speeds is not sufficient, the rotation speeds corresponding to those beyond the fundamental frequency may becomeimportant due to the onset of the instability zones related to the higher modes and the system may be unsafe at thoserotation speeds.

As a third example, the shaft system in the first example is considered again, but this time for λ¼10 to exhibit the effectof λ on the Ω–β parameter plane; the results are obtained for two different values of damping ratio ζ (ζ¼0.0001 and

β

β

Ω

S11Ω

H12Ω

+ΩC131

H31Ω

H11Ω

S12Ω

S11Ω

H11Ω

+ΩC121

S21Ω

H31Ω

+ΩC131

Fig. 3. Stability charts; effect of misalignment angle β (α¼1, γ¼1, μ¼1, ν1¼1, ν2¼1, λ¼1, m¼2, n¼2); (a) for ζ¼0.001 and (b) for ζ¼0.01.

β

β

Ω

H11Ω

S12Ω

+ΩC132

+ΩC142

S11Ω

+ΩC232 H

31ΩS21Ω

+ΩC121

+ΩC252

+ΩC131

+ΩC133

H32Ω

+ΩC133

H32Ω

+ΩC132 +ΩC

121

+ΩC232 +ΩC

142H31Ω

+ΩC131

Fig. 4. Stability charts; effect of misalignment angle β (α¼1, γ¼1, μ¼1, ν1¼1, ν2¼1, λ¼10, m¼2, n¼2); (a) for ζ¼0.0001 and (b) for ζ¼0.001.

G. Bulut / Journal of Sound and Vibration 333 (2014) 3691–37013698

ζ¼0.001) and presented in Fig. 4a and b. It can be seen from this figure that, along with an increase in the value of λ, thewidth of the parametric resonance zones related to the first mode decreases, while some combination resonance zones (e.g.,ΩCþ

121; ΩCþ132) enlarge. Furthermore, all the instability zones shift toward the lower velocity range, due to the fact that the

vibration frequencies of the system decrease with an increase of λ (see Table 1). This effect is more pronounced for theparametric resonance regions related to the fundamental mode, and these regions become closer to each other in a narrowrange below the fundamental frequency (ΩS

11). This implies that as the value of λ increases, the range below the fundamentalfrequency becomes more unsafe. As can be seen from Table 1, this phenomenon is expected to also be observed for highervalues of γ and for lower values of μ. On the other hand, most of the comments made in the previous examples on the effectof the parameters β and ζ also hold for the present one, and thus there is no need to elaborate further here. However, itshould be noted that the narrow stable band seen at Ω¼1.85 in Fig. 4a disappears in Fig. 4b, owing to the fact that thevibration frequencies of the system slightly decrease (so that the instability zones slightly move toward lower values of Ω)with an increase of damping ratio ζ.

An overall assessment of the stability analysis results presented above, in conjunction with those of the literature,indicates the following: (i) Unlike the stability analysis results of a 2-dof lumped-mass shaft system consisting of twomassless elastic shafts, each of which carries an inertia disk at one end (e.g., see Refs. [8,12]), the results of the distributedparameter shaft system show that, in the velocity range beyond the fundamental frequency, the most pronounced and largezones in the lower range of β (where is practically important) are the sum-type combination resonance regions, not the sub-harmonic resonance ones. (ii) Numerous parametric (e.g., ΩH

31 in Fig. 3, and ΩH31, Ω

H32 in Fig. 4) and sum-type combination

resonance (e.g., ΩCþ132, Ω

Cþ142, Ω

Cþ143, Ω

Cþ152, Ω

Cþ153, Ω

Cþ163, Ω

Cþ242 in Fig. 3, and ΩCþ

132, ΩCþ133, Ω

Cþ142, Ω

Cþ232 in Fig. 4) zones related to the third

and higher modes are seen to exist in the velocity range corresponding to below the region ΩS21 (that corresponds to the

second torsional frequency). Note that such instability zones cannot be predicted through 1- or 2-dof massless shaft modelsas considered so far in the literature. It is apparent from the above that continuous modeling is necessary for slender andlighter shaft systems that operate at rotation speeds corresponding to those beyond the fundamental frequency of thesystem.

Next, the effects of the disk moment of inertia λ with β¼0.3, γ¼1, μ¼1, ζ¼0.0001 and of the inertia ratio of the shafts γwith β¼0.3, μ¼1, λ¼10, ζ¼0.0001 are examined and the results are presented in Fig. 5a and b, respectively. Upon inspectionof Fig. 5, one notes that the following: (i) As the disk moment of inertia λ increases, the right end of the driven shaft tends toremain stationary, and so the vibration frequencies of the considered shaft system approach those of a system in which thedriven shaft has a fixed-end condition at its right end, except the lowest one that approaches zero. Thus, the instabilityzones shift toward a lower range of Ω, and as expected the amount of this shifting begins to decrease at higher values of λ. Itcan be said that an increase in the value of λ makes the lower rotation speeds unsafe. (ii) The parameter γ has similar effectsas those of λ. Particularly, as γ goes to infinity, 2m-1 frequencies approach those of a fixed-fixed torsional shaft, while thelowest 2nþ1 frequencies approach zero. That is, as γ increases, the instability zones shift rapidly toward a lower range of Ω.As a result of this, quite a few new unstable zones related to higher modes begin to appear, and thus the number of theseinstability zones in the considered velocity range significantly increases. It is concluded that the considered parameter rangebecomes more unsafe, while γ increases.

λ

γ

Ω

H11Ω

S11Ω

+ΩC132

+ΩC121 S

21Ω

+ΩC131

S11Ω

H11Ω

S21Ω

S31Ω

H21Ω

+ΩC121 +ΩC

131

Fig. 5. Stability charts; (a) effect of λ (α¼1, β¼0.3, γ¼1, μ¼1, ν1¼1, ν2¼1, ζ¼0.0001, m¼2, n¼2) and (b) effect of γ (α¼1, β¼0.3, μ¼1, ν1¼1, ν2¼1, λ¼10,ζ¼0.0001, m¼2, n¼2).

S21Ω

+Ω

Ω

C121

+ΩC131

S11Ω

+ΩC231

S31Ω

μ

μ S11Ω

H11Ω

+ΩC121

H11Ω

Fig. 6. Stability charts; effect of μ (α¼1, β¼0.3, γ¼1, λ¼10, ν1¼1, ν2¼1, ζ¼0.0001, m¼2, n¼2). (a) in the range of 0rμr10 and (b) in the range of0rμr0.8.

G. Bulut / Journal of Sound and Vibration 333 (2014) 3691–3701 3699

Finally, the effect of the stiffness ratio μ with β¼0.3, γ¼1, λ¼10, and ζ¼0.0001 is examined and the results are presentedin Fig. 6a and b, where Fig. 6b is a zoom to lower range of μ. An inspection of this figure indicates that as μ decreases, allnatural frequencies of the shaft system also decrease. Specifically, while μ tends to zero, the lowest 2n natural frequenciesapproach zero. As a result of this, numerous new unstable zones begin to appear and the number of these zones increasesrapidly, especially in the range μo1. It can be said that, contrary to the effect of γ, a decrease in the value of μ makes thesystem unsafe in the lower range of μ.

As seen from Figs. 5 and 6, as γ increases and μ decreases, many thin unstable zones related to the higher modes begin toappear. These unstable zones cannot be predicted through a massless 2-dof shaft model (e.g., see Ref. [8]). The system maybecome unsafe due to these numerous instability zones at certain ranges of parameters γ and μ. It can be seen from thefigures that a shaft system with a small value of γ and with a large value of μ is safer in terms of torsional instability of theshaft vibrations. On the other hand, the first-order harmonic resonance region of the first mode ΩH

11 exists in both figures.This implies that the considered shaft system may be unsafe at a rotation speed corresponding to the half of thefundamental torsional frequency for β¼0.3.

5. Conclusions

The dynamic stability of torsional vibrations of a shaft system composed of two torsionally elastic shafts interconnectedthrough a Hooke's joint is examined. For this purpose, the shafts are considered to be distributed parameter systems and thewhole shaft system is modeled using a finite element scheme. Then, the stability analysis of the parametrically excitedtorsional vibrations is performed via the monodromy matrix method and the results are presented in the form of a Strutt–Ince diagram. These results show that the following:

i)

Misalignment angle β has a considerable destabilizing effect on the torsional vibrations of the shaft system. This effectis increasingly intensified with increasing β.

ii)

While the disk moment of inertia (in other words, moment of inertia of the mechanism excited by the shaft system) λincreases, the instability zones shift toward a lower velocity range, but this effect of λ is not as severe as that of theinertia ratio γ. This behavior makes the shaft system unsafe at lower rotation speeds.

iii)

All types of instability zones exist. Unlike the other types of zones, difference-type combination resonance ones appearat higher values of β, which are not practically important.

iv)

Higher-order harmonic and sub-harmonic parametric resonance zones exist in a sequence in the velocity range belowthe fundamental frequency; that is, sub-fundamental resonance speeds may be unsafe.

v)

Together with an increase in damping ratio, most of the thin instability zones (especially higher-order ones) areeliminated, while the others become substantially smaller in size. That is, an increase in damping ratio provides astrong stabilizing effect in the considered shaft system.

vi)

Unlike the stability analysis results of the shaft systems composed of massless shafts, the present study shows that themost pronounced and large instability zones in the practically important range of β are the sum-type combinationresonance regions, not the sub-harmonic parametric resonance ones.

vii)

As the dimension of the problem increases, higher-order instability zones related to the third and higher modes appear,

G. Bulut / Journal of Sound and Vibration 333 (2014) 3691–37013700

especially in the form of sum-type combination resonance zones, below the second natural frequency. This implies that1 or 2-dof massless shaft models are inadequate to obtain a thorough stability analysis of the considered shaft system.

viii)

As the inertia ratio γ increases, and as the stiffness ratio μ decreases, the torsional vibration frequencies of the shaftsystem decrease, and thus instability zones shift toward a lower velocity range. As a result of this, numerous instabilityzones related to higher modes begin to appear in the lower velocity range. This causes the system to be more unsafe interms of stability at low rotation speeds, and hence the shaft system should be designed using a small value of γ and alarge value of μ (e.g., γo1 and μ41 should be taken for the shaft systems considered in the last two numericalexamples in the fourth section). This phenomenon cannot be observed in the stability analysis of the massless shaftsystems considered in the literature.

By considering the above results together with the current trend toward more slender and lighter machine elementsoperating at higher rotation speeds, it is concluded that parametric and combination resonance conditions should becertainly taken into consideration during the design process of a reliable shaft system interconnected through a Hooke'sjoint. Furthermore, it is clear from some of the above results that modeling the shafts as continuous systems is crucial toaccurately predict the instability characteristics of such shaft systems.

Appendix A

A finite element modeling procedure for the analysis of torsional (or longitudinal) vibrations of a circular cross-sectionbar (shaft) can be found in many textbooks (see, e.g., Ref. [15]). Non-dimensional parts of the global consistent mass andstiffness matrices, which correspond to a quadratic finite element scheme, of a shaft divided into equal-length elementswith three nodes, two of which are at the ends and the third is in the middle, are given below for different end conditions.

M1 ¼1

30mU

16 22 8 2 �1 0

2 16 2�1 2 8 2 �1

⋱�1 2 8 2 �1

0 2 16 2�1 2 4

266666666666664

377777777777775

; K1 ¼m3U

16 �8�8 14 �8 1 0

�8 16 �81 �8 14 �8 1

⋱1 �8 14 �8 1

0 �8 16 �81 �8 7

266666666666664

377777777777775

; (A.1)

where M1 and K1 are the 2m�2m-dimensional matrices and are derived for a fixed-free shaft divided into m elements, thenodal torsional displacements of which are defined in absolute coordinates.

M2 ¼1

30nU

5 2 �120 16 2 010 2 8 2 �120 2 16 210 �1 2 8 2 �1⋮ ⋱10 �1 2 8 2 �120 0 2 16 25 �1 2 4

266666666666666664

377777777777777775

; K2 ¼n3U

0 �8 10 16 �8 00 �8 14 �8 10 �8 16 �80 1 �8 14 �8 1⋮ ⋱0 1 �8 14 �8 10 0 �8 16 �80 1 �8 7

266666666666666664

377777777777777775

;

(A.2)

where M2 and K2 are the (2nþ1)� (2nþ1)-dimensional matrices and are derived for a free-free shaft divided into nelements, the nodal torsional displacements of which are defined in relative coordinates with respect to the left end ofthe shaft.

Appendix B

The structures of the coefficients matrices of the Eq. (11) are given below. Here, Z is a 2(mþn)�2(mþn)-dimensionalzero matrix, I is the (2nþ1)-dimensional identity matrix, z is a (2m-1)-dimensional zero vector and the operators ZUL(.) andZLR(.) denote that the matrices in the parentheses are embedded in the upper left-hand corner and in the lower right-handcorner, respectively, of the matrix Z. The matrices M1, K1, M2, and K2 are defined in Appendix A, and the definition of thevector fi is given in the Section 2.

M¼ ZUL M1 þZLR M2

; (B.1)

G. Bulut / Journal of Sound and Vibration 333 (2014) 3691–3701 3701

where M1 ¼M1 and M2 ¼ ½ðγη0�1ÞUf1fT1þI�½M2þλUf2nþ1ðfT1þfT2nþ1Þ�½ðη0�1ÞUf1fT1þI�;CM ¼ ZUL CM1

þZLR CM2

; (B.2)

where CM1 ¼ ν1 UM1 and CM2 ¼ ðαμ=γÞν2 U ðγη0�1ÞUf1fT1þIh i

UM2 U ðη0�1ÞUf1fT1þIh i

;

CK ¼ ZULðCK1 ÞþZLRðCK2 Þ; (B.3)

where CK1 ¼K1 and CK2 ¼ ðαμ=γÞU ðγη0�1ÞUf1fT1þIh i

UK2;

Cn ¼ ZLRðCnÞ; (B.4)

where Cn ¼ η00 U ð1=2Þðγη0�1Þf1fT1þI

h i2UM2þ2λUf2nþ1f

T1

h if1f

T1

h i;

K¼ ZULðK1ÞþZLRðK2Þ; (B.5)

where K1 ¼K1 and K2 ¼ ðμ=γÞU ðγη0�1ÞUf1fT1þIh i

UK2;

Kn ¼ ZLRðKnÞ; (B.6)

where Kn ¼ ðαμ=γÞν2 U ð2γη0�1ÞUf1fT1þI

h iUM2 U η00 Uf1f

T1

h i;

Knn ¼ ZLRðKnnÞ; (B.7)

where Knn ¼ η″0 U γ

η020η″0þη0

� ��1

h iUf1f

T1þI

h iM2þλUf2nþ1 Uf

T1

h if1f

T1

h i;

p¼ fzT pTgT; (B.8)

where p¼ αμγ ν2η0 U ðγη0�1ÞUf1fT1þI

h iUM2 Uf1 ;

and

q¼ fzT qTgT; (B.9)

where q¼ η00 U ðγη0�1ÞUf1fT1þIh i

M2þλUf2nþ1 UfT1

h iUf1:

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