Vibration in gear systems

Grzegorz Litak a,*, Michael I. Friswell b

a Department of Applied Mechanics, Technical University of Lublin, Nadbystrzycka 36, PL-20-618 Lublin, Polandb Department of Aerospace Engineering, Queens Building, University of Bristol, Bristol BS8 1TR, UK

Accepted 30 September 2002

Communicated by T. Kapitaniak

Abstract

This paper simulates the non-linear vibration of a gear pair including shaft flexibility. Backlash is included because

of the clearance between the teeth and the periodic variation in the stiffness of the mesh produces a parametric exci-

tation. This paper shows the different behaviour that may be observed when the additional degree of freedom is added

to the single degree of freedom model previously analysed.

� 2002 Elsevier Science Ltd. All rights reserved.

1. Introduction

Gear box dynamics is characterised by a periodically changing stiffness. In real gear systems a backlash also exists

that can lead to a loss in contact between the teeth. Due to this loss of contact the gear has piecewise linear stiffness

characteristics, and the gears can vibrate regularly and chaoticaly [1–6]. The theoretical description of this phenomenon

has been based mainly on single degree of freedom models [1–7] or multi degree of freedom models neglecting backlash

[8,9]. This paper examines the effect of adding an additional degree of freedom to account for shaft flexibility on one

side of the gearbox. This may also be regarded as a single mode approximation to the torsional system dynamics, or

alternatively as a simple model for a vibration neutraliser or absorber installed in one of the gears.

2. Gear system model

Here the motion of a single stage transmission gear, where one gear is coupled to a flexible shaft, will be modelled.

Fig. 1 shows a schematic of the physical system. The gear wheels are shown with moments of inertia I1 and I2 and are

coupled by the stiffness and damping of the teeth mesh, represented by kZ and cZ . The additional degree of freedom is

modelled as a disk of moment of inertia I3 that is coupled to the gearbox via a shaft of torsional stiffness kS . Externalmoments, M1, M2 and M3 are applied to the gear wheels and the disk as shown in Fig. 1.

The equations of motion of the system may be written in terms of the three degrees of freedom, w1, w2 and w3, that

represent the rotational angles of the gear wheels and disk. These angles are those that remain after the steady rotation

of the system is removed [6]. Thus, if the backlash is neglected initially,

I1 €ww1 þ ½kZðr1w1 � r2w2Þ þ cZðr1 _ww1 � r2 _ww2Þ�r1 ¼ M1 ð1Þ

I2 €ww2 � ½kZðr1w1 � r2w2Þ þ cZðr1 _ww1 � r2 _ww2Þ�r2 � kSðw3 � w2Þ ¼ �M2 ð2Þ

* Corresponding author. Tel.: +48-81-525-9061; fax: +48-81-525-0808.

E-mail address: litak@archimedes.pol.lublin.pl (G. Litak).

0960-0779/03/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.

PII: S0960-0779 (02 )00452-6

Chaos, Solitons and Fractals 16 (2003) 795–800

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I3 €ww3 þ kSðw3 � w2Þ ¼ �M3 ð3Þ

where r1 and r2 are the radii of gear wheels and the overdot represents differentiation with respect to time.

The gear system is able to rotate with a constant relative displacement at the gear mesh and with no strain in the

flexible shaft. It is possible to decouple this motion and reduce the number of equations from three to two using the new

relative displacement coordinates x ¼ r1w1 � r2w2 and y ¼ r2ðw3 � w2Þ. These new coordinates represent the relative

displacement of the gear wheels at the teeth and the torsional displacement in the flexible shaft. The equation of motion

for x is obtained by subtracting r2=I2 times Eq. (2) from r1=I1 times Eq. (1), and for y by subtracting r2=I2 times Eq. (2)

from r2=I3 times Eq. (3). Thus,

€xxþ 2f _xxþ kðtÞgðx; gÞ þ b1kSy ¼ BðtÞ ¼ B0 þ B1 cosðxt þ HÞ ð4Þ

€yy þ b2kSy þ 2b3f _xxþ b3kðtÞgðx; gÞ ¼ DðtÞ ¼ D0 þ D1 cosðxt þ UÞ ð5Þ

where the parameters are easily derived from Eqs. (1)–(3), as

b1 ¼ 1=I2 ð6Þ

b2 ¼ 1=I2 þ 1=I3 ð7Þ

b3 ¼ ðr22=I2Þ=½r21=I1 þ r22=I2� ð8Þ

2f ¼ cZ ½r22=I2 þ r22=I2� ð9Þ

BðtÞ ¼ r1M1=I1 þ r2M2=I2 ð10Þ

DðtÞ ¼ r2M2=I2 � r2M3=I3 ð11Þ

f, kðtÞ, gðx; gÞ and BðtÞ have the same meaning as given in [6] to allow easy comparison with the single degree of freedom

model. Note that the backlash and time dependent meshing stiffness have been included by rewriting the meshing force

as [6]

kZ ½r21=I1 þ r22=I2�x ¼ kðtÞgðx; gÞ ð12Þ

Eqs. (4) and (5) have assumed that the moments on the gear system are composed of a sinusoidal moment at frequency

x with a constant offset.

Fig. 1. The physical model.

796 G. Litak, M.I. Friswell / Chaos, Solitons and Fractals 16 (2003) 795–800

Fig. 2. The meshing stiffness kðtÞ.

Fig. 3. Poincar�ee sections for various kS , with g ¼ 10 and _xx0 ¼ �0:5x.

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When there is no flexible shaft kS ¼ 0, and Eq. (4) gives a single degree of freedom equation. If the shaft stiffness

becomes large then y becomes very small, and a single degree of freedom system is obtained where I2 is replaced with

I2 þ I3. Alternatively Eqs. (4) and (5) may be combined by assuming that €yy is small to give

€xxþ 2f 1

�� b1b3

b2

�_xxþ 1

�� b1b3

b2

�kðtÞgðx; gÞ ¼ BðtÞ � b1

b2

DðtÞ ð13Þ

showing that the two degrees of freedom system is indeed equivalent to the one degree of freedom system of [6].

The meshing stiffness is periodic and the backlash is described by a piece-wise linear function. Fig. 2 shows a typical

time dependent mesh stiffness variation kðtÞ [6] and the backlash is modelled for a clearance g as

gðx; gÞ ¼x xP 0

0 �g < x < 0

xþ g x6 g

8<: ð14Þ

3. Vibrations of a gear system

Numerical simulations of Eqs. (4) and (5) have been performed to highlight the effect of the addition of the flexible

shaft. System parameters have been used that reproduce the single degree of freedom results from Warmi�nnski et al. [6]

Fig. 4. Bifurcation diagrams for v ¼ _xx versus B for various kS , with g ¼ 10 and _xx0 ¼ �3:5x.

798 G. Litak, M.I. Friswell / Chaos, Solitons and Fractals 16 (2003) 795–800

for high shaft stiffness. Thus x ¼ 1:5, B0 ¼ 1, B1 ¼ 8, D0 ¼ 1, D1 ¼ 8, H ¼ 0, U ¼ 0 and (1� ðb1b3=b2Þ)f ¼ 0:08. Forthe initial simulations b1 ¼ 1, b2 ¼ 2, b3 ¼ 0:5 and the initial conditions of y and _yy are zero. Various clearance pa-

rameters g and shaft stiffnesses kS will be used.

Fig. 3 shows the Poincar�ee sections for various kS , with g ¼ 10 and initial conditions x0 ¼ �12:0 and v0 ¼ �0:5x. The

x is included in the initial velocity to compare with the results of Warmi�nnski et al. [6] who used the non-dimensional

single degree of freedom equation. Note that due to the relatively large backlash (g ¼ 10) x0 ¼ �12:0 is not a big initial

displacement (see Eq. (14)). In Fig. 3 only x and v ¼ _xx are shown, and for kS ¼ 10000 the shaft is essentially rigid and

the results of the single degree of freedom model are reproduced [6]. As the shaft stiffness reduces the y and _yy com-

ponents of chaotic attractor (Fig. 2 for kS ¼ 10000, 100, 5, 2) become significant and result in a reduction in the

structure of the attractor. When kS ¼ 1:25 the dynamics reduce to a period 6 oscillation.

Consider now the bifurcation diagrams with respect to B for various kS . Fig. 4 shows the results for an initial velocity

of v0 ¼ �3:5x. Note the different initial velocity from that used for Fig. 3. Such a change of initial conditions can lead

to different attractors as the system is non-linear [6]. Although for kS > 5 the character of the solution is similar to the

single degree of freedom model, as the shaft stiffness reduces the character of the solution changes considerably.

The effect of shaft stiffness may be highlighted by plotting the bifurcation diagrams with respect to kS for various

values of g, a system parameter which introduces the non-linearity. Fig. 5 shows the results for B ¼ 8 and x0 ¼ �0:5x.

Once again the character of the solution changes significantly with the shaft stiffness showing a region of relatively large

amplitude velocity for small stiffness kS < 2.

4. Conclusions

This paper has examined the effect of adding an additional degree of freedom to a simple model of gear vibration

(Fig. 1). The extra degree of freedom, which may represent a flexible shaft or a vibration neutraliser, has a considerable

effect on the dynamics. This is particularly visible in Fig. 2 where we present the evolution of attractor Poincar�ee sectionswith respect to the shaft stiffness kS showing a number of chaotic and regular attractors. The detailed discussion of the

character of the transition from one chaotic attractor to another is left to a future publication. Interestingly, the

maximum vibration velocity is obtained at a relatively small kS (Fig. 5). Also note that both limits of small (kS ¼ 0) and

large (kS ! 1) stiffness lead to a reduction in the dimensionality of the model, which effectively leaves a one degree of

Fig. 5. Bifurcation diagrams for v ¼ _xx versus kS for various values of g, with B ¼ 8 and _xx0 ¼ �0:5x.

G. Litak, M.I. Friswell / Chaos, Solitons and Fractals 16 (2003) 795–800 799

freedom system (Eqs. (4) and (5)). The intermediate coupling kS is a proper bifurcation parameter giving a wide range of

system behaviour. Proper changing of its value can be used to control of system vibrations.

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