Mechanism and Machine Theory 46 (2011) 466–478

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Mechanism and Machine Theory

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Nonlinear dynamic characteristics of geared rotor bearing systems withdynamic backlash and friction

Chen Siyu⁎, Tang Jinyuan⁎, Luo Caiwang, Wang QiboKey Laboratory of Modern Complex Equipment Design and Extreme Manufacturing (Central South University), Ministry of Education, School of Mechanical andElectrical Engineering, Central South University, Changsha Hunan, 410083, China

a r t i c l e i n f o

⁎ Corresponding authors. Tel.:+86 731 88877746;E-mail addresses: csyd5053@yahoo.com.cn (C. Siy

0094-114X/$ – see front matter. Crown Copyright ©doi:10.1016/j.mechmachtheory.2010.11.016

a b s t r a c t

Article history:Received 19 April 2010Received in revised form 24 November 2010Accepted 28 November 2010Available online 24 December 2010

Effects of the friction and dynamic backlash on the multi-degree of freedom nonlinear dynamicgear transmission system, which incorporate time varying stiffness, are investigated. Firstly,the relationship between gear central distance error and backlash is deduced and the dynamicbacklash is defined, subsequently a multi-degree of freedom nonlinear dynamic geartransmission system is developed with dynamic backlash, friction and time varying stiffness.The nonlinear dynamic system is solved by the Runge–Kutta method. The results show that thefriction force may enlarge the displacement magnitude and affect the high frequency partssignificantly in frequency domain at low speed. But the RMS of the steady response is reducedon the effect of friction. The difference between the constant backlash and the dynamicbacklash models is also discussed. The system may enter into previous chaotic motion due tothe effect of dynamic backlash. Finally, no impact motion, single-side impact motion anddouble-side impact motion are also predicted in the new dynamic backlash model.

Crown Copyright © 2010 Published by Elsevier Ltd. All rights reserved.

Keywords:Geared rotor bearing systemDynamic backlashFrictionTime varying stiffnessDynamic meshing force

1. Introduction

Instability, vibration and noise are the essential problems of the gear transmission system. Many researchers have dedicated tothese problems theoretically and experimentally [1,2]. In general, the nonlinear dynamic behaviors of gear transmission areconditioned by the time varying stiffness, static transmission error, backlash and sliding friction.

The periodic variation in tooth mesh stiffness is due to the change of the number of conjugate teeth pairs in contact during theconvolute action. Accordingly, the system may be excited parametrically by the static transmission error introduced bymanufacture errors and tooth deformation. In this case, the equation of motion of the gear pair essentially reduces to Mathieu's orHill's equation with a periodic external forcing function [3].

Bollinger and Harker [4] investigated the effect of the contact ratio and stiffness on the stability. Amabili and Rivola [5]developed a dynamic model combining with time varying stiffness, viscous damping and static transmission error. The steady-state response and stability of the single degree of freedom (SDOF) model of a pair of low contact ratio spur gears were studied.Transition curves separating stable and unstable regions were computed by Hill infinite determinant; different kinds of meshingstiffness were considered and the influences of both damping and contact ratios were investigated. Chen and Tang [6] studied theparameter stability and global bifurcations of a strong nonlinear system with parametric excitations and external excitationsevolved from the gear transmission system by using the method of Multiple Scales. In addition, in order to gain a qualitative andquantitative understanding of the dynamics identified by parametric and external excitations, many researchers have devotedthemselves to studying the dynamics of similar parametric systems. Esmailzadeh and Nakhaie [7] pointed out the necessary andsufficient conditions for the existence of at least one periodic solution for the Mathieu–Duffing equation. Sanchez and Nayfeh [8]have researched on the global behavior of non-linear oscillators under external and parametric excitations. Jackel and Mullin [9]

fax: +86 731 88877746.u), jytangcsu@yahoo.com.cn (T. Jinyuan).

2010 Published by Elsevier Ltd. All rights reserved.

Nomenclature

a′ instantaneous distancean the pressure angle,Δa the difference between actual and ideal operating center distancesbtbc backlashb′ dynamic backlashcm viscous meshing dampingcjk(j=p,g,k=x,y) bearing viscous damping coefficientsDMF dynamic meshing forcese(t) static transmission errorFpi dynamics mating forceFf friction forceg1(t),h1(t) the functions of displacement of the mating points.g2(t),h2(t) the functions of the direction coefficient of the friction forcesIp,Ig gear inertiaskm nonlinear meshing stiffnessk0 averaged nonlinear meshing stiffnesskjk(j=p,g,k=x,y) bearing stiffnessmp,mg gear masses,Rp,Rg base circle radiusωh gear rotation speedti the tooth thickness on the pitch circle of ideal gearing (no backlash),ta the actual tooth thicknessTp,Tg input torque and output torqueε the contact ratio of the gear pairs,[ε] rounds the elements of ε to the nearest integers towards plus infinityxi,yi ,θi (i=p,g) the bearings displacement and the dynamic angular displacements of the gears.λi the direction coefficient of the friction forceμi the nonlinear friction coefficient

467C. Siyu et al. / Mechanism and Machine Theory 46 (2011) 466–478

studied the parametric excitation system of a double pendulum. Arrowsmith and Mondragon [10] exhibited common features ofthe size of parametric regions of stability for the Mathieu equation.

Backlash or clearance, which is used to compensate for the thermal expansion in the design of gears is one of the sources whichinduces the nonlinear dynamic behaviors. A great deal of research effort has been devoted to the study of the nonlinear responses andthe transition to chaos of gear transmission system with backlash. Wang [11,12]defined a backlash function as the angular distancebetween reverse tooth flanks,while the forward active toothflank remains in contact; since the backlash depends on the gear angularposition, it was considered a time varying function. In the nonlinearmodel developed by Cai [13] the dynamic loads are forced to zerowhen tooth pairs are not in contact. Kahraman and Singh [14–16] considered the effect of backlash and time varying mesh stiffnessusing the harmonic balance method. Theodossiades and Natsiavas [17–19] predicted chaotic behaviors: intermittent chaos andboundary crises. Ozgüven [20] extended the nonlinear spur gear model considering both shaft and bearing dynamics.

Furthermore, the existing backlash model just considers the compensate error and is modeled as a constant backlash function.Actually, the backlash of the gear tooth will be varying in motion due to the deformation of the supporting bearing and axle, which isone of the main motivations of the paper and we give a detailed introduction about this dynamic backlash in the next section.

The friction between gear teeth and their cyclic nature has been either ignored (see the above mentioned literature) orincorporated as an equivalent viscous damping term [1]. Such an approach is clearly inadequate since viscous damping isessentially a passive characteristic and it cannot act as the external excitation to the governing system [21]. Houser et al. [22–24]experimentally demonstrated that the friction forces play a pivotal role in determining the load transmitted to the bearings andhousing in the OLOA direction; this effect is more pronounced at higher torque and lower speed conditions. Vaishya and Singh[21,25–29]developed a spur gear pair model with sliding friction and rectangular mesh stiffness by assuming that load is equallyshared among all the teeth in contact. They also solved the SDOF system equations in terms of the dynamic transmission error(DTE) by using the Floquet theory and the harmonic balancemethod.While the assumption of equal load sharing yields simplifiedexpressions and analytically tractable solutions, it may not lead to a realistic model. Using the period-enlargement method, Tangand Chen [30] developed a new nonlinear dynamic model combining with friction, time varying stiffness and backlash, andvalidated the models proposed by Kahraman [14] on the frequency responses without friction.

Although these researchers have brought out the issue of friction on the gear tooth, a detailed analysis of the vibration andnoise relationship with friction, especially incorporating backlash, should have given a deeper insight into gear dynamics, just assuggested in He's thesis [21].

468 C. Siyu et al. / Mechanism and Machine Theory 46 (2011) 466–478

Hence, in this paperwewill focus on theeffect of the frictionanddynamicbacklashon themulti-degreeof freedomnonlineardynamicgear transmission system incorporating time varying stiffness. The rest of this paper is organized as follows: in the next section, therelationship between gear central distance error and backlash is given and the dynamic backlash is defined and a five degrees of freedomnonlinear dynamic gear transmission system is developed with dynamic backlash, friction and time varying stiffness. The statictransmission error is also considered in the systemas aparametric excitation. Thedynamic responses are compared on the conditionwithand without friction in the first part of section 3. Subsequently, the constant backlash function and dynamic backlash are studied withbifurcation map, phase portrait, dynamic meshing forces respectively. In the last section, concluding remarks are presented.

2. System model

2.1. Dynamic backlash model with effect of central distance error

Backlash, in a pair of gears, is the amount of clearance between mated gear teeth as shown in Fig. 1. In other words, it is thedifference between the tooth space and the tooth thickness, as measured along the pitch circle. Theoretically, the backlash should bezero, but in actual practice somebacklashmust be allowed to prevent jammingof the teethdue to tooth errors and thermal expansion.This gapmeans that when a gear-train is reversed, the driving gearmust be turned a short distance before all the driven gears start torotate. Although gear backlashmay causemanyundesirable problems in applications, especially precision positioning, it is required toallow for lubrication, manufacturing errors, deflection under load and differential expansion between the gears and the housing.

Backlash, which is created when the tooth thickness of either gear is less than the tooth thickness of an ideal gear, or the zerobacklash tooth thickness, can be defined as,

here acenterstiffneFig. 2 aO’2 resnew a

bt = ti−ta ð1Þ

tiis the tooth thickness on the pitch circle of ideal gearing (no backlash), ta is the actual tooth thickness. Additionally, backlash can

wherealso be created when the operating center distance of the gear pair is less than that for two ideal gears, and this part can be calculatedas,

bc = 2Δa tan an ð2Þ

nis the pressure angle,Δa is the difference between actual and ideal operating center distances. It's noted that the difference of thedistance can be caused by direct assembly errors or the instantaneous varying of the axle center trajectory when the bearingss and damping are considered in the geared bearing system. The axle center of the static gear is denoted byO1 andO2 as shown innd the center distance is a. Considering the effect of the support of bearing in themotion, the axle centermay be changed toO’1 andpectively. Usually the origin points are O1 and O2 in themathematical model of gear dynamic system and letting the coordinate asxle center (x1, y1) and (x2, y2). The instantaneous distance a′ can be calculated,

a′ =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia cos an + x2−x1ð Þ2 + y2−a sin an−y1ð Þ2

qð3Þ

By using Eq. (2), one can obtain the backlash due to dynamic changing of axle center distance as,

2Δb = 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia cos an + x2−x1ð Þ2 + y2−a sin an−y1ð Þ2

q−a

� �tan an ð4Þ

Fig. 1. Sketch of backlash, gears profile are denoted by blue lines and magenta dot line is pitch circle.

and th

Fig. 2. Physical model of gear pairs system. Point O is the origin of the fixed coordinate, the axle center of the static gear is denoted by O1 and O2 and they will varyto new point O'1 and O'2 respectively when consider the effect of load.

469C. Siyu et al. / Mechanism and Machine Theory 46 (2011) 466–478

The new dynamic half backlash b′ with the effect of the center distance error can be defined as,

b′ = b + Δb ð5Þ

e backlash function in the nonlinear dynamic system can be rewritten as,

f xð Þ =x−b′ x N b′0 xj j≤ b′

x + b′ x b b′

8<: ð6Þ

Here it considers the effect of the dynamical motion on the changing of backlash, which is the motive of the paper.

2.2. Equations of motion

The nonlinear model of the geared bearing systemwith gear inertias Ip and Ig, gear massesmp andmg, and base circle radius Rpand Rg as shown in Fig. 3, is considered here. The gear mesh is described by a nonlinear stiffness km and viscous damping cm. Notethat the nonlinear meshing stiffness km is deduced from a curve fitted tooth stiffness equation in Ref. [31] and developed using themethod proposed in Ref. [30] to adapt the friction force in gear transmission system. Then themeshing stiffness can be representedas,

km = k0 1 + k tð Þ½ � ð7Þ

0 is the averaged tooth stiffness, k(t) is the variant part of the stiffness, in general it can be expanded in Fourier series.

here kFriction forces at the mesh point are considered as mentioned in Ref. [30]. Bearings are modeled by equivalent elements with

viscous damping coefficients cjk( j=p,g,k=x,y)and linear springs kjk( j=p,g,k=x,y). The high-frequency internal excitations dueto the static transmission error e(t)are considered in the formulation like the previous gear pair model [15]. But the input torqueand output torque are assumed to be constant, i.e. Tp and Tg.

The generalized coordinates vector of the nonlinear dynamic model includes 6 degrees of freedom and can be defined as

δf g = θp xp yp θg xg yg� � ð8Þ

xi and yi (i=p,g) are the bearings displacement, θi is the dynamic angular displacements of the gears. The dynamic mating

whereforces acting on the meshing points are

Fpi = ∑ε½ �

i=1kmi f yp + Rpθp−yg + Rgθg−e

� �+ cmi yp + Rp θp− yg + Rg θg− e

� �h ið9Þ

Fig. 3. Six degrees of freedom nonlinear models of the geared bearing system. Bearings are modeled by equivalent elements with viscous damping coefficients cjk(j=p,g,k=x,y)and linear springs kjk(j=p,g,k=x,y).

470 C. Siyu et al. / Mechanism and Machine Theory 46 (2011) 466–478

Here, ε is the contact ratio of the gear pairs, [ε] rounds the elements of ε to the nearest integers towards plus infinity. εb2 isconsidered in the paper. The friction forces are

Ff = ∑ε½ �

i=1λiμiFpi ð10Þ

λi is the direction coefficient of the friction force which is determined by the nominal sliding velocity, and μi is the nonlinear

wherefriction coefficient as proposed in Ref. [30]. The friction arms spi、sgi of tooth on any point of the actual line of contact relative to thedrive pinion and driven gears can be expressed as

spi tð Þ = Rg + Rp

� �tan an−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2ap−R2

g

q+ Rpωht sgi tð Þ =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2ap−R2

g

q−Rpωht ð11Þ

Here, Rap is the addendum circle radius of pinion. Then equations of the coupled transverse torsional motion of the nonlineargeared bearing system, as shown in Fig. 3, are

Ip θp + Rp∑ε½ �

i=1Fpi + Tp−∑

ε½ �

i=1λiμiFpispi = 0

Ig θg−Rg∑ε½ �

i=1Fpi−Tg + ∑

ε½ �

i=1λiμiFpisgi = 0

mpxp + cpx xp + kpxxp + ∑

ε½ �

i=1λiμiFpi = 0

mp yp + cpy yp + kpyyp−∑ε½ �

i=1Fpi = 0

mgxg + cgx xg + kgxxg−∑

ε½ �

i=1λiμiFpi = 0

mg yg + cgy yg + kgyyg + ∑ε½ �

i=1Fpi = 0

8>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

ð12Þ

471C. Siyu et al. / Mechanism and Machine Theory 46 (2011) 466–478

Here the dot on each variable means derivative with respect to time t. To proceed with our analysis, two assumptions could bemade: (i) the viscous damping acting on the mating gear teeth is constant, i.e. cm1=cm2=cm. (ii) the two gears with the samebase radius are selected, and the support stiffness and damping are the same in the X- and Y-direction as shown in Fig. 3, namely,kix=kiy=kp, i=p,g.

The first two functions of Eq. (12) can be simplified further by defining a new variable

x = Rpθp−Rgθg−e tð Þ ð13Þ

dimensionless form of Eq. (12) is obtained by letting Tg/m2Rg+Tp/m1Rp=F，Ip/Rp2=m1，Ig/Rg2=m2，m2m1/(m1+m2)=me,

and ax=x1bn，xp=x2bn，yp=x3bn，xg=x4bn，yg=x5bn，τ=ϖt，cm1= cm2= cm，ωh /ϖ=ω，cm/2ϖme=ς1，ϖ2=k0 /me，F /ϖ 2 = F 1，ω 2e 0 /ϖ 2b n = F 2，me =mp = m1，kpx = k0 = k1 cpx = cm = c1，kpy = k0 = k2， cpy = cm = c2，me =mg = m2，kgx = k0 = k3，cgx = cm = c3， kgy = k0 = k4，cgy = cm = c4，ωh/ϖ=ω，cm/2ϖme=ς1. Here the dimensionless excitation isconsidered for e(t) defined as e0 sin(ωt). The five-degrees-of-freedom governing equations of gear motion are concluded as

x″1 +1k0

g1 τð Þkm1 + h1 τð Þkm2ð Þf x3−x5 + x1ð Þ + 2ς1 g1 τð Þ + h1 τð Þð Þðx′3−x′5 + x′1Þ = F1 + F2 sinωτ

x″2 + 2m1 c1ς1 x2 + 2m1ς1 g2 tð Þ + h2 tð Þð Þðx′3−x′5 + x′1Þ + m1 k1xp + m11k0

km1g2 tð Þ + km2h2 tð Þð Þf x3−x5 + x1ð Þ = 0

x″3 + 2 c2m1ς1x′3 + m1 k2x3−m11k0

km1 + km2ð Þf x3−x5 + x1ð Þ−4m1ς1ðx′3−x′5 + x′1Þ = 0

x″4−m21k0

km1g2 tð Þ + km2h2 tð Þð Þf x3−x5 + x1ð Þ + 2m2 c3ς1x′4 + m2 k3x4−2m2ς1 g2 tð Þ + h2 tð Þð Þðx′3−x′5 + x′1Þ = 0

x″5 + 2m2 c4ς1x′5 + k4m2x5 + m21k0

km1 + km2ð Þf x3−x5 + x1ð Þ + 4m2ς1 x′3−x′5 + x′1Þ = 0

ð14Þ

Here ()′means derivative with respect to time τ. g1(t) and h1(t) are the functions of displacement of the mating points. g2(t)and h2(t) are the functions of the direction coefficient of the friction forces. They are defined as

g1 tð Þ = 1 =m1Rp + 1=m2Rg

� �meμ1λ1sg1 + 1 ð15Þ

h1 tð Þ = 1 =m1Rp + 1 =m2Rg

� �meμ2λ2sg2 + 1 ð16Þ

g2 tð Þ = μ1λ1 ð17Þ

h2 tð Þ = μ2λ2 ð18Þ

Here sg1and sg2 are the arms of friction forces. The backlash function in Eq. (14) is

f x3−x5 + x1ð Þ =x3−x5 + x1−b′ x3−x5 + x1 N b′

0 x3−x5 + x1j j ≤ b′x3−x5 + x1 + b′ x3−x5 + x1 b −b′

8<: ð19Þ

Note that when considering the effect of the operating center distance error or the instantaneous axle center changing in themotion, as mentioned in section 2.1, the half backlash can be represented as

b′ = b +ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia cos an + x4−x2ð Þ2 + x5−a sin an−x3ð Þ2

q−a

� �tan an ð20Þ

3. Numerical simulation and discussion

The gear pair parameters are listed in Table 1. Substituting these parameters into Eq. (14) with dimensionless equation, thenonlinear dynamic system can be solved by the Runge-Kutta method. Let frequency parameter ω, backlash b and frictioncoefficient μ be control parameters in the following analysis. The displacement, velocity and meshing forces are all calculated withunits after dimensionless process.

3.1. Effect of sliding friction with backlash

Friction forces act orthogonally to the line of action, and the resulting dynamic force and moment are governed by a number ofparameters, such as the relative surface speed, instantaneous load and spatial location of the point of contact. Sliding resistance isinherently non-linear in nature, and due to the additional presence of periodic meshing properties like stiffness and viscous

Table 1Gear system parameters.

Number of tooth zg , zp 40

Modulus m 3 mmCentral distance a 120 mmPressure angle an 20 °CContact ratio ε 1.718Rotation inertia Ig Ip 2230 kg×mm2

Mass mg mp 1.0 kgDamping of bearing cg cp 1.4 N s/mmStiffness of bearing kg kp 3.0×105 N/mmDamping ratio of gear cm 0.7 N s/mmBacklash (half) bn 0.05 mm

472 C. Siyu et al. / Mechanism and Machine Theory 46 (2011) 466–478

damping, its dynamic interactions occur between friction and system parameters [23,24,32,33]. The influence of sliding friction onthe spectral composition of the dynamic response was studied by Vaishya and Singh [33]. They found that friction is able to reducethe large oscillations at certain resonant conditions. But their works are just based on the single degree of freedom gear system,which negate the effect of the stiffness and damping of bearing. A higher degree of freedom model could conceivably bring outmore pronounced effect of friction, especially when the system includes the translational dynamics in the sliding direction. He andSingh's work [25] overcame the deficiency of Vaishya and Singh's work [23,24,32,33] by employing realistic tooth stiffnessfunctions and the sliding friction over a range of operational conditions, and a new multi-degree of freedom, linear time-varyingmodel is developed. Based on this newmulti-degree of freedommodel, they found that one of the main effects of sliding friction isthe enhancement of the dynamic transmission error magnitude at the second gear mesh harmonic. A key question whether thesliding friction is indeed the source of the OLOA motions and forces was answered renewedly by their model. But the aboveliterature did not incorporate the backlash nonlinearity, hence, it would be of interest to incorporate the backlash nonlinearity andexamine the influence of the loss of contact on the sliding characteristics.

The root mean square (RMS) of displacement x1 ,x1rms, and maximum dynamic meshing forces,DMFmax, with respect tofrequency ω (belong to region (0.1, 1.5)) are illustrated in Fig. 4. Then the response of multi-degree system without friction(μ=0.0) and with friction (μ=0.1) are indicated by red “-·-” line and black “-o-” line respectively in Fig. 4. The system undergoestwo resonances atω1 (=0.25 orω/4) andω2 (=0.5 orω/2). The enlargement illustration is shown in the right up corner of Fig. 4a.It is observed that x1rms without friction is slightly higher than with friction in region (0.1, ω3). The maximum dynamic meshingforces illustrated in Fig. 4b also indicate the resonance phenomenon. But the maximum dynamic mashing forces are not as wepredicted in which the maximummeshing forces without friction are larger than with friction in region (0.1, 0.14) and reverse inregion (0.14, ω3). For the detailed effect of the friction, the displacement x1 and corresponding Power Spectral Density Magnitude(PSM) are illustrated in Fig. 5. Here one should note that the frequency in X-axis is mesh order which corresponds to the frequencyω in the dimensionless Eq. (14). The magnitude has a very small distinction for different friction cases at the first twomesh orderswith low frequency ω=0.1 (or low speed) but there are significant differences at high mesh orders. In contrast, when thefrequency ω=0.7, the displacement x1 and PSD are shown in Fig. 6 respectively. It shows that there is no significant difference

Fig. 4. The RMS of displacement x1 and maximum DMF with respect to frequency ω with constant backlash b=1, (a) x1rms vs. ω, (b) DMFmaxvs. ω. The red linedenotes μ=0.0 and back line denotes μ=0.1.

0 50 100 150-0.5

0

0.5

1

1.5

t

x 1

0 2 4 6 8 10-100

-50

0

50

Frequency

PS

M /d

B

(a)

(b)

Fig. 6. Effect of friction coefficient on the x1 on the multi-degree of freedommodel with constant backlash b=1,ω=0.7. Subplot (a) is in time domain and (b) is infrequency domain. The red line denotes μ=0.0 and back line denotes μ=0.1.

0 100 200 300 400 500

0.93

0.935

0.94

0.945

0.95

t

x 1

1 2 3 4 5 6 7 8 9 10-150

-100

-50

0

Frequency

PS

M /d

B

(a)

(b)

Fig. 5. Effect of friction coefficient on the x1 on themulti-degree of freedommodel with constant backlash b=1,ω=0.1. Subplot (a) is in time domain and (b) is infrequency domain. The red line denotes μ=0.0 and back line denotes μ=0.1.

473C. Siyu et al. / Mechanism and Machine Theory 46 (2011) 466–478

between without friction case and with friction case on the magnitude of the power at mesh order nω/2 (n=1,2,3,⋯ ). But thepower at mesh order nω/6( except nω/2) without friction is much higher than it with friction.

Above all, one can find that the friction force may reduce the magnitude of RMS of the gear transmission system at a largefrequency ω region, which is not consistent with the results gained by Vaishya and Singh [33] with the single degree of freedomgear transmission system. The system undergoes two resonance at ω=0.25 and 0.50 which is obviously different from thepredicted phenomenon in the single degree of freedom system. The friction forces affect the high frequency parts seriously at lowspeed and at high speed, which may induce the nω/6 part.

3.2. Effect of the constant backlash

Much of the research has concentrated on the nonlinear dynamic responses, such as frequency jump, sub- harmonic resonance,and chaos and bifurcation of gear transmissions system incorporating time varying mesh stiffness and backlash. The piecewisenonlinear systems having periodically parameters have attracted significant attention [15,16,18,19,34,35]. This section devotes

474 C. Siyu et al. / Mechanism and Machine Theory 46 (2011) 466–478

itself to studying the effect of the backlash on the dynamic responses of the multi-degree of freedom gear pairs system combiningwith friction and time varying stiffness. In this case, the backlash function (Eq. (19) ) could be rewritten as

f x3−x5 + x1ð Þ =x3−x5 + x1−b x3−x5 + x1 N b

0 x3−x5 + x1j j≤ bx3−x5 + x1 + b x3−x5 + x1 b−b

8<: ð21Þ

To illustrate the foregoing discussion, let us start with the bifurcation and the dynamic meshing forces (DMF) of the piecewisenonlinear system with respect to the backlash parameter b, which may be typical for understanding the complicatedcharacteristics. As for the bifurcation characteristics, we use the step invariant Runge-Kutta method to solve the nonlinear system(14) with the backlash function (21). Then the DMF at a given time τ is defined as [36]

DMF =1k0

g1 τð Þkm1 + h1 τð Þkm2ð Þf x3−x5 + x1ð Þ ð22Þ

Here it may be noted that the magnitude of the damping force is very small compared to the mesh force, hence it may beneglected in the calculation of the total gear mesh force. In addition, we illustrate the dimensionless dynamicmeshing force in thispaper, and compare it with the RMS of the displacement x1.

Fig. 7 presents the bifurcation map, DMF and RMS of the multi-degree of freedom systemwith respect to backlash parameter brespectively. It is observed that, with the increase of parameter b as shown in Fig. 7a, the original system undergoes periodicmotion and chaotic motion, but the route to chaos is a “blue sky catastrophe” type [37], which is different from the period doublingbifurcation and quasiperiodic route to chaos [38]. So this is still unclear. As shown in Fig. 7a, in the parameter region BI={bbb1(1)},the bifurcation map indicates period one motion of the system generally. But it may be imprecise when we have a glance at thephase portrait as shown in Fig. 8a with backlash b=0.39. It is a asymptotic stable motion and the black line indicates the solutionobtained by integrating 600 T (T=2π/ω) and plotting the last 100 T, and the red line indicates the solution by integrating 1500 Tand plotting the last 100 T. The shape of the portrait is changing completely with the time increasing, which is different from thegeneral case. In this region the maximum dynamic meshing force is almost constant but the RMS x1

rmsincreases. The system jumpsinto chaotic motion regionBII={b1(1)bbbb1(2)} when the parameter b exceeds b1(1). The meshing force and the vibration magnitude(RMS) increase rapidly, which seriously imperils the gear pair system, as shown in Fig. 7b and the portrait at b=0.7 is illustrated atFig. 8b. In regionBIII={bNb1(2)}, the system still undergoes chaoticmotion but themaximummeshing force is constant and the RMSvalues increase asymptotically. The phase portrait is shown in Fig. 8c.

3.3. Effect of the dynamic backlash on system nonlinear characteristics

For the nonlinear system with dynamic backlash (Eqs. (19) and (20)), we will first compare the global characteristics by usingbifurcation map with respect to the frequency parameter ω between the constant backlash and dynamic backlash as shown inFig. 9. Because the frequencyω is very important andmuch of the existing literature is devoted to studying the nonlinear dynamic

Fig. 7. Effect of the constant backlash on the dynamic response of the multi-degree of freedom model.

0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.06

-0.04

-0.02

0

0.02

0.04

0.06

-8 -6 -4 -2 0 2 4 6 8-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

dx 1

/dt

dx 1

/dt

dx 1

/dt

-1 -0.5 0 0.5 1 1.5 2 2.5-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

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(c)

(b)

x1x1

x1

Fig. 8. Phase portrait of multi-degree of freedom model (14) at ω=0.5, μ=0.05, and with different backlash parameters (a) b=0.39, (b) b=0.70, (c) b=1.2

Fig. 9. Bifurcation map of the multi-degree of freedom system with (a) constant backlash model, (b) dynamic backlash model. The frequency ω is the controparameter.

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.

l

476 C. Siyu et al. / Mechanism and Machine Theory 46 (2011) 466–478

characteristics of the gear transmission system with the varying of this parameter, such as Kahraman and Singh [15],Theodossiades and Natsiavas [18,19] , Parker et al. [39]. The period doubling bifurcation of the gear systemwas observed by Parkeret al. [39] and is validated by comparing multi-degrees of freedom lumped mass dynamic model with discrete dynamic model. Inthis section, we will first discuss the difference of the predicted responses with constant backlash and dynamic backlash model,where the method and system parameters used are the same as in the above section.

The bifurcation maps show that both backlash models have some common ground on the prediction of the bifurcationcharacteristic and resonance of the multi-degree of freedom system. The two cases all induce period doubling bifurcation route tochaotic motion, but the system with dynamic backlash enters into chaotic motion previously.

Next, letting backlash parameter b as control parameter, the bifurcation map, x1rmsand maximum dynamic meshing force areobtained as shown in Fig. 10. Other parameters are located at ω=0.5, μ=0.05. It is observed that the system undergoes periodicmotion in region I (I={b∈(0.3,b2(1))}), the RMSof the displacement x1 is linearly increasingwith slope L1 but themaximumdynamicmeshing forces are always horizontal with constant level denoted byDMF1 as shown in Fig. 10b. In region II (II={b∈(b2(1),b2(2))}), theperiodic and chaoticmotions coexist andwith the varyingof the parameter b, the systemmotion jumps frequently among L1, L2 and L3,as denoted by the blue arrows in Fig. 10a. The dynamic meshing forces also jump between DMF1, DMF2, DMF3 and DMF4. In region III(III={b∈(b2(2),1.5)}), the system undergoes weak chaotic motion and the dynamic meshing forces are located at DMF3. The relativevelocity dx1/dt=x′ and the dynamic meshing force are shown in Fig. 11 with different backlash parameter b. When b=0.42, as inFig. 11a, the system occurs single side impact motion as predicted in Ref. [14] and the maximum meshing force is located in DMF2.When b=0.43 shown in Fig. 11b, there is no impact between the two mating teeth. When b=0.72, the system undergoes chaoticmotion and the doubling impact motion occurs, the dynamic meshing forces are negative, as shown in Fig. 11c.

4. Conclusion

In this study, the nonlinear dynamic responses of multi-degree of freedom gear transmission system, which incorporatesbacklash, friction and time varying stiffness, are investigated. Firstly, the relationship between gear central distance error andbacklash is explained and the dynamic backlash, considering the effect of the supporting bearingmisalignment, is defined and fivedegrees of freedom nonlinear dynamic equations are developed. By means of bifurcation maps, effects of the friction and backlashon the system are discussed by bifurcation maps, RMS and dynamic meshing forces. The main results are:

(i) The system occurs 1/4 and 1/2 resonance. The friction force may enlarge the displacement magnitude and affect the highfrequency parts seriously in frequency domain at low speed. But the RMS of the steady response is reduced on the effect offriction. In the high speed case, friction force can induce nω/6 parts obviously.

(ii) Bifurcation map, RMS of displacement and maximum dynamic meshing forces with respect to backlash parameter b areobtained numerically.With the varying of the backlash parameter, the gear transmission system undergoes periodicmotionand chaotic motion, but the maximum meshing forces change less at each kind of motion.

Fig. 10. Effect of the dynamic backlash b on the multi-degree of freedom system with ω=0.5, μ=0.05, and (a) bifurcation map; (b) dynamic meshing force(denoted by red -N- line) and RMS of the displacement ×1 (by blue -o- line) with respect to parameter b.

Fig. 11. Dynamic meshing force and velocity dx1/dt=x′ with backlash parameter. (a) b=0.42, (b) b=0.43, (c) b=0.72.

477C. Siyu et al. / Mechanism and Machine Theory 46 (2011) 466–478

(iii) The difference between the constant backlash and dynamic backlash model is discussed. The systemmay enter into chaoticmotion previously due to the effect of dynamic backlash. And

(iv) The backlash's effect on the dynamic meshing force is analyzed and the no impact motion, single side impact motion anddouble side impact motions are also predicted in the new dynamic backlash model.

In the end, by introducing the dynamic backlash, the detailed analysis of effect of the dynamic backlash on gear transmissionsystem nonlinear characteristics could give us a deeper insight into the gear nonlinear dynamics. It would be very desirable todesign precisely controlled experiments which can aid in identifying the effect of dynamic backlash.

Acknowledgements

The authors gratefully acknowledge the support of the National Science Foundation of China (NSFC) through Grants Nos.50875263, National Basic Research Program of China (2011CB706800). The authors also gratefully acknowledge the helpfulsuggestions of the reviewers.

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