Dynamic analysis of multi-stage parallel-axis

external-meshing gear with ilder system

Li Wei

Professor

Correspondence information: Jiapeng Yu, Postgraduate,

College Road 20 #, Beijing, China, liwei@me.ustb.edu.cn,

0086-13691255573

Dynamic analysis of multi-stage parallel-axis

external-meshing gear transmission system with ilder

and temperature

Wei Li*, Jiapeng Yu

School of Mechanical Engineering, University of Science and

Technology of Beijing, College Road, Beijing, China

Abstract

Multi-stage gear transmission system is widely used in modern machinery

such as diesel engines and retarder, therefore, researches on the vibrational

characteristics of these systems are of great significance and can help optimize

the operating conditions. In this paper, a non-linear dynamic model of a

parallel-axis external-meshing reduction gear transmission system with idler for

a diesel engine was established, which is influenced by time-varying meshing

stiffness, tooth backlash and meshing error. The thermal stiffness is proposed,

and it is smaller than the normal meshing stiffness. And the dynamic response of

the system is obtained. The paper also investigated the effects of rotational

speed, torsional damping, meshing damping ratio and modulus on the system

and compared the vibrational characteristics of the same gear pair in the single

stage gear and the multi-stage gears. In addition, relevant conclusions to reduce

the noise of gear system are given according to the study.

Keywords: Multi-stage gear system; Time-varying thermal meshing stiffness;

Dynamic response; Sensitivity analysis

1. Introduction

Gear systems are of wide application in the modern mechanical industry.

Parallel external meshing reduction gears with idler gear increase the number of

gears, but the size of the overall structure is small and the output torque is large.

Therefore, multi-stage gear transmission is widely used in a variety of mechanical

equipment, like in the majority of diesel engine deceleration starters and

retarder.

Various models of gear system analysis have been proposed. The linear

time-invariant model(LTIM) does not consider the non-linear factors such as the

time-varying characteristics of meshing stiffness, the ulnar side clearance and the

meshing error[1].These models are used in the calculation of the natural

frequency of the gear system [2-4]. The linear time-varying model(LTVM) only

considers the time-varying stiffness in the system, such as the meshing stiffness

[5-7].The nonlinear time-invariant model(NTIM) considers only the nonlinearity

of the gap in the system, regardless of the time-dependent stiffness

[8-10].Nonlinear time-varying model taking into account time-varying factors

such as meshing stiffness and tooth-side gap, and nonlinear factors such as gap

[11,12].

The research on single stage gear transmission system is more. Bonori

[13]introduced the gear manufacturing error into the gear rotor dynamics model.

Kim [14]studied the effect of translation on the gear system due to bearing

deformation. Kang [15]established a finite element rotor model that considers

the effects of gear eccentricity, transmission errors and residual axis bending.

Ma[16]considered the dynamic performance of the gear system when the tooth

was disengaged. Moradi [17]performed a vibration analysis of a gear system with

a nonlinear gap. Han [18]studied the dynamical behavior of the gear rotor

system considering the angular displacement of the foundation. Inalpolat

[19]studied the effect of exponential error on the dynamic response of a gear

system.

The mathematical model of the multi-stage gear transmission system is the

differential equation group whose coefficient changes cyclically [20-22].In recent

years, more research has been done on the vibration characteristics of planetary

gears. Guo and Parker [23,24] developed a purely rotational model of general

compound planetary gears and classified all vibration modes into two types and

then studied the sensitivity of general compound planetary gear natural

frequencies and vibration modes to inertia and stiffness parameters. Sondkar and

Kahraman [25]proposed a linear, time-invariant model of a double-helical

planetary gear set. And they investigated influence of key design parameters on

the system response. Other scholars [26,27] have also carried out some research

on the dynamic characteristics of planetary gears.

The above-mentioned researches have obtained many valuable results, but

the research object is mostly a single gear pair and didn’t consider the influence

of the temperature on the stiffness. The research on the nonlinear vibration

characteristics of parallel-axis external-meshing reduction gears with idlers is

relatively limited. In this paper, the torsional vibration model of the gear system is

established under the influence of time-varying meshing thermal stiffness,

meshing damping, gear transmission error and backlash. The thermal stiffness is

introduced and calculated. The system was solved by numerical method. The

influence of parameters on the dynamic characteristics of the gear system is

studied. And the vibration characteristics of single stage gear and multi stage

gear are analyzed and compared.

2. Modeling

The nonlinear analysis of the parallel-axis external meshing gear system with

the idler is complex, only the torsional vibration is considered in order to simplify

the calculation and analysis, and the following assumptions are satisfied: motor

connection shaft and output shaft center fixed, only torsional movement; elastic

deformation of all bearings ignored; the friction between the teeth ignored; the

engaging force always acts in the meshing line direction; the gears are simplified

as cylinders connected by damping and springs.

(a) The Prototype

() ()

, , , ,

(b) The Model

Fig. 1 The gear transmission system

In figure 1, 、、、 and are respectively the rotational inertia of the

prime mover, gear a, gear b, gear c and load. 、 、 、 and

respectively are respectively the rotation angle of the prime mover, gear a, gear

b, gear c and load. and are respectively the torsional rigidity coefficient

the axes a and the axes b. and are respectively the reverse meshing

stiffness coefficient of gear pairs. and are the damping coefficient of

the mesh. () and () are the errors of gear teeth. and are

respectively the drive torque and load torque. According to the basic theory of

dynamics, we can establish the following torsional vibration equation:

̈ + (̇ − ̇ ) + ( − ) =

̈ + (̇ − ̇) + ( − ) + = 0

̈ − + = 0 (1)

̈ + (̇ − ̇) + ( − ) − = 0

̈ + (̇ − ̇ ) + ( − ) = }

In Equation (2), ， and are the base circle radius. and

are the dynamic meshing force between the teeth.

= (̇ − ̇ − ̇ ) + ( − − ) (2)

= (̇ − ̇ − ̇ ) + ( − }

− )

In Equation (3), is the torsional damping ratio of the material. And tests

showed a value of 0.005-0.075 [28].

= 2 √

+

(3)

= 2 √

+ }

Finishing matrix form, get formula (4)

M̈ + ̇ + = (4)

Wherein,

q = [] (5)

Formula (5) is the generalized coordinate array of the system.

0 0 0 0 (6)

0 0 0 0

M = 0 0 0 0

0 0 0 0

[ 0 0 0 0 ]

Formula (6) is the quality matrix of the system.

− 0 0 0 (7)

− 0 0 0 (8)

K= 0 0 0 00

0 0 0 −

[ 0 0 0 − ]

− 0 0 0

− 0 0 0

C= 0 0 0 00

0 0 0 −

[ 0 0 0 − ]

Formula (7) and (8) are the stiffness matrix and the damping matrix of the

system respectively.

(9)

−

P = −

[ ]

Formula (9) is the generalized force array of the system.

3. The Dynamic Parameters

(1) Time-varying thermal meshing stiffness

According to the definition of stiffness, the force which is required to deform

the unit deflection of the pair of engagement teeth is defined as the stiffness of

the pair of gears. But in the meshing process, due to the different position of

gear teeth, and the sum of the minor deformation amounts of the drive and

driven gear teeth in the direction of the load force acting is different, the stiffness

changes from momentary engagement to disengagement. And because the

coincidence degree is bigger than 1, the front and rear gear pairs will also

participate in the engagement during the meshing process, which makes its

stiffness needs to be superimposed. In this paper, the stiffness is calculated

according to the Ishikawa model.

For involute gear meshing, because the shape is more complex, some

reasonable simplification is required. The equivalent tooth shape method is to

simplify the gear into a combination of trapezoidal and rectangular, as shown in

figure 2.

The mathematical formula for the gear stiffness is shown in equation (10).

K = (10)

Wherein, F is the load acting on the face. b is the meshing tooth width. δis

the total tooth deformation.

The amount of the meshing pair deformation in the direction of the meshing

line can be expressed as

δ = + + + (11)

Wherein, δ is the deformation amount in the meshing line direction. is

the bending deformation of the rectangular portion. is deformation of

trapezoidal section. is deformation caused by the shear force. is the

deformation caused by the base section inclination.

A pair of gear teeth in the contact process not only the above-mentioned

deformation, as well as the elastic deformation generated by the contact force,

so the total deformation of a pair of teeth can be expressed as

= 1 + 2 + (12)

(13)

Wherein, is the elastic deformation due to contact.

= 4(1−2)

Fig. 2 Equivalent of gear tooth shape

The instantaneous flash temperature of its two gears is expressed as:

f ufmwf v1 v2 B

( g11c1v1 g22c2v2 )

(14)

In the formula, u is the temperature rise coefficient, in this case the planetary gear is

cylindrical gear, u 0.7858 . fm is the friction coefficient of the two contact teeth. w f is

the tooth surface normal load on the unit tooth width（N/m）. v1 , v2 is the tangential

velocity of the two teeth (m/s). g1 , g 2 is the two-tooth surface heat transfer coefficient (J /

m·s·℃). 1 , 2 is the density of the two teeth (kg/m³). c1 , c2 is the specific heat capacity

of the two gears. B is the bandwidth of the two-tooth contact zone.

In this paper, the average body temperature is given for the estimation, using ISO rough

calculation of body temperature formula.

M toil 0.47 X s X mp fp (15)

In the formula, toil is the oil temperature, X s is the lubrication coefficient X s 1.0 ;

for a small wheel with np large wheel meshing X mp 1 np , in this model, fp is a

2

meshing cycle of the average flash temperature distribution, the expression is:

fp T

0 fi(t)dt

T (16)

The calculation formula of the thermal distortion error of the tooth profile curve due to

the temperature change in the gear teeth was given:

(t) 2 rb B (t)rb (rb ub ) cos ) l 2(invk inv (17)

ub cosk (t)rb (1 rb )

k

In the formula, (t) is the difference between the contact temperature of the contact

surface before and after entering the steady working state: (t) B (t) 0 . is the

linear expansion coefficient of the material. ub is the amount of thermal deformation of the

base circle when the gear is stable. r0 is the shaft radius of the gear. is Poisson's ratio.

The expression for ub is:

ub

rbt (r0 ) (1 ) rb rb2 (1 2) r02 t(rb ) t(r0 ) (18)

(1 ) (rb2 r02 )

The corresponding thermal deformation due to temperature effects is . Thermal

stress due to thermal deformation is calculated as:

FT k (19)

In the formula, k is the instantaneous engagement stiffness of the two meshing tooth

surfaces.

In the process of meshing, the elastic stress produced by the tooth engagement and the

thermal stress produced by the contact temperature of the gears are present at the same

time on the tooth surface.Therefore, the stress generated by the mutual coupling of the two

is called the thermo-elastic coupling stress.Because the calculation of the thermo-elastic

coupling is very complicated, it is assumed that there is a linear relationship between the

thermo-elastic coupling stress and the thermal stress. The thermal stress is modified by the

correction coefficient X oh of the thermo-elastic coupling stress, and the stress after the

correction is the thermo-elastic coupling stress Foh . Therefore, the thermo-elastic coupling

stress expression obtained by using the correction coefficient is:

Foh X oh FT (20)

The meshing thermal stiffness of the gear pair is equivalent to the series of single and

outer thermal stiffness of the main and driven wheels.

wf ki

（1 X oh）k t1 t2

kTi wf kiti (21)

kT kT1kT 2 wf wf k (22)

kT1 kT 2 X ohk(t1 t2 )

kT wf (23)

k wf X ohk(t1 t2 )

In the formula, k is the meshing stiffness, ki (i 1,2) is the representative of the drive

wheel, driven wheel single tooth stiffness.

The Fig.3 is the time-varying thermal meshing stiffness of the system. And

the thermal stiffness is larger than the stiffness at the normal temperature.

(a)First Stage Gear (b) Second Stage Gear

Fig. 3 Time-varying thermal meshing stiffness

(2) The nonlinear backlash function

In gear transmission, due to the processing and installation errors and

operating wear and other reasons, there is a backlash in the gear pair meshing,

which is called backlash. The backlash which exists between the gear pairs makes

the gear pair in the non-impact, unilateral impact, bilateral impact of three states

in the meshing process. If assumes that the backlash is 2b, the nonlinear

segmentation function of tooth engagement force can be expressed as:

x b x b

b x b

f x 0

x b

x b （24）

Its graphical form is in Fig. 4:

Fig. 4 Nonlinear Segment Backlash Function

(3) Meshing Damping

In the meshing gear pair work process, meshing damping is not very

convenient to obtain, and not fixed. But in the conventional calculation, the

general calculation by the following formula:

= 2√ 212212 (25)

211+222

In Equation (25), is the meshing damping ratio, and experiments show

that the value of 0.03 ~ 0.17 [28].

(4) Gear Errors

In the gear manufacturing and installation process, the error can not be

eliminated. When the driving wheel turns a circle, due to the error, the driven

wheel angle is not necessarily a circle. The corner error curve at this time is shown

in figure 5.

Fig. 5 the angle error of the driven gear caused by gear errors

4. The Nonlinear Vibration Characteristics Analysis

(1) Solution of System Vibration

The system parameters are shown in table 1. According to the vibration

model of the transmission system, the vibration characteristics of the system can

be obtained by MATLAB programming.

Table 1. Gearing system parameters

number of teeth 25

number of teeth 50

number of teeth 70

2

Modulus m

Tooth width/mm 20

Input speed /r/min 1000

Moment of inertia /（kg ∙ 2）

Motor rotor 0.004

load 0.010

Torsional stiffness of the shaft /（N ∙ m/rad）

Axle a 1.2 × 107

Axle b 1.6 × 107

Figure 6 shows the angular displacement of the three gears. The angular

displacements of the input, intermediate and output wheels are generally similar,

but the maximum angle of rotation is gradually decreasing and their phase

angles are also different. As can be seen in Figures 9 and 10, the rotational speed

of the prime mover is in general agreement with the angle of the load, but the

peak of the prime mover rotation angle is about 3 times than that of the load.

This is due to the presence of the backlash. The rotation angle of the second

stage gear is slightly smaller than thatof the first stage gear. And the peak value

of the rotation angle as a whole is decreasing and the phase angle is also

different.

(a) inputwheel

(b) intermediate wheel

(c) output wheel

Fig. 6 Angular displacement

It can be seen from Figure 7 how the angular velocity of the three gears

varies. The angular velocity at the beginning of the vibration phase is the

maximum amplitude of the whole stage, then began to decrease, and finally

stabilized. From Fig. 9 and Fig. 10, it can be seen that the angular speed of the

prime mover and the load changes. The peak values of the angular velocities of

the prime mover and the load are slightly increased in the partial region, but then

decreasing as a whole and tend to be stable.

(a) inputwheel

(b) intermediate wheel

(c) output wheel

Fig. 7 Angular velocity

Figure 8 shows the angular acceleration of the three gears. The angular

acceleration at the beginning of the vibration is also the largest amplitude of the

whole phase, then begins to decrease and eventually stabilizes. As can be seen in

Figures 8, 9 and 10, The peak value of the acceleration from the prime mover to

the input wheel angle is increasing, the peak value of the acceleration from the

input wheel to the output wheel angle is gradually decreasing, and the peak

value of the acceleration from the output wheel to the load angle is increasing.

The peak of the acceleration from the prime mover to the load angle is generally

decreasing.

(a) inputwheel

(b) intermediate wheel

(c) output wheel

Fig. 8 Angular acceleration

Due to the presence of tooth gap, the gear will be meshing shock, which

makes the initial gear meshing have a greater shock, the gear angular velocity

and angular acceleration will be larger. When the tooth movement is stable, the

angular velocity and angular acceleration of the gear will be a sinusoidal function

of the regular changes.

(a) Angular displacement

(b) Angular velocity

(c) Angular acceleration

Fig. 9 Dynamic characteristics of prime mover

(a) Angular displacement

(b) Angular velocity

(c) Angular acceleration

Fig. 10 Dynamic characteristics of load

Taking the output wheel as an example, the angular displacement is taken as

the horizontal axis, and the angular velocity is re-drawn as the vertical axis. The

angular displacement-angular velocity of the output wheel can be obtained. As

shown in Fig 11, similarly, angular displacement - angular acceleration and

angular velocity - angular acceleration are available.

(a) Angular displacement - angular velocity

(b) Angular displacement - angular acceleration

(c) Angular velocity - angular acceleration

Fig. 11 Output wheel phase diagram

(2) Comparison between single-stage and multistage gear

In order to study the difference between the multi-stage and the

single-stage gear transmission system, the input and output wheels in the model

shown in Fig. 1 are removed, the other system parameters are unchanged, a new

one-stage gear transmission system is established. The vibration characteristics

of the system before and after the change are compared to study the difference

between the multi-stage and the single-stage gear transmission system.

(a) Angular displacement

(b) Angular velocity

(c) Angular acceleration

Fig. 12 Comparison of vibration characteristics

Figure 12 shows the comparison of the vibration characteristics of the

output wheel of the prototype without the idler gear and the output wheel of the

original model (transmission ratio unchanged). As can be seen from the figure, at

the beginning, the angular displacement of the multi-stage gear is 1.40 times

than that of the single stage gear. The angular speed of the multi-stage gear is

1.09 times than that of the single-stage gear, but the angular speed of the

single-stage gear decreases rapidly and sinusoidally, but the change of the

multi-stage gear is not obvious. The angular acceleration of the multi-stage gear

is 1.23 times than that of the single-stage gear, and the angular acceleration of

the single stage gear decays rapidly, and the angular acceleration of the

multi-stage gear slows down slowly compared with the single stage gear.

Therefore, in the same transmission ratio, multi-stage gear vibration noise is

greater.

(3) The influence of modulus on the system vibration

In a gear system, the modulus of the gear also affects the dynamic behavior

of the system. Therefore, it is possible to study the influence of the modulus ratio

on the dynamic response of the system by changing the size of the module.

Taking the output wheel as an example, the dynamics of the system are

solved when the modulus numbers are respectively 2, 2.5 and 3, and the dynamic

response is shown in Fig 13. It can be seen from the figure, as the modulus

increases, the peak of output wheel angle gradually decreases, but the cycle is

basically the same. And with the increase of modulus, the amplitude of angular

velocity and angular acceleration also sharply reduced, the cycle is basically the

same. In the case of angular velocity, the peak values are respectively 11220,

5148 and 4294 rad / s, which are reduced by 54.1% and 16.6% respectively under

different modulus.

(a) Angular displacement

(b) Angular velocity

(c) Angular acceleration

Fig. 13 Dynamic response of output wheel

(4) The influence of torsional damping ratio on the system vibration

The torsional damping ratio of the two axes has a certain influence on the

solution of the system. Therefore, it is assumed that the two shafts are made of

the same material with the same shape and size, to study the effect of torsional

damping ratio on the dynamic characteristics of the system.

Taking the output wheel as an example, the dynamic response of the system

is solved, and the dynamic response is shown in Fig 14. It can be seen from the

figure, with the torsional damping ratio increases, the vibration noise gradually

reduced, and the vibration tends to be smooth. In the figure, the vibration curve

becomes smoother as the torsional damping ratio increases. The peak values of

angular velocity and angular acceleration decrease drastically with increasing

torsional damping ratio. Therefore, as the torsional damping ratio increases, the

amplitude of the system gradually decreases, and the vibration gradually

becomes stable.

(a) The angular displacement

(b) The angular velocities

(c) Angular acceleration

Fig. 14 Dynamic response of the output wheel

(5) The influence of meshing damping ratio on the system vibration

In gear transmission, the meshing damping between gear teeth will also

affect the dynamic characteristics of the system. The effect of meshing damping

ratio on the dynamic response of the system is investigated by changing the

meshing damping ratio.

Taking the output wheel as an example, the dynamic characteristics of the

system are solved, and the dynamic response shown in Fig 15. As can be seen

from the figure, with the increase of the meshing damping ratio, the amplitude of

the angular velocity and the angular acceleration of the gear are slightly reduced,

but the cycle is almost constant. And the cycle and amplitude of the angular

displacement of the gear are almost constant. Therefore, with the increase of the

meshing damping ratio, the cycle of the system is almost constant, but the peak

value decreases slightly.

(a) The angular displacement

(b) The angular velocities

(c) Angular acceleration

Fig. 15 Dynamic response of the output wheel

(6) The influence of rotating speed on the system vibration

In real life, the diesel engine will work at different speeds, so the dynamic

characteristics of the system under different speed analysis is necessary.

Taking the output wheel as an example, the dynamic characteristics of the

system are solved, and the results shown in Fig 16. As can be seen from the

figure, in the range of 500-1500r/min, different rotational speeds have some

influences on the angular displacement of the system, but have little effect on

angular velocity and angular acceleration. With the increase of rotational speed,

the cycle of the angular displacement is shortened, and the cycle of angular

displacement is linear with the rotational speed, but the amplitude is almost the

same. Angular velocity and angular acceleration are not sensitive to the change

of rotational speed, and their amplitude and period will not change too much

with the increase of rotational speed. Therefore, the rotational speed has a linear

relationship with the angular displacement cycle of the system in a certain speed

range, but has little effect on the amplitude, angular velocity and angular

acceleration amplitude and cycle of the system.

(a) Angular displacement

(b) Angular velocity

(c) Angular acceleration

Fig. 16 Dynamic response of the output wheels

5. Conclusion

The model of the parallel-axis eternal-meshing reduction gears is

established. The time-varying thermal meshing stiffness is calculated, and the

thermal stiffness is lower than the normal stiffness. And the motion differential

equations were solved through the numerical method. The vibrational

characteristics is obtained. The angular displacement, the angular velocity and

angular acceleration curves at different stages are given in the paper.

The effects of system speed, torsional damping, meshing damping ratio and

modulus on the dynamic characteristics of the system are also researched. The

vibrational noise of the same gear pair in the multi-stage gear is larger than that

in the single stage gear. As the modulus increases, the vibrational peak value

decreases gradually, but the cycle remains the same. With the increase of the

torsional damping ratio, the amplitude of the system decreases and the vibration

gradually becomes stable. When the meshing damping ratio increases in a

certain range, the cycle of the system is nearly constant but the peak decreases.

Within a certain speed rage, there is a linear relationship between the rotational

speed and the cycle of the system.

Therefore, in order to reduce the vibration of the gear, it is possible to

reduce the rotational speed, use a shaft with a large torsional damping ratio,

increase the system damping, and use a gear with a large modulus.

Acknowledgments

The authors would like to acknowledge the financial support from the NSFC,

the research is funded by National Natural Science Foundation of China (contract

No. 51775036), these supports are gracefully acknowledged.

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