Optimization Design and Performance Analysis of Vehicle Powertrain Mounting System
Zhou et al. Chin. J. Mech. Eng.
Optimization Design and Performance Analysis of Vehicle Powertrain Mounting System
Han Zhou 0 2
Hui Liu 0 1 2
Pu Gao 0 2
Chang‑Le Xiang 0 1 2
0 School of Mechanical Engineering, Beijing Institute of Technology , Beijing 100081 , China
1 National Key Lab of Vehicular Transmission, Beijing Institute of Technology , Beijing 100081 , China
2 Authors' Information Han Zhou, born in 1995, is currently a master candidate at School of Mechanical Engineering, Beijing Institute of Technology, China. His research interests include vibration reduction technology of vehicle transmission system. E‐mail: Hui Liu, born in 1975, is currently a professor at Beijing Institute of Technol- China, in 2003. Her research interests include vehicle dynamics and electrome‐ chanical drives. E‐mail:
The design strategies for powertrain mounting systems play an important role in the reduction of vehicular vibration and noise. As stiffness and damping elements connecting the transmission system and vehicle body, the rubber mount exhibits better vibration isolation performance than the rigid connection. This paper presents a complete design process of the mounting system, including the vibration decoupling, vibration simulation analysis, topology optimization, and experimental verification. Based on the 6‑ degrees‑ of‑ freedom vibration coupling model of the powertrain mounting system, an optimization algorithm is used to extract the best design parameters of each mount, thus rendering the mounting system fully decoupled and the natural frequency well configured, and the optimal parameters are used to design the mounting system. Subsequently, vibration simulation analysis is applied to the mounting system, considering both transmission and road excitations. According to the results of finite element analysis, the topological structure of the metal frame of the front mount is optimized to improve the strength and dynamic characteristics of the mounting system. Finally, the vibration bench test is used to verify the availability of the optimization design with the analysis of acceleration response and vibration transmissibility of the mounting system. The results show that the vibration isolation performance of the mounting system can be improved effectively using the vibration optimal decoupling method, and the structural modification of the metal frame can well promote the dynamic characteristics of the mounting system.
Mounting system; Optimization algorithm; Vibration simulation analysis; Topological structure; Acceleration response; Vibration transmissibility
The vibration isolation design of vehicle powertrain
systems is important for improving the noise, vibration,
and harshness performance of vehicles, and it has been
widely studied. The rubber mount structure design of the
transmission system, which is the main vibration source,
directly affects the vibration coupling state of each degree
of freedom (DOF) of the powertrain system, and also
affects the vehicle ride and handling performance [
Therefore, vibration optimal decoupling is an effective
way to improve the vibration isolation effect of the
rubber mount [
Johnson et al. [
], for the first time, performed the
optimization of the mounting system design. He considered
the allocation of system natural frequency and the
vibration decoupling between DOFs as the objective
function and the stiffness and position of mounts as design
variables to perform the optimization. Consequently,
vibration coupling between each translational DOF was
reduced and the system natural frequencies were ensured
beyond the desired range. For the vibration of the engine
mounting system, Hata et al. [
] pointed out that
damping effect achieved by optimizing the mount position is
better than that obtained by optimizing the mount
stiffness. Demic [
] considered the response force and torque
of the mounting points as the objective function to
optimize the positions of the mounting system. Furthermore,
this method was suitable for both rubber mounts and
Sun et al. [
] decoupled the stiffness matrix of the
powertrain mounting system and achieved the
vibration decoupling of the system. Hu et al. [
] proposed a
method for investigating a powertrain system with
spectrally varying mount properties, especially for torque
roll axis decoupling. Refs. [
] established the 12-DOF
model of the mounting system using the ADAMS
software, and researched its vibration isolation performance.
] utilized the multi-objective topology
optimization to modify the structure of the mounting
system, considering the static and dynamic loads of the
system. Shangguan et al. [
] established the 13-DOF
dynamic model, which includes 6 DOFs of the
powertrain, 3 DOFs of the body, and 4 DOFs of the unsprung
mass. Considering the vibration and left ear noise as
the objective function, parameter matching and
optimization of the mounting system were carried out to
achieve better vibration isolation performance. In
addition, he also investigated the effect of different damage
parameters on the prediction of fatigue life of rubber
isolators. Angrosch et al. [
] investigated the dynamic
performance of hydraulic mounting systems, considering
engine torque and road excitation comprehensively. Zhen
et al. [
] established 1-DOF and 3-DOF models,
considering the influence of the stiffness ratio of the mount
frame on the vibration isolation material. To ensure that
the natural frequency of the system is lower than the
first-order harmonic frequency of the engine excitation,
the stiffness of the mount frame should be 1–10 times
that of the vibration isolation material. Wang et al. [
considered the generalized force transmissibility (GFT)
and sum of GFT integrals as the vibration isolation index,
and proposed an optimization approach for powertrain
In recent years, active and semi-active mounts have
become a hot topic in the research on powertrain
mounting systems. Fan et al. [
] analyzed the configuration of a
new semi-active hydraulic mount with a variable-stiffness
decoupling membrane and tested its dynamic
characteristics. Chen et al. [
] researched the dynamic model
and experimental testing of magnetorheological fluid
mounts, especially at a wide frequency. Ladipo et al. [
presented the simulation of magnetorheological
elastomers (MREs) as engine mounts. A four-parameter model
was used to model the MRE mounts and the performance
was compared with those of passive or rubber mounts.
Zheng et al. [
] used the aforementioned model to
control the engine mount system with concurrent
consideration of random road input and engine excitation.
Pan et al.  designed the
fuzzy-proportional-integralderivative switching control strategy for the
magnetorheological semi-active mounting system. Farjoud et al. [
developed a detailed mathematical model of semi-active
magnetorheological engine and transmission mounts
using multi-physics modeling techniques for physical
systems with various energy domains.
In previous studies, researchers only optimized the
mounting system according to the results of the
vibration decoupling, or only applied the structural topology
optimization to the mount frame, and subsequently
verified the validity of the design scheme via simulation or
experiment. A comprehensive and complete process of
optimization design and performance analysis of vehicle
powertrain mounting systems has not been studied yet.
This paper presents a complete design process of the
mounting system, including the vibration decoupling,
vibration simulation analysis, topology optimization, and
experimental verification. According to the actual state
of the mounting system, the 6-DOF vibration model is
established. The natural frequency and vibration mode
are obtained using eigenvalue analysis. Considering the
low-frequency vibration decoupling rate as the
objective function, the design parameters (including mount
hardness, thickness, and stiffness in all directions) that
render the mounting system fully decoupled and the
natural frequency well configured are extracted, and the
optimal parameters are used to design the mounting
system. Subsequently, the mounting system is simulated and
analyzed to study the vibration response. The torsional
excitation of the shafting is calculated as the
transmission excitation using the equivalent model, and the road
excitation is simulated using the sinusoidal scanning
signal based on the data of acceleration measured by the
actual vehicle. A comprehensive analysis shows that the
strength and dynamic characteristics of the front mount
should be improved, and hence, topology optimization is
applied to the metal frame. Finally, the vibration bench
test of the mounting system is carried out to extract
the acceleration signals in time domain and frequency
domain of the upper and lower parts of the mounts. The
data is analyzed and the vibration transmissibility is
calculated using the corresponding root-mean-square
values of acceleration. The results showed that the design
process of the mounting system could well improve the
vibration isolation performance.
2 Dynamic Model of Powertrain Mounting System
The mounts of the integrated transmission system are
divided into left, right, and front mounts (see Figure 1).
The powertrain device is regarded as a rigid body with
elastic support, and a simplified model of the three-point
mounting system is established as shown in Figure 2. A
Cartesian coordinate system G0-xyz is defined as the
global coordinate. G0 is the center of mass in static
balance; the x-axis is parallel to the output axis of the
integrated transmission system; the positive y-axis indicates
the vehicle driving direction; the z-axis is determined
using the right hand rule.
The vibration response of the system in the global
coordinate system is set as u = x, y, z, θx, θy, θz T,
where x, y, and z are the translational motions along the
three coordinate axes; θx, θy, and θz are the rotational
motions around the three coordinate axes. For the
powertrain mounting system simplified as a rigid body, the
undamped free vibration equation in the global
coordinate system is obtained as
Mu+Ku = 0,
Jxx −Jxy −Jxz
−Jxy Jyy −Jyz
−Jxz −Jyz Jzz
where m is the mass parameter, and Jxx, Jyy, Jzz, Jxy, Jxz, Jyx,
and Jzx are the inertia parameters. The 3D model of the
integrated transmission system is established, the
material density of the parts is set, the inertial properties are
analyzed using software, and finally the system quality,
position of mass center, moment of inertia, inertia
product, etc. are obtained:
EiTT iTkiT iEi,
1 0 0 0 zi −yi
Ei = 0 1 0 −zi 0 xi ,
0 0 1 yi −xi 0
cos αui cos βui cos γui
T i = cos αvi cos βvi cos γvi ,
cos αwi cos βwi cos γwi
where ki is the stiffness matrix of the mount i, and kui,
kvi, and kwi indicate the stiffness along the elastic axes of
the mount i. According to the empirical formula, Ei is
the coordinate matrix of the mount i obtained from the
position of each point. Further, Ti is the orientation angle
matrix of mount i, and the elements in the main diagonal
of the matrix are 1.
The eigenvalues and eigenvectors obtained from
eigenvalue analysis are employed in the vibration decoupling
as described below.
3 Vibration Optimal Decoupling of the Mounting
The rigid-body vibration of the mounting system is
decoupled and the comprehensive vibration isolation
performance of the mounting system is improved by
using the vibration energy decoupling method and
nonlinear optimization method.
3.1 Design Variables
The vibration coupling of the mounting system is closely
related to the supporting position, installation angle, and
stiffness of the mounts. Owing to the limitation of the
vehicle arrangement, the supporting position and
installation angle of the mounting cannot be changed easily.
The stiffness parameters of the mounts can be
independent variables of the optimization method, mainly
determined by the rubber hardness and size. To facilitate the
application of the optimization results to the system
design, the hardness of the left and right rubber mounts
and the hardness and thickness of the front rubber mount
are considered as design variables. Therefore, there are
three design variables in the optimization model:
of the natural vibration of the system can be calculated
using the quality matrix and mode shapes, and when it is
expressed in the form of a matrix, it can be defined as the
energy distribution matrix.
The maximum kinetic energy of the integrated
transmission system in the nth natural vibration is
Hsr , Hf , Hsf ,
where Hsr is the hardness of the left and right mounts; Hs
is the hardness of the front mount; Hf is the thickness of
the front mount.
3.2 Constraint Conditions
3.2.1 Configuration Range of Natural Frequency
The natural frequency of the mounting system is
guaranteed to be in a reasonable range when matching the
The natural frequency of the direction around the
y-axis should be less than 1/√2 of the engine idling
vibration frequency, and hence, the natural frequency of the
y-axis should be
fθy ≤ Z · 60 √2 ,
where n is the number of cylinders; Nmin is the engine
idle speed; Z is the number of strokes. The engine of the
vehicle is a V type 12-cylinder engine, and the idle speed
is 800 r/min. The calculated natural frequency in the
torsional direction should be less than 57 Hz.
In order to ensure the service life of the mounting
system, the natural frequencies of the system are
generally greater than 5 Hz; in order to avoid the resonance
of the mounting system, the general requirement of the
minimum difference between the natural frequencies is
approximately 1 Hz. According to the engine speed and
sensitive area for the vibration of human body, the
vibration frequency ranges of each direction are given by
5 Hz ≤ fx ≤ 57 Hz, 5 Hz ≤ fy ≤ 57 Hz,
6 Hz ≤ fz ≤ 57 Hz,
5 Hz ≤ fθx ≤ 57 Hz, 5 Hz ≤ fθy ≤ 57 Hz,
5 Hz ≤ fθz ≤ 57 Hz.
3.2.2 Range of Rubber Hardness and Thickness
According to the mechanical design manual and related
literature, we can determine the range and initial value of
the rubber hardness. According to the vehicle
arrangement space, we can determine the range and initial value
of the rubber thickness of the front mount, as listed in
3.3 Objective Function
According to the 6-DOF dynamic equation of the
system, the natural frequencies and vibration modes can
be calculated. Subsequently, the energy distribution
Tmn ax = ωn2[φn]T[M][φn] 2,
Tmn ax = 2
where ωn is the nth natural frequency of the mounting
system, mkl is the element in the kth row and lth column
of the mass matrix, φnl is the lth element of the mode
shapes [φn], and φnk is the kth element.
The energy allocated to the kth generalized coordinates
is obtained as
In the nth natural vibration, the percentage of energy
allocated to the kth generalized coordinates in the total
energy of the system is
3.4 Optimization Results
Based on the integrated analysis program of iSIGHT
and MATLAB, the mounting system parameters are
optimized using a nonlinear method. Finally, the
objective function is convergent and evidently reduced by 34%,
as presented in Table 2.
Before and after optimization, the frequency
configuration of the mounting system is given in Table 3.
Table 3 indicates that the frequency configuration
before optimization is not reasonable because the
vibration frequencies in the θx and θy directions are close to
the engine idle frequency and prone to resonance. The
frequencies after optimization are within the frequency
range and not close to the engine idle frequency.
Before and after optimization, the vibration decoupling
rate of the mounting system is presented in Table 4.
Table 4 indicates that, before optimization, the
decoupling rate of the system is between 31.2% and 99%; the
decoupling rate in the x, θy, and θz directions is relatively
high, and can reach more than 85%; the decoupling rate
in the other directions is lower than 70%. After
optimization, except the decoupling rate in the θx direction, which
is 61.5%, the rest are more than 70%. The decoupling rate
of the optimized system is evidently increased, and the
overall isolation performance of the mounting system is
By using the above method, the optimal design
parameters of the system can be obtained, and the stiffness of
each mount in different directions after optimization is
listed in Table 5. The hardness and thickness of the
rubber mount after optimization are listed in Table 6.
3.5 Mounting Design
The mounting cushion adopts a combination design
method of vulcanized rubber, anti-aging agent, and
metal frame. The appropriate choice of vulcanized
rubber, proportion of anti-aging agent, and shape and size of
the metal frame can result in the hardness and thickness
parameters of the rubber mount presented in Table 6.
The combination of the rubber, agent, and metal frame
enables the achievement of a strong bond and facilitates
deformation in the directions of stretching, compression,
4 Vibration Simulation Analysis of the Mounting
In this paper, the integrated transmission box and the
mounts are analyzed separately. First, the transient
dynamic response of the integrated transmission box is
analyzed. Subsequently, the dynamic response force of
the front mount is extracted, and finally, the static
analysis of the mounts is carried out.
4.1 Vibration Response Analysis of Integrated
In order to obtain the torsional excitation of the shafting
as the boundary condition to calculate the forced
vibration response of the system, the normalized equivalent
model of the torsional vibration of the integrated
transmission is established.
The shafting of the transmission is a continuous and
complex multi-DOF quality system. The mass and
elastic distributions of the system are uneven, and hence,
the discrete approximation model of multi-DOF
concentrated mass could be adopted. The actual system is
transformed into a system with rigid bodies without
elastic deformation and elastic shaft sections without
the moment of inertia [
]. The sixth-gear equivalent
model of transmission system is shown in Figure 3.
The engine torque model is established by
considering the excitation torque of the engine gas pressure and
reciprocating inertia force. Subsequently, the torque
signal of the sixth-gear output shaft when the engine speed
is 2200 r/min is calculated using the normalized
equivalent model, as shown in Figure 4.
The vertical force of left bearing housing in one period
is calculated according to the force analysis of gears and
shifts. The signal can be used as the boundary condition
of the mounting system to calculate the forced
vibration response of the system. The deformation and stress
nephograms of the box calculated using finite element
analysis are shown in Figure 5.
Subsequently, the reaction force at six fixed bolt holes
of the front mount is extracted, as shown in Figure 6.
4.2 Vibration Response Analysis of the Mounts
4.2.1 Response Analysis of the Transmission Excitation
The root-mean-square values of the reaction forces at
six bolt holes extracted as described above are applied to
the corresponding six bolt holes in the front mount. The
vertical force at the upper bearing is extracted, and the
root-mean-square value is calculated to be loaded on the
left and right mounts in the form of uniform pressure.
The deformation and stress nephograms of the mounts
under the transmission excitation are shown in Figure 7.
4.2.2 Response Analysis of the Road Excitation
Road excitation is a significant excitation for driving a
vehicle. Normally, road excitation is expressed using road
surface roughness, and when loaded into the mounting
model, the spatial signal must be transformed into time
domain. In practical applications, we usually use the
harmonic superposition method to simulate the road
roughness model [
]. However, this method is too complex
and difficult for the simulation of the actual situation
of the road, and hence, we use the method of harmonic
response analysis to analyze the vibration characteristics
of each mount . The maximum amplitude of vibration
acceleration collected in the actual vehicle is considered
as the amplitude of the input signal, and the sinusoidal
scan is carried out in the range of 0.5–10 Hz for each
mount at an interval of 0.01 Hz.
The deformation and stress nephograms of the mounts
under the road excitation are shown in Figure 8.
The maximum deformation and stress of the mounts
under the transmission excitation and road excitation are
given in Table 7.
It can be observed from Figure 8 and Table 7 that the
front part of the front mount frame has a maximum
deformation of 0.82 mm, mainly because there are no
rubber blocks. The stress in the front mount is mainly
flexibility and weighted natural frequency. The static
multi-load condition and dynamic vibration
characteristics of the mounting system are considered, and the
expression is given by
distributed in the six bolt holes and two sides of the
vertical part of the metal frame, and the bolt hole of the
horizontal part has a maximum stress of 82.017 MPa.
Compared with the left and right mounts, the front
mount has larger deformation and stress, and is more
easily damaged during use, and hence, it is necessary to
improve the reliability of the front mount.
5 Optimization Design of the Metal Frame
We only modify the metal frame of the front mount as
the left and right mounts satisfy the structural strength
and performance requirements. Considering the
flexibility (static) and vibration natural frequency (dynamic)
of the components as the objective function, the
topological structure of the metal frame of the front mount is
In this paper, a multi-objective optimization model
of stiffness and low-order natural frequencies is
established. The objective function is a combined index of the
global response of the structure, including the weighted
Ck (ρ) − Ckmin 2
Ckmax − Ckmin
Λimax − Λ(ρ) 2
Λimax − Λimin
where Ckmaxand Ckmin are the maximum and minimum
values of flexibility in the three working conditions,
respectively, which can be obtained from the topology
optimization by considering the minimum flexibility as
the objective function; Λmax and Λmin are the maximum
and minimum values of natural frequency of each order,
respectively, which can be obtained from the
topology optimization by considering the maximum natural
frequency as the objective function; wk and wi are the
weighting factors of flexibility and modal frequency,
The corresponding data is provided as input to
Hyperworks and the Optistruct module is used for topology
optimization. Finally, we obtain the optimized metal
frame of the front mount, as shown in Figure 9.
The mass after optimization is 1.31 kg less than that
of the original model, accounting for 14.2% of the
original mass. Subsequently, analysis of the optimized metal
frame shows that the maximum stress of the three
working conditions [static, undulating road (40 km/h), and
cement road (50 km/h)] is reduced to a certain extent,
and the strength of the structure is enhanced. The
firstorder natural frequency is 538 Hz, which is away from
the resonant sensitive region of the integrated
transmission. In summary, the dynamic characteristics of the
mounting system are improved.
6 Vibration Bench Test
The vibration acceleration signal of the upper and lower
parts of each mount in time domain and frequency
domain are extracted. The schematic diagram of the
vibration table test is shown in Figure 10.
6.1 Analysis of Test Results
The gearbox remains in the sixth gear, and the engine
speed is 2200 r/min. From the test results, we can
observe that the vibration acceleration of the upper and
lower parts of each mount is reduced to a certain extent,
and the attenuation effect of the lower part of the front
mount is more evident than that of the others, as shown
in Figure 11.
Table 7 Maximum deformation and stress of the mounts
Maximum deformation (mm) Maximum stress (MPa)
Front mount 0.82
Left mount 0.00162
Right mount 0.00212
It can be observed from Figure 11 that the
root-meansquare values of the vibration acceleration before and
after the optimization are 4.84g and 0.82g, respectively,
indicating a reduction of 83% and hence, the vibration is
evidently attenuated. This is because we optimize both
the rubber and metal frame of the front mount, which
changes the ratio of stiffness of the metal frame to that
of the rubber. The figure of frequency domain also shows
that the vibration optimal decoupling and topology
optimization can evidently attenuate the vibration of the
front mount, especially in the frequency band of 1250 Hz.
Subsequently, the vibration acceleration signal of the
upper and lower parts of each mount is extracted when
the gearbox is in different gears, and the engine speed
remains 2200 r/min. The root-mean-square values of the
acceleration and vibration transmissibility are calculated.
The vibration transmissibility of the front mount before
and after the optimization is shown in Figure 12.
Figure 12 shows that the vibration transmissibility of
the front mount does not fluctuate when the gearbox is
in different gears, which indicates that the vibration
isolation performance of the mount is not related to the
transmission ratio. The vibration transmissibility after
the optimization is approximately 1/3 of that before the
optimization. It shows that the optimization method can
significantly reduce the vibration transmissibility and
effectively improve the vibration isolation performance
of the mounting system.
The vibration transmissibility of the left and right
mounts before and after the optimization is shown in
From Figure 13, it can be observed that the vibration
transmissibility of the left mount after the optimization
decreases slightly and that of the right mount before
and after the optimization is not significantly changed.
The vibration isolation performance is not significantly
improved because only the parameters of the rubber are
optimized, whereas the other parts remain unchanged.
In summary, the mounting system has an
apparent attenuation effect on the vibration generated by the
transmission system. The vibration optimal decoupling
and topology optimization have significantly improved
the vibration isolation performance of the front mount
and proved the effectiveness of the design scheme.
In this study, based on the 6-DOF coupling vibration
model of the mounting system, an optimization
algorithm was used to extract the best design parameters
of the mounts, thus rendering the mounting system
fully decoupled and frequency well configured, and the
optimal parameters were used to design the
mounting system. Subsequently, the vibration response of the
mounting system was simulated and analyzed,
considering the influence of transmission excitation and road
excitation. The analysis showed that the strength and
dynamic characteristics of the front mount were to be
improved, and the topological structure of the metal
frame was optimized. Finally, the vibration bench test
was used to verify the availability of vibration decoupling
and structural topology optimization, with the analysis of
acceleration signal and vibration transmissibility of the
This paper presented a complete optimization design
process of the mounting system, including the vibration
decoupling, vibration simulation analysis, topology
optimization design, and experimental verification, and the
process is instructive for the actual engineering design of
HL was in charge of the whole trial; HZ wrote the manuscript; PG and
C‑LX assisted with sampling and laboratory analyses. All authors have read
and approved the final manuscript.
reduction technology of vehicle transmission system. E‑mail: gaopu1989@126.
Chang‑Le Xiang, born in 1963, is currently a professor at Beijing Institute
of Technology, China. He received his PhD degree from Beijing Institute of
Technology, China, in 2001. His research interests include vehicle dynamics and
electromechanical drives. E‑mail: .
The authors declare that they have no competing interests.
Ethics Approval and Consent to Participate
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 X Y Pan . An investigation on calculation and modeling methods for dynamic properties of a rubber isolator . Hangzhou: Zhejiang University of Technology, China, 2009 . (in Chinese)
 Z H Lv , R L Fan . Design method for vibration uncoupling of powerplant mounting system . Chinese Journal of Mechanical Engineering , 2005 , 41 ( 4 ): 49 ‑ 54 . (in Chinese)
 S R Johnson , J W Subhedar . Computer optimization of engine mounting systems . No. 790974. SAE Technical Paper , 1979 .
 H Hata , H Tanaka . Experimental method to derive optimum engine mount system for idle shake . No. 870961. SAE Technical Paper , 1987 .
 M Demic. A contribution to the optimization of the position and the characteristics of passenger car powertrain mounts . International Journal of Vehicle Design , 1990 , 11 ( 1 ): 87 ‑ 103 .
 B B Sun , Q J Zhang , Q H Sun , et al. Study on decoupled engine mounting system . Journal of Vibration Engineering , 1994 , ( 03 ): 240 ‑ 245 . (in Chinese)
 X Z Li , K Chen . ADAMS based research on vibration isolation performance of powertrain mounting system . Proceedings of 2015 International Industrial Informatics and Computer Engineering Conference (IIICEC 2015 ), 2015 : 4 .
 J F Hu , W W Chen , H Huang . Decoupling analysis for a powertrain mounting system with acombination of hydraulic mounts . Chinese Journal of Mechanical Engineering , 2013 , 26 ( 4 ): 737 ‑ 745 .
 K Chen , P Lv . Simulation method for vibration isolation performance of vehicle powertrain mounting system . China Mechanical Engineering , 2014 , 25 ( 20 ): 2830 ‑ 2834 . (in Chinese)
 Q H Zhao , X K Chen , L Wang , et al. Simulation and experimental validation of powertrain mounting bracket design obtained from multi‑ objective topology optimization . Advances in Mechanical Engineering , 2015 , 7 ( 6 ): 1687814015591317 .
 J F Zhu , Y Lin , G B Shi , et al. Topology optimization of engine mount bracket with consideration of engineering constraints . Automotive Engineering . 2014 ( 12 ): 1508 ‑ 1512 . (in Chinese)
 L C Zhang , Q H Zhao , H X Zhang , et al. Multi ‑ objective topology optimization for the mount bracket of vehicle powertrain . Automotive Engineering , 2017 , 5 : 551 ‑ 555 . (in Chinese)
 W B Shangguan , X A Liu , Z P Lv , et al. Design method of automotive powertrain mounting system based on vibration and noise limitations of vehicle level . Mechanical Systems and Signal Processing , 2016 , 76 : 677 ‑ 695 .
 W B Shangguan , X C Duan , Q K Liu , et al. Study on the effect of different damage parameters on the predicting fatigue life of rubber isolators . Journal of Mechanical Engineering , 2016 , 52 ( 2 ): 116 ‑ 126 . (in Chinese)
 B Angrosch , M Plöchl , W Reinalter . Mode decoupling concepts of an engine mount system for practical application . Proceedings of the Institution of Mechanical Engineers , Part K : Journal of Multi-body Dynamics , 2015 , 229 ( 4 ): 331 ‑ 343 .
 J Zhen , S Fredrickson. The effect of mounting structure stiffness on mounting system isolation performance on off-highway machines . No. 2015‑01‑2350. SAE Technical Paper , 2015 .
 Y N Wang , Z H Lv . Optimal design method of power‑train mounting system for generalized force transmissibility reduction . Journal of Mechanical Engineering , 2011 , 50 ( 11 ): 52 ‑ 58 . (in Chinese)
 R L Fan , X L Zhang. Study on semi‑active hydraulic mount with variable ‑ stiffness decoupling membrane . Journal of Mechanical Engineering , 2015 , 51 ( 14 ): 108 ‑ 114 . (in Chinese)
 S W Chen , P F Du , R Li , et al. Dynamic parametric modeling and identification of magnetorheological fluid engine mounts . Journal of Mechanical Engineering , 2016 , 52 ( 8 ): 29 ‑ 35 . (in Chinese)
 I L Ladipo , J D Fadly , W F Faris . Characterization of magnetorheological elastomer (MRE) engine mounts . Materials Today: Proceedings , 2016 , 3 ( 2 ): 411 ‑ 418 .
 L Zheng , Z X Deng , J Pang , et al. Semi‑active vibration control of a vehicle featuring magneto‑rheological engine mount . Automotive Engineering , 2016 , 2 : 221 ‑ 228 . (in Chinese)
 L Zheng , Q B Liu , Z L You , et al. Development of modified lumped parameter model involving amplitude‑ dependence characteristics on semiactive engine mount and experimental verification . Journal of Mechanical Engineering , 2017 , 53 ( 14 ): 98 ‑ 105 . (in Chinese)
 D Y Pan , Z Tang , P C Shi , et al. Applying fuzzy‑PID switching control to magnetorheological semi‑active suspension system . Mechanical Science and Technology for Aerospace Engineering . 2017 , 2 : 292 ‑ 297 . (in Chinese)
 A Farjoud , R Taylor , E Schumann , et al. Advanced semi‑active engine and transmission mounts: tools for modelling, analysis, design, and tuning . Vehicle System Dynamics , 2014 , 52 ( 2 ): 218 ‑ 243 .
 C L Fang , Z D Feng , Z H Lv . An investigation into the natural characteristics and structural modification control of torsional vibration of automotive power train system . Automotive Engineering , 1993 , 15 ( 1 ): 9 ‑ 18 . (in Chinese)
 P Schwibinger , D Hendrick , W Wu , et al. Reduction of vibration and noise in the powertrain of passenger cars with elastomer dampers . No. 910616. SAE Technical Paper , 1991 .
 Z Zhou . Study on virtual test method based on real road spectrum for virtual fatigue prediction . Hunan University, 2013 . (in Chinese)
 X F Han. Several researches of vehicle NVH test methods . Hefei University of Technology, 2008 . (in Chinese)