*Corr. Author’s Address: Universidad Autónoma del Estado de Morelos, Av. Universidad 1001, Cuernavaca, Mor., México, jcgarcia@uaem.mx 101
Strojniški vestnik - Journal of Mechanical Engineering 67(2021)3, 101-113, Received for review: 2020-12-01
© 2021 Journal of Mechanical Engineering. All rights reserved.  Received revised form: 2021-01-18
DOI:10.5545/sv-jme.2020.7051 Original Scientific Paper Accepted for publication: 2021-02-10
Reduction of Stresses and Mass of an Engine Rubber  
Mount Subject to Mechanical Vibrations
Dávalos, O. – Caldiño-Herrera, U.– Cornejo-Monroy, D. – Tenango-Pirin, O. – García, J.C. – Basurto-Pensado, M.A.
Omar Dávalos
1
 – Uzziel Caldiño-Herrera
1
 – Delfino Cornejo-Monroy
1
 –  
Oscar Tenango-Pirin
1
 – Juan Carlos García
2,
* – M.A. Basurto-Pensado
2
1
Universidad Autónoma de Ciudad Juárez, México 
2
Centro de Investigación en Ingeniería y Ciencias Aplicadas, Universidad Autónoma del Estado de Morelos, México
A rubber engine mount (EM) is a mechanical coupling between the engine and the chassis, and its main function is to diminish, in the chassis, 
the amplitude of vibrations caused for the engine operation. Such vibrations cause discomfort for vehicle passengers and reduce the EM 
lifetime. To increase the comfort of vehicle passengers and the lifetime of the EM, this paper presents an EM optimization by means of 
reducing three main criteria: the EM mass, the displacements transmitted to the chassis, and the mechanical stress in the EM rubber core. 
For carrying out the EM optimization, the optimum global determination by linking and interchanging kindred evaluators (GODLIKE), assisted 
by artificial neural networks (ANN) and finite element method (FEM), was used. Because of the optimization process, a reduction greater than 
10 % was achieved in the three criteria in comparison with a baseline design. The frequency responses were compared and showed that 
although the optimization was carried out for the range of 5 Hz to 30 Hz the trend of reduced responses continues beyond this range. These 
results increased the comfort of vehicle passengers and the lifetime of the EM; in addition, the reduction of mass diminishes its production 
costs.
Keywords: multi-objective optimization, vehicle engine mount, ANN, FEM, global optimum determination
Highlights
• 	 An engine mount (EM) was optimized using the integration of optimization algorithms, ANN, and FEM.
• 	 For the EM optimization, the global optimum determination by linking and interchanging kindred algorithm was used.
• 	 The optimization is focused on the reduction of EM mass, displacements, and stress responses under mechanical vibration.
• 	 The engine operation frequencies were obtained via experimental measurements.
• 	 The errors of ANN model predictions were less than 5 %.
• 	 The FEM model was validated by the experimental measurements of natural frequencies.
0  INTRODUCTION
A current trend in the automotive industry is the 
tendency to downsize the components to increase 
vehicle power capacity, to simplify the manufacturing 
process and reduce production costs, among other 
features. This downsizing is only possible if the 
modifications do not compromise passenger comfort 
or affect the performance of the automotive systems. 
Thus, there is a simultaneous need to find automotive 
components susceptible to improvements and 
strategies capable of optimizing new designs. One 
of these automotive parts is the EM, whose main 
function includes the attenuation of engine vibrations, 
by means of the reduction of stiffness, and to support 
the weight of the engine [1] and [2].
The EM consists of a cylindrical steel structure 
fixed to a rubber core, and it is exposed to forced 
vibrations caused by the engine operation [3] to [5]. 
The forced vibrations could cause failure due to 
fatigue of the EM rubber core, limiting the EM lifetime 
to five or six years at most. It should be noted that the 
damage caused by the cyclic load on the steel section 
of the EM parts is minimal [6] and [7]. Under these 
conditions, a good design of an EM mainly involves 
the reduction of the displacements transmitted from 
engine to chassis, the increasing of the lifetime 
through the reduction of mechanical stresses and the 
reduction of weight. This is a complicated task due 
to the interaction of the design variables. Heuristic 
optimization techniques are a good option when 
solving this type of engineering problems.
Researchers in different engineering fields have 
analysed these types of problems using optimization 
techniques. Pérez-Carabaza et al. [8] optimized 
the trajectory of an unmanned aerial vehicle using 
search algorithms of minimum time. Daróczy 
et al [9] optimized the aerofoil geometry of an 
H-rotor employing computational fluid dynamics 
and genetic algorithms; they increased the H-rotor 
power coefficient from 0.40 to 0.48. Cheng et al. 
[10] improved a cuckoo search algorithm applied 
to vibration fault diagnosis. In another study [11], a 
soft optimization based on differential evolution was 
applied to attenuate the vane-rotor shock interaction 
in high-pressure turbines. They achieved attenuation 
above 60 % without stage-efficiency abatement. Rai 
and Barman [12] applied simulated annealing and real 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)3, 101-113
102 Dávalos, O. – Caldiño-Herrera, U.– Cornejo-Monroy, D. – Tenango-Pirin, O. – García, J.C. – Basurto-Pensado, M.A.
coded genetic algorithms to optimize the design of 
a spur gear. They obtained reductions of 14.1 % and 
16.6 % of material by using the simulated annealing 
algorithm and the real coded genetic algorithm, 
respectively. In [13], the non-dominated sorting genetic 
algorithm was used to improve the efficiency, and 
the output power of a piston compressed air engine. 
They obtained, as an optimized result, an efficiency of 
31.17 % when the output power was 2 kW. However, 
for the case of multi-objective optimization problems, 
GODLIKE has been used to find one common optimal 
solution [14].
Regarding EM optimization, several works have 
employed heuristic techniques. In Ahn et al. [15] an 
optimization of an EM, by means of an enhanced 
genetic algorithm with a simplex method and 
sequential quadratic programming, was employed. The 
aim of the study was to reduce both the notch depth 
and the resonance peaks. Both algorithms converged 
in the presence of selected constraints for the design 
parameters which improved the forces of transmission 
to the vehicle by about 30 % due to reduction of notch 
depth and resonance peaks. Lee and Kim [16] used a 
micro-genetic algorithm, ANN, and FEM to reduce 
the mechanical stresses and to increase the life cycles 
of an elastomeric EM. They obtained a 24 % reduction 
of stresses and an increment greater than 100 % in 
the fatigue life cycles. Furthermore, they compared 
their results against a simple genetic algorithm (GA), 
showing that a micro-GA performs better than a simple 
GA. In Zhao et al. [17], a topological optimization was 
performed to reduce the weight of an EM, maximizing 
the natural frequencies and increasing the life cycles. 
They obtained 1.5 × 10
6
 cycles, more than three times 
the initial target. Alvarado-Iniesta et al. [18] used 
memetic genetic programming to optimize an engine 
mount under static load conditions, using a surrogate 
method and FEM. They reported a reduction of EM 
weight and stresses.
In this work, a multi-objective design 
optimization of an EM is proposed using GODLIKE 
assisted by ANN and FEM. The optimization aim 
was to reduce the mass of the EM and to reduce 
the displacements and stresses under mechanical 
vibrations. Three objective functions were defined, 
including a target and a penalty factor for each 
objective function. Experimental measurements were 
accomplished to identify the mechanical vibration 
frequency range of the engine; later, this range 
was used as a boundary condition in commercial 
FEM software to perform frequency response 
simulations. ANN was used as a surrogate method 
within GODLIKE to predict the rubber core mass, 
displacement, and stress responses. The ANN was 
trained with a database generated from a central 
composite design of the experiments. Measurement 
of the EM natural frequency was used to validate 
the numerical model, and this was used to compute 
the EM frequency response using FEM. This multi-
objective optimization is a novel way to link three 
different algorithms (GODLIKE, ANN, and FEM) to 
improve its performance and reach a fast optimization 
methodology of EM, considering stress and 
displacement responses under mechanical vibrations. 
Using another way for the optimization process, the 
calculation of these responses requires a large amount 
of time for the numerical computations. However, in 
this multi-objective optimization algorithm, to reduce 
the computation time, a surrogated method, like ANN, 
is employed. Thus, the cases of FEM computations 
are reduced according to the size of the database used 
in ANN training. Furthermore, the ANN database was 
constructed using a design of experiments which is 
helpful in attaining convergence, further reducing 
the time of optimization. The proposed optimization 
methodology is applied to a specific EM model, 
however; the methodology can be applied to other EM 
types considering its characteristics.
1  METHODS
1.1  Optimization Methodology
This work is focused on the numerical optimization of 
an EM, but the manufacturing of the optimized EM 
is out of the scope. The EM used in this investigation 
is located on the front right side of a 4-cylinder, 2.4 
L Toyota internal combustion engine. One of its sides 
is attached to the chassis, and the other side supports 
the engine. The materials of which it is made are steel 
and rubber. The steel portion of the EM includes a 
cylindrical cover, a ring between the cylindrical cover 
and the rubber core, and a metal bushing between the 
rubber core and the bolt that fixes the engine. The 
rubber core has two lobe-shaped holes, one lateral 
tip at each side and a conical-shape around the metal 
bushing. The typical failure of the EM occurs in the 
rubber core. The crack propagation path extends along 
the neck of the lobe and around the conical shape, as 
shown in Fig. 1. Generally, the damage in the EM 
steel parts is negligible.
The sequence of the optimization methodology 
is shown in Fig. 2. First, the frequencies of the 
forced vibration caused by the engine operation are 
gotten through experimental measurements. These 
measured frequencies will show the range of the EM 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)3, 101-113
103 Reduction of Stresses and Mass of an Engine Rubber Mount Subject to Mechanical Vibrations 
excitation frequencies in which responses will be 
computed. Then, the design parameters are defined 
based on the EM geometry and the optimization 
requirements, which are the reduction of three criteria: 
EM mass, displacement response, and stress response. 
Afterwards, a database is generated using a central 
composite design of experiments and the values of 
stress and displacement to complete the database are 
computed using frequency response analysis trough 
FEM simulations. Once the database was generated, 
it was used for the training and the validation of an 
ANN model, which could accurately predict the 
rubber core mass and both displacement and stress 
responses. The final part is the application of the 
GODLIKE algorithm in the optimization of the design 
of the EM. The detailed description of each part of 
the optimization methodology is presented in the next 
sections.
1.2  Measurement of Engine Mechanical Vibrations
A triaxial G-Link-200 Microstrain
®
 wireless 
accelerometer was placed over the engine block (Fig. 
3) to measure the acceleration amplitude of engine 
mechanical vibrations. The acquired signal from the 
accelerometer was transmitted to a WSDA
®
-200-
USB Microstrain
®
 receiver connected to a personal 
computer. The acceleration measurements were 
carried out at 1500 rpm, 2000 rpm, 2500 rpm, 3000 
rpm, and ralenti conditions (around 800 rpm to 900 
rpm).
Fig. 3.  Mounting of the accelerometer over the engine block
Fig. 4.  Engine mechanical vibration spectrum at ralenti, 1500 
rpm, 2000 rpm, 2500 rpm, and 3000 rpm
The acceleration signal of engine mechanical 
vibration in the time domain was converted to the 
frequency domain using the fast Fourier transform. 
The results of measured frequencies are presented 
in Fig. 4. At ralenti, the measured engine frequency 
was 5.7 Hz and increased with increasing engine 
revolutions. The maximum frequency was 25.6 Hz 
Fig. 1. Failed of the rubber core in an EM
Fig. 2.  Optimization methodology

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)3, 101-113
104 Dávalos, O. – Caldiño-Herrera, U.– Cornejo-Monroy, D. – Tenango-Pirin, O. – García, J.C. – Basurto-Pensado, M.A.
at 3000 rpm. Based on these findings, the frequency 
range for the FEM computations and optimization was 
set from 5 Hz to 30 Hz.
1.3  Parameterization
For the EM optimization, only the part that holds the 
engine was considered since it is where failure occurs 
within the rubber core element. The metal parts were 
considered in the optimization process to reduce the 
total mass of the EM.
A total of eight geometrical variables were 
selected as design parameters: the outer diameter of 
the metal ring (V
1
), the external diameter of the rubber 
core (V
2
), the internal diameter of the rubber core (V
3
), 
the external diameter of the cylindrical cover (V
4
), the 
separation between the internal diameter of the rubber 
core and the lateral tip (V
5
), the internal diameter of the 
thickness of rubber wall (V
6
), the separation between 
tips of rubber lobes (V
7
) and the base diameter of a 
rubber cone (V
8
) around the metal bushing. Fig. 5 
illustrates the location of the design parameters.
Fig. 5.  EM design parameters:  
V
1
, V
2
, V
3
, V
4
, V
5
, V
6
, V
7
, and V
8
The lower and upper limits of the design variables 
are presented in Table 1. These limits were defined to 
avoid EM geometry interferences and available space 
to install EM into the engine bay.
1.4  Database and Frequency Response Analyses
To complete the database for ANN training and 
validation, the EM geometrical values of Table 1 were 
used to compute the frequency harmonic response 
using FEM by means of Ansys software. In this way, 
the stress, based on the von Mises stress theory, and 
displacements responses caused by engine mechanical 
vibrations were computed. Firstly, a load, considered 
as sinusoidal, with a frequency range between 5 Hz 
and 30 Hz with a step of 1 Hz was used. To compute 
the EM response to higher harmonics excitation, the 
frequency range was extended until 100 Hz. 
Table 1.  Design variables bounds
Variable Lower limit [mm] Upper limit [mm]
V
1
64 68
V
2
60 64
V
3
20 24
V
4
72 76
V
5
1 3
V
6
57 61
V
7
36 40
V
8
34 40
The frequency response analysis is based on the 
equation of motion for a mechanical vibration which 
is written as [19]:
 M+ Cu+K u=f u        
  t , (1)
where M is the mass matrix, u is the nodal 
displacement vector, C is the damping matrix, K is the 
stiffness matrix, and f(t) is the forcing vector or the 
dynamic load applied to EM. This dynamic load is a 
harmonic function:
 fF tt   sin,  (2)
where ω is the frequency and F is the force vector.
The FEM computations assumed the rubber 
material as linear elastic to simplify the frequency 
response study. However, this assumption is limited 
to small deformations. For this research, to guarantee 
the validity of the FEM computations, experimental 
validation with the EM natural frequencies was 
implemented.
The property of materials, rubber, and metal, are 
described in Table 2.
The baseline EM model was discretized using 
tetrahedral elements with sizes between 1.2 mm 
and 1.8 mm, resulting in 401,613 finite elements 
(Fig. 6). For each node, six degrees of freedom were 
considered. For the rest of the EM models of the 
database, a slight variation in the number of elements 
is expected due to changes in the design parameters.

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)3, 101-113
105 Reduction of Stresses and Mass of an Engine Rubber Mount Subject to Mechanical Vibrations 
Table 2.  Properties of materials
Property Steel Rubber
Elastic modulus [Pa] 200×10
9
1.5×10
6
Poisson’s ratio 0.29 0.45
Density [kg/m
3
] 7850 1100
Fig. 6.  Meshed domain for the EM geometrical model 
The total engine weight is distributed and applied 
as a force among the three EM that support the 
engine in the engine bay. So, a force of 457.83 N was 
calculated using the engine mass of 140 kg, and then 
it was applied at the internal face of the EM bushing. 
Furthermore, the harmonic response requires applying 
a dynamic force, which was considered as three times 
the force applied to EM bushing resulting in a final 
force of 1373.5 N. Two displacement restrictions were 
applied, one at the inner face of the bushing of the part 
bolted to chassis, the second one at the external surface 
of the cylindrical cover. No sliding or separation was 
considered for the contact between the EM parts.
The harmonic response analysis was performed 
in a range of 5 Hz to 30 Hz with frequency steps of 1 
Hz. At each frequency step, resultant deformation and 
stress responses were calculated.
1.5  Artificial Neural Networks
An ANN works as an interpolator in the classification 
process and static or time-series predictions. A type 
of feed-forward ANN, widely used in prediction 
is the multilayer perceptron (MLP), which has an 
architecture of layers arranged in input, hidden, and 
output layers. Each layer is composed of a defined 
number of neurons which, in the case of input and 
output layers, corresponds to the number of inputs 
and predicted outputs, respectively. In the case of 
the hidden layer, the number of neurons is defined 
by trial and error, until it reaches an acceptable 
reliability prediction. Each layer is linked to the 
next one through weighted connections. To establish 
nonlinear relationships between inputs and outputs, 
transfer functions must be added to ANN. A good 
performance in prediction could be found using the 
hyperbolic tangent (HT) function between input and 
hidden layer, whereas between the hidden and output 
layers, a linear function could be used [20] to [23]. The 
predicted output is calculated through:
 yl wH Tb
j
n
jn
 











1
22
, (3)
where l is the linear function, w is the weight 
connection, HT is the hyperbolic tangent transfer 
function, b is the bias, n is the total number of hidden 
neurons. The HT is:
 HT
e
n x
n




2
1
1
2
, (4)
where x is defined as:
 xw in b
n
i
m
ii j
 


1
11
, (5)
where m is the number of neurons in the input 
layer. The superscripts 1 and 2 indicate inputs to the 
hidden layers and outputs from the hidden layers, 
respectively. Weights and biases coefficients are 
obtained from the ANN training process and inserted 
in the above equations to calculate the desired output. 
The adequate selection of these coefficients is made 
by a training algorithm. To train, validate, and test the 
ANN, the database of section 1.4 was used.
In this work, three ANN models were 
implemented to calculate stress and displacements 
response at a specified frequency range and, due to 
the complex shape, the mass of the rubber core of the 
EM. The architecture of all three models consists of 
three layers with eight and one neurons at input and 
output layers, respectively. The model to predict the 
stress response area has six neurons in the hidden 
layer, whereas the model for displacements has three 
neurons, and just two neurons in the hidden layer 
for the mass prediction. The database to train the 
ANN consists of 80 design combinations based on 
a central composite design of experiments and an 
additional baseline design. This design of experiments 
was implemented to avoid duplicated information 
which difficult the ANN learning. The database was 
constructed by FEM simulations of the harmonic 
response of EM subjected to excitation frequencies. 
All design combinations of the database show the same 
trend increasing displacement and stress responses as 
frequency increases. The net was trained using the 
Levenberg-Marquardt algorithm, which has shown 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)3, 101-113
106 Dávalos, O. – Caldiño-Herrera, U.– Cornejo-Monroy, D. – Tenango-Pirin, O. – García, J.C. – Basurto-Pensado, M.A.
a good prediction performance compared with other 
training algorithms [24]. The transfer functions used 
in these models were HT, between input and hidden 
layer, and linear, between hidden and output layer. 
Due to the use of HT, the input data were normalized 
between 0.1 and 0.9 using the next expression [25]:
 n
In In
In In
i
i









 08 01 .. .
min
maxm in
 (6)
Here n
i
 is the normalized variable, In
i
 is the not-
normalized variable, and In
min
 and In
max
 are the lower 
and upper range of the design variable.
The performance of the ANN models was 
evaluated using the estimation of both root mean 
square error, RMSE, and the correlation coefficient, 
R
2
, calculated as follows:
 RMSE
yy
T
t
T
ANN FEM

 


1
2
, (7)
 R
yy
yy
t
T
ANN FEM
t
T
FEM ave
2 1
2
1
1 
 
 




, (8)
where y
ANN
 is the output predicted by ANN, y
FEM
 is 
the output predicted by FEM, y
ave
 is the average of 
actual values. From the database, 80 % of data were 
randomly selected to train the net, whereas the rest of 
the data were used in the validation process. To test 
the ANN models, eight additional simulations (10 % 
of the database) were computed to evaluate ANN 
predictions against FEM calculations.
1.6  Optimization
The proposed optimization is based on the 
GODLIKE algorithm developed by Oldenhuis and 
Vandekerckhove [26]. GODLIKE uses four single 
meta-heuristic algorithms: differential evolution (DE), 
genetic algorithm (GA), particle swarm optimization 
(PSO) and adaptive simulated annealing (ASA). 
In GODLIKE, each algorithm performs the first 
approach from an initial population. Before starting 
a second approach, a defined number of members of 
the firstly optimized population is randomly selected 
from one algorithm (e.g., DE) and inserted in the 
population of the remaining algorithms (e.g., GA, 
PSO, and ASA). The second approach starts until all 
the algorithms have shared members among them. 
This process is repeated until a stop condition is 
met. In this way, the possible poor performance of 
each algorithm is improved due to the integration of 
population members from other algorithm approaches. 
This link between algorithms is intended to find the 
global optimum due to the use of populations with 
the fittest individuals. The use of this methodology 
requires many objective function evaluations, which 
increase the computational cost; however, it is offset 
by the robustness of the algorithm [27].
1.7  Objective Function
A multi-objective optimization works by minimizing 
or maximizing several objective functions that satisfy 
a defined set of constraints [28]. The problem can be 
mathematically written as:
 minimize fX fX fX
N 12
 
 

 
,, ,, (9)
subject to; 
gX
j


 0
, hX
k


 0, and x
lb
 ≤  x
M
 ≤  x
ub
. 
Here f is the objective function, 

X is a vector that 
contains the design variables x, g and h are the 
inequality and equality constraint functions, 
respectively. The subscript N is the total of objective 
functions; j, k and M are the amounts of inequalities 
constraints, equalities constraints, and design variables 
respectively, whereas l
b
 and u
b
 are the lower and upper 
bounds of the corresponding design variable. When a 
multi-objective optimization is performed, more than 
one solution is obtained; these feasible solutions lie on 
the Pareto-optimal front [29].
In this work, the total mass reduction of the EM 
was calculated by adding the mass of all its parts: 
rubber core, ring, metal cover, and the bushing. Due 
to the complex shape of the rubber core, its mass was 
computed using an ANN model, whereas the mass of 
the rest of the components was evaluated as follows:
 mm
ru ru ANN
=
,
, (10)
 mV V
ri


62 8
1
2
2
2
., (11)
 mV V
mc


62 8
4
2
1
2
., (12)
 mV
bu


90 275 0 010
3
22
.. . (13)
Here m
ru
 is the mass of the rubber part, and the 
subscript ANN indicates the rubber mass predicted by 
means of the neural net. Deformations and stresses 
vary as functions of the mechanical vibrations and 
their excitation frequencies. Since the engine works 
at different frequencies, a method was implemented 
to evaluate both displacement and stress as response 
areas considering a frequency step of S = 1 Hz. The 
method aims to calculate every response area as 
an area of a trapezoidal shape formed between two 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)3, 101-113
107 Reduction of Stresses and Mass of an Engine Rubber Mount Subject to Mechanical Vibrations 
consecutive frequency steps. For this purpose, the 
coordinates of four points must be in a plane where 
frequencies versus response (displacement or stresses) 
are plotted at the x-axis and y-axis, respectively. 
The coordinates of the four points are defined by 
their position at x-axis and y-axis, respectively. For 
example, in Fig. 7, the displacement response area 
A is formed by points 1, 2, 3, and 4. The coordinates 
of point 1 and 2 at the x-axis are the frequency ω at 
step 1 and frequency ω+1 respectively, whereas at the 
y-axis its coordinate is 0. The coordinates of points 3 
and 4 at the x-axis are the same as in points 1 and 2; 
at the y-axis, the coordinates correspond to response 
magnitude at ω and ω+1, respectively. Then, the four 
points are connected, closing the profile forming a 
trapezoidal area, which evaluates the response through 
two consecutive frequencies. 
Fig. 7.  Definition of coordinates at the response area
The response areas for displacement and stress 
are calculated using Eqs. (14) and (15) respectively:
 A
dd
s
d



1
2
, (14)
 As





1
2
, (15)
where A
dω
 and A
σω
 are the areas for displacement 
and stress responses, respectively, and d and σ are 
the displacement and stress responses obtained at a 
specific excitation frequency, ω.
For the optimized design of the EM, a multi-
objective optimization was raised. Three objective 
functions were selected with a set target for each 
one of them. Penalty factors were added to objective 
functions to define the relevance of each variable in 
the optimization process.
Objective functions are expressed in Eqs. (16) to 
(18). Eq. (16) minimizes the expression to evaluate 
the mass of the EM, which involves the mass of 
rubber part, ring, metal cover, and bushing. Eqs. 
(17) and (18) are the average of the response areas 
of displacement and stress, respectively, during the 
excitation frequencies range.
 min, f
mmmm
m
P
ru ri mc bu
tar
f 11

  





 (16)
 min,
,
f
A
A
P
d
avg
dt ar
avg f 22




 (17)
 min.
,
f
A
A
P
ave
tar
ave f 33




 (18)
Here, AA S
d
avg
i
S
di 
 



1
1
1
,
 is the average area 
response of displacement, AA S
avg
i
S
i  
 



1
1
1
,
 is 
the area average response of stresses, whereas S is the 
total number of excitation frequencies. The subscript  
tar indicates the targets which were set to reach the 
convergence to the optimized values. The targets 
were, m
tar
 = 0.45 kg, A
d,tar
 = 0.0035 m·Hz and A
σ,tar
 = 
0.5 MPa·Hz. The penalty factors are: P
f 1
, P
f 2
 and P
f 3
 
for mass, displacement response and stress response, 
respectively. In this work, it was assumed that 
passenger comfort (which is related to the 
displacement) is the main criterion to consider 
followed by the resistance of components and finally 
the mass of the component. Based on these 
assumptions, the penalty factors were set as follows: 
0.15 for mass, 0.60 for displacement and 0.25 for 
stresses.
1.8  Validation of FEM Model
The FEM model was validated through the 
measurement of the first natural frequency of the EM. 
The accelerometer utilized was the same used in the 
measurements of engine mechanical vibrations. The 
device was mounted over the EM inside the engine 
bay. The excitation signal was provided by an impact 
hammer.
The experimental measurement of EM first 
natural frequency is shown in Fig. 8. The value of 
the first natural frequency was 75.015 Hz. The first 
three natural frequencies from FEM computations 
are shown in Table 3. The comparison of both the 
experimental and numerical first natural frequencies 
shows a good agreement between measured and 
predicted frequencies with a difference of around 7.5 
%. These results guarantee the accuracy of the FEM 
model and its validity for the optimization of the EM.

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)3, 101-113
108 Dávalos, O. – Caldiño-Herrera, U.– Cornejo-Monroy, D. – Tenango-Pirin, O. – García, J.C. – Basurto-Pensado, M.A.
Table 3.  Natural frequencies calculated by FEM
Natural frequency Frequency [Hz]
1
st
80.68
2
nd
89.68
3
rd
104.98
Fig. 8.  Experimental measurement of the first natural frequency 
2  RESULTS AND DISCUSSION
The results of the ANN computations are presented in 
Fig. 9. For A
avg

 training and validation, the correlation 
coefficient was greater than 0.98, whereas the cases of 
A
d
avg
ω
 and m
ru
 is greater than 0.99. To test the ANN 
models, eight additional simulations (10 % of the 
database) were carried out to compare ANN 
predictions against FEM calculations. In the cases of 
displacement response and rubber mass computation, 
the ANN testing predictions have an error below 5%, 
whereas for stress response prediction the error is 
slightly greater than 5 %. These results show the 
capability of the ANN models to predict reliably and 
ensure its use in the optimization of the EM.
The weights and biases to predict stress response, 
displacement response and rubber mass are presented 
in Tables 4 to 6. These coefficients were obtained 
during ANN training and correspond to the best fit for 
the three predictions.
In contrast with other works, such as [16] and 
[18], that have used similar optimization techniques 
assisted by surrogated methods, this research includes 
experimental measurements, which guarantee 
that optimization was performed over operating 
conditions. Furthermore, considering displacement 
as an objective function instead of a constraint (as 
in [16]) allows reducing displacement along with the 
whole range of engine frequencies.
a) 
b) 
c) 
Fig. 9.  Results of ANN predictions for  
a) A
avg

, b) A
d
avg
ω
 and c) m
ru
In Fig. 10, the results of the optimization process 
are presented. Around 25 % of the Pareto points 
improve the baseline design. Possible candidates were 
found with a reduced displacement; nevertheless, 
their mass was around 0.5 kg which makes them non-
feasible candidates. As the A
avg

 and A
d
avg
ω
 reduce their 
magnitude, more options can be found with a lower 
mass.

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)3, 101-113
109 Reduction of Stresses and Mass of an Engine Rubber Mount Subject to Mechanical Vibrations 
Table 4.  ANN coefficients for prediction of area average of stress response
w
11
1
,
-1.45854
w
12
1
,
0.96315
w
13
1
,
-1.82456
w
14
1
,
-0.49916
w
15
1
,
-0.45800
w
16
1
,
5.08531
w
17
1
,
2.05659
w
18
1
,
-3.16593
w
21
1
,
0.98225
w
22
1
,
2.23806
w
23
1
,
-1.75895
w
24
1
,
-1.17739
w
25
1
,
-2.32274
w
26
1
,
-1.14974
w
27
1
,
1.65331
w
28
1
,
-2.22049
w
31
1
,
1.39160
w
32
1
,
-5.58502
w
33
1
,
-2.39672
w
34
1
,
-0.12927
w
35
1
,
-0.05249
w
36
1
,
1.15514
w
37
1
,
-0.59857
w
38
1
,
-1.20014
w
41
1
,
0.63538
w
42
1
,
3.30013
w
43
1
,
-1.19526
w
44
1
,
-0.09034
w
45
1
,
-1.41403
w
46
1
,
-1.35004
w
47
1
,
-3.69895
w
48
1
,
-2.78710
w
51
1
,
-2.83462
w
52
1
,
-1.74426
w
53
1
,
1.65473
w
54
1
,
1.03900
w
55
1
,
3.66780
w
56
1
,
-0.30788
w
57
1
,
-0.69853
w
58
1
,
2.18040
w
61
1
,
0.17086
w
62
1
,
-1.81009
w
63
1
,
0.84667
w
64
1
,
0.12948
w
65
1
,
-2.15127
w
66
1
,
4.82269
w
67
1
,
0.90327
w
68
1
,
-0.18045
w
11
2
,
0.03196
w
12
2
,
-0.07535
w
13
2
,
0.16155
w
14
2
,
0.06778
w
15
2
,
-0.53198
w
16
2
,
0.12223
b
11
1
,
1.15072
b
21
1
,
2.51253
b
31
1
,
0.89235
b
41
1
,
2.00106
b
51
1
,
-5.08572
b
61
1
,
-3.36807
b
11
2
,
0.33196
Table 5.  ANN coefficients for prediction of area average of displacement response
w
11
1
,
-0.11114
w
12
1
,
-0.06945
w
13
1
,
6.66939
w
14
1
,
-0.11978
w
15
1
,
0.05504
w
16
1
,
0.22898
w
17
1
,
0.31355
w
18
1
,
-0.27377
w
21
1
,
0.01930
w
22
1
,
-0.08350
w
23
1
,
-0.02500
w
24
1
,
0.04518
w
25
1
,
0.03851
w
26
1
,
-0.53731
w
27
1
,
0.52415
w
28
1
,
0.18607
w
31
1
,
0.01093
w
32
1
,
3.65897
w
33
1
,
-1.03528
w
34
1
,
0.72872
w
35
1
,
0.91272
w
36
1
,
-0.74952
w
37
1
,
0.14733
w
38
1
,
-0.06379
w
11
2
,
-0.00018
w
12
2
,
-0.00162
w
13
2
,
0.00010
b
11
1
,
-2.58080
b
21
1
,
0.32367
b
31
1
,
-2.47434
b
11
2
,
0.00464
Table 6.  ANN coefficients for rubber mass prediction
w
11
1
,
0.00051
w
12
1
,
-0.47488
w
13
1
,
0.21294
w
14
1
,
0.00173
w
15
1
,
0.03700
w
16
1
,
0.31568
w
17
1
,
-0.11993
w
18
1
,
-0.12761
w
21
1
,
1.11450
w
22
1
,
2.60450
w
23
1
,
-0.06103
w
24
1
,
1.59204
w
25
1
,
0.57664
w
26
1
,
-0.56463
w
27
1
,
-0.80459
w
28
1
,
-1.36578
w
11
2
,
-0.03618
w
12
2
,
0.000068
b
11
1
,
0.05320
b
21
1
,
-0.93912
b
11
2
,
0.05983

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)3, 101-113
110 Dávalos, O. – Caldiño-Herrera, U.– Cornejo-Monroy, D. – Tenango-Pirin, O. – García, J.C. – Basurto-Pensado, M.A.
Individual results are presented in Table 7. For 
the optimized case, total reductions of 21.5 %, 12.46 
%, and 15.3% were obtained for A
avg

, A
d
avg
ω
 and 
mass, respectively, regarding the baseline design. 
These reductions are attributable to the application 
of targets and penalty factors. A
d
avg
ω
 is considered 
the most important parameter, and it was set with a 
penalty factor of 60 %; however, the magnitude of 
its reduction was lower than others resulting in the 
hardest parameter to optimize.
Fig. 10.  Pareto points, optimal design found and baseline design
Table 7.  Comparison of optimization results of A
avg

, A
d
avg
ω
 and 
mass
Baseline design Optimal design
A
avg

 [MPa·Hz]
0.53042 0.41632
A
d
avg
ω
 [m·Hz]
0.00385 0.00337
mass [kg] 0.49148 0.41628
Table 8.  Magnitudes of the baseline and optimized design 
variables
Variable Baseline design [mm] Optimal design [mm]
V
1
66 67.404
V
2
62 61.683
V
3
22 22.003
V
4
74 72.189
V
5
2 1.683
V
6
59 57.151
V
7
38 39.63
V
8
37 38.736
In Table 8, the optimized dimensions of EM are 
presented. The loss of mass is due to the reduction 
of the thickness of the wall of the cylindrical cover 
and the internal diameter of ring. The increment 
of V
1
 combined with the reduction of V
2
 causes the 
increment of internal ring wall thickness. Due to the 
increment of the ring wall thickness, the movement 
of the rubber core is restricted, thereby reducing 
the displacements. The length of V
7
 increases the 
strengthening of the rubber core thus achieving a 
reduction of the stress response.
a)             
b) 
Fig. 11.  Comparison of responses through the frequency range, a) 
stress, b) displacement
With the optimized design variables, a new 
geometrical model, which corresponds to optimized 
EM, was generated. For such optimized geometry, 
new computations of the frequency response were 
performed and compared against the baseline design. 
In Fig. 11, stress and displacements are plotted 
against the typical (5 Hz to 30 Hz) and extended 
(30 Hz to 100 Hz) excitation frequency of the 
internal combustion engine. The improvement of the 
optimized EM is outstanding due to the difference 
in stress and displacement responses between 
baseline and the optimal design. For the case of the 
baseline design, the difference between the computed 
amplitude of displacement response for the initial (5 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)3, 101-113
111 Reduction of Stresses and Mass of an Engine Rubber Mount Subject to Mechanical Vibrations 
Hz) and the last (30 Hz) frequencies was 0.0896 ×10
-3
 
m. While for the case of the optimized design, such a 
difference was reduced to 0.06839 ×10
-3
 m. Also, the 
stress response amplitude had a reduction of 35.9 % 
between the baseline design case and the optimized 
case. These results are significant when a vehicle is 
accelerating between this frequency range because, 
in the case of displacements, a smaller movement 
will be transmitted to vehicle chassis affecting 
passenger comfort, whereas, in the case of stresses, its 
amplitude is minor, reducing the effects of fatigue in 
the EM. Extending the frequency excitation range, the 
difference between displacement and stress responses 
(Fig. 11) for the case of baseline and optimized EM 
is increased. Also, Fig. 11 shows that the peak which 
corresponds to the first natural frequency is decreased 
for the case of the optimized model. This contributes 
to reducing the amplitude of vibrations of the vehicle 
chassis in case of resonance at that frequency.
The distribution of stresses at 21 Hz, 2500 rpm, is 
presented in Fig. 12. Stresses are concentrated in the 
region near the rubber neck. Fig. 12a shows how the 
magnitude of stresses is greater than the optimized EM 
for which a reduction is observed at this zone. Around 
the external surface of the rubber core, the stresses are 
lower than in the centre region. A stress line extends 
out from the front of the neck to the back. Thereby, if 
a failure occurs, it is expected to be in this zone. These 
findings are consistent with those presented in Fig. 1, 
which shows the cracks of a failed EM.
Fig. 12.  Comparison of stress distributions at 21 Hz (2500 rpm),  
a) baseline design, b) optimized design
A comparison of displacements is presented in 
Fig. 13 at 21 Hz equivalent to 2500 rpm. Maximum 
displacements are observed in the surface around 
the bushing and in the cone. In the optimized EM, 
the displacements are reduced mainly at the sides of 
the cone and, in a smaller proportion, at the top and 
bottom.
Fig. 13.  Comparison of displacement at 21 Hz at 2500 rpm,  
a) baseline design, b) optimal design
3  CONCLUSIONS
An EM with a rubber core was optimized using a 
methodology that includes optimization algorithms, 
ANN and FEM. The optimization is focused on the 
reduction of EM mass, and both displacements and 
stresses responses under the excitation of mechanical 
vibration due to the engine operation. The EM mass 
has a reduction of 15.3 %, the  and  were reduced by 
21.5 % and 12.46 %, respectively. The target values 
added in the objective functions were reached, leading 
the optimization process toward better solutions than 
a baseline design. A lower magnitude of displacement 
and stress responses was reached, increasing the 
ring wall thickness and the distance between lobes 
in the rubber core, respectively. The reduction of 
the  and  means lowering the amplitude of stress and 
displacements levels in the whole frequencies range, 
which, in turn, reduces the displacement transmitted 
from engine to chassis and increases the lifetime 
of EM. In contrast, the mass reduction decreases 
production costs. The results from training, testing, 
and validation of the ANN assure a high confidence 
level in the predictions. The good definition of ANN 
models contributes to simplifying the optimization 
process in this research. The integration of the 
optimization methodology here presented could 
be applied for many optimization problems in 
engineering. The frequency response computation 
with the extended frequency range shows that the 
proposed optimization method of averaging both 
displacement and stress responses is effective because 
the trend of displacement and stresses response 
improvement remains continual throughout the 
extended range.
4  REFERENCES
[1] Shangguan, W.-B., Liu, X.-A., Lv, Z.-P., Rakheja, S. (2016). 
Design method of automotive powertrain mounting system 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)3, 101-113
112 Dávalos, O. – Caldiño-Herrera, U.– Cornejo-Monroy, D. – Tenango-Pirin, O. – García, J.C. – Basurto-Pensado, M.A.
based on vibration and noise limitations of vehicle level. 
Mechanical Systems and Signal Processing, vol. 76-77, p. 
677-695, DOI:10.1016/j.ymssp.2016.01.009.
[2] Azammi, A.M.N., Sapuan, S.M., Ishak, M.R., Sultan, M.T.H. 
(2018). Conceptual design of automobile engine rubber 
mounting composite using triz-morphological chart-analytic 
network process technique. Defence Technology, vol. 14, no. 
4, p. 268-277, DOI:10.1016/j.dt.2018.05.009.
[3] Liu, X.-A., Shangguan, W.-B., Lv, Z.-P., Ahmed, W., Zhu, W. 
(2017). A study on optimization method of a powertrain 
mounting system with a three-cylinder engine. Proceedings 
of the Institution of Mechanical Engineers, Part C: Journal of 
Mechanical Engineering Science, vol. 231, no. 12, p. 2235-
2252, DOI:10.1177/0954406216631004.
[4] Idrees, M., Rajendra Prasad, V.B.S. (2018). Response of 
rubber based engine mounts with sbr as the core rubber. IOP 
Conference Series: Materials Science and Engineering, vol. 
455, p. 012108, DOI:10.1088/1757-899x/455/1/012108.
[5] Ramachandran, T., Padmanaban, K.P. (2013). Minimization of 
IC engine rubber mount displacement using genetic algorithm. 
The International Journal of Advanced Manufacturing 
Technology, vol. 67, p. 887-898, DOI:10.1007/s00170-012-
4533-1.
[6] Tang, N., Soltani, P., Pinna, C., Wagg, D., Whear, R. (2018). 
Ageing of a polymeric engine mount investigated using digital 
image correlation. Polymer Testing, vol. 71, p. 137-144, 
DOI:10.1016/j.polymertesting.2018.08.036.
[7] Kim, W.D., Lee, H.J., Kim, J.Y., Koh, S.-K. (2004). Fatigue 
life estimation of an engine rubber mount. International 
Journal of Fatigue, vol. 26, no. 5, p. 553-560, DOI:10.1016/j.
ijfatigue.2003.08.025.
[8] Pérez-Carabaza, S., Scherer, J., Rinner, B., López-Orozco, 
J.A., Besada-Portas, E. (2019). UAV trajectory optimization 
for minimum time search with communication constraints 
and collision avoidance. Engineering Applications of 
Artificial Intelligence, vol. 85, p. 357-371, DOI:10.1016/j.
engappai.2019.06.002.
[9] Daróczy, L., Janiga, G., Thévenin, D. (2018). Computational 
fluid dynamics based shape optimization of airfoil geometry 
for an H-rotor using a genetic algorithm. Engineering 
Optimization, vol. 50, no. 9, p. 1483-1499, DOI:10.1080/0305
215X.2017.1409350.
[10] Cheng, J., Wang, L., Xiong, Y. (2018). An improved cuckoo 
search algorithm and its application in vibration fault 
diagnosis for a hydroelectric generating unit. Engineering 
Optimization, vol. 50, no. 9, p. 1593-1608, DOI:10.1080/0305
215X.2017.1401067.
[11] Joly, M.M., Verstraete, T., Paniagua, G. (2013). Differential 
evolution based soft optimization to attenuate vane–rotor 
shock interaction in high-pressure turbines. Applied Soft 
Computing, vol. 13, no. 4, p. 1882-1891, DOI:10.1016/j.
asoc.2012.12.005.
[12] Rai, P., Barman, A.G. (2018). Design optimization of spur 
gear using SA and RCGA. Journal of the Brazilian Society of 
Mechanical Sciences and Engineering, vol. 40, art. ID 257, 
DOI:10.1007/s40430-018-1180-y.
[13] Yu, Q., Cai, M., Shi, Y., Fan, Z. (2014). Optimization of the 
energy efficiency of a piston compressed air engine. Strojniški 
vestnik - Journal of Mechanical Engineering, vol. 60, no. 6, p. 
395-406, DOI:10.5545/sv-jme.2013.1383.
[14] Sbayti, M., Bahloul, R., BelHadjSalah, H., Zemzemi, F. (2018). 
Optimization techniques applied to single point incremental 
forming process for biomedical application. The International 
Journal of Advanced Manufacturing Technology, vol. 95, p. 
1789-1804, DOI:10.1007/s00170-017-1305-y.
[15] Ahn, Y.-K., Kim, Y.-C., Yang, B.-S., Ahmadian, M., Ahn, K.-K., 
Morishita, S. (2005). Optimal design of an engine mount using 
an enhanced genetic algorithm with simplex method. Vehicle 
System Dynamics, vol. 43, no. 1, p. 57-81, DOI:10.1080/0042
3110412331290400.
[16] Lee, J.S., Kim, S.C. (2007). Optimal design of engine mount 
rubber considering stiffness and fatigue strength. Proceedings 
of the Institution of Mechanical Engineers, Part D: Journal 
of Automobile Engineering, vol. 221, no. 7, p. 823-835, 
DOI:10.1243/09544070JAUTO433.
[17] Zhao, Q., Chen, X., Wang, L., Zhu, J., Ma, Z.-D., Lin, Y. (2015). 
Simulation and experimental validation of powertrain 
mounting bracket design obtained from multi-objective 
topology optimization. Advances in Mechanical Engineering, 
vol. 7, no. 6, p. 1-10, DOI:10.1177/1687814015591317.
[18] Alvarado-Iniesta, A., Guillen-Anaya, L.G., Rodríguez-Picón, 
L.A., Ñeco-Caberta, R. (2020). Multi-objective optimization 
of an engine mount design by means of memetic genetic 
programming and a local exploration approach. Journal of 
Intelligent Manufacturing, vol. 31, p. 19-32, DOI:10.1007/
s10845-018-1432-9.
[19] Thorby, D. (2008). Structural Dynamics and Vibration in 
Practice, Thorby, D. (ed.), Butterworth-Heinemann, Oxford, p. 
77-97.
[20] Bartecki, K. (2013). A general transfer function representation 
for a class of hyperbolic distributed parameter systems. 
International Journal of Applied Mathematics and Computer 
Science, vol. 23, no. 2, p. 291-307, DOI:10.2478/amcs-2013-
0022.
[21] Bonakdari, H., Khozani, Z.S., Zaji, A.H., Asadpour, N. (2018). 
Evaluating the apparent shear stress in prismatic compound 
channels using the genetic algorithm based on multi-layer 
perceptron: A comparative study. Applied Mathematics 
and Computation, vol. 338, p. 400-411, DOI:10.1016/j.
amc.2018.06.016.
[22] Kong, Y.S., Abdullah, S., Schramm, D., Omar, M.Z., Haris, S.M. 
(2019). Optimization of spring fatigue life prediction model for 
vehicle ride using hybrid multi-layer perceptron artificial neural 
networks. Mechanical Systems and Signal Processing, vol. 
122, p. 597-621, DOI:10.1016/j.ymssp.2018.12.046.
[23] Durodola, J.F., Ramachandra, S., Gerguri, S., Fellows, N.A. 
(2018). Artificial neural network for random fatigue loading 
analysis including the effect of mean stress. International 
Journal of Fatigue, vol. 111, p. 321-332, DOI:10.1016/j.
ijfatigue.2018.02.007.
[24] Hamzaoui, Y.E., Hernández, J.A., Silva-Martínez, S., 
Bassam, A., Álvarez, A., Lizama-Bahena, C. (2011). Optimal 
performance of cod removal during aqueous treatment of 
alazine and gesaprim commercial herbicides by direct and 
inverse neural network. Desalination, vol. 277, no. 1-3, p. 325-
337, DOI:10.1016/j.desal.2011.04.060.

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)3, 101-113
113 Reduction of Stresses and Mass of an Engine Rubber Mount Subject to Mechanical Vibrations 
[25] Despagne, F., Massart, D.L. (1998). Neural networks 
in multivariate calibration. Analyst, vol. 123, no. 11, p. 
157R-178R, DOI:10.1039/A805562I.
[26] Oldenhuis, R., Vandekerckhove, J. (2009). Godlike: A 
Robust Single-& Multi-Objective Optimizer. from http://
www. mathworks.co.uk/matlabcentral/fileexchange/24838-
godlike-a-robust-single-multi-objective-optimizer, accessed on 
2013-04-19.
[27] Sbayti, M., Bahloul, R., BelHadjSalah, H., Zemzemi, F. (2018). 
Optimization techniques applied to single point incremental 
forming process for biomedical application. The International 
Journal of Advanced Manufacturing Technology, vol. 95, p. 
1789-1804, DOI:10.1007/s00170-017-1305-y.
[28] Coello Coello, C., Lamont, G.B., Van Veldhuizen, D.A. (2007). 
Evolutionary Algorithms for Solving Multi-Objective Problems. 
Springer, New York, DOI:10.1007/978-1-4757-5184-0.
[29] Lim, J., Jang, Y.S., Chang, H.S., Park, J.C., Lee, J. (2020). 
Multi-objective genetic algorithm in reliability-based 
design optimization with sequential statistical modeling: 
An application to design of engine mounting. Structural 
and Multidisciplinary Optimization, vol. 61, p. 1253-1271, 
DOI:10.1007/s00158-019-02409-1.