INTRODUCTION
The major functions of the engine mounting system are to support
the weight of the engine and to isolate the unbalanced engine disturbance
force from the vehicle structure. For an internal combustion engine, there
exist two basic dynamic disturbances: (a) the firing pulse due to the
explosion of the fuel in the cylinder and (b) the inertia force and torque
caused by the rotating and reciprocating parts (piston, connecting rod
and crank). The firing pulses will cause a torque to act on the engine
block about an axis parallel to the crank. The directions of the inertia
forces are both parallel to the piston axis and perpendicular to the crank
and piston axes. For a multicylinder engine, the components of the engineunbalanced
disturbance depend on the number and arrangement of the cylinders in the
engine. These engine disturbances will excite the engine six Degree of
Freedom (DOF) vibration modes as shown in Fig. 1 (Yu
et al., 2000).
In order to obtain a low transmissibility, the natural frequency of the
mounting system in a certain direction must be below the engine disturbance
frequency of the engine idle speed to avoid excitation of mounting system
resonance during normal driving conditions. This means that the engine
mount stiffness coefficient should be as low as possible to obtain a low
transmissibility. If the elastic stiffness of the engine mount is too
low, then the transient response (of the engine mount system) can be problematic
for the shock excitation. Shock excitation would be a result of sudden
acceleration and deceleration, braking and riding on uneven roads.

Fig. 1: 
Engine six DOF modes 
So from this point of view, high
stiffness and high damping are required to minimize the engine motion
and absorb engine shake and resonance.
From the above discussion, it can be easily inferred that to isolate
the engine vibration in a relatively high frequency range, the engine
mounts are required to be softlow elastic stiffness and low damping and
to prevent engine bounce in the low frequency range, engine mounts should
be hardhigh elastic stiffness and high damping (Yu et al., 2000).
Elastomeric (or rubber) mounts have been widely used to isolate vehicle
structure from engine vibration since the 1930s (Browne and Taylor, 1939).
Since then, much significant advancement, were made to improve the performance
of the elastomeric mounts (Browne and Taylor, 1939; Bucksbee, 1987). Elastomeric
mounts can be designed for the necessary elastic stiffness rate characteristics
in all directions for proper vibration isolation. They are compact, costeffective
and maintenance free. The elastomeric mount can be modeled by Voigt model
which consists of a spring and a viscous damper (Swanson, 1992).
The objective of the engine mount optimization should be clear in advance
of realizing any optimization procedure. Different objectives of optimization
have been considered in the literatures. One objective of optimization
is to tune the natural frequency of the engine mounting system to some
desired range to avoid resonance and to improve the isolation of vibration
and noise and shock excitations (Arai et al., 1993; Johnson and
Subhedar, 1979). Johnson and Subhedar (1979) proposed a design objective,
which tunes all of the system natural frequencies to 616 Hz and decouples
each mode of vibration through dynamic analysis and optimization. Geck
and Patton (1984) proposed a lower roll natural frequency for the consideration
of torque isolation and a relatively higher vertical natural frequency
for the consideration of avoiding shock excitation. This is because the
requirements for shock prevention and vibration isolation are conflicting
ones. Hence, the selection of natural frequency for a mounting system
is only a compromise for the linear elastomeric mounts. Bernard and Starkey
(1983) attempted to move the system natural frequency away from an undesired
frequency range to reduce the large transferred forces. Swanson et
al. (1993) showed that the transmitted force could be directly minimized
in order to determine a truly optimal design of the mounts. Ashrafiuon
(1993) also used these criteria to minimize the dynamic force transmitted
from the engine to the body.
The aim of this research is to present analytical as well as finite element
model of Budsan truck engine mount system and verify the analytical model
to achieve the best stiffness and hystersis damping and reduce the system
vibration.
MODELING OF ENGINE MOUNT SYSTEM
For the modeling of engine mount system, the engine is connected
to the rigid base structure by rubber mounts and is modeled as a sixDOF
rigid body free to translate along and rotate about the three independent
Cartesian axes. The center of this coordinate system is located at the
center of gravity of the engine and gearbox.

Fig. 2: 
Configuration of an engine with four mounts 
The rubber mounts are modeled
as springs with the stiffness coefficient and a corresponding hysteresis damping value in each of three
principal directions. Figure 2 shows the configuration
of a typical enginemount system. Using the assumptions above and the
Newton`s second law, the equation of the motion of system may be written
as
where, are the 6x1 displacement, velocity and acceleration vectors, respectively;
M is simply the 6x6 engine`s rigid mass matrix; is the system`s 6x6 complex stiffness matrix; and C is the 6x6 viscous
damping matrix which is present only when dampers are also installed between
the engine and its base. The righthand side of Eq. 1
represents a 6x1 vector of harmonic forces and their resulting moments
where ω is the forcing frequency. These forces are normally due to
the rotating unbalance.
The majority of mounts used in engine mounting applications are a rubber
bonded to metal or elastomeric construction. Elastomeric materials behave
viscoelastically and for this reason, a complex spring stiffness is used
to model the dynamic behavior of the mount; as:
Where:
K`, K" 
= 
Resistive and dissipative spring rates 
η 
= 
Loss factor 
Another interesting characteristic of elastomeric mounts is that they
generally possess a higher dynamic stiffness than static stiffness. To
account for this, a dynamictostatic spring ratio equal to K`/K" is defined.
A mount constructed from a purely elastic material will have a loss factor
of zero and a dynamicto static spring ratio of unity.
Based on Eq. 2, the stiffness matrix of an elastic
or viscoelastic mounting in its local coordinate system can be written
as:
In this study, in spite of the existence of dampers in the Eq.
1, they don`t account for the analytic modeling, because the damping
is exerted in complex term of stiffness (Ashrafiuon and Nataraj, 1992).
Therefore Eq. 1 can be rewritten as:
The mass matrix M is shown:
Here, m is the mass of engine and gearbox and I_{ii} is tensor
elements of inertia.
DYNAMIC ANALYSIS OF ENGINE MOUNT
Vibration is a characteristic of all machinery such as IC engines
which have rotating and reciprocating components. It is brought by gas
or inertia forces causing stretch in elastic materials, thus energy stored
in the elastic deformation strain energy. The dissipation of this energy
and the return to equilibrium is the source of vibration. The way it is
stored and/or return to equilibrium is dependent on the characteristics
of the engine mount system.
Calculation of the harmonic forces on the cylinder block Gas
pressure forces: The gas pressure forces exerted on the piston area
are replaced with one force acting along the cylinder axis and applied
to the piston pin axis in order to make the dynamic analysis easier. It
is determined for each angle of crank rotation φ. The piston pressure
force is:

Fig. 3. 
Combustion pulse in a fourcylinder engine 
where, F_{g} is piston pressure force (MN), A_{p} is
piston area (m^{2}), p_{g} and p_{o} are gas pressure
at any moment of time and atmospheric pressure (MPa), respectively (Kolchin
and Demidov, 1980).
The gas pressure due to the fuel combustion is in such a way that it
is maximized in top dead center of cylinder and decreases by moving the
piston downward. In this study, gas pressure curve is assumed to be a
sawtooth. This form can simulate the actual curve of pressure with respect
to the crank shaft angle. The combustion pulse in a fourcylinder engine
is shown in Fig. 3.
In Fig. 3, the function p_{g} (φ) can
be written as:
where, p_{g} is the maximum mean effective pressure in the combustion
chamber. Because this function is a periodic function with the period
of ,
it can be represented by expansion of this function by writing its FS
as:
In a fourcylinder engine the primary forces and moments are balanced
whereas the secondary ones are unbalanced. Thus the inertia force for
a fourcylinder engine is
where, m_{j} is system of concentrated masses at piston pin (kg),
R is crank radius (m), ω is angular velocity (rad sec^{1}),
λ is crank radius to connecting rod length ratio and φ crank
angle.
Table 1: 
Phaser 135Ti properties of crank mechanism 

The mass m_{j} in Eq. 9 is obtained
from m_{j} = m_{p}+m_{crp}, where, m_{p} is piston mass and m_{crp} = 0.275 m_{cr} in most existing automobile engines,
where, m_{cr} is crankshaft mass.
The total forces acting in a crank mechanism are determined algebraically
by adding up the gas pressure forces to the forces of reciprocating masses.
According to the properties of crank mechanism of Perkins engine, phaser
135Ti, which is shown in Table 1 (Mireei, 2004) and
Eq. 8 and 9, the total force on
crank mechanism can be written as:
Where:
F_{0} = 5352.8 N, F_{1} = 0.1845 ω^{2}N,
F_{2} = 3409.4 N, F_{3} = 1704.7 N, F_{4} = 1136.5
N, ω_{1} = 2ω, ω_{2} = 4ω, ω_{3}
= 8ω, ω_{4} = 12ω
Modal analysis of Budsan truck engine mount: Generally, for determination
of natural frequencies in the modal analysis, the equation of motion of the
system should be solved under homogonous condition, i.e., the solving Eq.
1 with f = 0. To find the natural frequencies of the Budsan track engine
mount, first, the mass and stiffness matrices should be determined using Eq.
3 and 5 and then the Eq. 1 must be solved. Table 2 shows
the mass and moment of inertia of engine and gearbox for phaser 135Ti engine
(Anonymous, 1995).
Using Table 2, mass matrix of equation of motion may
be written as:
Table 2: 
Mass and moment of inertia of engine and gearbox for
Perkins engine (phaser 135Ti) 

Table 3: 
Static stiffness values in three principal axes 

To find the stiffness matrix, static stiffness at engine and gearbox
loading is determined. Although dynamic to static stiffness ratio may
be achieved from manufacturing of engine mounts that is equal to 1.2.
Static stiffness values of rear and front engine mounts in three principal
axes are shown in Table 3. As mentioned, dissipative
or hystersis coefficient of engine mounts must be considered in Eq.
2 to calculate the complex stiffness. This value actually is the difference
between loading and unloading curves and is 0.335 for front and 0.242
for rear engine mounts.
Dynamic stiffness of engine mounts in each direction can be determined
by multiplying the static stiffness by the dynamic to static stiffness
ratio. The total dynamic stiffnesses in three principal directions are:
Using the directional stiffness, geometric properties of engine and gearbox
and moment about each axis, we can write the rotational stiffness about
three principal axes. The rotational stiffness about x, y and z axes are:
Therefore is:
Using MATLAB software Ver. 6.5 and writing the mass and stiffness matrices
we can get the natural frequencies of the system.
Harmonic analysis of Budsan truck engine mount system: Generally,
the displacement of the system can be found by using the mass, damping
and stiffness matrices. Hence, for the system with complex stiffness,
this displacement can be written by solving the Eq. 4
as:
where, U is displacement vector, f is disturbance force vector and (Eq.
2) is complex stiffness matrix. We use the MATLAB software to determine
the system response to the frequency changes from 0 to 200 Hz.
Harmonic analysis of the system using finite element method: In
this study, the engine and gearbox is modeled by MPC184 element which
is a rigid element in ANSYS software. This element is placed between center
of gravity of engine and gearbox and engine mount locations and between
engine mounts too. Engine mounts is simulated by Voigt model using spring
and damper elements. The spring and damper elements that is used here,
is called COMBIN14 element in ANSYS software. The stiffness and damping
values are obtained from Budsan engine mount manufacturer. Mass21 element
in ANSYS is used to model the mass of the system. Finite element model
of Budsan engine mount system is shown in Fig. 4.
Since the total forces on the crank mechanism of engine is a force in
y direction, the spring and damper elements are in the y direction. COMBIN14
element is an element with two nodes which define the direction of spring
and damper in the model. This element is shown in Fig. 5.
To apply the boundary conditions to the model, the end of spring and
damper elements assumed to be fixed (because of rigid chassis assumption)
and the connection nodes between spring and damper elements and rigid
elements are constrained to the y displacement.
A program is written in ANSYS in such a way that the amplitude of harmonic
force is requested from user and then it is applied to the center of mass (Fig.
6).

Fig. 4: 
Finite element model of Budsan engine mount system in ANSYS 

Fig. 5: 
COMBIN14 element (Anonymous, 2003) 

Fig. 6: 
Applying the boundary conditions and harmonic force 
RESULTS AND DISCUSSION
Dynamic forces on a crank mechanism: Since the minimum and
maximum of engine speed in this study are 750 and 2600 rpm, respectively,
the harmonic frequency range is 12.5 to 43 Hz. Figure 7
shows the harmonic force versus the crank angle at four frequencies 12,
24, 36 and 48 Hz.
As can be seen from Fig. 7, because of increasing the
effect of reciprocating part, the harmonic force on cylinder block increases
by increasing the engine speed.
Natural frequencies of system: Modal analysis of system was done
to determine the natural frequency of the system. Table
4 shows the results obtained for the natural frequency of Budsan truck
engine mount system.
Displacement of the system due to the harmonic forces: The harmonic
forces were applied to the model to determine the displacement and response
of the system. Figure 8 shows the displacement of the
system versus the crank angle at four frequencies of 12, 24, 36 and 48
Hz. Figure 9 shows the system response to the frequency.
It is seen from Fig. 8 that the total displacement
increases by increasing the frequency (from 12 to 48 Hz). In Fig.
9, at frequency of about 59 Hz, the displacement amplitude increase
abruptly. As indicated in Table 4, this frequency is
the natural frequency in z direction. The control of displacement in natural
frequency is due to the complex term existence in stiffness matrix that
expresses the hystersis damping in elastomeric engine mounts (Ashrafiuon
and Nataraj, 1992; Ashrafiuon, 1993; Swanson et al., 1993).
Harmonic analysis results using FEM: Three terms of harmonic force
of Eq. 8 including, which
have sinusoidal waveforms were applied to the finite element model. Figure
10 shows the result of finite element model response to the harmonic
force
As can be seen in Fig. 10, natural frequency value
in FEM is 58.91 Hz whereas this value in analytical model is 59.07.
Table 4: 
Natural frequencies of system (Hz) 


Fig. 7: 
Total dynamic force in three rotational speed (a) 720,
(b) 1440, (c) 2160 and (d) 2880 rpm 

Fig. 8: 
Displacement of system versus the crank angle in frequencies
(a) 12, (b) 24, (c) 36 and (d) 48 Hz 

Fig. 9: 
Displacement versus frequency curve 
Table 5 shows FEM as well as analytic results
of system response. The comparison between analytical and numerical solutions
shows the fitness of results to state the natural frequency and displacement
of system in each analysis and verifies the analytical model. As mentioned,
analytical model was based on complex stiffness and zero linear damping
whereas the spring and damper model was used in the FEM.

Fig. 10: 
System response to the harmonic force 
Optimization: The best stiffness and hystersis coefficients in
the z direction should be determined in such a way that the natural frequency
of system be far rom the harmonic frequency range (Ashrafiuon, 1993; Shoureshi et al., 1986).

Fig. 11: 
Optimized engine mount system, a) stiffness 5500 kN
and hystersis 0.285 and b) stiffness 5500 kN and hystersis 0.5 
Thus to design optimization of a system
it is necessary to note: (1) Natural frequency in direction of applied
force and (2) Harmonic frequency range. According the modal analysis results,
natural frequency of the system in z direction is 59 Hz. Since the engine
speed is between 750 to 2600 rpm (12.5 to 43 Hz) and inertia force is
,
thus harmonic frequency range is between 25 to 85 Hz and resonance frequency
is:
As shown in Fig. 11a, by increasing the stiffness
from 2943 to 5500 kN, the natural frequency increases from 59 to 90 Hz
and the total displacement reduces from 7 to 3.5 mm. The shaded area indicates
the working range of engine. The comparison between Fig.
9 and 11a illustrate that by increasing the stiffness
from 2943 to 5500 kN, maximum displacement decreases from 7 to 3.5 mm.
However, by increasing the hystersis damping in z direction from 0.285
to 0.5 and stiffness to 5500 kN, the total displacement could be reduced from 3.5 to 2 mm as shown in Fig.
11b.
Table 5: 
Comparison between analytical and numerical results 

Thus, despite of limitations and trial and error method we can
conclude that the optimized stiffness and hystersis damping for the Budsan
engine mount system are 5500 and 0.5 kN, respectively. With these values,
the total displacement could be reduced from 7 to 2 mm.
CONCLUSION
It is concluded that analytical engine mount model based on complex
stiffness and zero linear damping has a good fitness in results by finite
element model based on spring and damper model. After verification of
analytical model by finite element model we applied it to optimize the
Budsan truck engine mount system. In this way, by increasing the stiffness
from 2943 to 5500 kN and hystersis damping in z direction from 0.285 to
0.5, it is possible to reduce the total displacement from 7 to 2 mm.
ACKNOWLEDGMENT
The authors would like to thank Research Deputy of University of
Tehran for its financial support.