Optimal Seat Suspension Design Using Genetic Algorithms

Optimal Seat Suspension Design Using Genetic Algorithms
Journal of Mechanics Engineering and Automation 1 (2011) 44-52
Optimal Seat Suspension Design Using Genetic
Algorithms
Wael Abbas1, Ossama B. Abouelatta2, Magdy El-Azab3, Mamdouh El-Saidy3 and Adel A. Megahed 4
1. Engineering Physics and Mathematics Department, Faculty of Engineering, (Mataria), Helwan University, Cairo11718, Egypt
2. Production Engineering and Mechanical Design Department, Faculty of Engineering, Mansoura University, Mansoura 35516, Egypt
3. Mathematics and Engineering Physics Department, Faculty of Engineering, Mansoura University, Mansoura 35516, Egypt
4. Mathematics and Engineering Physics Department, Faculty of Engineering, Cairo University, Cairo 12211, Egypt
Received: March 29, 2011 / Accepted: April 28, 2011 / Published: June 25, 2011.
Abstract: The linear seat suspension is considered due to the low cost consideration therefore, the optimal linear seat suspension
design method can be used for this purpose. In this paper, the design of a passive vehicle seat suspension system was handled in the
framework of linear optimization. The variance of the dynamic load resulting from the vibrating vehicle operating at a constant speed
was used as the performance measure of a suspension system. Using 4-DOF human body model developed by Abbas et al., with linear
seat suspension and coupled with half car model. A genetic algorithm is applied to solve the linear optimization problem. The optimal
design parameters of the seat suspension systems obtained are kse = 3 012.5 N/m and cse = 1 210.4 N.s/m, respectively.
Key words: Biodynamic response, seated human models, simulation, genetic algorithms.
1. Introduction
In the last fifty years, many people become more
concerned about the ride quality of vehicle which is
directly related to driver fatigue, discomfort, and safety.
As traveling increases, the driver is more exposed to
vibration mostly originating from the interaction
between the road and vehicle. Vibration will make
them feel discomfort and fatigue sometimes along with
injury. It is important to know how the vibration is
transmitted through the human body before we try to
Ossama B. Abouelatta, associate professor, research fields:
surface roughness and its application in industry, computer
aided measurements and simulation, biomedical engineering.
Magdy El-Azab, professor, research fields: engineering
mathematics, numerical analysis.
Mamdouh El-Saidy, associate professor, research fields:
engineering mathematics, numerical analysis.
Adel A. Megahed, professor, research fields: applied
mathematics, computational fluid mechanics.
Corresponding author: Wael Abbas, Ph.D., research fields:
applied mathematics, vehicle suspension and dynamics, genetic
algorithms, simulator technology, system modeling and
analysis. E-mail: wael_abass@hotmail.com.
manage it.
Many researchers discussed a various biodynamic
models that have been developed to depict human
motion from a single-DOF to multi-DOF models.
These models can be divided as distributed (finite
element) models, lumped parameter models and
multibody models. The distributed model treats the
spine as a layered structure of rigid elements,
representing the vertebral bodies, and deformable
elements representing the intervertebral discs by the
finite element method [1-2]. Multibody human models
are made of several rigid bodies interconnected by pin
(two-dimensional)
or
ball
and
socket
(three-dimensional) joints, and can be further separated
into kinetic and kinematic models. The kinetics is
interested in the study of forces associated with motion,
while kinematics is a study of the description of motion,
including considerations of space and time, and are
often used in the study of human exercise and injury
assessment in a vehicle crash.
Optimal Seat Suspension Design Using Genetic Algorithms
The lumped parameter models consider the human
body as several rigid bodies and spring-dampers. This
type of model is simple to analyze and easy to validate
with experiments. However, the disadvantage is the
limitation to one-directional analysis. These models
can be summarized as: 1-DOF model [3], 2-DOF
human body [4], 3-DOF analytical model [5], 4-DOF
human model [6-8], 6-DOF nonlinear model [9], and
7-DOF model [10]. A complete study on
lumped-parameter models for seated human under
vertical vibration excitation has been carried out by
Liang and Chiang [11], based on an analytical study
and experimental validation. So, it is known that the
lumped parameter model is probably one of the most
popular analytical methods in the study of biodynamic
responses of seated human subjects, though it is limited
to one-directional analysis. However, vertical vibration
exposure of the driver is our main concern.
Some lumped-parameter models were further
modified to represent seated human vehicle’s driver
with seat and integrated with a vehicle model to assess
the biodynamic responses of seated human body
expose to vertical vibrations in driving conditions
[12-14]. On the other hand, a genetic algorithms (GA)
method increases the probability of finding the global
optimal solution and avoids convergence to a local
minimum which is a drawback of gradient-based
methods [15-17]. Therefore, genetic algorithms
optimization is used to determine both the active
control and passive mechanical parameters of a vehicle
suspension system and to minimize the extreme
acceleration of the passenger’s seat, subjected to
constraints representing the required road-holding
ability and suspension working space.
In this paper presents 4-DOF human body with
linear seat suspension and coupled with half car model.
With this model, the genetic algorithm is applied to
search for the optimal parameters of the seat in order to
minimize seat suspension deflection and driver’s body
acceleration to achieve the best comfort of the driver.
The paper is organized as follows: section 2 discusses
the mathematical model formulations; section 3
45
introduces the optimal linear seat suspension design;
section 4 presents results and discussions; section 5
gives conclusions and recommendations.
2. Mathematical Model Formulations
2.1 Proposel Model
This section is devoted to the mathematical modeling
of proposed model, including the biodynamic lumped
human linear seat model coupled with half model of
ground vehicles as illustrated in Fig. 1.
The human-body, has a 4-DOF that proposed by
Abbas et al. [18]. In this model, the seated human
body was constructed with four separate mass
segments interconnected by five sets of springs and
dampers, with a total human mass of 60.67 kg. The
four masses represent the following body segments:
head and neck (m1), upper torso (m2), lower torso (m3),
and thighs and pelvis (m4). The arms and legs are
combined with the upper torso and thigh, respectively.
The stiffness and damping properties of thighs and
pelvis are (k5) and (c5), the lower torso are (k4) and (c4),
upper torso are (k2, k3) and (c2, c3), and head are (k1)
and (c1). The schematic of the model is shown in Fig.
2a, and biomechanical parameters of the model are
listed in Table 1.
Fig. 1 Schematic diagram of biodynamic lumped human
linear seat models coupled with half car model.
46
Table 1
model..
Optimal Seat Suspension Design Using Genetic Algorithms
The biomechanical parameters of the Abbas
Mass
(kg)
m1 = 4.17
Damping coefficient
(N.s/m)
C1 = 310
Spring constant
(N/m)
k1 = 166,990
m2 = 15
C2 = 200
k2 = 10,000
m3 = 5.5
C3 = 909.1
k3 = 144,000
m4 = 36
C4 = 330
k4 = 20,000
-
C5 = 2,475
k5 = 49,340
7
8
A half car model with four degrees of freedom is
shown in Fig. 1, taking into consideration the pitch
motion of the vehicle's body. The degrees of freedom
are; vertical body displacement xb, vehicle body pitch
angle θ, front wheel displacement xwf and rear wheel
displacement xwr. The front wheel of the vehicle is
represented by the mass mwf, the damping coefficient
Ctf and the spring coefficient Ktf. Similarly the rear
wheel is represented by the mass mwr, the damping
coefficient Ctr and the spring coefficient Ktr. The
suspensions of the front and rear wheels are described
by the damper's coefficients Csf and Csr and the spring's
coefficients Ksf and Ksr, respectively. The mass mb and
the inertia I represent the vehicle body sprung mass.
The location of the centre of gravity is given by L1 and
L2. Typical design parameters for the half car and seat
suspension are listed in Table 2.
The equations of motion of the resulting coupled
model can be put in the classical form:
1
–
2
9
2.2 Input Profile Excitations
In this work, the sinusoidal road profiles excitation is
adopted to evaluate the proposed system. In simple
analytical ride studies the sinusoidal road is considered
to have only one spatial frequency of bumps at a time.
The sinusoidal road equations are listed below:
sin .
sin
.
where ω the radian frequency of the road, is
Mathematical model of road profile can be derived
as the follows: vehicle with wheelbase Lw passing over
each hump with speed V will have front ground
displacement xof. The rear ground xor follows the same
track as the front with a given time delay , (wheelbase
correlation) and that is equal to the wheelbase divided
by vehicle speed
. In this study, assuming that
the vehicle model travels with the constant velocity
⁄
of20
5.55 ⁄ , HB (0.035 m) is the hump
height, and LB (1 m) is the width of the hump.
3. Optimal Linear Seat Suspension Design
3
4
5
6
3.1 Numerical Simulations
The displacement, velocity, and acceleration for the
model in terms time domain are obtained by solving
equations of motion represented by (1-9). Using
MATLAB software ver. 7.10 (R2010a), dynamic
system simulation software, Simulink. The initial
conditions are assumed at equilibrium position. In this
assumption, the driver is seated but the input excitation
Journal of Mechanics Engineering and Automation 1 (2011) 44-52
Table 2 System parameters of the half car and seat suspension model.
Parameter
Symbol
Value
Front and rear tire stiffness (N/m).
Ktf ,Ktr
155,900
Front and rear axle masses (Kg).
mwf ,mwr
28.58, 54.3
Linear front and rear suspension damping Coefficients (N.s/m).
Csf, Csr
1,828
Front and rear tire damping Coefficients (N.s/m).
Ctf, Ctr
0
Front and rear suspension stiffness (N/m).
Ksf ,Ksr
15,000
Distance between the C.G and front axle (m).
L1
1.098
Distance between the C.G and rear axle (m).
L2
1.468
Distance between the C.G and seat (m).
a
0.7
Body mass "sprung mass" (Kg).
mb
505.1
Body mass moment of inertia (Kg.m2).
I
651
Seat mass (Kg).
mse
35
Seat damping Coefficients (N.s/m).
Cse
150
Seat suspension stiffness (N/m).
Kse
15,000
has not been provided to the seat. Therefore, the initial
velocity and displacement for each mass is equal to zero.
When the spring is free, no force is generated. At
equilibrium position, the spring is compressed by the
weight of the human and seat. In this work the
maximum static displacement is selected to de 0.157m
(this value depend on the cab space restriction for the
vehicle). Therefore, the initial stiffness of the spring
can be determined:
.
.
3.2 Optimization Via Genetic Algorithms
In this section, optimization software based on
stochastic techniques search methods, Genetic
algorithms (GAs), is employed to search for the
optimal linear parameters of the seat to achieve the best
comfort of the driver. The upper boundaries of seat
suspension parameters are selected based on previous
studies. Table 3 shows the genetic algorithms
parameters and its selected values.
(1) Objective function
Since the health of the driver is as important as the
stability of the car, the desired objective is proposed as
the minimization of a multi-objective function formed
by the combination of not only seat suspension
Table 3 Genetic algorithm parameters.
GA parameters
Population size
No of generations
Fitness scaling
Crossover technique
Probability of crossover
Mutation technique
Generation gap
Lower boundary
Upper boundary
Objective function accuracy
Value
50
200
Rank
Heuristic
0.8
Uniform
0.9
130-3,000
1,600-200,000
1
working space (seat suspension deflection ‘ssws’) but
also the head acceleration
, and seat mass
acceleration (
.
This study used the classical weighted sum
approaches to solve a multi-objective optimization
problem as equation
.
.
.
where, w1, w2 and w3 are weighting factors to
emphasize the relative importance of the terms. Table 4
shows weighting factors used in excitation inputs.
(2) Optimization procedure
First, the bounds of the design variables and
initialize suspension design variables ks and cs. Then ks
and cs are passed into the proposal model to solve for
the dynamic response (displacement, velocity, and
accelerations values) of the system. The population is
then coded into chromosomes, a binary representation
of a solution (consisting of the components of the
48
Optimal Seat Suspension Design Using Genetic Algorithms
Table 4 Weighting factors used in excitation inputs.
Weight
w1
w2
w3
Excitation input
0.5
0.3
0.2
decision variables known as genes in the genetic
algorithm). The whole population of chromosomes
represents a generation. An evaluation function rates
solutions in terms of their fitness. Here, fitness is a
numerical value describing the probability for a
solution (genome) to survive and reproduce. Only a
portion of the population (survivors or solutions with
higher fitness values) is selected for creating a new
population (offspring production). This new population
is created by using a crossover operator.
Crossover is a procedure for exchanging pieces of
chromosome data with one another. Crossover allows
genes that generate good fitness to be preserved and
enlarged in a new generation of the population.
Mutation is a genetic operator and it randomly flips the
bits of an offspring’s genotype. This is equivalent to
perturbing the mated population stochastically.
Mutation prevents the population from homogenizing
in a particular set of genes such that any gene in a
generation has a certain probability (determined by the
mutation rate) of being mutated in future generations.
The new population is being mixed up to bring some
new information into this set of genes, and this needs to
happen in a well-balanced way.
Once the new generation is created, the
aforementioned steps are repeated until some
convergence criteria are satisfied, such as running time
or fitness. The overall technique is summarized in the
flowchart given in Fig. 2.
4. Results and Discussion
The optimal linear seat parameters for the present
model were determined by genetic algorithms method,
and compared with current passive parameters are
tabulated in Table 5.
Fig. 3 shows the displacement histories obtained for
head, upper torso, lower torso, pelvis, and seat
respectively. Fig. 4 present the acceleration histories
Fig. 2 Design process using GA.
Table 5 The design results from the GA program.
Seat suspension setting
Passive
GA optimization
Kse (N/m)
15 000
3 012.5
Cse (N.s/m)
150
1 210.4
obtained at the coupled human seat with half car model.
The figure presents four human components, head,
upper torso, lower torso, and pelvis, and seat
acceleration respectively. The obtained results by
genetic algorithms method were compared with
passive model.
The results obtained by GA were compared to those
passive results in terms of RMS values is given in
Table 6. From the table, the percentage improvement
for head acceleration is 37.09%, for upper torso
acceleration is 37.22%, for lower torso acceleration is
37.29%, and for pelvic acceleration is 37.09. On the
other hand the percentage improvement for seat
acceleration is 36.33%.
Table 7 indicates that the reduction of the human’s
vertical acceleration and displacement peak. It can be
49
Optimal Seat Suspension Design Using Genetic Algorithms
Road input (m)
0.04
Lower torso displacement (m)
Front road input
Rear road input
0.02
0
-0.02
-0.04
0
1
2
3
Time (sec)
(a)
4
0.1
0
-0.1
-0.2
-0.3
1
Passive
Genetic
0
-0.1
-0.2
-0.3
1
2
3
Time (sec)
4
0.2
2
3
Time (sec)
(d)
4
5
0
-0.1
-0.2
-0.3
1
0.3
Passive
Genetic
0.1
0
-0.1
-0.2
-0.3
0.2
seat displacement (m)
Upper torso displacem ent (m )
5
Passive
Genetic
(c)
-0.4
0
4
0.1
-0.4
0
5
0.3
0.2
2
3
Time (sec)
(b)
Pelvic displacement (m)
Head displacement (m)
Passive
Genetic
0.3
0.1
-0.4
0
0.2
-0.4
0
5
0.3
0.2
0.3
Passive
Genetic
0.1
0
-0.1
-0.2
-0.3
-0.4
2
3
4
5
0
1
2
3
4
Time (sec)
Time (sec)
(e)
(f)
Fig. 3 Displacement histories obtained (a) road input, (b) head, (c) upper torso, (d) lower torso, (e) pelvic and (f) seat.
1
observed that the reduction of the human’s vertical peak
acceleration is approximately 54.76-60.46% in case of
GA suspension as compared with passive suspension.
The reduction of the human’s vertical displacement
peak is approximately 31.21-31.236% in case of GA
5
suspension as compared with passive suspension. Also,
the reduction of the seat vertical peak acceleration is
57.02 % and the reduction of the seat vertical
displacement peak is 31.29 % of GA suspension as
compared with passive suspension.
50
Optimal Seat Suspension Design Using Genetic Algorithms
20
Pelvic acceleration (m/s2)
Passive
Genetic
10
0
-10
1
2
3
Time (sec)
(a)
4
10
0
-10
1
2
3
Time (sec)
(b)
4
Passive
Genetic
20
10
0
-10
1
2
3
Time (sec)
4
Passive
Genetic
15
10
5
0
-5
-10
-15
0
5
1
2
3
Time (sec)
Lower torso acceleration (m/s2)
(c)
4
(d)
20
Passive
Genetic
10
0
-10
-20
0
1
2
3
Time (sec)
(e)
4
5
Fig. 4 Acceleration histories obtained: (a) head, (b) upper torso, (c) lower torso, (d) pelvic, and (e) seat.
Table 6
RMS percentage improvement results for half car model.
Passive
GA
% Improvement
5
20
30
-20
0
Passive
Genetic
-20
0
5
2
Upper torso acceleration (m /s )
-20
0
seat acceleration (m /s2 )
Head acceleration (m/s2)
20
RMS for
head acc.
8.0766
5.08
37.09
RMS for upper torso RMS for lower torso RMS for
acc.
acc.
pelvic acc.
8.0671
8.1102
7.9901
5.0643
37.22
5.0857
37.29
5.0258
37.09
RMS for
seat acc.
7.4788
4.7613
36.33
5
Optimal Seat Suspension Design Using Genetic Algorithms
Displacement
(m)
Acceleration
(m/s2)
Table 7
51
Reduction in peak values for half car model.
Head
Upper torso
Lower torso
Pelvic
Seat
Head
Upper torso
Lower torso
Pelvic
Seat
Passive
19.7899
20.3072
19.6788
19.2235
17.3471
0.2672
0.2670
0.2680
0.2654
0.2528
Maximum overshoot
GA
8.9513
8.0281
7.9230
7.8414
7.4546
0.1838
0.1836
0.1843
0.1825
0.1737
5. Conclusions and Recommendations
This paper the genetic algorithm is applied for
obtaining the optimal linear and nonlinear parameters
of the seat in order to minimize seat suspension
deflection and driver’s body acceleration to achieve the
best comfort of the driver. It can be concluded that:
(1) Optimal linear seat suspension by using genetic
algorithms has successfully managed improving for all
the dynamic performance parameters. Genetic
algorithms explore the entire space to search for the
optimal solutions from a population of solutions to
another population of solutions, rather than from one
solution to another, this characteristic makes GAs
uniquely suited to multi-objective optimization;
(2) The results of optimal linear seat suspension
characteristics show that, to obtain the best vibration
isolation, the stiffness of the spring near to the lower
boundary;
(3) The results and the plots indicate that optimal
linear seat suspension system is less oscillatory, and
have lower values of maximum over shoots than
passive suspension system. This is directly related to
driver fatigue, discomfort, and safety;
(4) The optimal linear seat suspension has limitation
on improving the vibration isolation.
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