Engineering, 2010, 2, 710-719
doi:10.4236/eng.2010.29092 Published Online September 2010 (http://www.SciRP.org/journal/eng)
Optimization of Biodynamic Seated Human Models Using
Genetic Algorithms
Wael Abbas1*, Ossama B. Abouelatta2, Magdi El-Azab3, Mamdouh Elsaidy4, Adel A. Megahed5
1Engineering Physics and Mathematics Department, Faculty of Engineering (Mataria), Helwan University,
Cairo, Egypt
2Production Engineering and Mechanical Design Department, Faculty of Engineering, Mansoura University,
Mansoura, Egypt
3,4Mathematics and Engineering Physics Department, Faculty of Engineering, Mansoura University,
Mansoura, Egypt
5Mathematics and Engineering Physics Department, Faculty of Engineering, Cairo University, Cairo, Egypt
Received May 16, 2010; revised July 21, 2010; accepted August 4, 2010
Abstract
Many biodynamic models have been derived using trial and error curve-fitting technique, such that the error
between the computed and measured biodynamic response functions is minimum. This study developed a
biomechanical model of the human body in a sitting posture without backrest for evaluating the vibration
transmissibility and dynamic response to vertical vibration direction. In describing the human body motion, a
three biomechanical models are discussed (two models are 4-DOF and one model 7-DOF). Optimization
software based on stochastic techniques search methods, Genetic Algorithms (GAs), is employed to deter-
mine the human model parameters imposing some limit constraints on the model parameters. In addition, an
objective function is formulated comprising the sum of errors between the computed and actual values (ex-
perimental data). The studied functions are the driving-point mechanical impedance, apparent mass and seat-
to-head transmissibility functions. The optimization process increased the average goodness of fit and the
results of studied functions became much closer to the target values (Experimental data). From the optimized
model, the resonant frequencies of the driver parts computed on the basis of biodynamic response functions
are found to be within close bounds to that expected for the human body.
Keywords: Biodynamic Response, Seated Human models, Simulation, Genetic algorithms
1. Introduction
Recently, many people have focused their attention on
the ride quality of vehicle which is directly related to
driver fatigue, discomfort, and safety. As traveling in-
creases, the driver is more exposed to vibration mostly
originating from the interaction between the road and
vehicle. Whole-body vibration occurs in transportation
and when near heavy machinery [1]. The vibrations
cause the operator’s whole body to vibrate, as opposed to
just one part of their body, says their hand or foot. Harm-
ful effects of whole-body vibration are experienced when
the exposure time is longer than the recommended stan-
dard set by ISO 2631-1 [2].
Biodynamic responses of seated human occupant ex-
posed to vibration have been widely characterized to
define frequency-weightings for assessment of exposure,
to identify human sensitivity and perception of vibration,
and to develop seated body models [3]. The biodynamic
response of the human body exposed to vibration have
been invariably characterized through measurement of
force motion relationship at the point of entry of vibra-
tion “To-the-body response function”, and transmission
of vibration to different body segments “Through-the-
body response function”. Considering that the human
body is a complex biological system, the “To-the-body”
response function is conveniently characterized through
non-invasive measurements at the driving point alone.
The vast majority of the reported studies on biodynamic
response to whole-body vibration have considered vibra-
tion along the vertical axis alone.
The reported studies on biodynamic responses under

W. ABBAS ET AL.
711
vertical vibration are thoroughly reviewed, specifically
their response characteristics, experimental conditions,
and the measured data. The biodynamic response char-
acteristics reported in terms of either the driving-point me-
chanical impedance (DPMI) or apparent mass (APMS),
and the seat-to-head transmissibility (STHT) are classi-
fied under different experimental conditions used in the
study.
In early studies, various biodynamic models have been
developed to depict human motion from single-DOF to
multi-DOF models. These models can be divided as dis-
tributed (finite element) models, lumped parameter mod-
els and multi-body models. The distributed model treats
the spine as a layered structure of rigid elements, repre-
senting the vertebral bodies, and deformable elements
representing the intervertebral discs by the finite element
method. Multi-body 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 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. Coermann [4],
measured the driving-point impedance of the human
body and suggested 1-DOF model. Suggs et al. [5] de-
veloped a 2-DOF human body. It was modeled as a
damped spring-mass system to build a standardized ve-
hicle seat testing procedure. A 3-DOF analytical model
for a tractor seat suspension system is presented by Te-
wari et al. [6]. It was observed that the model could be
employed as a tool in selection of optimal suspension
parameters for any other type of vehicles. Boileau et al.
[7] used an optimization procedure to establish a 4-DOF
human model based on test data. In addition, Zong and
Lam [8] validated a 4-DOF nonlinear model originating
from Liu et al. [9].
Furthermore, Muksian and Nash [10] presented a 6-
DOF nonlinear model dedicated to the analysis of vibra-
tion imposed on a seated human. This model was modi-
fied by Patil et al. [11], who suggested a 7-DOF model.
This model was further incorporated with a tractor model
to evaluate vibration responses of an occupant-tractor
system. A complete study on lumped-parameter models
for seated human under vertical vibration excitation has
been carried out by Liang and Chiang [12], based on
analytical study and experimental validation.
On the other hand, GA optimization is used by Bau-
mal et al. [13] to determine both active control and pas-
sive mechanical parameters of a vehicle suspension sys-
tem, to minimize the extreme acceleration of the passen-
ger’s seat, subjected to constraints representing the re-
quired road holding ability and suspension working
space. The GA is used to solve the problem and results
were compared to those obtained by simulated annealing
technique and found to yields similar performance mea-
sures.
It is clear 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 con-
cern. Therefore, this paper carries out a thorough survey
of literature on the lumped-parameter models for seated
human subjects exposed to vertical vibration.
This work aims to develop a biomechanical model of
the human body in a sitting posture without backrest for
evaluating the vibration transmissibility and dynamic
response to vertical vibration direction.
2. Biodynamic Response of the Human Body
The biodynamic response of a seated human body ex-
posed to whole-body vibration can be broadly catego-
rized into two types. The first category “To-the-body”
force motion interrelation as a function of frequency at
the human-seat interface, expressed as the driving-point
mechanical impedance or the apparent mass. The second
category “Through-the-body” response function, gener-
ally termed as seat-to-head transmissibility for the seated
occupant.
The DPMI relates the driving force and resulting ve-
locity response at the driving point (the seat-buttocks
interface), and is given by [3]:
() ()
() () ()
F
jFj
Zj Vj
X
j





 (1)
where, ()
Z
j

is the complex DPMI, ()
F
j

and
()Vj

or ()
X
j

 are the driving force and response
velocity at the driving point, respectively.

is the an-
gular frequency in rad/s , and j = 1 is the complex
phasor.
In a similar manner, the apparent mass response re-
lates the driving force to the resulting acceleration re-
sponse, and is given by [14]:
()
() ()
F
j
APMS jaj



 (2)
where, ()aj

is the acceleration response at the driv-
ing point. The magnitude of APMS offers a simple
physical interpretation as it is equal to the static mass of
the human body supported by the seat at very low fre-
quencies, when the human body resembles that of a rigid

W. ABBAS ET AL.
712
mass. The above two functions are frequently used in-
terchangeably, due to their direct relationship that given
by:
()
()DPMI j APMS jj

  (3)
The biodynamic response characteristics of seated oc-
cupants exposed to whole body vibration can also be
expressed in terms of seat-to-head transmissibility, which
is termed as “through-the-body” response function. Un-
like the force-motion relationship at the driving-point,
the STHT function describes the transmission of vibra-
tion through the seated body. The STHT response func-
tion is expressed as:
()
() ()
H
aj
Hj aj



 (4)
where, ()
H
j

is the complex STHT, ()
H
aj

is the
response acceleration measured at the head of seated
occupant, and ()aj

is the acceleration response at the
driving point. The above three functions have been
widely used to characterize the biodynamic responses of
the seated human subjects exposed to whole body vibra-
tion.
3. Experimental Data
Many mathematical models on the study of biodynamic
responses of seated human subjects have been published
based on individual test data over the years. However,
significant variation is known to exist between various
data sets. Such variation may be partly attributed to the
differences associated with the methodology, experi-
mental conditions or subject population used by different
investigators.
3.1. Basic Assumptions on Experimental Data
The biodynamic of seated human subjects exposed to
vertical vibration has been widely assessed in terms of
STHT, DPMI, and APMS. The first function refers to the
transmission of motion through the body, while the other
two relate the force and motion at the point of vibration
input to the body. A variety of test data used to charac-
terize these response functions has been established us-
ing widely varied test conditions. This has resulted in
considerable discrepancies among the data. To avoid
these discrepancies, a preliminary conclusion was reach-
ed that any attempt to define generalized values might
not be appropriate unless it could be defined specifically
for a particular application or within a limited and well-
defined range of situations [12].
Data sets satisfying the following requirements are se-
lected for the synthesis of biodynamic characteristics of
the seated human subjects [15-17].
 A human subject is considered to be sitting erect
without backrest support, irrespective of the hands’
position.
 Body masses will be limited within 49-94 kg.
 Feet are supported and vibrated.
 Analysis is constrained to the vertical direction.
 Vibration excitation amplitudes are below 5 m/s2,
with the nature of excitation specified as being si-
nusoidal wave.
 Excitation frequency range is limited to 0.5-20 Hz.
3.2. Experimental Results
While vertical DPMI, APAS, and STHT characteristics
were not measured as part of this study, applicable target
values were defined on the basis of a synthesis of pub-
lished data Boileau [14], Liang et al. [12,17] and Wu [18].
Figure 1 shows upper, lower, and target values of DPMI,
APMS, and STHT magnitude established as target values
within 0.5-20 Hz frequency range, respectively.
4. Biomechanical Modeling
The human body in a sitting posture can be modeled as a
mechanical system that is composed of several rigid
bodies interconnected by springs and dampers. In this
study, three types of biomechanical models are discussed
to describe the vertical response: 4-DOF Wan and
Schimmels model, 4-DOF Boileau and Rakheja model,
and 7-DOF Patil and Palanichamy model as shown in
Figure 2.
4.1. Wan and Schimmels 4-DOF Model
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 [19]. 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
Figure 2(a), and biomechanical parameters of the model
are listed in Table 1.
The equations of motion of the human-body can be
obtained as follows:

W. ABBAS ET AL.
713
111 121 12
221 121 12223
22 332 432 4
332 232 23
434434
444 344343 24
32 4
()()
()()()
()()()
()()
()()
()()()
(
mxc xxkxx
mxc xxk xxcxx
kx xcx xkx x
mxc xxkxx
cxxkx x
mxcx xkxxcxx
kx x
 
  

 

 
 



 

 54 54
)( )( )
se se
cx xkx x












 

(5)
0 24 6 810 1214 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Frequency (Hz)
STHT
:Experimental Upper limit
:Experimental Lower limit
:Experimental Target value
(a)
0 2 4 6810 1214 16 1820
0
500
1000
1500
2000
2500
3000
Frequency (Hz)
DPMI (N s/m)
:Experimental Upper limit
:Experimental Lower limit
:Experimental Target value
(b)
0 24 6 810 12 14 16 18 2
0
0
500
1000
1500
2000
2500
3000
Frequency (Hz)
APMS (kg)
:Experimental Upper limit
:Experimental Lower limit
:Experimental Target value
(c)
Figure 1. Experimental data of (a) STHT; (b) DPMI; and (c)
APMS.
(a) (b)
(c)
Figure 2. Biomechanical models. (a) Wan and Schimmels
4-DOF model; (b) Boileau and Rakheja 4-DOF model; and
(c) Patil and Palanichamy 7-DOF model.
4.2. Boileau and Rakheja 4-DOF Model
The human-body consists of four mass segments inter-
connected by four sets of springs and dampers with a
total mass of 55.2 kg [7], as shown in Figure 2(b). The
four masses represent the following four body segments:
the head and neck (m1), the chest and upper torso (m2),
the lower torso (m3), and the thighs and pelvis in contact
with the seat (m4). The mass due to lower legs and the
feet is not included in this representation, assuming their
negligible contributions to the biodynamic response of
the seated body. The stiffness and damping properties of

W. ABBAS ET AL.
714
thighs and pelvis are (k4) and (c4), the lower torso are (k3)
and (c3), upper torsos are (k2) and (c2), and head are (k1)
and (c1). The biomechanical parameters of the model are
listed in Table 2. The equation of motion of the human
body can be obtained as follows:
111 121 12
221 121 12223
22 3
332 232 23334
33 4
443 243 2444
44
()()
()()()
()
()()()
()
()()()
()
s
e
se
mxc xxkxx
mxcx xkxxcx x
kx x
mxcxxkx xcx x
kx x
mxcx xkx xcx x
kx x
 
 
 
 
 












(6)
4.3. Patil and Palanichamy 7-DOF Model
Based on Muksian’s 6-DOF model, a 7-DOF nonlinear
model was developed by Patil and Palanichamy [11]. In
this model, the human body consists of seven mass seg-
ments interconnected by eight sets of springs and damp-
ers, with total mass of 80 kg. The seven masses represent
the following body segments: head and neck (m1), back
(m2), upper torso (m3), thorax (m4), diaphragm (m5), ab-
domen (m6) and thighs and pelvis (m7). The arms and
legs are combined with the upper torso and thigh, respec-
tively. The stiffness and damping properties of thighs
and pelvis are (k8) and (c8), abdomen are (k6) and (c6),
the diaphragm are (k5) and (c5), the thorax are (k4) and
(c4), the torso are (k2, k3) and (c2, c3), back are (k7) and
(c7), and head are (k1) and (c1). The schematic of the
model is shown in Figure 2(c), and biomechanical pa-
rameters of the model are listed in Table 3.
The equation of motion of the human-body can be
obtained as follows:
111 121 12
221 121 12727
72 722 322 3
332 232 23334
33 4
443343 344 45
445
()()
()()()
()()()
()()()
()
()()()
(
mxc xxkxx
mxcx xkxxcx x
kx xcx xkx x
mxcxxkxxcxx
kx x
mxc xxk xxcxx
kx x
 
  

  
 
 

 




554 454 455 56
55 6
665 565566 67
66 7
776 676 67727
72 78787
)
()()()
()
()()()
()
()()()
()()( )
sese
mxcxxkxxc xx
kxx
mxcxxk xxc xx
kx x
mxc xxkxxc xx
kx xcx xkx x

 
 
















 







(7)
5. Estimation of Biodynamic Response
Characteristics
There are two methods to solve system equations of mo-
tion; time domain and frequency domain. Frequency
domain solutions are often of more interest than the time
history and can usually be performed more conveniently
than in the time domain. However, for the solutions to be
applicable, the equations must either be linear, or lin-
earized. Frequency domain analysis employs the Fourier
transformation.
The system equations of motion, Equations (5-7), for
the model can be expressed in matrix form as follows:










 
M
xCxKxf
 
 (8)
where,


M
,


C and


K
are n × n mass, damping,
and stiffness matrices, respectively;


f
is the force
vector due to external excitation.
By taking the Fourier transformation of equation (8),
the following matrix form of equation can be obtained:
   
1
2
() ()
X
jKMjCFj

 


 

(9)
where,


()
X
j

and


()
F
j

are the complex Fourier
transformation vectors of


x
and


f
respectively.
ω is the excitation frequency. Vector

()
X
j

contains
complex displacement responses of n mass segments as a
function of ω
(


123
( ),(),( ),......()
n
x
jxjxj xj

  ).


()
F
j

, consists of complex excitation forces on the
mass segments as a function of ω as well.
The driving-point mechanical impedance is defined as
the ratio of driving force (summation of spring and
damping forces between pelvis and seat) to the driving-
point velocity (input velocity of the seat). Accordingly,
DPMI can be represented as follows (e.g. Boileau and
Rakheja, model):
44 4
44
0
()
()
() ()
()
DPMI j
kxjk
cc
jx j






(10)
Seat-to-head transmissibility is defined as the ratio of
output responses (head) to input excitation.
1
0
()
() ()
x
j
STHTj
x
j



 (11)
Apparent mass, can be expressed in terms of DPMI,
Equation (3), as follows (e.g. Boileau and Rakheja
model):
44444
22
0
()
()
()
()()
()
DPMI j
APMS jj
ckxj ck
jxjj






 

(12)

W. ABBAS ET AL.
715
Table 1. The biomechanical parameters of the Wan and
Schimmels model (Before and after optimization).
Damping coefficient
(N.s/m) Spring constant (N/m)
Mass
(kg) Before After Before After
m1 =
4.17 c1 = 250 c1 =310 k1 = 134400
k1 =
166990
m2 = 15 c2 = 200 c2 =200 k2 =
10000
k2 =
10000
m3 =
5.5 c3 = 909.1 c3 = 909.1
k3 =
192000
k3 =
144000
m4 = 36 c4 = 330 c4 =330 k4 =
20000
k4 =
20000
- c
5 = 2475 c5 =2475 k5 = 49340 k5 =49340
6. Development of Models
Many biodynamic models have been derived using trial
and error curve-fitting technique, such that the error be-
tween the computed and measured biodynamic response
functions is minimum. Such curve-fitting methods may
lead to a proper fit over a specific frequency range, but
rarely provide good results when extended over a broad
frequency range. Alternatively, nonlinear programming
based optimization techniques may be effectively em-
ployed to determine the model parameters, involving the
use of a constrained optimization algorithm in conjunc-
tion with well defined biodynamic response function [14].
A constrained objective function may be defined to
minimize the error between the computed and the target
values of specific biodynamic response function over a
specific frequency range.
Optimization software based on stochastic techniques
search methods, Genetic algorithms (GAs), is employed
to determine the human model parameters imposing
some limit constraints on the model parameters. An ob-
jective function is formulated comprising the sum of
errors between the computed and of the driving-point
mechanical impedance, apparent mass and seat-to-head
transmissibility functions. The model thus derived can
provide reasonable correlation with the impedance, ap-
parent mass and transmissibility characteristics.
Starting with an assumed set of model parameters, the
differential equations of motion are solved for unit dis-
placement excitation to drive the driving-point mechani-
cal impedance using Equation (1), apparent mass using
Equation (3), and seat-to-head transmissibility using
Equation (4). At each iteration of search, the sum of
square errors defined by an objective function over the
entire frequency range is examined, and the procedure is
re-initiated with modified parameter values when the
error exceeds that from the previous search. The search
is terminated when the computed error approaches the
minimum value.
6.1. Objective Function
The objective function is selected to comprise the squar-
ed sum errors associated with driving-point mechanical
impedance 1 ()U apparent mass 2 ()Uand seat-to-head
transmissibility functions 3 ()U to minimize the error
between the computed and the target values. This study
used the classical weighted sum approaches to solve a
multi-objective optimization problem as follows:
112 23 3
.( ).().()OBJWUWUWU

 (13)
where,
2
1
1
() (),
n
iti
i
UDPMIjDPMIj 


 



2
2
1
() (),
n
iti
i
UAPMS jAPMSj 






2
3
1
() ()
n
iti
i
USTHT jSTHTj 






In the above equations ()
ti
DPMI j

, ()
ti
APMS j

and ()
ti
STHT j

are the target values of driving-point
mechanical impedance, apparent mass, and seat-to-head
transmissibility, respectively. The target values of DPMI,
APMS, and STHT are illustrated in Figure 1. 1 W, 2 W
and 3 W are weighting factors to emphasize the relative
importance of the terms.
The limit constraints are calculated as ±25% variations
about the biomechanical parameters of the three models.
6.2. Evaluation of Biodynamic Seated Human
Models
To evaluate the prediction accuracy of each human mo-
del in comparison with experimental results from litera-
ture, the ratio of the root-mean-square error to the mean
value is calculated with the following equation:
Goodness of fit (

)
2
()/(2)
/
ec
e
N
N



 (14)
where e

is the test datum, c

is the calculated result
from each model, and N is the number of test data points
used in the comparison. The fit of predicted results to test
data is perfect when

is equal to 1. The predictions on
seat-to-head transmissibility, the driving-point mechani-
cal impedance, and apparent mass for each lumped-pa-
rameter model will be compared using Equation (14) to
obtain prediction accuracy.
7. Results and Discussion
1) Wan and Schimmels modified model
The solution of the constrained optimization problem,

W. ABBAS ET AL.
716
Equation (5), resulted in an optimized model parameters
are listed in Table 1. Figure 3 presents a comparison of
the driving-point mechanical impedance, apparent mass,
and seat-to-head transmissibility with the target data,
respectively. It is obvious that the developed model bet-
ter fits target values compared with Wan and Schimmels
model. The calculated goodness of fit for seat-to-head
transmissibility is 92% compared to 90.9% for Wan and
Schimmels model.
On the other hand, the developed model matches the
target values better with a goodness of fit of 82.1% for
05 10 15 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Frequency (Hz)
STHT
Wan and Schimmels model (

= 90.9 %)
Developed model (

= 92 %)
:Wan and Schimmels model
:Developed model
:Experimental Upper limit
:Experimental Lower limit
:Experimental Target value
(a)
05 10 15 20
0
500
1000
1500
2000
2500
3000
Frequency (Hz)
DPMI (N s/m)
Wan and Schimmels model (

= 80.1 %)
Developed model (

= 82.1 %)
:Wan and Schimmels model
:Developed model
:Experimental Upper limit
:Experimental Lower limit
:Experimental Target value
(b)
05 10 15 20
10
20
30
40
50
60
70
80
90
100
Frequency (Hz)
APMS (kg)
Wan and Schimmels model (  = 86.8 % )
Developed model (

= 87.1 % )
:Wan and Schimmels model
:Developed model
:Experimental Upper limit
:Experimental Lower limit
:Experimental Target value
(c)
Figure 3. Comparison of biodynamic response characteris-
tics for Wan model, and optimized model with the target
data.
driving-point mechanical impedance and 87.1% apparent
mass compared to 80.1% and 86.8%, for Wan and
Schimmels model, respectively.
In addition, the peak values of the Wan and Schim-
mels model occur at 4 Hz for seat-to-head transmissibil-
ity, 3.7 Hz for apparent mass, and 7.2 Hz for driving-
point mechanical impedance, whereas for the optimized
model, they occur at 4.05, 3.8 and 6.9 Hz, respectively.
2) Boileau and Rakheja modified model
The solution of the constrained optimization problem,
Equation (6), resulted in optimized model parameters
which are listed in Table 2.
Simulation results are illustrated in Figure 4. This fig-
ure presents a comparison of the driving-point mechani-
cal impedance, apparent mass and seat-to-head transmis-
sibility with the target data, respectively. It is obvious
that the optimized model better fits target values than the
Boileau and Rakheja model; with a goodness of fit for
seat-to-head transmissibility is 80.6%, compared to
76.8% for the Boileau and Rakheja model.
In addition, the developed model matches the target
values better with a goodness of fit of 84% for driv-
ing-point mechanical impedance compared to 80.1% for
Boileau and Rakheja model. Optimized model matches
the target values better with a goodness of fit of 87% for
apparent mass compared to 86.7% for the Boileau and
Rakheja model.
On the other hand, the peak values of the Boileau and
Rakheja model occur at 4.7 Hz for seat-to-head trans-
missibility, 4.6 Hz for apparent mass, and 5.5 Hz for
driving-point mechanical impedance, whereas for the
optimized model, they occur at 4.95, 4.55 and 5.9 Hz,
respectively.
3) Patil and Palanichamy modified model
In a similar way, the solution of the constrained opti-
mization problem, Equation (7), resulted in an optimized
of the following model parameters are listed in Table 3.
Figure 5 presents a comparison of the driving-point me-
chanical impedance, apparent mass, and seat-to-head
transmissibility with the target data, respectively. It was
observed that the developed model better fits target val-
ues compared the Patil and Palanichamy model, with a
goodness of fit for seat-to-head transmissibility is 35%
compared to 22% for Patil and Palanichamy model. On
the other hand, the developed model matches the target
values better with a goodness of fit of 13% for driving-
point mechanical impedance compared to 1% for Patil
and Palanichamy model. The developed model matches
the target values with a goodness of fit of 34% for appar-
ent mass compared to 0% for Patil and Palanichamy model.

W. ABBAS ET AL.
717
Table 2. The biomechanical parameters of the Boileau and
Rakheja model (Before and after optimization).
Damping coefficient
(N.s/m) Spring constant (N/m)
Mass (kg)
Before After Before After
m1 =
5.31 c1 =400 c 1 =460 k1 = 310000
k1 =
356370
m2 =
28.49 c2 = 4750 c 2 =5400 k2 = 183000
k2 =
208570
m3 =
8.62 c3 = 4585 c 3 =5190 k3 = 162800
k3 =
187110
m4 =
12.78 c4 = 2064 c 4 =2370 k4 =90000 k4 = 103480
05 10 15 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Frequency (Hz)
STHT
Boileau and Rakheja model (

= 76.8 %)
Developed model (

= 80.6 %)
:Boileau and Rakheja model
:Developed model
:Experimental Upper limit
:Experimental Lower limit
:Experimental Target value
(a)
05 10 1520
0
500
1000
1500
2000
2500
3000
Frequency (Hz)
DPMI (N s/m)
Boileau and Rakheja model (

= 80.1 %)
Developed model (
 = 84 %)
:Boileau and Rakheja model
:Developed model
:Experimental Upper limit
:Experimental Lower limit
:Experimental Target value
(b)
05 10 15 20
10
20
30
40
50
60
70
80
90
100
Frequency (Hz)
APMS (kg)
Boileau and Rakheja model (

= 86.7 % )
Developed model (
 = 86 % )
:Boileau and Rakheja model
:Developed model
:Experimental Upper limit
:Experimental Lower limit
:Experimental Target value
(c)
Figure 4. Comparison of biodynamic response characteris-
tics for Boileau model, and optimized model with the target
data.
05 10 15 20
0
0.5
1
1.5
2
2.5
3
Frequency (Hz)
STHT
Patil et al. model (

= 22 %)
Developed model (

= 35 %)
:Patil et al. model
:Developed model
:Experimental Upper limit
:Experimental Lower limit
:Experimental Target value
(a)
05 10 15 20
0
500
1000
1500
2000
2500
3000
3500
4000
Frequency (Hz)
DPMI (N s/m)
Patil et al. model (  = 01 %)
Developed model (

= 13 %):Patil et al. model
:Developed model
:Experimental Upper limit
:Experimental Lower limit
:Experimental Target value
(b)
2 46 810 12 1416 18 20
0
50
100
150
200
250
Frequency (Hz)
APMS (kg)
Patil et al. model (

= - % )
Developed model (

= 34 % )
:Patil et al. model
:Developed model
:Experimental Upper limit
:Experimental Lower limit
:Experimental Target value
(c)
Figure 5. Comparison of biodynamic response characteris-
tics for Patil model, and optimized model with the target
data.
In addition, the peak values of Patil and Palanichamy
model occur at 2.5 Hz for seat-to-head transmissibility,
2.5 Hz for apparent mass, and 2.6 Hz for driving-point
mechanical impedance, whereas for the developed model,
they occur at 2.1, 2.1 and 2.3 Hz, respectively.
The simulations of the three lumped-parameter models
listed in this study for seated human subjects exposed to
vertical vibration exposure are listed in Table 4. It is
observed that the 4-DOF optimization of Wan and
Schimmels can give the best estimation on seat-to-head
transmissibility with goodness of fit values of 92%. The

W. ABBAS ET AL.
718
Table 3. The biomechanical parameters of the Patil and
Palanichamy model (Before and after optimization).
Damping coefficient
(N.s/m) Spring constant (N/m)
Mass (kg)
Before After Before After
m1 = 5.55 c1 =3651 c 1 =3542 k1 =
53640
k1 =
41978
m2 = 6.94 c2 =3651 c 2 =2685 k2 =
53640
k2 =
40943
m3 =
33.33 c3 =298 c 3 =351 k3 =8941 k3 =1001
m4 =
1.389 c4 =298 c 4 =237 k4 =8941 k4 =845
m5 =
0.4629 c5 =298 c 5 =354 k5 =8941 k5 =1052
m6 = 6.02 c6 =298 c 6 =225 k6 =8941 k6 =1035
m7 = 27.7 c7 =3651 c 7 =2929 k7 =
53640
k7 =
39575
c
8 =378 c 8 =463 k8 =
25500
k8 =
19325
goodness of fit is 82.1% for driving-point mechanical
impedance. On the other hand, the development model
matches the target values better with a goodness of fit of
87.1% for apparent mass compared to all models.
8. Conclusions and Recommendations
A study on the biodynamic models of seated human sub-
jects exposed to vertical vibration is carried out. A three
lumped-parameter models from literature have also been
analyzed and optimized using genetic algorithms to
match an experimental data in terms of STH transmissi-
bility, DPM impedance, and AP mass. It is shown that
the optimized 4-DOF Wan and Schimmels model can
give the best estimation on STH transmissibility, DPM
impedance, AP mass with goodness of fit values of
91.2%, 82.1%, and 87.1%, respectively. In addition, it
Table 4. Result of STHT, DPMI, and APMS for different models.
STHT DPMI APMS
Model Name Peak fre-
quency (Hz)
Goodness of
fit
(%)
Peak fre-
quency (Hz)
Goodness of
fit
(%)
Peak fre-
quency (Hz)
Goodness of
fit
(%)
Goodness of
fit average
(%)
DOF
Target values 5.1 4.8 4.4
Wan model 4 90.9 7.2 80.1 3.7 86.8 85.8
4 Optimized Wan model 4.05 92 6.9 82.1 3.8 87.1 87
Boileau model 4.7 76.8 5.5 80.1 4.6 86.7 81.2
4 Optimized Boileau 4.95 80.6 5.9 84 4.55 87 83.86
Patil model 2.5 22 2.6 1 2.5 0 7.67
7 Optimized Patil model 2.1 35 2.3 13 2.1 34 27.7
has the highest average of goodness of fit (87%). So, this
model is recommended for the study of biodynamic re-
sponses of seated human subjects exposed to vertical
whole body vibration. The biomechanical parameters of
the Wan and Schimmels model that match the experi-
mental data was changed for the head as c1 = 310 N.s/m
and k1 = 166990 N/m, respectively; and for upper torso
stiffness as k3 = 144000 N/m. From the model, the main
body resonant frequencies computed on the basis of both
biodynamic response functions are found to be within
close bounds to that expected for the human body.
This research provides a comprehensive understanding
of the aforementioned biodynamic responses. Future
research may be extended to the following:
1) Therefore, further research can be conducted on the
other lumped-parameter models from literature. This
work will be done after applying an optimization proc-
esses to determine the much closer one that match ex-
perimental data to STH transmissibility, DPM impedance,
and AP mass values.
2) Quarter, semi, and full car suspension system in-
cluding seat-human car suspension system should be
analyzed and validated to find the actual frequency re-
sponses of driver parts.
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