Biodynamic modelling of seated human subjects exposed to whole-body
vibration in both vertical and fore-and-aft directions
Zengkang Gan1*, Andrew J. Hillis1, Jocelyn Darling1
1
Centre for Power Transmission and Motion Control, Department of Mechanical Engineering,
University of Bath, Bath, United Kingdom
{z.gan, a.j.hillis, j.darling}@bath.ac.uk
Keywords: Biodynamics, Seated human subjects, Whole-body vibration, Lumped-parameter
models.
Abstract: In this article, the biodynamic responses of seated human subjects (SHS) exposed
to whole-body vibration (WBV) in both vertical and fore-and-aft directions are modelled. The
mathematical model can be used to obtain a better insight into the mechanisms and
biodynamic behaviour of the SHS system. The main limitation of some previous SHS models
is that they were derived to satisfy a single biodynamic response function. Such an approach
may provide a reasonable fit with the function data being considered but uncertain matches
with the others. The model presented in this study is based on all three types of biodynamic
response functions: seat-to-head transmissibility (STHT), driving-point mechanical
impedance (DPMI) and apparent mass (APM). The objective of this work is to match all three
functions and to represent the biodynamic behaviour of SHS in a more comprehensive way.
Three sets of synthesized experimental data from published literature are selected as the target
values for each of the three transfer functions. A curve fitting method is used in the parameter
identification process which involves the solution of a multivariable optimization function
comprising the root mean square errors between the computed values using the model and
those target values measured experimentally. Finally, a numerical simulation of the frequency
response of the model in terms of all three biodynamic functions has been carried out. The
results show that an improved fit is achieved comparing with the existing models.
Zengkang Gan, Andrew J. Hillis, Jocelyn Darling
1
INTRODUCTION
The harmful effects on human performance and health caused by low frequency wholebody vibration are of increasing concern. It is well know that the transmission of unwanted
vibration to the human body may lead to fatigue and discomfort. In addition, severe vibration
may cause neck and spine injuries. Much evidence has shown that there is a positive
relationship between lower back pain (LBP) and continuous exposure to whole-body
vibration [1-5].
Backman [6] conducted a health survey comprising 633 male drivers showed that the most
common complaints among the drivers were shoulder and neck pain and back trouble. The
study also found that 40 per cent of bus drivers had LBP with prevalence increasing with age.
Bovenzi et al. [7] carried out a study of LBP in port machinery operators and it was found
that the prevalence of low back symptoms increased with the increase of WBV exposure
expressed as duration of exposure (driving years), equivalent vibration magnitude, or
cumulative vibration exposure. A cross-sectional study by Rehn et al. [8] showed that
occupational drivers of various categories of all-terrain vehicles (ATVs), such as forest
machines, snowmobiles and snowgroomers, exhibited significantly increased risks of
symptoms of musculoskeletal disorders primarily in the neck and shoulder regions. From the
study it was suggested that the increased risk was related to exposure to whole-body vibration
and shock. Seidel et al. [9] also concluded that people who sit in a vibrating environment that
is close to or exceeds the ISO Exposure Limit [10, 11], place their musculoskeletal system at
risk, particularly for LBP.
In order to gain a better understanding of seated human subject biodynamic response and
adverse effects under low frequency whole-body vibration, a variety of statistical and
analytical studies have been carried out by various researchers. Statistical studies usually
involve measuring the kinetic and biodynamic responses of human subjects. There are three
types of generalized biodynamic responses functions - seat-to-head transmissibility (STHT),
driving-point mechanical impedance (DPMI) and apparent mass (APM), which are widely
used to characterize biodynamic response of the seated human subject under most commonly
encountered vibration environment. The STHT function is defined as the complex ratio of the
output vibration level on the head to the input vibration level on the seat in the frequency
range of interest [12]. This can be obtained in different forms by examining different response
quantities, such as displacement, velocity and acceleration. The DPMI function is defined as
the complex ratio between the transmitted dynamic force to which the subject is exposed and
the input driving-point velocity [13]. The APM function is defined similarly to the DPMI
function. It specifies the complex ratio of driving force to the driving-point acceleration.
While the STHT function can provide indications on the dynamic behaviour of human body
parts which are distant from the driving-point, the DPMI and APM functions can show the
biodynamic characteristics of the human body load at the input point. All three functions can
be evaluated by calculating the magnitude and phase responses in the frequency range of
interest.
The measured biodynamic responses data from the statistical study can be used to identify
mechanical-equivalent properties of the human body and to help in developing and validating
mathematical models for analytical study in order to obtain a better insight of the human body
behaviour under vibration. These mathematical models can be further used to help in
developing anthropodynamic manikins for vibration assessment and to design anti-vibration
seats and devices.
In the past few decades, a number of mathematical models have been developed based
upon diverse measurements. Suggs et al. [14] developed a two-DOF damped spring-mass
2
Zengkang Gan, Andrew J. Hillis, Jocelyn Darling
model based on measurements which closely approximated the major dynamic characteristics
of a seated man to vertical modes of vibration below 10 Hz. The model was used to build a
standardised vehicle seat testing procedure. Wei and Griffin [15] suggested a two-DOF
human and seat model to predict car seat vibration transmissibility. It was found that the
predicted seat transmissibilities were close to those measured in a group of eight subjects over
the entire frequency range. Boileau and Rakheja [13] proposed a four-DOF human body
model considered the typical vehicle driving positions, such as erect without backrest support
(ENS) posture, feet supported and low frequency excitation below 4 m/s2. The model
parameters were estimated by attempting to match the magnitude and phase characteristics of
the vertical DPMI function. Rosen and Arcan [16] constructed a multi-DOF lumpedparameter model incorporating global dynamics in terms of apparent mass and local dynamics
of the body/seat contact interaction. The model parameters were optimised based on the APM
experimental data.
Some other human subject models have also been developed by Muskian and Nash [17],
Patil and Palanichamy [18], Qassem and Othman [19], Wan and Schimmels [20], Qiu and
Griffin [21], and Stein et al. [22]. However, a concern regarding the majority of human
subject models is raised by the fact that they were derived to satisfy a single biodynamic
response function. Such an approach may provide a reasonable fit with the function data being
considered but uncertain matches with the others. Since the biodynamic behaviour of the
seated human body is equally characterized by all the three types of functions, it would be
more useful for a mathematical model to fit all three biodynamic functions.
The objective of this work is to develop a mathematical model which can portray the
essential biodynamic behaviour of seated human subject exposed to low frequency WBV in a
more comprehensive way. In the present paper a synthesis of experimental data of seated
human subject responses in both vertical and fore-and-aft directions in terms of STHT, DPMI
and APM is generated. The mean values of the synthesized are selected as the target values
for the model parameters identification. Model equations of motion are derived and the
solutions in frequency domain are given. In the end, a numerical simulation of the magnitude
and phase responses of the STHT, DPMI and APM biodynamic functions are carried out. A
comparison with the existing models is made and the results are analysed and discussed.
2
MEASUREMENT DATA OF SEATED HUMAN SUBJECTS
The biodynamic responses data of seated human subjects was obtained from a variety of
field and experimental measurements which were carried out under widely varying test
conditions [12-13, 23]. The variation of test conditions for individual measurements may
involve both intrinsic and extrinsic variables, such as subjects mass and populations, seat
postures, feet and hand positions, vibration excitation types and levels, seat backrest angle and
measurement locations on the subjects. In order to avoid significant discrepancies among the
measurements data associated with the above variable conditions, the following requirements
were specified for the synthesis of the biodynamic characteristics of seated human subjects
[13, 23]:
(a) Studies presenting measurements results based on at least six subjects;
(b) The measured subjects are considered to be sitting erect or upright posture, irrespective
of the hands’ position;
(c) Feet are supported and vibrated on the same excitation base;
(d) Subject mass are limited in the range of 45-100 kg;
(e) Excitation levels are below 5m/s2 and magnitude and phase data are reported in the 020 Hz frequency range;
3
Zengkang Gan, Andrew J. Hillis, Jocelyn Darling
(f) Either sinusoidal or random vibration excitation is used in the measurements.
Based upon the above requirements, the following published measurement data were
selected for the synthesis of seated human body biodynamic properties: Paddan and Griffin
[12], Boileau and Rakheja [13] and Hinz et al. [24] for the seat-to-head transmissibility
(STHT) data; Boileau and Rakheja [13], Fairley and Griffin [25], Hinz and Seidel [26] and
Holmlund and Lundström [27] for the driving-point mechanical impedance (DPMI) data;
Mansfield and Griffin [28], Toward and Griffin [29], Fairley and Griffin [30] and Qiu and
Griffin [31] for the apparent mass (APM) data. The synthesized data, shown in Table 1 and 2,
is derived by averaging the above data sets and applying proper smoothing within the
frequency range of interest.
STHT (abs)
Frequency
(Hz)
Magnitude Phase (deg)
0.5
1.01
-0.2
0.75
1.00
-0.7
1.0
1.01
-0.8
2.0
1.10
-6.0
3.0
1.16
-10.0
4.0
1.28
-17.5
4.5
1.37
-29
5.0
1.45
-40
5.5
1.43
-50
6.0
1.30
-61
6.5
1.18
-62
7.0
1.09
-60
8.0
0.99
-62
9.0
0.94
-70
10.0
0.95
-76
12.0
0.86
-85
14.0
0.76
-97
16.0
0.67
-105
18.0
0.60
-113
20.0
0.56
-121
DPMI (N*s/m)
Magnitude Phase (deg)
95
89.5
175
89.0
310
88.5
754
87.5
1255
82
2252
66
2704
45
2605
31
2254
23
2105
23
1865
20
1892
22
1998
21
2002
20
2015
16
1905
17
1770
18
1625
19
1585
20
1605
20
APM (kg)
Magnitude Phase (deg)
59
-2.2
60
-2.3
60
-3.5
61
-4.5
71
-10
81
-15
80
-23
76
-31
67
-43
53
-55
48
-60
44
-64
39
-68
36
-70
32
-72
31
-80
25
-83
18
-82
14
-81
11
-81
Table 1: Synthesized data of STHT, DPMI and APM mean values in the vertical direction.
4
Zengkang Gan, Andrew J. Hillis, Jocelyn Darling
STHT (abs)
Frequency
(Hz)
Magnitude Phase (deg)
0.5
1.26
-1
0.75
1.44
-16
1.0
1.62
-40
1.25
1.59
-64
1.5
1.41
-85
1.75
1.23
-103
2.0
1.10
-119
2.5
0.86
-135
2.75
0.71
-144
3.0
0.56
-159
3.5
0.41
-168
4.0
0.29
-175
4.5
0.26
-189
5.0
0.23
-198
6.0
0.18
-211
7.0
0.13
-231
8.0
0.10
-247
9.0
0.08
-253
10.0
0.08
-255
11.0
0.08
-254
12.0
0.08
-252
DPMI (N*s/m)
Magnitude Phase (deg)
55
83.0
148
80.8
257
76.6
392
71.55
515
65.4
655
59.5
850
48.4
1010
33.6
1068
25.0
1095
17.1
1083
1.8
1061
-7.5
1028
-15.5
974
-21.5
912
-25.8
853
-29.0
746
-29.8
647
-29.3
555
-28.0
521
-27.2
487
-25.7
APM (kg)
Magnitude Phase (deg)
53.0
-8.0
57.0
-10.6
59.5
-12.8
63.1
-16.0
67.2
-20.2
69.6
-24.0
70.2
-28.5
62.5
-39.4
58.6
-46.0
54.0
-53.5
45.1
-62.5
37.7
-75.0
35.3
-80.5
30.6
-81.0
20.5
-86.2
17.4
-91.0
12.9
-90.2
10.3
-87.0
7.8
-85.0
7.2
-84.0
6.0
-84.0
Table 2: Synthesized data of STHT, DPMI and APM mean values in the fore-and-aft direction.
The lower and upper limits of each data are not included in the above tables. However,
they are shown in Table 6. It is noted that sufficient measurement data of STHT, DPMI and
APM responses in the fore-and-aft direction is only available up to 12 Hz.
3
3.1
MATHEMATICAL MODEL DEVELOPMENT
Model description
The proposed model is a lumped-parameter linear spring and damper system. The model
includes segments representing appropriate anatomical parts of the body and is capable of
accommodating translational and rotational (head and neck joint) movements of these
segments, which enable it to represent the measured STHT, DPMI and APM function data
under low frequency whole-body vibration.
This model is composed of two sub-models: the vertical model and the fore-and-aft model,
as shown in Figure 1. The vertical model consists of five segments: head and neck (m5), upper
torso (m4), arms (m3), viscera (m2), and lower torso (m1). The spring (k41) and damper (c41)
connecting the upper and lower torsos represent the body spine. In the fore-and-aft model, the
main body mass (m1+ m2+ m3+ m4) is treated as a single limped mass. The head and neck (m5)
and the main body are connected by a rotational degree. These rigid masses are coupled by
linear elastic and damping elements. The masses of the lower legs and the feet are not
incorporated in the model representation, assuming their negligible contributions to the
whole-body biodynamic response. This assumption is in agreement with the evidence that the
5
Zengkang Gan, Andrew J. Hillis, Jocelyn Darling
contribution of the supported legs to the whole-body vibration is relatively insignificant when
both the seat and the footrest are vibrated under the same excitation.
x1(t)
x0(t)
Head and neck-m5
z5(t)
k5
c5
θ(t)
kr, cr
cb
z4(t)
Upper torso -m4
k3
c3
k4
Arms -m3
kb
k41
Viscera -m2
k2
z3(t)
c4
c41
z2(t)
Spine
c2
Lower torso -m1
z1(t)
k1
c1
z0(t)
Seat -m0
Figure 1: Schematic of model for seated human subjects in both vertical and fore-and-aft directions.
3.2
Equations of motion (EOMs)
EOMs of the model were derived from the free-body diagram of each part. The vertical
model has five degrees of freedom in total: the vertical displacement of each segment {𝑧1 (𝑡),
𝑧2 (𝑡), 𝑧3 (𝑡), 𝑧4 (𝑡), 𝑧5 (𝑡)}. The vertical model EOMs can be expressed by the following
coupled differential equations:
𝑚1 𝑧1̈ + 𝑐1 (𝑧1̇ − 𝑧0̇ ) + 𝑘1 (𝑧1 − 𝑧0 ) − 𝑐2 (𝑧2̇ − 𝑧1̇ ) − 𝑘2 (𝑧2 − 𝑧1 )
−𝑐41 (𝑧4̇ − 𝑧1̇ ) − 𝑘41 (𝑧4 − 𝑧1 ) = 0
(1)
𝑚2 𝑧2̈ + 𝑐2 (𝑧2̇ − 𝑧1̇ ) + 𝑘2 (𝑧2 − 𝑧1 ) − 𝑐4 (𝑧4̇ − 𝑧2̇ ) − 𝑘4 (𝑧4 − 𝑧2 ) = 0
(2)
𝑚3 𝑧3̈ − 𝑐3 (𝑧4̇ − 𝑧3̇ ) − 𝑘3 (𝑧4 − 𝑧3 ) = 0
(3)
𝑚4 𝑧4̈ + 𝑐3 (𝑧4̇ − 𝑧3̇ ) + 𝑘3 (𝑧4 − 𝑧3 ) + 𝑐4 (𝑧4̇ − 𝑧2̇ ) + 𝑘4 (𝑧4 − 𝑧2 )
+𝑐41 (𝑧4̇ − 𝑧1̇ ) + 𝑘41 (𝑧4 − 𝑧1 ) − 𝑐5 (𝑧5̇ − 𝑧4̇ ) − 𝑘5 (𝑧5 − 𝑧4 ) = 0
(4)
𝑚5 𝑧5̈ + 𝑐5 (𝑧5̇ − 𝑧4̇ ) + 𝑘5 (𝑧5 − 𝑧4 ) = 0
(5)
The above differential equations can be expressed in matrix form:
[𝑴]{𝒛̈ } + [𝑪]{𝒛̇ } + [𝑲]{𝒛} = {𝒇𝒏 }
(6)
where [𝑴], [𝑪] and [𝑲] are mass, damping and stiffness matrices with a size of 5 × 5,
respectively; {𝒛̈ }, {𝒛̇ } and {𝒛} are acceleration, velocity and displacement vectors, respectively,
6
Zengkang Gan, Andrew J. Hillis, Jocelyn Darling
with a size of 5 × 1; {𝒇𝒏 } is an 5 × 1 excitation force vector. All the above matrices and
vectors can be expressed as follows:
0
0
0
0
0
𝑚2
0
0
0
[𝑴] = 0
0
𝑚3
0
0 ;
0
0
0
𝑚4
0
[0
0
0
0
𝑚5 ]
𝑐1 + 𝑐2 + 𝑐41
−𝑐2
0
−𝑐41
0
−𝑐2
𝑐2 + 𝑐4
0
−𝑐4
0
0
0
𝑐3
−𝑐3
0
−𝑐41
−𝑐4
−𝑐3
𝑐3 + 𝑐4 + 𝑐41 + 𝑐5
−𝑐5
0
0
0
−𝑐5
𝑘1 + 𝑘2 + 𝑘41
−𝑘2
0
−𝑘41
0
−𝑘2
𝑘2 + 𝑘4
0
−𝑘4
0
0
0
𝑘3
−𝑘3
0
−𝑘41
−𝑘4
−𝑘3
𝑘3 + 𝑘4 + 𝑘41 + 𝑘5
−𝑘5
0
0
0
−𝑘5
[𝑪] =
[
[𝑲] =
[
{𝒛̈ } =
𝑚1
𝑐5 ]
𝑧1̇
𝑧1
𝑐1 𝑧0̇ + 𝑘1 𝑧0
𝑧2̈
𝑧2̇
𝑧2
0
𝑧4̈
; {𝒛̇ } =
𝑧3̇
; {𝒛} =
𝑧3
; {𝒇𝒏 } =
0
𝑧4
𝑧4̇
{ 𝑧5 }
;
𝑘5 ]
𝑧1̈
𝑧3̈
;
.
0
}
0
The fore-and-aft model has two degrees of freedom in total: the fore-and-aft displacement
of the main body part {𝑥1 (𝑡)} and the head and neck rotational degree {𝜃(𝑡)}. The EOMs of
the fore-and-aft model can be expressed by the following differential equations:
{ 𝑧5̈ }
{ 𝑧5̇ }
{
M𝑥1̈ + 𝑚5 𝑙𝑛 𝜃̈ cos 𝜃 − 𝑚5 𝑙𝑛 𝜃̇ 2 sin 𝜃 + 𝑘𝑏 (𝑥1 + 𝑥0 ) + 𝑐𝑏 (𝑥1̇ + 𝑥0̇ ) = 0
(7)
𝑚5 𝑙𝑛 2 𝜃̈ + 𝑚5 𝑙𝑛 𝑥1̈ cos 𝜃 − 𝑚5 𝑙𝑛 𝑔 sin 𝜃 + 𝑘𝑡 𝜃 + 𝑐𝑡 𝜃̇ = 0
(8)
where M is the mass of the whole body (M= m1+ m2+ m3+ m4+ m5), 𝑙𝑛 is the average distance
between the shoulder and the gravity centre of the head, 𝑔 is the acceleration due to gravity
(𝑔 = 9.81 𝑚/𝑠 2). 𝑘𝑡 and 𝑐𝑡 are the rotational spring and damper coefficients of the neck, 𝑘𝑏
and 𝑐𝑏 are the spring and damper coefficients between the main body and the backrest,
respectively.
7
Zengkang Gan, Andrew J. Hillis, Jocelyn Darling
3.3
EOMs solution in the frequency domain
By taking Fourier transforms of the above EOMs the models can be analysed in the
frequency domain. The Fourier transform of the vertical model EOMs (Eq. (6)) results in:
−1
{𝒁(𝑗𝜔)} = [−𝜔2 [𝑴] + 𝑗𝜔[𝑪] + [𝑲]] {𝑭𝒏 (𝑗𝜔)}
(9)
where {𝒁(𝑗𝜔)} and {𝑭𝒏 (𝑗𝜔)} are the complex Fourier transform vectors of {𝒛} and {𝒇𝒏 } ,
respectively, 𝑗 is the imaginary unit and 𝜔 is the angular frequency. The vector {𝒁(𝑗𝜔)}
contains the complex displacement responses of the 5 mass segments as a function of angular
frequency, and they can be represented by {𝒁𝟏 (𝑗𝜔), 𝒁𝟐 (𝑗𝜔), 𝒁𝟑 (𝑗𝜔), 𝒁𝟒 (𝑗𝜔), 𝒁𝟓 (𝑗𝜔)}.
{𝑭𝒏 (𝑗𝜔)} contains the complex excitation forces as a function of angular frequency as well,
which is {(𝑘1 + 𝑗𝜔𝑐1 )𝒁𝟎 (𝑗𝜔), 0, 0, 0, 0} , where 𝒁𝟎 (𝑗𝜔) is the complex displacement of
excitation. The EOMs of the fore-and-aft model contain some nonlinear terms −𝑚5 𝑙𝑛 𝜃̈ cos 𝜃,
−𝑚5 𝑙𝑛 𝜃̇ 2 sin 𝜃, 𝑚5 𝑙𝑛 𝑥1̈ cos 𝜃, −𝑚5 𝑙𝑛 𝑔 sin 𝜃. Small oscillations were assumed (i.e. around
𝜃 = 0), and the following linearization were used: cos 𝜃 = 1, sin 𝜃 = 𝜃, 𝜃̇ 2 sin 𝜃 = 0. The
Fourier transform of the linearized equations can be expressed as follows:
−𝜔2 M𝐗 𝟏 (𝑗𝜔) − 𝜔2 𝑚5 𝑙𝑛 𝜽(𝑗𝜔) + (𝑘𝑏 + 𝑗𝜔𝑐𝑏 )(𝐗 𝟏 (𝑗𝜔) − 𝐗 𝟎 (𝑗𝜔)) = 0
(10)
−𝜔2 𝑚5 𝑙𝑛 2 𝜽(𝑗𝜔) − 𝜔2 𝑚5 𝑙𝑛 𝐗 𝟏 (𝑗𝜔) + 𝑚5 𝑙𝑛 𝑔𝜽(𝑗𝜔) + (𝑘𝑡 + 𝑗𝜔𝑐𝑡 )𝜽(𝑗𝜔) = 0
(11)
Based on the preceding definitions, the STHT, DPMI and APM biodynamic functions for
the vertical model can be derived as follows:
STHT_v =
(12)
(𝑘1 + 𝑗𝜔𝑐1 )[𝒁𝟎 (𝑗𝜔) − 𝒁𝟏 (𝑗𝜔)]
|
𝑗𝜔𝒁𝟎 (𝑗𝜔)
(13)
(𝑘1 + 𝑗𝜔𝑐1 )[𝒁𝟎 (𝑗𝜔) − 𝒁𝟏 (𝑗𝜔)]
DPMI_v
|=|
|
𝑗𝜔
−𝜔 2 𝒁𝟎 (𝑗𝜔)
(14)
DPMI_v = |
APM_v = |
𝒁𝟓 (𝑗𝜔)
𝒁𝟎 (𝑗𝜔)
Considering the Eqs. (10) and (11) of the fore-and-aft model in a similar manner, the
STHT, DPMI and APM biodynamic functions for the fore-and-aft model can be derived as
follows:
STHT_f =
(15)
(𝑘𝑏 + 𝑗𝜔𝑐𝑏 )(𝐗 𝟏 (𝑗𝜔) − 𝐗 𝟎 (𝑗𝜔))
|
𝑗𝜔𝐗 𝟎 (𝑗𝜔)
(16)
(𝑘𝑏 + 𝑗𝜔𝑐𝑏 )(𝐗 𝟏 (𝑗𝜔) − 𝐗 𝟎 (𝑗𝜔))
DPMI_f
APM_f = |
|=|
|
𝑗𝜔
−𝜔 2 𝐗 𝟎 (𝑗𝜔)
(17)
DPMI_f = |
4
𝑙𝑛 𝜽(𝑗𝜔) + 𝐗 𝟏 (𝑗𝜔)
𝐗 𝟎 (𝑗𝜔)
MODEL PARAMETER IDENTIFICATION
Model parameters were identified using curve fitting methods formulated in Matlab
(version 2011b). The Least Absolute Residual (LAR) method and the 'Trust-Region'
algorithm are used. The fitting process involves the solution of a multivariable optimization
function comprising the root mean square errors between the computed values using the
8
Zengkang Gan, Andrew J. Hillis, Jocelyn Darling
model and those target values measured experimentally (Tables 1 and 2). In the vertical model
there are 17 unknown parameters in total, which can be represented in a vector as: pv =[c1 , c2 ,
c3 , c4 , c41 , c5 , k1 , k 2 , k 3 , k 4 , k 41 , k 5 , m1 , m2 , m3 , m4 , m5 ]T. Since the mass (m1 , m2 , m3 ,
m4 , m5 ) are shared parameters, in the fore-and-aft model the unknown parameters vector can
be expressed as: pf =[cb , ct , k b , k t , ln ]T. The vectors 𝑝𝑣 and 𝑝𝑓 were identified separately by
fitting the biodynamic functions in Eqs. (12-17).
In order to make the fitting procedure more effective, a set of initial, upper and lower limits
values was estimated by referring to previously published studies. 73.6% (percentage of body
mass supported by the seat for erect seating posture) of the whole body weight (75 kg) is used
for the total model mass (i.e. 73.6% of 75kg =55.2kg). The damping and stiffness coefficients
of the human body segments are not known precisely, therefore, the ranges are relatively large.
The estimated initial, upper and lower limits values are listed in Table 3. The identified model
parameters are listed in Table 4.
Model parameters
Head and neck mass 𝑚5 (kg)
Upper torso mass 𝑚4 (kg)
Arms mass 𝑚3 (kg)
Viscera mass 𝑚2 (kg)
Lower torso mass 𝑚1 (kg)
Average distance 𝑙𝑛 (m)
Damping coefficient 𝑐1 (Ns/m)
Damping coefficient 𝑐2 (Ns/m)
Damping coefficient 𝑐3 (Ns/m)
Damping coefficient 𝑐4 (Ns/m)
Damping coefficient 𝑐41 (Ns/m)
Damping coefficient 𝑐5 (Ns/m)
Damping coefficient 𝑐𝑏 (Ns/m)
Damping coefficient 𝑐𝑡 (Ns/m)
Stiffness coefficient 𝑘1 (N/m)
Stiffness coefficient 𝑘2 (N/m)
Stiffness coefficient 𝑘3 (N/m)
Stiffness coefficient 𝑘4 (N/m)
Stiffness coefficient 𝑘41 (N/m)
Stiffness coefficient 𝑘5 (N/m)
Stiffness coefficient 𝑘𝑏 (N/m)
Stiffness coefficient 𝑘𝑡 (N/m)
Initial values
5.5
22
6
10.2
11.7
0.16
2000
1000
300
4000
4000
400
200
200
120000
6000
10000
7000
160000
300000
10000
1000
Lower limits
5
20
5
8
10
0.13
500
400
100
400
500
300
10
10
5000
5000
5000
5000
5000
5000
500
500
Upper limits
7
25
8
12
15
0.19
5000
5000
2000
5000
5000
2000
3000
2000
200000
100000
200000
100000
250000
500000
200000
200000
Table 3: Estimated initial, upper and lower limits values for model parameter identification.
9
Zengkang Gan, Andrew J. Hillis, Jocelyn Darling
Model parameters
Head and neck mass 𝑚5 (kg)
Upper torso mass 𝑚4 (kg)
Arms mass 𝑚3 (kg)
Viscera mass 𝑚2 (kg)
Lower torso mass 𝑚1 (kg)
Average distance 𝑙𝑛 (m)
Damping coefficient 𝑐1 (Ns/m)
Damping coefficient 𝑐2 (Ns/m)
Damping coefficient 𝑐3 (Ns/m)
Damping coefficient 𝑐4 (Ns/m)
Damping coefficient 𝑐41 (Ns/m)
Identified
values
5.6
20.3
8.0
9.2
10.0
0.19
2376.4
675.8
145.8
1797.7
4023.2
Model parameters
Damping coefficient 𝑐5 (Ns/m)
Damping coefficient 𝑐𝑏 (Ns/m)
Damping coefficient 𝑐𝑡 (Ns/m)
Stiffness coefficient 𝑘1 (N/m)
Stiffness coefficient 𝑘2 (N/m)
Stiffness coefficient 𝑘3 (N/m)
Stiffness coefficient 𝑘4 (N/m)
Stiffness coefficient 𝑘41 (N/m)
Stiffness coefficient 𝑘5 (N/m)
Stiffness coefficient 𝑘𝑏 (N/m)
Stiffness coefficient 𝑘𝑡 (N/m)
Identified
values
977.4
621.9
18.9
120123.3
5300.3
13177.7
9151.1
128198.6
292010.0
9925.7
772.4
Table 4: Identified parameter values for the vertical and fore-and-aft human body models.
5
SIMULATION AND COMPARISON RESULTS
After all the model parameters have been identified, the magnitude and phase responses of
the STHT, DPMI and APM biodynamic functions were simulated in Matlab. The simulation
results are listed in Table 6. To evaluate the goodness-of-fit (GOF) of the presented models,
the ratio of the root-mean-square error to the mean value was calculated using the following
equation [23]:
𝐺𝑂𝐹 = 1 −
√∑(𝒚𝒎 −𝒚𝒄 )2 /(𝑁−2)
∑ 𝒚𝒎 /𝑁
(18)
where 𝒚𝒎 and 𝒚𝒄 are the measured target data and calculated value, respectively. 𝑁 is the
number of the measured target data points. The 𝐺𝑂𝐹 statistic can take on any value less than
or equal to 1, with a value closer to 1 indicating a better fit. The 𝐺𝑂𝐹 values of the seated
human subject models in the comparison are summarized in Table 7. In table 6 and 7, any
𝐺𝑂𝐹 value less than 0 is marked by ~.
As shown in the simulation results, four previous seated human subject models were
selected for comparison: a four-DOF linear vertical model developed by Wan and Schimmels
[20], which has been found providing the highest average of goodness-of-fit in [23] and a
four-DOF vehicle driver model proposed by Boileau and Rakheja [13] were chose for the
vertical model comparison; a two-DOF fore-and-aft model developed by Stein et al. [22] and
a four-DOF fore-and-aft apparent mass model presented by Qiu and Griffin [21] were selected
for the fore-and-aft model comparison. The schematics and parameters of the comparison
models are listed in Table 5.
10
Zengkang Gan, Andrew J. Hillis, Jocelyn Darling
Model
Name
Model parameters
Mass
(kg)
m1
36.0
k1
Schematic of model
Damping
(Ns/m)
Stiffness
(N/m)
49340
c1
2475.0
Head and neck -m4
z4(t)
c5
k5
Wan and
Schimmels
[20]
m2
5.5
k2
20000
c2
330.0
m3
15.0
k41 192000
c41
909.1
m4
4.17
k4
10000
c4
200.0
k5
134400
c5
250.0
Upper torso -m3
z3(t)
k4
Viscera -m2
k41
c4
c41
z2(t)
k2
c2
Lower torso -m1
k1
z1(t)
c1
Seat -m0
Boileau and
Rakheja
[13]
z0(t)
m1
12.78 k1
90000
c1
2064
Head & neck -m4
m2
8.62
162800
c2
4585
Chest & upper
torso -m3
m3
m4
k2
28.49 k3
5.31
k4
183000
310000
c3
c4
4750
z4(t)
k4
c4
k3
c3
z3(t)
Lower torso -m2
z2(t)
400
k2
c2
k1
c1
Thighs & pelvis -m1
z1(t)
Seat -m0
m1
54
k1
39322
c1
465.9
m2
10.4
k2
9
c2
8.0
k3
1054
c3
113.1
z0(t)
x2(t)
x1(t)
k2
Stein et al.
[22]
k3
x3(t)
m2
c2
k1
c3
c1
m1
mb
Qiu and
Griffin [21]
8
k1
39886
c1
x3(t)
359
ms
10
k2
10924
c2
542
m1
20
kb
24610
cb
0.0
m2
35
kt
10
ct
112
ks
26646
cs
0.0
kb
mb
x2(t)
k2
m2
c2
cb
ks
kt, ct
k1
m1
cs
ms
c1
Table 5: Schematics and parameters of comparison models of seated human subject.
11
x1(t)
Zengkang Gan, Andrew J. Hillis, Jocelyn Darling
Wan’s model (Vertical)
Presented model (Vertical)
STHT (Seat-to-head transmissibility)
Presented model GOF=90.84%
1.5
1
0.5
10
15
20
Presented model GOF=82.71%
0
-50
-100
2000
1000
Phase (deg)
0
5
80
15
20
Presented model GOF=78.33%
60
20
5
10
15
Frequency (Hz)
APM (Apparent mass)
100
Presented model GOF=86.85%
0
5
10
15
Frequency (Hz)
Upper limits,
0
5
1000
80
10
15
20
Wan’s model GOF=74.87%
60
40
20
10
15
Frequency (Hz)
APM (Apparent mass)
20
100
Wan’s model GOF=84.91%
-50
-100
5
2000
APM (Kg)
20
20
10
15
20
Frequency (Hz)
DPMI (Driving-point mechanical impedance)
4000
Wan’s model GOF=76.10%
3000
50
0
0
0
Phase (deg)
15
0
5
Fig.2-2(c)
Fig.2-1(c)
10
15
-100
20
50
5
10
Wan’s model GOF=77.48%
Presented model GOF=87.80%
0
0
0
5
0
0
40
0
APM (Kg)
10
Fig.2-2(b)
Fig.2-1(b)
0
0.5
-200
5
DPMI (N*s/m)
10
15
20
Frequency (Hz)
DPMI (Driving-point mechanical impedance)
4000
Presented model GOF=81.33%
3000
DPMI (N*s/m)
0
1
Phase (deg)
-150
Wan’s model GOF=89.27%
1.5
0
0
100
Phase (deg)
Phase (deg)
5
Fig.2-2(a)
Fig.2-1(a)
0
0
50
Phase (deg)
STHT (Seat-to-head transmissibility)
2
STHT (abs)
STHT (abs)
2
Lower limits,
10
20
-50
0
5
10
15
Frequency (Hz)
Mean values.
Table 6: Simulation and comparison results of the seated human subject models.
12
15
Wan’s model GOF=86.11%
-100
20
5
20
Zengkang Gan, Andrew J. Hillis, Jocelyn Darling
Boileau’s model (Vertical)
Presented model (Fore-and-aft)
STHT (Seat-to-head transmissibility)
2.5
STHT (abs)
Boileau’s model GOF=75.30%
1
0.5
15
-50
-100
1000
Phase (deg)
Boileau’s model GOF=68.81%
50
0
5
10
15
Frequency (Hz)
APM (Apparent mass)
100
15
20
Boileau’s model GOF=73.57%
0
5
10
15
Frequency (Hz)
Upper limits,
0
2
10
12
Presented model
GOF=80.92%
2
4
6
8
10
12
Presented model GOF=52.68%
50
0
-50
4
6
8
Frequency (Hz)
10
12
Presented model GOF=91.99%
50
2
4
6
8
10
12
Presented model GOF=81.33%
-50
-100
Lower limits,
13
12
500
-150
Table 6 (continued)
10
1000
0
0
0
20
8
4
6
8
Frequency (Hz)
100
-50
-100
6
-200
2
Fig.2-4(c)
Fig.2-3(c)
10
4
-100
20
50
5
2
Presented model GOF=36.57%
0
0
100
Boileau’s model GOF=80.19%
0
0
0
0
0
APM (Kg)
APM (Kg)
20
Fig.2-4(b)
Fig.2-3(b)
15
0.5
1500
2000
10
1
0
5
5
1.5
-300
10
15
20
Frequency (Hz)
DPMI (Driving-point mechanical impedance)
4000
Boileau’s model GOF=79.36%
3000
Presented model GOF=37.09%
2
100
DPMI (N*s/m)
0
0
0
100
Phase (deg)
20
Phase (deg)
Phase (deg)
10
Boileau’s model GOF=65.98%
0
-150
DPMI (N*s/m)
5
Fig.2-4(a)
Fig.2-3(a)
0
0
50
Phase (deg)
1.5
Phase (deg)
STHT (abs)
2
0
2
4
6
8
Frequency (Hz)
Mean values.
10
12
Zengkang Gan, Andrew J. Hillis, Jocelyn Darling
4
6
8
10
12
Stein’s model GOF=29.86%
0
-100
-200
-300
0
2
4
6
8
Frequency (Hz)
4000
10
Phase (deg)
4
6
8
10
DPMI (N*s/m)
12
Stein’s model GOF=30.32%
50
0
-50
2
4
6
8
Frequency (Hz)
10
12
Stein’s model GOF=42.53%
-50
-100
-150
0
2
4
6
8
Frequency (Hz)
Upper limits,
10
2
4
6
8
Frequency (Hz)
0
100
12
Qiu’s model
GOF=79.59%
0
2
4
8
10
12
Qiu’s model GOF ~
0
-50
2
4
6
8
Frequency (Hz)
10
12
Qiu’s model GOF=80.85%
50
2
4
6
8
10
12
Qiu’s model GOF=61.80%
-50
-100
-150
Lower limits,
Table 6 (continued)
6
50
0
0
0
12
14
10
500
APM (Kg)
10
12
1000
Phase (deg)
APM (Kg)
Phase (deg)
8
0
100
Fig.2-6(c)
Fig.2-5(c)
0
6
10
Qiu’s model GOF=39.53%
-100
50
4
8
-200
12
100
2
6
-100
Stein’s model GOF ~
0
4
0
150
0
2
1500
Fig.2-6(b)
Fig.2-5(b)
100
2
1
0.5
-300
Stein’s model GOF ~
0
1.5
0
0
100
12
2000
0
Qiu’s model GOF ~
2
STHT (abs)
2
Phase (deg)
1
Fig.2-6(a)
Fig.2-5(a)
Phase (deg)
2.5
Stein’s model GOF ~
2
0
0
100
DPMI (N*s/m)
Qiu’s model (Fore-and-aft)
Phase (deg)
STHT (abs)
Stein’s model (Fore-and-aft)
0
2
4
6
8
Frequency (Hz)
Mean values.
10
12
Zengkang Gan, Andrew J. Hillis, Jocelyn Darling
STHT GOF (%)
Direction
DPMI GOF (%)
APM GOF (%)
Model Name
Magnitude Phase Magnitude Phase Magnitude Phase
Vertical
Fore-and-aft
Presented model
90.84
82.71
81.33
78.33
87.80
86.85
Wan’s model
89.27
77.48
76.10
74.87
84.91
86.11
Boileau’s model
75.30
65.98
79.36
68.81
80.19
73.57
Presented model
37.09
36.57
80.92
52.68
91.99
81.33
Stein’s model
~
29.86
~
30.32
~
42.53
Qiu’s model
~
39.53
79.59
~
80.85
61.80
Table 7: Summary of comparison results of the seated human subject models.
6
DISCUSSION
Since there is some variation between the synthesized target data used in these models, the
calculated GOF of the comparison models may not be the original presented values. The
measurement condition of the target data used in Stein’s model varies somewhat from those
of the synthesized target data in the fore-and-aft direction. However, the target data used in
Wan, Boileau and Qiu’s models are very close to the synthesized target data in this study.
From the simulation results, the vertical seated human models show a higher average
goodness-of-fit than the fore-and-aft models in both the magnitude and phase responses of the
STHT, DPMI and APM functions. The presented vertical model provides 90.84% GOF and
82.71% GOF for the STHT magnitude and phase responses, 81.33% GOF and 78.33% GOF
for the DPMI magnitude and phase responses , 87.80% GOF and 86.85% GOF for the APM
magnitude and phase responses, respectively. This indicates that a better overall GOF is
achieved for predicting the above biodynamic functions for the seated human subject under
vertical vibration. The results also show that very close peaks occur at about 5 Hz in the
magnitude responses of all the three functions, which indicates the reliabilities of predicting
identical primary resonant frequencies are validated by each other. In addition, the presented
vertical model predicts a second resonant frequency around 8 Hz which is observed in the
target data. Wan’s model has been found providing a generally good fit of STHT and APM
functions. However, the fit of DPMI function is relatively poor, with 76.10% GOF for
magnitude response and 74.87% GOF for phase response. The peak values occur at about 4
Hz for the STHT and APM functions while the peak value is around 7.5 Hz for the DPMI
function. Besides, both Wan’s and Boileau’s models fail to predict the second resonant
frequency.
The fore-and-aft model simulation results show that a relatively large deviation is exhibited.
The presented model provides the highest match for the APM function, with 91.99% GOF for
magnitude response and 81.33% GOF for phase response. The GOF values for the DPMI
function are relatively lower, with 80.92% for magnitude response and 52.68% for phase
response. However, these values still outperform the comparison models. The prediction for
the STHT function is the poorest for all the three models. The GOF values for both magnitude
15
Zengkang Gan, Andrew J. Hillis, Jocelyn Darling
and phase responses are below 40%. Because of the variation between the measurement data,
Stein’s model shows poor matches for all the three functions. Qiu’s model provides a
reasonably good fit for the APM function, but poor fits for the other two functions. One of the
reasons is that the model was developed based only on the APM measurement data.
It is noted that the quantity of reported experimental data for the seated human subject
responses in the fore-and-aft direction is considerably less than the data in the vertical
direction. More measurement data is needed to guide and validate the human body modelling
in the fore-and-aft direction. It is also noted that the phase responses of the three biodynamic
functions are usually measured in experimental studies; however, they are rarely evaluated
and analysed in human body modelling studies. The phase responses of the biodynamic
functions are evaluated in this study since the phase responses can be equally as important as
the magnitude responses, if not more so, when it comes to human body vibration cancelation.
7
CONCLUSION
A lumped-parameter biodynamic model of a seated human subject exposed to low
frequency whole-body vibration in both the vertical and fore-and-aft directions is developed.
Model parameters were identified using curve fitting methods and the STHT, DPMI and APM
biodynamic magnitude and phase response functions are simulated in Matlab. The goodnessof-fit of the presented model is evaluated graphically and statistically, a comparison with the
existing models is carried out and the results show that an improved fit with the synthesized
experimental data is achieved. Through the model, the biodynamic behaviour of the seated
human subjects can be observed in a more comprehensive way.
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