The 33rd International Congress and Exposition on Noise Control Engineering
A Nonlinear Dynamic Model of Human Body Seated on Vehicles
K.S. Kima, J.H. Leeb, K.J. Kimc and H-K. Jangd
a,b,c
Center for Noise and Vibration Control (NOVIC), Dept. of ME, KAIST, Science Town, Daejeon, Republic of Korea
dInstitude for Advanced Engineering (IAE), Ko-an 633-2, Baegam, Yong-in, Gyeonggi, Republic of Korea
a,c[kimkisun; kjkim]@kaist.ac.kr; b[email protected];
Abstract [226] Ride comfort assessment of a vehicle is generally performed by test drivers, which is very costly and inherently subjective. With realization of unmanned vehicle driving especially on proving ground, roles of test drivers are being reduced. Yet, one problem of unmanned driving is that vibrations of an unmanned vehicle are not the same as those of a vehicle with a driver in it. That is, for a correct assessment of unmanned vehicle, there should be a dummy seated on it which behaves dynamically in the same way as a human body. The purpose of this paper is to develop a dynamic model for the human body so that it may be used for design of dummies at a later stage.
From previous research results, it is known that a predominant natural frequency of seated human body decreases as the input level at the seat increases. That is, human response to the seat vibration shows nonlinear characteristics, to be more specific, a softening spring, which is often observed in typical viscoelastic materials. This fact shows the possibility of implementation of such materials for developing dummies for the test driver. In this study, nonlinear dynamic behaviors of seated human bodies are presented and analyzed to derive mathematical models.
1. INTRODUCTION
The ride comfort assessment for a vehicle running on proving ground is conducted typically by professional test drivers. Sincethis method is costly and yet inherently subjective, unmanned driving is becoming popular, which generates another problem. For an example, vibration behavior without the test driver may not be the same as the one with the driver on the seat. In such a case, a dummy could substitute the test driver. In order for such a substitution to be successful, the dynamics of a human body on seat must be well understood and represented by the dummy.
From previous studies [1, 2], it is known that predominant natural frequency of seated human body exposed to vertical vibration decreases as the input level increases. This softening nonlinearity is believed to be caused by the skin and the muscle at the buttocks, which is also observed in the behavior of viscoelastic materials [3]. Up to now, only linear models were derived from measurements on seated human body [1]. But, a linear model may be useful at a given input level. In this research, nonlinear dynamic models are developed for seated human body based on the test at various levels of input excitation.
2. DYNAMIC MODEL OF SEATED HUMAN BODY
Before proceeding to the dummy development, a mathematical model of seated human body should be derived. To do the end, some candidate models for a seated human body will be suggested with the experimental results which were obtained by Song [1]. Then, an appropriate model will be selected from observing the curve fitting results.
2.1. Experimental Result
A set of measured apparent mass of seated human body has been quoted from the study by Song [1] as shown in Fig.2.1. Here, the apparent mass is defined as the ratio of cross spectral density function (CSD) between the input acceleration and the force at the base, Gaf( )f , to power spectral density function (PSD) of acceleration at base, Gaa( )f , as Eq.(2.1). The subject was exposed to random vibration over the frequency range between 1.0 and 30Hz in the vertical direction. And the input acceleration levels of 0.5, 1.0, 2.0 m/s2 (RMS) were chosen for each experiment.
( ) af( ) aa( )
M f =G f G f (2. 1)
In Fig.2.1, we can observe that the predominant natural frequency moves from 5Hz to 4.2Hz. Therefore, it is reasonable to make a conclusion that the equivalent stiffness of seated human body shows a softening behavior.
5 10 15 20 25 30
0 20 40 60 80 100 120
140 4.2 Hz
5 Hz
|M(f)| [kg]
Frequency [Hz]
Input level : 0.5 m/s2 (RMS) Input level : 1.0 m/s2 (RMS) Input level : 2.0 m/s2 (RMS)
5 10 15 20 25 30
-100 -80 -60 -40 -20 0 20
Phase [Deg]
Frequency [Hz]
Fig.2.1. Apparent mass of seated human body
2.2. Description of Apparent Mass Models
As mentioned previously, some suggested models on trial will employ the model of viscoelastic(or rubber-like) materials, because typical rubber-like materials show the softening behaviors which also could be observed from the equivalent stiffness of seated human body.
However, the stiffness of rubber-like materials is also dependent on the exciting frequency, static pre-load and temperature, etc. Since the modal mass of the model has constant value, we are free from the dependency on static pre-load. And if we keep the temperature to be roughly constant, we can neglect the temperature dependency. Furthermore, if we ignore the frequency dependency of rubber-like materials in the low frequency range where ride comfort assessment is concerned, the structure damping model with the dependency on dynamic amplitude might be proposed for the description of rubber-like materials as follows.
( )
*( ) ( ) 1 ( )
k x =k x + jη x (2. 2)
With the prescribed complex stiffness model above, the single DOF(degree-of-freedom) model as shown in Fig.2.2.(a) has been suggested as a simplest model. However, this model is not
proper because a seated human body has two natural frequencies in the interested frequency range as shown in Fig.2.1. And, the two DOF model has been suggested as shown in Fig.2.2.(b), but this model does not give satisfactory curve fitting result which will be shown later. Therefore, the three DOF, but two different types of model have been proposed for the investigation with the effect of different model configuration. One is formed by mass in series, called the series three DOF model(Fig.2.2(c)), and the other is done by mass in parallel, called the parallel three DOF model(Fig.2.2(d)). Moreover, the four DOF model(Fig.2.2.(e)) has been considered for judging the effect of increasing the number of degrees of freedom.
2.3. Curve Fitting Result
The extraction of model parameters was conducted by minimizing the follow function,
2 2
1
2 2
1
( ){ ( ) ( )} ( ){ ( ) ( )}
(%) 100(%)
{ ( )} { ( )}
=
=
− + −
= ×
+
∑
∑
N
b rf rm b if im
k
N
rm im
k
w k M k M k w k M k M k error
M k M k
(2.3)
where Mrf( )k (or Mrm( )k ) is real part of the apparent mass from the curve fit(or real part of the apparent mass from the measurement data at the input level
of 0.5 m/s2 (RMS) ) at the k-th frequency point. Similarly,
if( )
M k and Mim( )k correspond to the imaginary ones. And the weighting function, wb( )f defined by British Standards [5] was applied for the reflection of human response to vibration.
Optimized model parameters were obtained by non- linear parameter search method, provided within MATLAB (MathWorks Inc.).
As represented in Fig.2.3, the error between the model and the experimental data shows a dramatic reduction
as the order of model is increased. And the results explain that the parallel three DOF model is superior to the series three DOF one. In addition, there is no significant improvement from the parallel three DOF model to the four DOF one. From the viewpoint of the dummy realization, a simpler model is more desirable than a complex one. Therefore, among the suggested candidate models, it is proper to employ the parallel threeDOF model for the description of the measured apparent mass.
3. A NONLINEAR MODEL FOR SEATED HUMAN BODY
Fig.2.3. Curve fitting results according to the order of model Fig.2.2. Candidate Models for the description of seated human body
m1 k*1
m2
k*2 k*
m
1 2 3 4
0 5 10 15 20 25 30 35
Parallel 3 D O F Series 3 D O F
Error [%]
D egree of Freedom
k*2
m2 m3
k*3
m1
k*1
k*2
m2 k*3
m3
k*1
m1 m1
k*1
k*3
m3
k*4
m4 m2
k*2
(a) (b) (c) (d) (e)
3.1. Insertion of a Nonlinear Element to the Selected Model
With the model selected in preceding section, it is necessary to investigate about the location where is the source of nonlinearity. Knowing the fact that excessive displacements cause a nonlinear behavior, illustration of the mode shape around the predominant natural frequency where the softening phenomenon has arisen would make possible to identify of that location.
Fig.3.1 shows the mode shape of the employed model at the predominant natural frequency, and it could be recognized that the relative displacement acting on k2* between m1 and m2 is dominant, straightforwardly. Therefore, it is reasonable to mention that k*2 could be regarded as a nonlinear component and described with Eq.(2.2), on the contrary, k1* and k3* could be treated as a linear component, that is, the dependency on the dynamic amplitude could be neglected. Then, while leaving all other model parameter as constant except k2*, a set of k*2 for each input level(i.e. 1.0, 2.0 m/s2 (RMS)) can be obtained. The fitted results are represented in Fig.3.2.
2 46 8 10 12 14 16 18 20 22 24 26 28 30 0
20 40 60 80 100
120 Input level: 2.0 m/s2 (RMS)
Frequency [Hz]
0 20 40 60 80 100
120 Input level: 1.0 m/s2(RMS)
Magnitude of apparent mass, [kg]
0 20 40 60 80 100 120
Input level: 0.5m/s2(RMS) Measured Fitted
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 -100
-80 -60 -40 -20 0 20
Input level: 2.0 m/s2(RMS)
Frequency [Hz]
-100 -80 -60 -40 -20 0 20
Input level: 1.0 m/s2(RMS)
Phase of apparent mass, [Deg]
-80 -60 -40 -20 0 20
Input level: 0.5 m/s2(RMS) Measured Fitted
Fig.3.2. Curve fitting results of the measured apparent mass with the parallel three DOF model
3.2. Derivation of the Nonlinear Mathematical Model for Seated Human Body For the estimation of the nonlinear relation between k2* and the displacement, the relative displacement, u=x2−x1, which acts on k*2 should be calculated with the input acceleration level at the base. To this end, if the displacement of the base could be known, the equation for the estimation of the relative displacement ,u, is easily derived from the parallel three DOF model(Eq.(3.1)),
2
1 2 1
1
( ) ( ) ( )
( ) ( )
( ) ( ) ( )
uu zz
X f X f X f
G f G f
Z f X f Z f
= − (3.1)
where, Guu( )f and Gzz( )f are PSD(Power Spectral Density) of the relative displacement and the displacement of the base, respectively. X1( )f ,X2( )f and Z f( ) are the Fourier transform of x1, x2, z all which were defined in the parallel three DOF model(Fig.3.1) However, we know the acceleration measured at the base, not the displacement. Thus, the double integration in the frequency domain would yield the displacement of the base(Eg.(3.2)).
4
( ) 1 ( )
zz aa
G f G f
=ω (3.2)
Fig.3.1. The Approximated first mode shape x1
k*2
m1
m2 m3
k*1
k*3
x2 x3
z, f
Where, Gaa( )f is PSD of acceleration at base. But, if we use the measured acceleration signals at the base, noisy signals dwelling in the low frequency ranges will be amplified by the double integration. Thus, it is appropriate to replace the measured signal with the band-limited ideal white noise for the prevention of noise amplification. Hereby, the estimated PSD of the relative displacement, Guu( )ω corresponding to each input level is represented in Fig.3.3. And, from Fig.3.3, it could be observed that the relative displacement has one dominant frequency content.
Thus, it is feasible to say that the RMS and the peak value of relative displacement, u have the relation of
peak 2 rms
u ≅ u (3.3)
From the approximated relation above, the nonlinear relation between k2* and the relative displacement(or dynamic amplitude) can be estimated at the three points corresponding to the three input levels(Fig.3.4).
5 10 15 20 25 30
1E-11 1E-10 1E-9 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3
G uu(f) [ m2 / Hz ]
Frequency [Hz]
Input level: 0.5 m/s2 (RMS) Input level: 1.0 m/s2 (RMS) Input level: 2.0 m/s2 (RMS)
0.0005 0.0010 0.0015 0.0020 0.0025 0.0030
0.34 0.35 0.36 0.37
η 2
Dynamic amplitude [m]
0.0005 0.0010 0.0015 0.0020 0.0025 0.0030
18k 19k 20k 21k 22k 23k 24k
k2 [N/m]
Fig.3.3. PSD of relative displacement, u Fig.3.4. The nonlinear relation between
*
k2and displacement
The fitted results with the equation defined in Eg.(3.4) are represented with solid lines in Fig.3.4, and the coefficients in Eg.(3.4) are given in Table 3.1.
/ 2
/ 2
( ) ( )
u ck
k k
u c
k u B A e
u Bη A eη η η
−
−
= +
= + (3.4)
Coefficient Ak Bk Ck Aη Bη Cη
Value 11059 17732 8.9785e-4 0.07151 0.3377 7.3897e-4 Table.3.1. Coefficients for the nonlinear component, k2*
Finally, with the aid of Eq.(3.4) which describes the behavior of the nonlinear component inserted in the parallel three DOF model, the equations of motion could be derived as follows,,
( )
( ) ( ) ( )
( )
( )
*
1 1 2 2 3 3 1 1
* * *
1 1 1 1 2 1 2 3 1 3
*
2 2 2 2 1
*
3 3 3 3 1
( )
( ) 0
( ) 0
0
m x m x m x k z x f t
m x k x z k u x x k x x
m x k u x x
m x k x x
+ + = − =
+ − + − + − =
+ − =
+ − =
(3.5)
where, the parameters in Eq.(3.5) are summarized in Table.3.2.
Parameter m1[kg] η1 k1[N/m] m2[kg] m3[kg] η3 k3[N/m]
Value 10.97 0.1043 697938.3 21.75 30.95 1.795 63561.7
Table.3.2. Model parameters for the nonlinear mathematical model of seated human body
4. SUMMARY
For the development of dummy to be used during unmanned driving for ride comfort assessment, vibration characteristics of seated human bodies measured in terms of apparent mass were analyzed. It was found that a resonance frequency of human bodies decreased with increase of the input level at the seat. Thus, it was thought that the nonlinearity should be included in the dynamic model, in order that the model may be useful for various levels of input excitations..
Besides, it was thought that rubber-like materials could be applied to the dummy development because they show similar nonlinear behavior.
Several lumped mass models were then considered for the seated human bodies. Among those, a three DOF model including a rubber-like element for the nonlinear behavior was decided to be most desirable from viewpoint of accuracy and easiness of realization. The behavior of the nonlinear element was defined as a function of dynamic amplitude between the adjacent two masses at each level of excitation at the seat. From analysis of these results, a dynamic model with a nonlinear complex stiffness element was derived to describe the nonlinearity of seated human body. With this nonlinear three DOF model, development of a dummy which can substitute the human body for various input levels would be realized rather easily.
REFERENCES
[1] Suyon Song, Dynamic Characterization of Human Body in Seated Posture in Vertical Direction and Its Application to Prediction of Transmissibility of Seat Foams, M.S. Thesis (2000), KAIST 2001.
[2] Neil J. Mansfield and Michael J. Griffin, Non-linearities in Apparent Mass and Transmissibility during Exposure to Whole-body Vertical Vibration, Journal of Biomechanics 33, pp. 933-941, (2000).
[3] Y. Matsumoto and M. J. Griffin, Effect of Muscle Tension on Non-linearities in the Apparent masses of Seated Subjects Exposed to Vertical Whole-body Vibration, Journal of Sound and Vibration 253, pp. 77-92, (2002).
[4] Ahid D. Nashif, David I. G. Jones and John P. Henderson, Vibration Damping, John Wiley &
Sons, the United States of America 1985, pp. 89-94.
[5] British Standards Institution, Measurement and Evaluation of Human Exposure to Whole-body Mechanical Vibration and Repeated Shock BS 6841, London 1987.
