This study focuses on the biodynamic responses of a seated human model to wholebody vibrations in a vehicle. Fivedegreeoffreedom nonlinear equations of motion for a human model were derived, and human parameters such as spring constants and damping coefficients were extracted using a threestep optimization processes that applied the experimental data to the mathematical human model. The natural frequencies and mode shapes of the linearized model were also calculated. In order to examine the effects of the human parameters, parametric studies involving initial segment angles and stiffness values were performed. Interestingly, mode veering was observed between the fourth and fifth human modes when combining two different spring stiffness values. Finally, through the frequency responses of the human model, nonlinear characteristics such as frequency shift and jump phenomena were clearly observed.
Vibrational characteristics of seated humans are an important consideration in the automotive industry because they play a major role in riding comfort. Furthermore, recent significant advances in electric and autonomous vehicles may affect the perception and emotions of occupants in a vehicle. For example, in conventional vehicles, combustion engineinduced vibration and noise act in concert to mask roadinduced vibration and noise; generally, this does not occur with electric vehicles. Thus, we can expect that drivers and passengers will be more sensitive to roadinduced vibrations, and that the importance of analyzing vibrational perception in vehicles will increase.
The characteristics of the dynamic responses of a seated human body are mainly affected by lowfrequency vibrations (below 50 Hz). A number of studies have also found that the fundamental frequency of a seated human exposed to wholebody vibration is lower than 10 Hz [
Various finite element models of the human have been designed to investigate the complex characteristics of a seated human. Vavalle et al. [
To overcome the disadvantages of finite element models, lumped parameter models consisting of masses, springs, and dampers have been widely emphasized in various studies. Wei and Griffin [
Lumped human models with rotational degreesoffreedom have been studied extensively to interpret foreandaft and pitch movements in the sagittal plane. Matsumoto and Griffin [
To develop the human model, we used a Lagrangian formulation to derive the nonlinear equations of motion for a fivedegreeoffreedom model. The spring constants and damping coefficients were extracted from experimental data in the literature using an optimization process. The natural frequencies and mode shapes were also calculated from the linearized human model. In addition, several parametric studies were performed. Finally, we calculated the frequency response curves of the nonlinear human model, and then compared those of the linear model, nonlinear model, and the experiment.
In order to investigate dynamic characteristics of a seated human, we developed a lumped parameter model consisting of masses, dampers, and springs. In the proposed model, we considered three rigid bodies—a head, trunk, and lower body (including the thighs and pelvis). The trunk and lower body are the suggested measurement points of vibrations in BS 6841, which is the standard for assessing human vibrations [
Based on these assumptions, we developed a fivedegreeoffreedom human model consisting of
The proposed fivedegreeoffreedom seated biodynamic human mathematical model (red colored symbols: generalized coordinates).
The combination of seat foam and human skin was considered as the translational springs and dampers. All springs assumed to be massless and frictionless were in nonstretched and nonrotated conditions at the initial configuration. Although seat foam has nonlinear viscoelastic properties [
It is necessary to define translational spring deformation values to reflect actual deformation characteristics of the seat foam and tissue. Unlike vertical human models comprising onedimensional motion in vertical directions, each segment of the proposed fivedegreeoffreedom model is able to represent horizontal and rotational motions. When the deformation value of the translational spring is defined by the distance between two connected points, the spring forces are generated in the diagonal direction under this definition. In the case of actual foam and tissue deformation, when the contact points are moved, the foam of the seat located at the moved position generates the force to support a human body in the normal direction. To this end, the tangential force of the translational spring was disregarded, and the deformation of the translational spring in the normal direction of the seat floor and seatback was used for calculating the dynamic responses (Figure
Directional definition of the translational spring deflections.
The nonlinear equations of motion for the fivedegreeoffreedom mathematical model were derived using the Lagrange equation. The locations of the center of mass at each segment are
The elements of each matrix are given in Appendix
Prior to examining the dynamic characteristics using nonlinear equations of motion, linearization was performed. It is not a simple task to simulate the nonlinear human model to obtain dynamic characteristics such as the natural frequency, mode shape, and parametric sensitivity, and excessive computational time would be required. Therefore, we carried out the linearization by expending the Taylor series at the initial configurations, which makes it possible to represent the equations of motion with a set of generalized coordinates. In the linearization of nonlinear equations of motion, the following approximations were considered:
The elements of each matrix are given in Appendix
Thus, we can calculate the natural frequency and mode shape of the human model using the mass and stiffness matrix. The seattohead (STH) transmissibility of the linear model can be computed from Equation (
The inertial and geometric properties, stiffness, and damping coefficients are crucial parameters that determine the dynamic responses of the proposed human model. In this study, we determined the human parameters through the following three steps:
Determination of inertial and geometric properties
Extraction of stiffness values
Extraction of damping coefficients.
The inertial and geometric parameters were measurable using the measurement tools; further, we chose the mass, moment of inertia, and human segment length data from the reported literature related to anthropometry. By contrast, the stiffness and damping coefficients are not easy to measure experimentally, and thus we extracted these parameters by using the optimization process from the experimental STH transmissibility results [
In the initial step, we determined the mass, moment of inertia, and segment length values. In the second and third step to obtain the stiffness and damping coefficients, the experimental subjects in the wholebody experiments were Korean males in their late 20s [
Inertial properties and length data of the fivedegreeoffreedom human model.
Property  Value  

Mass (kg) 

10.49 

33.98  

6.67  


Mass moment of inertia (kg·m^{2}) 

0.23 

2.05  

0.03  


Length (mm) 

598.60 

571.70  

217.10  

88.00  

459.80  

100.00  

478.90  

156.20  

224.00 
In this study, the springs and dampers underneath the lower body and the springs and dampers connected between the backrest and trunk were set to have different stiffness values and damping coefficients. The foam of the seatback shows the different forcedisplacement relationship from the seat pad [
To identify the stiffness and damping coefficients, the experimental STH transmissibility was obtained from the reported literature [
The stiffness of the fivedegreeoffreedom human model was determined based on the first and second natural frequencies (4.2 Hz and 7.5 Hz) from the experimental results. The following objective function was used in the process for determining the stiffness values:
Natural frequencies of the proposed human model and experiment.
Mode  Natural frequency (Hz)  

Proposed model  Experiment  
1  4.20  4.20 
2  7.50  7.50 
3  9.99  — 
4  19.70  — 
5  21.35  — 
Mode shapes of the fivedegreeoffreedom human model (solid line: initial configuration; dashed line: mode shape configuration; black color: head; blue color: trunk; red color: lower body).
The objective function for the extraction of the damping coefficient is as follows:
Initial and optimized values of the stiffness and damping coefficients.
Parameter  Parameter  Initial value  Optimized value 

Stiffness (kN/m, kNm/rad) 

80.00  66.60 

100.00  95.54  

2.00  1.42  

1.00  1.12  


Damping coefficient (kNs/m, kNms/rad) 

1.50  0.89 

1.50  0.94  

0.30  0.30  

0.20  0.20 
It can be seen that the first natural frequency and the maximum amplitude of the STH transmissibility of the human mathematical model calculated from the optimized parameters are consistent with those of the experimental STH transmissibility, as shown in Figure
STH transmissibility of experimental results [
Understanding how parameters affect the dynamic characteristics of the human model is an important issue in the mechanical approach to explaining the dynamic responses of the human body. Such understanding also requires parametric analysis of the human mathematical model, which is achieved by varying the human parameters. Thus, we analyzed the variations in the natural frequencies of the human model according to changes in the parameters, which mainly affect the dynamic responses of the proposed model. In order to conduct the parameter study, we selected the inclination angles of the human model, including seat pan, backrest and head angles, and the translational and torsional stiffness values.
Parameter studies focused on the angle of the seat pan, backrest, and head were performed in order to analyze the variations in natural frequencies caused by varying the sitting posture. In the linear human model, the angles of the seat pan, backrest, and head were defined as
Figure
Natural frequencies of the fivedegreeoffreedom linear model with respect to the seat pan, backrest, and head angle (dotted vertical green line: baseline angle; blue line in the graph in the second column of the first row: experimental results [
The changes in natural frequencies with respect to the seat pan angle are shown in the first column of Figure
The second column of Figure
The variations in the natural frequencies with respect to the head angle were also calculated, as shown in the third column of Figure
To investigate the effect of the translational and torsional spring constants, we analyzed the change in the natural frequencies according to the change in stiffness (Figure
Natural frequency of the human model with respect to the translational and torsional stiffness values (dotted black vertical and horizontal line: baseline values of the stiffness; the top through the bottom rows, respectively, represent the first through the fifth modes).
The first natural frequency increases with an increase in
The second natural frequency was sensitive to
Interestingly, the veering phenomenon was observed for variations in
Mode veering between the fourth and fifth modes for variations in
In order to investigate the veering quantitatively, a veering index value was calculated from the combination of the modal dependence factor (MDF) and the crosssensitivity quotient (CSQ) [
The MDF was also defined as
Figure
Veering index of the fourth and fifth modes with respect to
We carried out dynamic analysis of a fivedegreeoffreedom nonlinear human model in the frequency domain to investigate its nonlinearity. Because nonlinear dynamic systems may have multiple responses at the same excitation frequency, steadystate responses are needed to reach the solution for the frequency response function. We also used the optimized parameters of the human model in the analysis of the linear model. For the nonlinear human model, we applied a nonlinear forcedisplacement relationship to the translational springs to more accurately reflect the actual deformation behavior of seat foam and tissue. When the deflection of the translational spring is larger than the initial length of the corresponding spring, it means that the connection between the human body and the seat would be physically removed. Therefore, we considered that the tension force of the translational spring was set to zero when
In order to compute the frequency response function of the fivedegreeoffreedom nonlinear model near the fundamental frequency, the base floor was excited by harmonic excitation within a frequency range of 0.1 Hz to 5 Hz at intervals of 0.1 Hz. To extract the steadystate responses of the human model, a sufficient excitation time must be taken into account to ensure steadystate responses; we set the simulation time at each frequency step to 50 seconds. The excitation frequency was also increased and then decreased uniformly using 0.1 Hz intervals. The displacement excitation was applied to the base of the nonlinear human model, and the amplitude of the excitation was changed in a range from 3 mm to 12 mm in 3 mm increments to analyze the dynamic characteristics of the human model with respect to the amplitude of the excitation displacement.
Figure
Amplitude of the vertical displacement of the hip joint and the angular displacement of the head with four different excitation amplitudes using the nonlinear human model (the circle and asterisk symbols indicate the amplitude values with increasing and decreasing excitation frequency, respectively).
Importantly, it can be observed that a frequency softening phenomenon occurs in which the first natural frequency decreases with the increase in the excitation amplitude. The fundamental frequency of the vertical displacements of the hip joint and the angular displacements of the head was, respectively, reduced by approximately 15% and 14% while the excitation amplitude increased from 3 mm to 12 mm. The first natural frequency exhibited a minor variation as the excitation amplitude increased from 3 mm to 6 mm. The fundamental frequencies in the translational and angular displacements were rapidly decreased when the excitation amplitude exceeded 6 mm. In addition, the jump phenomenon (in which the amplitude dramatically changes) becomes clear in the high excitation amplitude. As an example study, Mansfield et al. reported that the fundamental frequency of experimental subjects decreased by 22.2% from 5.4 Hz to 4.2 Hz based on median data and presumed that this frequency shifting would be caused by various complex causes such as muscle and tissue responses [
STH transmissibility of the linear and nonlinear proposed mathematical model and the experimental results [
Figure
It can be seen that the STH transmissibility of the nonlinear model is in good agreement with that of the linear model. The maximum amplitude of the nonlinear model is slightly lower than the amplitude of the experimental data and linear human model. This illustrates the fact that if the amplitude of the vertical excitation is relatively small, the linear human model could sufficiently represent frequency response function. However, it is more difficult to express the frequency response characteristics of the human body using the linear human model alone for large excitation amplitudes that cause shifting of the first natural frequency, and thus the necessity of the nonlinear model becomes more important. Further, the nonlinear model enables one to describe rapid changes in amplitude, such as those caused by the jump phenomenon. Therefore, it could be expected that the nonlinear model is more suitable for representing the vibrational characteristics of a seated human subjected to wholebody vibrations generated from the road, in which the profiles are changed from a smooth to a rough surface and vice versa.
In this study, we first derived the equations of motion for a human model. The determination of the human parameters was performed in three steps. The inertial and geometrical parameters were selected on the basis of anthropometry reference data comprised of measurable properties, and the stiffness and damping coefficients were extracted from the experimental STH results using the optimization process. The mode shapes were also obtained using the mass and stiffness matrix of the linearized model.
In addition, we analyzed the variations in the natural frequencies of the linear human model when human parameters were varied. For this parametric study, the inclination angle and stiffness values were considered as the prominent parameters in the dynamic characteristics of the human model. As a result, the first natural frequency is most sensitive to the backrest angle, represented by spring stiffness value
The frequency response functions of the nonlinear human model were presented using the steadystate amplitude of translational and angular displacements caused by harmonic base excitation. We note that frequency shifting was observed in the first mode, and various studies have reported similar phenomenon. Thus, the proposed human model could be reasonably expected to accurately exhibit the dynamic response characteristics of a seated human.
The elements of the matrix in Equation (
The elements of the mass, damping, and stiffness matrix of the fivedegreeoffreedom linear model are given by
The length and thickness data of the Korean male are accessible at
The authors declare that they have no conflicts of interest.
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (Ministry of Education) (No. 2015R1D1A1A01060582).