## 2 Development of the 7-DOF Human Model
### Derivation of Model Equations of Motion  
The equations governing motions in the 7-DOF human model were derived through the Lagrangian dynamics approach in the form:  
\[\frac{d}{dt}\left(\frac{\partial(KE)}{\partial\dot{q}_{k}}\right)-\frac{\partial (KE)}{\partial q_{k}}+\frac{\partial(PE)}{\partial q_{k}}+\frac{\partial D}{ \partial\dot{q}_{k}}=0(k=1,\;2,\cdots,7) \tag{1}\]  
where \(q_{k}=\{x_{h},z_{h},\theta_{1},\theta_{2},\theta_{3},\theta_{4},\theta_{5}\}\) are the generalized coordinates corresponding to the horizontal and vertical displacement of the hip joint and rotational displacement of the center of gravity of each body segment of the 7-DOF model. The kinetic energy (\(KE\)) of the system is given as:  
\[KE=\frac{1}{2}\overset{5}{\underset{i=1}{\sum}}m_{i}\Big{(}\dot{x}_{i}^{2}+ \dot{z}_{i}^{2}\Big{)}+\frac{1}{2}\overset{5}{\underset{i=1}{\sum}}j_{i}\dot{ \theta}_{i}^{2}, \tag{2}\]  
where \(x_{i}\) and \(z_{i}\) are the position vectors of the center of gravity of the body segments in the x-z plane derived as  
\[x_{1}=x_{h}+\frac{L_{1}}{2}\text{cos}\;\theta_{1}+L_{2}\text{cos}\;\theta_{2}, \;x_{2}=x_{h}+\frac{L_{2}}{2}\text{cos}\;\theta_{2},\]  
\[x_{3}=x_{h}+\frac{L_{3}}{2}\text{cos}\;\theta_{3},\;x_{4}=x_{h}+L_{3}\text{cos }\;\theta_{3}+\frac{L_{4}}{2}\text{cos}\;\theta_{4} \tag{3}\]  
\[x_{5}=x_{h}+L_{3}\text{cos}\;\theta_{3}+L_{4}\text{cos}\;\theta_{4}+\frac{L_{5 }}{2}\text{cos}\;\theta_{5},\]  
\[z_{1}=z_{h}+\frac{L_{1}}{2}\text{sin}\;\theta_{1}+L_{2}\text{sin}\;\theta_{2},\;z_{2}=z_{h}+\frac{L_{2}}{2}\text{sin}\;\theta_{2},\]  
\[z_{3}=z_{h}+\frac{L_{3}}{2}\text{sin}\;\theta_{3},\;z_{4}=z_{h}+L_{3}\text{sin }\;\theta_{3}+\frac{L_{4}}{2}\text{sin}\;\theta_{4}, \tag{4}\]  
\[z_{5}=z_{h}+L_{3}\text{sin}\;\theta_{3}+L_{4}\text{sin}\;\theta_{4}+\frac{L_{5 }}{2}\text{sin}\;\theta_{5}\]  
Equations of the potential energy (PE) and Rayleigh's dissipation function (D) of the system are given as:  
\[PE=\frac{1}{2}\overset{5}{\underset{i=2,j=1}{\sum}}k_{ij}\dot{\theta}_{ij}^{ 2}+\frac{1}{2}\overset{4}{\underset{i=1}{\sum}}k_{i}^{T}\dot{\varphi}_{i}^{2} +\overset{5}{\underset{i=1}{\sum}}m_{i}g(\Delta z_{i}), \tag{5}\]  
\[D=\frac{1}{2}\overset{5}{\underset{i=2,j=1}{\sum}}c_{ij}\dot{\theta}_{ij}^{2} +\frac{1}{2}\overset{5}{\underset{i=1}{\sum}}c_{i}^{T}\dot{\varphi}_{i}^{2}, \tag{6}\]  
where \(\dot{\phi}_{ij}\) and \(\phi_{i}\) are the deflections of the translational and torsional springs; \(g\) represents gravity; and \(\Delta z_{i}\) is the change in the value of \(z\)-axis position vector of the center of gravity of body segments. The deformation of the 7-DOF model under the influence of the vertical excitation, \(z_{b}\) is presented in \[KE=\frac{1}{2}\overset{5}{\underset{i=1}{\sum}}m_{i}\Big{(}\dot{x}_{i}^{2}+ \dot{z}_{i}^{2}\Big{)}+\frac{1}{2}\overset{5}{\underset{i=1}{\sum}}j_{i}\dot{ \theta}_{i}^{2}, \tag{2}\]0 The deflection terms, \(\dot{\phi}_{ij}\) and \(\phi_{i}\) are taken as the displacement of the position of the translational and torsional spring-damper as the model vibrates from the undeformed to the deformed configurations. Similarly, the position of the center of gravity, \(P_{i}\) with coordinates, \(x_{i}\) and \(z_{i}\) during vibrations is considered. The deflection of the torsional spring damper, \(\phi_{i}\) is derived as the angular deflection during vibrations in the form:  
\[KE=\frac{1}{2}\overset{5}{\underset{i=1}{\sum}}m_{i}\Big{(}\dot{x}_{i}^{2}+ \dot{z}_{i}^{2}\Big{)}+\frac{1}{2}\overset{5}{\underset{i=1}{\sum}}j_{i}\dot{ \theta}_{i}^{2}, \tag{2}\]1\[KE=\frac{1}{2}\overset{5}{\underset{i=1}{\sum}}m_{i}\Big{(}\dot{x}_{i}^{2}+ \dot{z}_{i}^{2}\Big{)}+\frac{1}{2}\overset{5}{\underset{i=1}{\sum}}j_{i}\dot{ \theta}_{i}^{2}, \tag{2}\]2  
where * denotes the initial value of the angle before vibration. On the other hand, the deflection of the translational spring-damper, \(\delta_{ij}\) is taken as the parallel displacement of the ends of the translational spring-damper with respect to the undeformed configuration.