## 2 Integrated Seat Suspension and MR Damper Model  
The proposed integrated semi-active seat suspension model, as shown in Figure 1, consists of the quarter-car suspension model and the semi-active seat suspension with the MR damper mounted over it (see, Figure 1a). The human model with 4-DOF was also taken into account. The human body was modelled as a 4-DOF system consisting of thighs, lower torso, high torso and head (see Figure 1b). The arms and legs were integrated with the upper torso and thighs. In this paper, only the vertical motion of the system was considered for analysis. The notations used for the integrated model are mentioned in Table 1. The values of the parameters of the system for the numerical simulation were taken from [PERSON] (2016). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Parameter** & **Value** \\ \hline \(c_{u}\) & Damping of tire & 0 (Ns/m) \\ \hline \(c_{s}\) & Damping of suspension & 2000 (Ns/m) \\ \hline \(F_{mr}\) & MR damper force & - \\ \hline \(c_{c}\) & Damping of seat cushion & 200 (Ns/m) \\ \hline \(c_{1}\) & Damping of buttocks and thighs & 2064 (Ns/m) \\ \hline \(c_{2}\) & Damping of lumber spine & 4585 (Ns/m) \\ \hline \(c_{3}\) & Damping of thoracic spine & 4750 (Ns/m) \\ \hline \(c_{4}\) & Damping of cervical spine & 400 (Ns/m) \\ \hline \(k_{u}\) & Tyre stiffness & 180,000 (N/m) \\ \hline \(k_{\rm s}\) & Sprung mass stiffness & 10,000 (N/m) \\ \hline \(k_{\rm ss}\) & Seat suspension stiffness & 31,000 (N/m) \\ \hline \(k_{\rm c}\) & Seat cushion stiffness & 18,000 (N/m) \\ \hline \(k_{1}\) & Buttocks and thigh stiffness & 90,000 (N/m) \\ \hline \(k_{2}\) & Lumber spine stiffness & 162,800 (N/m) \\ \hline \(k_{3}\) & Thoracic spine stiffness & 183,000 (N/m) \\ \hline \(k_{4}\) & Cervical spine stiffness & 310,000 (N/m) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of integrated seat suspension system. Figure 1: Integrated model of quarter car semi-active seat suspension with the human body model: (**a**) integrated quarter car semi-active seat suspension model; (**b**) human body model. The dynamical equations of the integrated model were derived using [PERSON]'s second law of motion. \[m_{u}\ddot{z}_{u}=-k_{u}(z_{u}-z_{r})-c_{u}(z_{u}-z_{r})+k_{s}(z_{s}-z_{u})+c_{s} (z_{s}-z_{u}) \tag{1}\]  
\[m_{s}\ddot{z}_{s}=-k_{s}(z_{s}-z_{u})-c_{s}(z_{s}-z_{u})+k_{ss}(z_{ss}-z_{s})+F_{d} \tag{2}\]  
\[m_{f}\ddot{z}_{ss}=-k_{ss}(z_{ss}-z_{s})-F_{d}+k_{c}(z_{c}-z_{ss})+c_{c}\Big{(}z_ {c}-z_{f}\Big{)} \tag{3}\]  
\[m_{c}\ddot{z}_{c}=-k_{c}(z_{c}-z_{ss})-c_{c}\Big{(}z_{c}-z_{f}\Big{)}+k_{1}(z_{1 }-z_{c})+c_{1}(z_{1}-z_{c}) \tag{4}\]  
\[m_{1}\ddot{z}_{1}=-k_{1}(z_{1}-z_{c})-c_{1}(z_{1}-z_{c})+k_{2}(z_{2}-z_{1})+c _{2}(z_{2}-z_{1}) \tag{5}\]  
\[\ddot{m}_{2}\ddot{z}_{2}=-k_{2}(z_{2}-z_{1})-c_{2}(z_{2}-z_{1})+k_{3}(z_{3}-z_{ 2})+c_{3}(z_{3}-z_{2}) \tag{6}\]  
\[\ddot{m}_{3}\ddot{z}_{3}=-k_{3}(z_{3}-z_{2})-c_{3}(z_{3}-z_{2})+k_{4}(z_{4}-z_ {3})+c_{4}(z_{4}-z_{3}) \tag{7}\]  
\[\ddot{m}_{4}\ddot{z}_{4}=-k_{4}(z_{4}-z_{3})-c_{4}(z_{4}-z_{3}) \tag{8}\]  
Equations (1) and (2) represent the dynamics of the quarter car model. The dynamics of the seat suspension are represented using Equations (3) and (4). Human body dynamics are represented by Equations (5) to (8). Here, \(F_{d}\) is the force generated by the damper. The damper force is given by Equation (9):  
\[F_{d}=\left\{ \begin{array}{cc}c_{s}(z_{ss}-z_{s})&\qquad\text{(Passive suspension)}\\ F_{mr}&\qquad\text{(semi-active suspension)}\end{array} \right. \tag{9}\]  
In this model, the viscous damping, \(c_{ss}\), was considered as zero while analysing the performance of the MR damper. The MR damper generates the controlled force, \(F_{mr}\). However, to compare the performance of the semi-active seat suspension with the passive seat suspension, a non-zero value of \(c_{ss}\) was considered only while analysing the passive seat suspension.