## Appendix A Physical and Mathematical Model of a Human-Seat System with Two Degrees of Freedom  
After the parametric estimation of the suspension of the seat characteristics, a two-degree-of-freedom physical model was formulated as shown in Figure 11. The human anthropo-dynamic model is represented as a concentrated mass \(m_{c}\). The seat is modelled in the form of a concentrated mass \(m_{f}\), a passive deformable element in the form of a Voigt-Kelvin rheological model with the parameters \(k_{f1}\) and \(c_{f1}\) representing the suspension, and the cushions also in the form of a Voigt-Kelvin rheological model with parameters \(k_{f2}\) and \(c_{f2}\). The number of degrees of freedom resulting from the formulated structure of the model of a person sitting in a chair describing the perturbation of the fundamental motion in the vertical plane is two. An independent generalised coordinate \(y_{f1}\) is used to determine the vertical displacement of the concentrated mass \(m_{f}\) at any time. An independent generalised coordinate \(y_{c}\) is used to determine the vertical displacement of the concentrated mass \(m_{c}\) at each instant. The force acting on the seat suspension base from the support side is determined by the \(y_{f0}\) coordinate. The mathematical model is a discrete, physically non-linear, determined, time-varying, dissipative and non-sympathetic system. The equations of motion of the human-physical system in summation form are as follows:  
\[ \begin{array}{c}m_{f}\ddot{y}_{f1}+\Big{(}c_{f1}+c_{f2}\Big{)}\dot{y}_{f1}-c _{f2}\dot{y}_{c}+\Big{(}k_{f1}+k_{f2}\Big{)}\Big{(}y_{f1}+y_{f1 st}\Big{)}-k_{f2 }(y_{c}+y_{cst})\\ =c_{f1}\dot{y}_{f0}+k_{f1}y_{f0}-m_{f}\dot{g}\end{array}  \tag{11}\]  
\[ \begin{array}{c}m_{c}\ddot{y}_{c}-c_{f2}\dot{y}_{f1}+c_{f2}\dot{y}_{c}-k_{f2 }\Big{(}y_{f1}+y_{f1 st}\Big{)}+k_{f2}(y_{c}+y_{cst})=-m_{c}\dot{g}\end{array}  \tag{12}\]  
where:  
\(y_{f1 st}\)--static displacement of the seat,  
\(y_{cst}\)--static displacement of a person,  
\(g\)--the acceleration of gravity. The equilibrium equations are as follows:  
\[ \begin{array}{c}(k_{f1}+k_{f2}\Big{)}y_{f1 st}-k_{f2}y_{cst}=-m_{f}\dot{g}\\ -k_{f2}y_{f1 st}+k_{f2}y_{cst}=-m_{c}\dot{g}\end{array}  \tag{13}\]  
The natural frequencies are determined from the following characteristic equation:  
\[ \begin{array}{c}m_{f}m_{c}s^{4}+\Big{[}m_{f}k_{f2}+m_{c}(k_{f1}+k_{f2}) \Big{]}s^{2}+(k_{f1}+k_{f2}\Big{)}k_{f2}-k_{f2}^{2}=0\end{array}  \tag{14}\]  
Hence, conjugate imaginary elements: \(s_{1}=\pm\dddot{i}\omega_{01}\), \(s_{2}=\pm\dddot{i}\omega_{02}\). The mode-shape vectors of natural vibrations are:  
\[ \begin{array}{c}m_{f}m_{c}s_{*}^{4}+[m_{f}c_{f2}+m_{c}(c_{f1}+c_{f2})]s_{*} ^{3}+[m_{f}k_{f2}+m_{c}(k_{f1}+k_{f2})+(c_{f1}+c_{f2})\\ -c_{f2}^{2}]s_{*}^{2}+[(c_{f1}+c_{f2})k_{f2}+(k_{f1}+k_{f2})c_{f2}-(c_{f2}k_{ f2}t)^{2}]s_{*}\\ +(k_{f1}+k_{f2})k_{f2}-k_{f2}^{2}=0\end{array}  \tag{15}\]  
Hence, conjugate imaginary elements: \(s_{*1}=-h_{1}\pm\dddot{i}\omega_{*01}\), \(s_{*2}=-h_{2}\pm\dddot{i}\omega_{*01}\). The parameters for model Equation (15) are given in Table 1.  
The model of the force acting on the base of the seat suspension is analogous to that of the single-degree-of-freedom model. The course of variation as a function of time of the displacement of the human and seat masses relative to a given forcing for the initial values of pressure \(p\) = [3.75, 4.0, 4.3, 4.5, 5.0, 5.5, 6.25] bar was obtained after carrying out numerical simulations of the movement of the developed theoretical model. Figure A2 shows the variation as a function of time of the displacement of the human mass and seat against a given forcing for an initial pressure value of \(p\) = 3.75 and \(p\) = 4.3 bar. The graph shows the response of the system to the kinematic forcing. Results were obtained for initial pressure values of \(p\) = 4.5 bar and \(p\) = 5.0 bar (Figure A3). The results were obtained for initial pressure values of \(p\) = 5.5 bar and \(p\) = 6.25 bar (Figure A4).