## 3 Physical and Mathematical Model of a Single-Degree-of-Freedom Seat Suspension  
The operation of work machinery and vehicles necessitates the involvement of a human being, who serves as a subordinate decision-making element. The movement of the system is contingent upon the individual's behavioural responses, assessment of prevailing conditions, dexterity in performing manipulative actions and speed in making the appropriate decisions. Consequently, the introduction of an adequate vibration isolation system is intended to enhance human comfort and safety.  
The parametric estimation of the suspension characteristics of the seat, as illustrated in Figure 2, has led to the formulation of a physical model with one degree of freedom, as depicted in Figure 3. This model is deemed appropriate for the test bench structure. Consequently, the human anthropo-dynamic model has been reduced to a concentrated mass, \(\rm m_{c}\). The fundamental motion of the human with the seat is closely related to the fundamental motion of the carrier.  
The seat on which the human sits is modelled as a concentrated mass \(m_{f}\) of a passive deformable element in the form of a rheological Voigt-Kelvin model with the parameters \(k_{f1}\) and \(c_{f1}\), where \(k_{f1}\) is the stiffness coefficient and \(c_{f1}\) is the damping coefficient. The mass of the human-seat system is:  
\[m=m_{c}+m_{f} \tag{1}\]  
The human mass was assumed based on the parameters defined by [PERSON]  
The equilibrium equation is as follows:  
\[k_{f1}y_{f1 st}+mg=0 \tag{3}\]  
Equation (2) is the mathematical model describing the motion of the physical system shown in Figure 3. In this equation, there is a component mg on the right-hand side representing the weight of the passive deformable element in the form of a Voigt-Kelvin rheological model. Consequently, the static displacement \(y_{f1 st}\) of the mass m, obtained from the equilibrium Equation (3), must be included in the equation of motion (2). Without the \(y_{f1 st}\) component in Equation (2), the system model would be excited to oscillate even before the introduction of the force. The mass m is placed on the seat during the experimental tests and does not move. Only introducing an external force--in the model, these are the components \(y_{f0}\) and \(\dot{y}_{f0}\)--causes the system to move. Therefore, \(y_{f1}\) is the displacement of the mass m relative to the static displacement \(y_{f1 st}\).  
The frequency of free vibration is as follows:  
\[\omega_{*}=\sqrt{\omega_{0}^{2}-\delta^{2}} \tag{4}\]  
where:  
\[\omega_{0}=\sqrt{\frac{k_{f1}}{m}},\ \delta=\frac{c_{f1}}{2m}.\]  
Parameters of the model: \(m_{f}=8.0\) kg, \(m_{c}=69.6\) kg, \(g=9.81\) m/s\({}^{2}\).  
A mathematical model of the forcing mechanism is presented:  
\[y_{f0}=\frac{A_{w}}{\pi}\bigg{[}arctg\bigg{(}\frac{t-t_{H1}}{A_{H}\tau}\bigg{)} -arctg\bigg{(}\frac{t-t_{H2}}{A_{H}\tau}\bigg{)}\bigg{]} \tag{5}\]  
\[\dot{y}_{f0}=\frac{A_{H}\tau A_{w}}{\pi}\bigg{[}\frac{1}{A_{H}^{2}\tau^{2}+ \left(t-t_{H1}\right)^{2}}-\frac{1}{A_{H}^{2}\tau^{2}+\left(t-t_{H2}\right)^{ 2}}\bigg{]} \tag{6}\]  
where:  
\(A_{w}\)--amplitude of the forcing;  
\(A_{H}\)--aspect ratio;  
\(t_{H1},t_{H2}\)--times for which velocity reaches an extreme;  
\(\tau\)--duration of the extortion.  
A physical and mathematical model with two degrees of freedom was also formulated. It is presented in Appendix A.