## 2 Human-Seat Coupling Modelling
### Mechanical Model Analysis
During the plastic phase, the flow rule [31] and the hardening rule [32] are used to determine how the subsequent yield surface of the material changes. The relationship between the plastic strain increment and the current stress and strain increment is established on this basis. The detail of the flow rule is shown in Equation (12). The isotropic hardening rule states that the load surface spreads uniformly outward in all directions. At the same time, the shape, the center and the position of the surface remain unchanged. \[\mathrm{d}e_{ij}^{p}=\mathrm{d}\lambda\frac{\partial F}{\partial\sigma_{ij}} \tag{12}\]  
\(\mathrm{d}e_{ij}^{p}\) is the component of the plastic strain increment. \(\mathrm{d}\lambda\) is a certain quantity which is associated with the hardening rule. \(F\) is the subsequent yield function. \(\frac{\partial F}{\partial\sigma_{ij}}\) is the vector whose direction is the same as the normal direction of the subsequent yield surface. The energy absorbed by the key energy-absorbing structure through plastic deformation is shown in Equation (13). \[W^{e}=\int_{V}\sigma_{ij}e_{ij} \tag{13}\]  
The D'Alembert principle [33], which is also known as the Lagrange-d'Alembert principle, states that if at any given moment the inertia force is added to the active forces and the reactions of the constraints acting on a particle, the resultant force system will be in equilibrium. The D'Alembert principle provides a method of solving problems of dynamics by developing equations of motion in the form of equations of equilibrium. According to the above principle, the relationship among them satisfies Equation (14). \(\mathbf{F}_{A}^{a}\) and \(\mathbf{F}_{A}^{n}\) are the active force and the constraining force, respectively. \[\dot{\mathbf{u}}(\mathbf{U},t)=\frac{\partial\mathbf{u}(\mathbf{U},t)}{ \partial t} \tag{5}\]0  
The general kinetic equation of the system of the mass point can be obtained by making a vector dot product of both ends of Equation (14) and the virtual velocity of the mass and integrating the equation for each mass. See Equation (15) for details. A multibody system can be considered as a system of the mass point, which includes a system of the mass point corresponding to each rigid body in the whole system. The general kinetic equation of the multibody system is shown in Equation (16). \[\dot{\mathbf{u}}(\mathbf{U},t)=\frac{\partial\mathbf{u}(\mathbf{U},t)}{ \partial t} \tag{5}\]1  
\[\dot{\mathbf{u}}(\mathbf{U},t)=\frac{\partial\mathbf{u}(\mathbf{U},t)}{ \partial t} \tag{5}\]2  
When the generalized coordinates of the system are constrained by equations (i.e., \(\Phi(\mathbf{q},t)=0\)), Equation (16) is no longer appropriate. The kinetic equation under the restraint system is shown in Equation (17) [34]. \[ \begin{bmatrix}\mathbf{M}&\mathbf{\Phi}_{q}^{T}\\ \mathbf{\Phi}_{q}&0\end{bmatrix} \Bigg{\{} \begin{array}{c}\ddot{\mathbf{q}} \\ -\lambda\end{array} \Bigg{\}}=\Bigg{\{} \begin{array}{c}\mathbf{F}\\ -\dot{\mathbf{\Phi}}_{a}\dot{\mathbf{q}}-\dot{\mathbf{\Phi}}_{t}\end{array}  \Bigg{\}} \tag{17}\]  
\(\mathbf{F}\) is the generalized force matrix. \(\Phi_{q}\) is the restraint Jacobi matrix. \(\lambda\) is the Lagrange multiplier vector whose dimension is equal to the number of constraint conditions. The expression for the work enacted by the force on the ATD during the impact, as shown in Equation (18), can be obtained. \(\mathbf{F}_{p}\) is the compressive load between the lumbar column and the pelvis. \[W_{human}=\int_{s_{p}}\mathbf{F}_{p}d\mathbf{q}+\sum_{i}\int_{s_{i}}\mathbf{F} _{i}d\mathbf{q} \tag{18}\]  
Finally, according to above equations (i.e., Equations (1)-(3), (13) and (18)), a link between the spine load of the ATD and the energy absorbed by the key energy-absorbing structure of the helicopter seat is established through the law of conservation of energy. Additionally, \(\mathbf{F}_{p}\) will change over time.