## 2 Human-Seat Coupling Modelling
### Mechanical Model Analysis  
The aerodynamic forces have little effect on the dynamic behavior of the seat and ATD throughout the crash. Therefore, the effect of aerodynamic forces on both is ignored when the FEM (Finite Element Method) is used to complete the study of the collision. The established differential equation of motion is shown in Equation (4). **M**, **C**, **K** are the mass matrix, the damping matrix and the stiffness matrix, respectively. \(\ddot{\mathbf{u}}\), \(\dot{\mathbf{u}}\), \(\mathbf{u}\), \(\mathbf{U}\) are the acceleration, the velocity, the displacement and the position of the node in the system. The relationship among the three is shown in Equations (5) and (6). \[\mathbf{M}\ddot{\mathbf{u}}+\mathbf{C}\dot{\mathbf{u}}+\mathbf{K}\mathbf{u}= \mathbf{F}^{\mathrm{e}}(t) \tag{4}\]  
\[\dot{\mathbf{u}}(\mathbf{U},t)=\frac{\partial\mathbf{u}(\mathbf{U},t)}{ \partial t} \tag{5}\]  
\[\ddot{\mathbf{u}}(\mathbf{U},t)=\frac{\partial\dot{\mathbf{u}}(\mathbf{U},t)} {\partial t}=\frac{\partial^{2}\mathbf{u}(\mathbf{U},t)}{\partial t^{2}} \tag{6}\]  
Equation (8) can be derived from Equations (4) and (7). \(\mathbf{F}^{\mathrm{e}}\), \(\mathbf{F}^{\mathrm{f}}\), \(\mathbf{F}^{\mathrm{r}}\) are the external forces, the internal forces and the remaining force, respectively. \[\mathbf{M}\ddot{\mathbf{u}}=\mathbf{F}^{\mathrm{e}}-\mathbf{F}^{\mathrm{f}} \tag{7}\]  
\[\ddot{\mathbf{u}}=\mathbf{M}^{-1}(\mathbf{F}^{\mathrm{r}}) \tag{8}\]  
Equations (9) and (10) can be derived by the central difference method [29]. The value of \(\mathbf{u}_{t+\Delta t}\) can be obtained when \(\ddot{\mathbf{u}}_{t}\), \(\dot{\mathbf{u}}_{t}\), \(\mathbf{u}_{t}\) are known. \[\mathbf{u}_{t+\Delta t}=\mathbf{u}_{t}+\dot{\mathbf{u}}\Delta t+\frac{1}{2} \ddot{\mathbf{u}}_{t}\Delta t^{2} \tag{9}\]  
\[\ddot{\mathbf{u}}_{t+\Delta t}=\dot{\mathbf{u}}_{t}+\frac{1}{2}\big{(}\ddot{ \mathbf{u}}_{t}+\ddot{\mathbf{u}}_{t+\Delta t}\big{)}\Delta t \tag{10}\]  
In this paper, the following three principles were followed when establishing the constitutive equation [30]. 1. The principle of determinacy assumes that the stress \(\sigma\) of a point in an object is determined entirely by the state and deformation process of the neighborhood of the mass point. 2. The principle of local action assumes that the influence of each other point decreases as the distance increases when calculating the stress or another physical quality of a point. In other words, the material response at a given point in the continuum is determined entirely by the values of the field variables at that point. 3. The principle of material objectivity assumes that the constitutive equation is completely determined by the nature of matter and does not change as the observer changes. In coordinate systems for relative motion, the constitutive equation of the matter has the same form. There will be a large impact during the crash which will cause the structure to deform drastically. When the stress exceeds the yield limit of the material, the material moves from the elastic phase to the plastic phase. During the elastic phase, the relationship between the stress and the strain is linear and the elastic deformation is recoverable. During the plastic phase, the relationship between the stress and the strain is nonlinear and the plastic deformation is irreversible. When the key energy-absorbing structure of the seat absorbs energy, the energy dissipated by plastic deformation should be much larger than the energy stored by elastic deformation. In this paper, the generalized von Mises yield criterion wasselected to determine when the plastic deformation of the structure occurs. Its expression in the three-dimensional principal stress space is shown in Equation (11). \[F^{0}\big{(}\sigma_{ij}\big{)}=1/6\Big{[}\big{(}\sigma_{1}-\sigma_{2}\big{)}^{2}+ \big{(}\sigma_{2}-\sigma_{3}\big{)}^{2}+\big{(}\sigma_{1}-\sigma_{3}\big{)}^{2} \Big{]}-1/3\sigma_{s}^{2}=0 \tag{11}\]  
\(\sigma_{ij}\) is the stress tensor and \(\sigma_{s}\) is the initial yield stress of materials.