# Dynamic Characteristics Analysis of a Rigid Body System with Spatial Multi-Point Supports
## 3 Structural Parameters Determination and Model Verification
### The Main Parameters of the Structural System
#### 3.1.2 Determination of Support Stiffness  
Calculate the stiffness of each support under local coordinate system. Take the top-left support A for example, \(x_{\text{A}}\), \(y_{\text{A}}\), and \(z_{\text{A}}\) form a new cartesian coordinate system, which is the local coordinate system A\({}_{0}\)-X\({}_{\text{A}}\)Y\({}_{\text{A}}\)Z\({}_{\text{A}}\) of the top-left support A, as shown in Figure 1a, where the \(x_{\text{A}}\)-axis is along the axial direction of the casing, the \(y_{\text{A}}\)-axis is along the normal direction of the casing, and the \(z_{\text{A}}\)-axis is the tangential direction of the casing. To obtain support stiffness of the mounting structure, the static analysis of each mounting structure is carried out via Workbench software. The support is modeled using solid187 elements and bolt holes connecting the support and casing that is set to be fixed (as shown in Figure 4a). Meanwhile, 1000 N force are applied at the mounting hole along the local coordinate axis \(x_{\text{A}}\), \(y_{\text{A}}\), and \(z_{\text{A}}\), respectively, to obtain the axial displacement \(x_{\text{A}}=0.083486\) mm \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Property** & **Symbol** & **Value** & **Unit** \\ \hline Mass & \(m\) & 11.76 & kg \\ Moment of inertia about the \(x\)-axis & \(J_{x}\) & 264,706 & kg\(\cdot\)mm\({}^{2}\) \\ Moment of inertia about the \(y\)-axis & \(J_{y}\) & 309,764 & kg\(\cdot\)mm\({}^{2}\) \\ Moment of inertia about the \(z\)-axis & \(J_{z}\) & 255,256 & kg\(\cdot\)mm\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Inertial parameters of the simplified rigid body. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Location (mm)**} & \multicolumn{4}{c}{**Mount \#**} \\ \cline{2-5}  & **A** & **B** & **C** & **D** \\ \hline \(d_{x}\) & \(-211\) & \(-211\) & 211 & 211 \\ \(d_{y}\) & \(-13.08\) & \(-7.54\) & 13.15 & \(-24.15\) \\ \(d_{z}\) & \(-96.38\) & 108.3 & \(-77.66\) & 120.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Definition of each support position in the global coordinate system. Figure 3: A simplified model of a rigid body object with multi-point supports. (as shown in Figure 4b), normal displacement \(y_{\rm A}\) = 0.00128 mm, and tangential displacement \(z_{\rm A}\) = 0.013327 mm of the top-left support A in the local coordinate system. Using the formula K = F/\(x\), the axial stiffness \(k_{\rm A^{\prime}}\) = F/\(x_{\rm A}\) = 1000 N/0.083486 mm = 1.20 \(\times\) 10\({}^{7}\) (N/m), the normal stiffness \(k_{\rm Ay^{\prime}}\) = F/\(y_{\rm A}\) = 1000/0.00128 = 7.80 \(\times\) 10\({}^{8}\) (N/m), and the tangential stiffness \(k_{\rm Az^{\prime}}\) = F/\(z_{\rm A}\) = 1000 N/0.013327 mm = 7.50 \(\times\) 10\({}^{7}\) (N/m) of the top-left support A in the local coordinate system can be obtained. \(\mathbf{K^{{}^{\prime}}}_{\mathbf{A}}\), \(\mathbf{K^{{}^{\prime}}}_{\mathbf{B}}\), \(\mathbf{K^{{}^{\prime}}}_{\mathbf{C}}\), and \(\mathbf{K^{{}^{\prime}}}_{\mathbf{D}}\), respectively, represent the support stiffness under the local coordinate system of corresponding support. The support stiffness under the local coordinate system is shown in Table 3.  
The direction angles of \(x_{\rm A}\)-axis in the global coordinate system are defined as \(\theta_{x_{\rm A}x}\),\(\theta_{x_{\rm A}y}\), and \(\theta_{x_{\rm A}z}\), which are the included angles between the \(x_{\rm A}\)-axis and the \(x\), \(y\), and \(z\) axes, respectively. Further, their cosine values are defined as the direction cosine of \(x_{\rm A}\)-axis. The direction cosines of the \(x_{\rm A}\) axis can be described as:  
\[n_{x_{\rm A}x} =\cos\theta_{x_{\rm A}x} \tag{11}\] \[n_{x_{\rm A}y} =\cos\theta_{x_{\rm A}y}\] (12) \[n_{x_{\rm A}z} =\cos\theta_{x_{\rm A}z} \tag{13}\]  
Correspondingly, the direction cosines of the \(y_{\rm A}\)-axis is \(n_{y_{\rm A}x}\), \(n_{y_{\rm A}y}\), and \(n_{y_{\rm A}z}\), and the direction cosines of the \(z_{\rm A}\)-axis is \(n_{z_{\rm A}x}\), \(n_{z_{\rm A}y}\), and \(n_{z_{\rm A}z}\).