# Dynamic Characteristics Analysis of a Rigid Body System with Spatial Multi-Point Supports
## 2 Dynamic Model of a Rigid Body System with Multi-Point Elastic Supports
### Derivation of Frequency Equation Based on Energy Equations  
The overall displacement of the simplified rigid body is composed of the displacement \(\mathbf{r}=\left(r_{x},r_{y},r_{z}\right)^{\mathrm{T}}\) and the rotation \(\mathbf{\theta}=\left(\theta_{x},\theta_{y},\theta_{z}\right)^{\mathrm{T}}\) around the center of mass. According to the overall displacement of the rigid body, the vibration displacement of the n-th support can be obtained by:  
\[\mathbf{r}_{\mathrm{n}}=\mathbf{r}+\mathbf{\theta}\times\mathbf{d}_{\mathrm{n }}=\left(r_{x},r_{y},r_{z}\right)+\left(\theta_{x},\theta_{y},\theta_{z} \right)\times\left(d_{nx},d_{ny},d_{nz}\right) \tag{1}\]  
where \(r_{x},r_{y}\), and \(r_{z}\) represent the displacements of the center of mass in x, y, and z directions under the global coordinate system, accordingly. \(\theta_{x}\), \(\theta_{y}\), and \(\theta_{z}\) represent the rotation anglesof the center of mass \(x\), \(y\), and \(z\) directions under the global coordinate system, respectively. \(d_{mx}\), \(d_{ny}\), and \(d_{nz}\) denote distances in \(x\), \(y\), and \(z\) directions from each support to the center of mass. The vibrational kinetic energy of the rigid body can be calculated as follows:  
\[T=\frac{1}{2}\dot{\textbf{r}}\cdot\textbf{m}\cdot\dot{\textbf{r}}+\frac{1}{2} \dot{\textbf{o}}\cdot\textbf{J}\cdot\dot{\textbf{o}} \tag{2}\]  
The elastic potential energy provided by supports can be calculated as follows:  
\[V=\frac{1}{2}\sum_{n=1}^{\text{N}}\textbf{r}_{n}\times\textbf{k}_{n}\times \textbf{r}_{n} \tag{3}\]  
By substituting Equation (1) into Equation (3), the elastic potential energy of the rigid body can be written as:  
\[V=\frac{1}{2}\sum_{n=1}^{\text{N}}\left(\textbf{r}+\textbf{o}\times\textbf{d} _{n}\right)\cdot\textbf{k}_{n}\cdot\left(\textbf{r}+\textbf{o}\times\textbf{d} _{n}\right) \tag{4}\]  
Expand Equations (2) and (4), and the vibrational kinetic energy \(T\) and elastic potential energy \(V\) of the rigid body can be written as:  
Figure 1: Definition of a rigid body system with multi-point elastic supports and its motion: (**a**) the definition of the global coordinate system of a rigid body and the local coordinate system of each support; (**b**) motion definition of the rigid body. Figure 2: The definition of each support spatial location and its stiffness decomposition (take the top-left support A as an example): (**a**) definition of each support spatial location; (**b**) support stiffness decomposition. \[T=\frac{1}{2}m\dot{r_{x}}^{2}+\frac{1}{2}m\dot{r_{y}}^{2}+\frac{1}{2}m\dot{r_{z}}^{2 }+\frac{1}{2}\dot{r_{z}}^{2}+\frac{1}{2}\dot{r_{x}}\dot{\theta_{x}}^{2}+\frac{1} {2}\dot{J_{y}}\dot{\phi_{y}}^{2}+\frac{1}{2}\dot{J_{z}}\dot{\phi_{z}}^{2} \tag{5}\]  
where \(J_{x},J_{y}\), and \(J_{z}\) represent the moment of inertia components of the rigid body around in x, y, and z directions, under the global coordinate system, respectively. \[ \begin{array}{ll}V&=\sum\limits_{n=1}^{\text{N}}\frac{1}{2}k_{nx}\big{(}r_{ x}+d_{nz}\cdot\theta_{y}-d_{ny}\cdot\theta_{z}\big{)}^{2}+\sum\limits_{n=1}^{ \text{N}}\frac{1}{2}k_{ny}\big{(}r_{y}-d_{nz}\cdot\theta_{x}+d_{nx}\cdot\theta _{z}\big{)}^{2}+\\ &\sum\limits_{n=1}^{\text{N}}\frac{1}{2}k_{nz}\big{(}r_{z}+d_{ny}\cdot\theta_{ x}-d_{nx}\cdot\theta_{y}\big{)}^{2}\end{array}  \tag{6}\]  
where \(k_{mx}\), \(k_{ny}\), and \(k_{nz}\) denote the stiffness component of each support in \(x\), \(y\), and \(z\) directions under the global coordinate system. Substitute the above energy expressions into the Lagrange equations \(L=T-V\), and minimizing \(L\) yields the following eigenvalue equations:  
\[(\mathbf{K}-\omega^{2}\mathbf{M})\mathbf{u}=\mathbf{0} \tag{7}\]  
where \(\mathbf{K}\) is the stiffness matrix, \(\mathbf{M}\) is the mass matrix, and \(\mathbf{u}\) is the displacement matrix. \[\mathbf{u}=(r_{x},r_{y},r_{z},\theta_{x},\theta_{y},\theta_{z})^{\text{T}} \tag{8}\]  
\[\mathbf{M}=\left( \begin{array}{ccccc}m&&&&\\ &m&&&\\ &&m&&&\\ &&&J_{x}&&&\\ &&&J_{y}&\\ &&&&J_{x}\end{array} \right) \tag{9}\]  
\[\mathbf{K}=\left(