## 2 Elastodynamic Model of Parallel Mechanism with Tube Structure
### Tube Model in MSA  
The first case is the outer tube-inner tube, as shown in Figure 2. The small sliding and small deflection hypotheses of the inner tube and outer tube were used, considering the structural vibration condition. The kinematic pair equivalent method was used to construct the tube model. According to the tube's cross-section shape, we used either a prismatic pair or a cylinder pair to simulate the contact of the tube surface. If the tube cross-section is  
Figure 1: Parallel mechanism in matrix structural analysis. in a circle shape, the inner tube can both rotate and translate along the outer tube. Then, a series of cylinder pairs were used to simulate the contact behavior between them. As shown in Figure 2, the cylinder pairs are added at the red dotted lines. If the cross-section is rectangular, it can only translate a small distance and cannot rotate. Therefore, a series of prismatic pairs can be used. In the following, cylinder pairs are used as an example. Both the outer and inner tubes are considered 3D Euler-Bernoulli beam elements. They are connected by a series of passive cylinder pairs. We cut the tube into ten pieces, as shown in Figure 2, in which the cylinder pairs are placed at the red dotted lines. We call four adjacent elements near a cylinder pair a unit. For example, elements \(1,\ 2,\ 3\) and \(4\) comprise the first unit. The stiffness and mass matrices of these four elements are deduced as follows. The elements in the other units are the same. For the first unit, since the cylinder pair is passive, we choose nodes \(\mathbf{u}_{1}^{1},\mathbf{u}_{2}^{2},\mathbf{u}_{3}^{1},\mathbf{u}_{4}^{2}\) as independent nodes. Once this node displacement is determined, the elastic potential energy \(V\) of the first unit is determined. \(\mathbf{u}_{1}^{2},\mathbf{u}_{3}^{2}\) are dependent nodes, which are irrelevant to \(V\). Their displacement is determined by the independent node displacement. We first write the elastic potential energy \(V\) of the first unit with both the dependent and independent nodes in Equation (1). \(\mathbf{K}_{i}\) (\(i=1\sim 4\)) is the stiffness matrix of the Euler-Bernoulli beam in the global coordinate. \(\overline{\mathbf{K}}_{i}\) (\(i=1\sim 4\)) is the stiffness matrix in the local coordinate. \(\mathbf{R}\) is the rotation matrix of the local coordinate in the global coordinate. The expression of \(\overline{\mathbf{K}}_{i}\) is shown in Appendix A. \[ \begin{split} V=\frac{1}{2}\begin{bmatrix}\mathbf{u}_{1}^{1}\\ \mathbf{u}_{1}^{1}\end{bmatrix}^{T}\mathbf{K}_{1}\begin{bmatrix}\mathbf{u}_{1}^{1}\\ \mathbf{u}_{1}^{1}\end{bmatrix}+\frac{1}{2}\begin{bmatrix}\mathbf{u}_{1}^{2}\\ \mathbf{u}_{2}^{2}\end{bmatrix}^{T}\mathbf{K}_{2}\begin{bmatrix}\mathbf{u}_{2}^{1}\\ \mathbf{u}_{2}^{2}\end{bmatrix}+\frac{1}{2}\begin{bmatrix}\mathbf{u}_{3}^{1}\\ \mathbf{u}_{3}^{1}\end{bmatrix}^{T}\mathbf{K}_{3}\begin{bmatrix}\mathbf{u}_{3}^{1}\\ \mathbf{u}_{3}^{1}\end{bmatrix}+\frac{1}{2}\begin{bmatrix}\mathbf{u}_{3}^{2}\\ \mathbf{u}_{4}^{2}\end{bmatrix}^{T}\mathbf{K}_{4}\begin{bmatrix}\mathbf{u}_{3}^{2}\\ \mathbf{u}_{4}^{2}\end{bmatrix}\\ \mathbf{K}_{i}=\begin{bmatrix}\mathbf{K}_{1}^{11}\ \mathbf{K}_{i}^{12}\\ \mathbf{K}_{i}^{21}\ \mathbf{K}_{i}^{22}\end{bmatrix}(i=1\sim 4)\\ \end{split}  \tag{1}\]  
Figure 2: Case 1, the outer tube–inner tube. For the first unit, node \(\mathbf{u}_{3}^{2}\) and node \(\mathbf{u}_{1}^{2}\) are connected by a passive cylinder pair. Therefore, node \(\mathbf{u}_{3}^{2}\) can slide a small distance \(\theta_{1}\) and rotate a small angle \(\theta_{2}\) along the axis of the tube \(\mathbf{n}\). Their displacement has the following relationship:  
\[\mathbf{u}_{1}^{2}=\mathbf{u}_{3}^{2}+\mathbf{H}\mathbf{\theta} \tag{2}\] \[\mathbf{H}= \begin{bmatrix}\mathbf{n}&\mathbf{0}\\ \mathbf{0}&\mathbf{n}\end{bmatrix} ,~{}\mathbf{\theta}= \begin{bmatrix}\theta_{1}\\ \theta_{2}\end{bmatrix} \]  
Adding Equation (2) to Equation (1), the elastic potential energy \(V\) of the first unit becomes a function of six variables, as shown below.