# Research on the Dynamics of the Space Tubular Expandable Structure Driving Deployment Unit
## 2 Methods
### The Spatial Nodal Dynamic Description of the Thin-Walled Flexible Beam Structure
#### 2.2.3 The Dynamic Equations of the Thin-Walled Flexible Beam Structure  
First, derivative of time t of Equation (13) can be used to obtain the element kinetic energy expression of thin-walled flexible beam element:  
\[T=\frac{1}{2}\rho h\int_{0}^{x}\int_{0}^{t}q^{T}qdxdy=\frac{1}{2}q^{T}Mq \tag{14}\]  
where: \(\rho\) is the mass density, \(h\) is the thickness, \(q\) is the coordinates of element nodes, \(\mathbf{M}\) is the element mass matrix, and the element mass matrix is:  
\[\begin{array}{c}\mathbf{M}\&=\int\limits_{V}\rho\mathbf{S}^{\mathsf{T}} \cdot\mathbf{S}dV\\ \left[\begin{array}{cccccc}\frac{13}{35}m\mathbf{I}&\frac{11}{210}l \mathbf{m}&\frac{7}{20}\rho lQ_{2}\mathbf{I}&\frac{7}{20}\rho lQ_{3}\mathbf{I}& \frac{9}{420}l\mathbf{m}&-\frac{13}{420}l\mathbf{m}&\frac{3}{20}\rho lQ_{2} \mathbf{I}&\frac{3}{20}\rho lQ_{3}\mathbf{I}\\ \frac{11}{105}l^{2}\mathbf{m}&\frac{1}{20}\rho l^{2}Q_{3}\mathbf{I}&\frac{1}{2 0}\rho l^{2}Q_{3}\mathbf{I}&\frac{13}{420}l\mathbf{m}&-\frac{1}{140}l^{2} \mathbf{m}I&\frac{3}{30}\rho l^{2}Q_{2}\mathbf{I}&\frac{1}{30}\rho l^{2}Q_{3} \mathbf{I}\\ \frac{1}{3}\rho ll_{22}\mathbf{I}&\frac{1}{3}\rho ll_{22}\mathbf{I}&\frac{3}{ 20}\rho lQ_{3}\mathbf{I}&-\frac{1}{30}\rho l^{2}Q_{3}\mathbf{I}&\frac{1}{6} \rho ll_{22}\mathbf{I}&\frac{1}{6}\rho ll_{22}\mathbf{I}\\ \frac{1}{35}\rho ll_{39}\mathbf{I}&\frac{3}{20}\rho lQ_{3}\mathbf{I}&\frac{1}{ 30}\rho l^{2}Q_{3}\mathbf{I}&\frac{1}{6}\rho ll_{39}\mathbf{I}&\frac{1}{6} \rho ll_{39}\mathbf{I}\\ &\frac{13}{35}m\mathbf{I}&-\frac{11}{210}l\mathbf{m}&\frac{7}{20}\rho lQ_{2} \mathbf{I}&\frac{7}{20}\rho lQ_{3}\mathbf{I}\\ &\frac{11}{105}l^{2}\mathbf{m}&-\frac{1}{25}\rho l^{2}Q_{2}\mathbf{I}&-\frac{1} {20}\rho l^{2}Q_{3}\mathbf{I}\\ &\frac{1}{35}\rho ll_{22}\mathbf{I}&\frac{1}{35}\rho l_{39}\mathbf{I}\\ &\frac{1}{3}\rho l_{39}\mathbf{I}\end{array}\right] \tag{15}\]Strain energy of thin-walled flexible beam element:  
\[U=U^{\varepsilon}+U^{\kappa} \tag{16}\]  
where, \(U^{\varepsilon}\) is the membrane strain energy caused by the tension, compression, and other deformation of the middle surface of the thin-walled beam element [38], and \(U^{\varepsilon}\) is the bending strain energy caused by the bending and torsional deformation of the thin-walled beam element. They are:  
\[U^{\varepsilon}=\frac{Eh}{2(1-v^{2})}\int\limits_{0}^{w}\int\limits_{0}^{1} \varepsilon_{x}^{2}+\varepsilon_{y}^{2}+2v\varepsilon_{x}\varepsilon_{y}+ \frac{1}{2}(1-v)\gamma_{xy}^{2}\mathrm{d}x\mathrm{d}y \tag{17}\]  
\[U^{\varepsilon}=\frac{Eh^{3}}{24(1-v^{2})}\int\limits_{0}^{w}\int\limits_{0}^{ 1}\kappa_{x}^{2}+x_{y}^{2}+2v\kappa_{x}x_{y}+2(1-v)x_{xy}^{2}\mathrm{d}x\mathrm{ d}y \tag{18}\]  
where, \(E\) is [PERSON]'s modulus and \(v\) is [PERSON]'s ratio. The strain and curvature are:  
\[\varepsilon_{x}=\frac{1}{2}\Big{(}\mathbf{r}_{x}^{\mathrm{T}}\mathbf{r}_{x}- 1\Big{)},\varepsilon_{y}=\frac{1}{2}\Big{(}\mathbf{r}_{y}^{\mathrm{T}}\mathbf{ r}_{y}-1\Big{)},\gamma_{xy}=\mathbf{r}_{x}^{\mathrm{T}}\mathbf{r}_{y} \tag{19}\]  
\[\kappa_{x}=\frac{\mathbf{r}_{xx}^{\mathrm{T}}\mathbf{n}}{\parallel\mathbf{n} \parallel^{3}},\kappa_{y}=\frac{\mathbf{r}_{yy}^{\mathrm{T}}\mathbf{n}}{ \parallel\mathbf{n}\parallel^{3}},\kappa_{xy}=\frac{\mathbf{r}_{xy}^{\mathrm{ T}}\mathbf{n}}{\parallel\mathbf{n}\parallel^{3}} \tag{20}\]  
where: \(\mathbf{r}\) is the partial derivative of \(x\) or \(y\), and \(\mathbf{n}=\mathbf{r}_{x}\times\mathbf{r}_{y}\) is the normal vector of the mid plane of the thin-walled flexible beam element.  
Further obtaining the dynamic equations of the thin-walled flexible beam structure, we can derive them using the Lagrangian equation [39; 40] as shown in Formula (21).  
\[\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial T}{\partial\dot{\mathbf{q}} }\right)-\frac{\partial T}{\partial\mathbf{q}}+\frac{\partial U}{\partial \mathbf{q}}=\mathrm{F} \tag{21}\]  
where: \(T\) is the kinetic energy, \(U\) is the strain energy, and \(F\) is the generalized external force derived from the virtual work of the external force [41; 42].