# Large-Amplitude Vibrations of Spider Web Structures
## 2 Materials and Methods  
### Equations of Motion  
The concept of cable structure can be applied to spider silk since the structure of spider silk is naturally a special class of prestressed system called tensional integrity (tensegrity). Hence, to investigate the vibration responses of spider silk, the strain energy method considering only axial deformation was applied. The states of spider silk can be divided into three states: unstretched state, equilibrium state, and dynamic or displaced state. The first state is the initial unstretched state dS at (\(x\),\(y\)). After tension applied, the cable is elongated and the structure moves to the equilibrium state at dS0 at (\(x_{0}\),\(y_{0}\)). On account of free vibration at dynamic or displaced state \(d\overline{s}\), spider silk displaces to the position of (\(x_{0}+u,y_{0}+v,w\)), in which \(u,v\) and \(w\) are the hypothetic displacements in the direction of \(x,y\) and \(z\)-axis,respectively [15]. The length of \(ds_{0}\) is an infinitesimal spider silk element, and the equilibrium state can be represented as  
\[{ds_{0}}^{2}={dx_{0}}^{2}+{dy_{0}}^{2}\text{ or }ds_{0}=\sqrt{1+\dot{y}_{0}^{2}}dx_{0}, \tag{1}\]  
Similarly, the equation of displaced state \(d\overline{s}\) can be written as  
\[d\overline{s}=\sqrt{\left(1+\dot{u}\right)^{2}+\left(\dot{y}_{0}+\dot{v} \right)^{2}+\dot{w}^{2}}dx_{0}, \tag{2}\]  
where \(\dot{u}=\frac{du}{dx}\), \(\dot{v}=\frac{dv}{dx}\), \(\dot{w}=\frac{dv}{dx}\).  
Total strain due to stretching of the cable can be derived at the displaced state by  
\[\overline{\varepsilon}=\frac{d\overline{s}-ds}{ds}=\frac{\left(1+\varepsilon _{0}\right)}{\sqrt{1+\dot{y}_{0}^{2}}}\sqrt{\left(1+\dot{u}\right)^{2}+\left( \dot{y}_{0}+\dot{v}\right)^{2}+\dot{w}^{2}}-1, \tag{3}\]  
where \(\varepsilon_{0}\) is the initial static strain.  
The strain energy equation taking dissipated energy due to stretching of the cable can be written as  
\[\begin{array}{l}\delta U=\int_{0}^{X_{H}}((\frac{EA(1+\varepsilon_{0})\dot{ u}+EA}{\sqrt{1+\dot{y}_{0}^{2}}}-\frac{EA(1+\dot{u})}{\sqrt{(1+\dot{u})^{2}+ \left(\dot{y}_{0}+\dot{v}\right)^{2}+\dot{w}^{2}}})\delta u^{\prime}\\ +(\frac{EA(1+\varepsilon_{0})\dot{v}+EA\dot{y}_{0}}{\sqrt{1+\dot{y}_{0}^{2}}}- \frac{EA(\dot{y}_{0}+\dot{v})}{\sqrt{(1+\dot{u})^{2}+\left(\dot{y}_{0}+\dot{v} \right)^{2}+\dot{w}^{2}}})\delta v^{\prime}+(\frac{EA(1+\varepsilon_{0}) \left(\dot{w}\right)}{\sqrt{1+\dot{y}_{0}^{2}}}-\frac{EA\dot{w}}{\sqrt{(1+ \dot{u})^{2}+\left(\dot{y}_{0}+\dot{v}\right)^{2}+\dot{w}^{2}}})\delta w^{ \prime})dx_{0},\end{array} \tag{4}\]  
The virtual work done by inertial force can be derived by  
\[\delta W_{i}=-\int_{0}^{i}\frac{m\sqrt{1+\dot{y}_{0}^{2}}}{g(1+\varepsilon_{0 })}\Big{(}\ddot{u}\delta u+\ddot{v}\delta v+\ddot{w}\delta w\Big{)}dx_{0}, \tag{5}\]  
Total energy (\(\delta\Pi\)) can be derived by [15]  
\[\delta\Pi=\delta U-\delta W_{i}=0, \tag{6}\]  
where \(\delta\Pi\) is the total energy, \(\delta U\) is a variation of strain energy, \(\delta W_{i}\) is the virtual work done due to initial force.  
The pinned support conditions are applied at both ends of the silk member so that the boundary conditions are \(\delta u=\delta v=\delta w=0\) when \(x_{0}=0\) and \(x_{0}=1\). After substituting the boundary conditions and equilibrium conditions (\(\delta\dot{u}=\delta\dot{v}=\delta\dot{w}=\delta\ddot{u}=\delta\ddot{v}=\delta \ddot{w}=0\)) into the total energy equation, the governing equations of motions can be expressed as shown in Equations (7)-(9). These equations are used to analyse the three-dimensional large amplitude free vibration with the undamped system.  
\[\left[\frac{EA(1+\varepsilon_{0})\dot{u}+EA}{\sqrt{1+\dot{y}_{0}^{2}}}-\frac {EA\big{(}1+\dot{u}\big{)}}{\sqrt{\left(1+\dot{u}\right)^{2}+\left(\dot{y}_{0 }+\dot{v}\right)^{2}+\dot{w}^{2}}}\right]^{\prime}-\frac{m\sqrt{1+\dot{y}_{0}^{ 2}}}{g(1+\varepsilon_{0})}\ddot{u}=0, \tag{7}\]  
\[\left[\frac{EA(1+\varepsilon_{0})\dot{v}