# Dynamic and Stability Analysis of Multibolt Plane Joints under Normal Forces
## 3 Dynamic Model of the Bolt Joint Interface
Introducing the overall stiffness at static balance \(K_{\textit{hypre}}=K_{\textit{bases}}K_{\textit{sprre}}/\big{(}K_{\textit{bases}}+K_{ \textit{sprre}}\big{)}+K_{\textit{bolts}}\), where \(K_{\textit{sprre}}=K_{\textit{s}}(0)\) depends on the value of \(x_{\textit{s}0}=x_{\textit{sprre}}\), the dimensionless equation of the vertical vibration model can be obtained by defining \(\tau=\omega t\), \(\omega_{n}=\sqrt{K_{\textit{hypre}}/M}\), \(\overline{\omega}=\omega/\omega_{n}\), \(\zeta=c/(2M\omega_{n})\), \(\overline{F}=F/K_{\textit{hypre}}\), \(\overline{F}_{\textit{pre}}=F_{\textit{pre}}/K_{\textit{hypre}}\), \(\overline{K}_{\textit{bases}}=K_{\textit{bases}}/K_{\textit{hypre}}\), \(\overline{K}_{\textit{bolts}}=K_{\textit{bolts}}/K_{\textit{hypre}}\), and \(\overline{f}(\Delta x_{\textit{s}})=\big{[}\overline{P}_{\textit{e}}\big{(}x_ {\textit{sprre}}-\Delta x_{\textit{s}}\big{)}-\overline{P}_{\textit{e}} \big{(}x_{\textit{sprre}}\big{)}\big{]}/\overline{K}_{\textit{bases}}+\Delta x _{\textit{s}}=f(\Delta x_{\textit{s}})\), in which \(\overline{P}_{\textit{e}}=P_{\textit{e}}/K_{\textit{hypre}}\). The equation of motion is transformed into the dimensionless form  
\[\overline{\omega}^{2}\frac{d^{2}\Delta x}{d\tau^{2}}+2\zeta\overline{\omega} \frac{d\Delta x}{d\tau}+\overline{g}(\Delta x)=\overline{F}\sin\tau \tag{15}\]  
where \(\zeta=0.005\)[27] and \(\overline{g}(\Delta x)=\left\{ \begin{array}{ll}\big{(}\overline{K}_{ \textit{bases}}+\overline{K}_{\textit{bolts}}\big{)}\Delta x-\overline{K}_{ \textit{bases}}\overline{f}^{-1}(\Delta x),&\Delta x\leq\Delta x_{\textit{ hypre}}\\ \overline{K}_{\textit{bases}}\Delta x-\overline{K}_{\textit{bases}}\overline {f}^{-1}(\Delta x)+\overline{F}_{\textit{pre}},&\Delta x>\Delta x_{\textit{ hypre}}\end{array} \right..\)  
By contrast, if the roughness of contact surfaces is neglected, the dynamic model changes into an ideal model and \(\Delta x_{\textit{s}}\) can be expressed in the form  
\[\Delta x_{\textit{s}}=\left\{ \begin{array}{ll}\Delta x-\Delta x_{\textit{ apre}},&\Delta x\leq\Delta x_{\textit{apre}}\\ 0,&\Delta x>\Delta x_{\textit{apre}}\end{array} \right. \tag{16}\]  
in which \(\Delta x_{\textit{apre}}=-\frac{Mg+F_{\textit{pre}}}{K_{\textit{bases}}}\). Equation (14) can be transformed into  
\[\left\{ \begin{array}{ll}M\Delta\tilde{x}+c\Delta\dot{x}+K_{\textit{bolts}} \Delta x=F\sin(\omega t)+Mg+F_{\textit{pre}},&\Delta x\leq\Delta x_{\textit{ apre}}\\ M\Delta\tilde{x}+c\Delta\dot{x}+(K_{\textit{bases}}+K_{\textit{bolts}})\Delta x =F\sin(\omega t),&\Delta x_{\textit{apre}}<\Delta x\leq\Delta x_{\textit{ hypre}}\\ M\Delta\tilde{x}+c\Delta\dot{x}+K_{\textit{bases}}\Delta x=F\sin(\omega t)-F_{ \textit{pre}},&\Delta x>\Delta x_{\textit{hypre}}\end{array} \right.. \tag{17}\]  
As \(\Delta x_{\textit{s}}\)\(=0\) in a small neighborhood of \(\Delta x\)\(=0\), the contact stiffness at the static equilibrium position \(K_{\textit{spre}}\) is infinite, and then \(K_{\textit{hypre}}=K_{\textit{bases}}+K_{\textit{bolts}}\) is achieved. Introducing the parameter \(\overline{G}=Mg/K_{\textit{hypre}}\), the dimensionless equation is as follows  
\[\overline{\omega}^{2}\frac{d^{2}\Delta x}{d\tau^{2}}+2\zeta\overline{\omega} \frac{d\Delta x}{d\tau}+\overline{h}(\Delta x)=\overline{F}\sin\tau \tag{18}\]  
where \(\overline{h}(\Delta x)=\left\{ \begin{array}{ll}\overline{K}_{\textit{bolts}} \Delta x-\overline{G}-\overline{F}_{\textit{pre}},&\Delta x\leq\Delta x_{ \textit{apre}}\\ \big{(}\overline{K}_{\textit{bases}}+\overline{K}_{\textit{bolts}}\big{)} \Delta x,&\Delta x_{\textit{apre}}<\Delta x\leq\Delta x_{\textit{ hypre}}\\ \overline{K}_{\textit{bases}}\Delta x+\overline{F}_{\textit{pre}},&\Delta x> \Delta x_{\textit{hypre}}\end{array} \right..\)  
The force between bases is obtained as  
\[\overline{P}(\Delta x)=\left\{ \begin{array}{ll}0,&\Delta x\leq\Delta x_{ \textit{apre}}\\ \overline{K}_{\textit{bases}}\Delta x+\overline{G}+\overline{F}_{\textit{pre}},& \Delta x>\Delta x_{\textit{apre}}\end{array} \right.. \tag{19}\]