# Study on Vibration Characteristics of Multi-Beam Structures with Stick and Slip at Joints
## 2 Theoretical Modeling
### Energy Equation of Beam Structure  
The expressions of the kinetic energy \(T_{i}\) and the strain potential energy \(V_{i}\) of each beam are as follows:  
\[T_{i}=\frac{1}{2}\int_{0}^{L_{i}}\rho_{i}S_{i}\bigg{(}\frac{\partial u_{i}}{ \partial t}\bigg{)}^{2}dx_{i} \tag{1}\]  
Figure 1: A beam structure model considering connection. The length of the beam is \(L_{i}\); the translational and rotational stiffness levels of the constraints at the edges are \(k_{1L}\), \(k_{12}\), \(k_{11}\), and \(k_{r2}\); and the connection stiffness levels at the joints are \(k_{ui}\) and \(k_{ii}\), respectively.  
\[V_{i}=\frac{1}{2}\int_{0}^{L_{i}}EI\Bigg{(}\frac{\partial^{2}u_{i}}{\partial x_{i} ^{2}}\Bigg{)}^{2}dx_{i} \tag{2}\]  
where \(\rho_{i}\) represents the mass density of the beam, \(S_{i}\) denotes its cross-sectional area, \(u_{i}\) signifies the vibration displacement, \(E\) represents the Young's modulus of the material, and \(I\) is the moment of inertia of the section; \(i=1\) or \(2\) represents the kinetic energy of beam 1 or 2, and the same is true for \(i\) below. The spring stiffness is introduced to simulate the boundary conditions at the boundary end of the beam, where \(k_{l1}\) and \(k_{r1}\) are the boundary translational stiffness, while \(k_{l2}\) and \(k_{r2}\) are the rotational stiffness, as shown in Figure 1. Therefore, the energy descriptions of the boundary constraints \(V_{1I}\), \(V_{l2}\), \(V_{r1}\), \(V_{r2}\) are, respectively, as follows:  
\[\left\{\begin{array}{l}V_{I1}=\frac{1}{2}k_{I1}\Big{(}\frac{\partial u_{1}} {\partial x_{1}}\Big{|}_{x_{1}=0}\Big{)}^{2}\\ V_{I2}=\frac{1}{2}k_{I2}\Big{(}\frac{\partial u_{1}}{\partial x_{1}}\Big{|}_{ x_{1}=0}\Big{)}^{2}\\ V_{r1}=\frac{1}{2}k_{r1}\Big{(}\frac{u_{r1}}{u_{r1}}\Big{|}_{x_{m}=L_{m}} \Big{)}^{2}\\ V_{r2}=\frac{1}{2}k_{r2}\Big{(}\frac{\partial u_{r1}}{\partial x_{m}}\Big{|}_{ x_{m}=L_{m}}\Big{)}^{2}\end{array}\right.. \tag{3}\]  
Disregarding the interconnections, the total kinetic energy and potential energy can be expressed as \(T=T_{1}+T_{2}+\ldots+T_{n}\) and \(V=V_{1}+V_{1}+\ldots+V_{n}+V_{11}+V_{l2}+V_{r1}+V_{r2}\). The Dirac delta function \(\delta\) is used to express the external excitation force as a distributed force, and its mathematical expression is as follows:  
\[f(x_{i},t)=\delta(x_{i}-x_{0})F_{0}e^{(i\omega_{0}t)}. \tag{4}\]  
The excitation amplitude is denoted as \(F_{0}\), the coordinates of the operating point are designated as \(x_{0}\), and the excitation frequency is represented by \(\omega_{0}\). The expression of the work term \(W_{ext}\) can be obtained as follows:  
\[W_{ext}=\int_{0}^{L_{i}}f(x_{i},t)u_{i}(x_{i},t)dx=F_{0}e^{i\omega_{0}t}\Phi_ {i}(x_{0})T_{i}(t). \tag{5}\]  
Considering the friction at the connection during transverse vibrations, the equivalent stiffness \(K_{eq}\) and damping \(C_{eq}\) are utilized to characterize the relationship between the interface connection force, damping, and stiffness at the connection:  
\[\left\{\begin{array}{l}f_{u1}=K_{eq}(u_{1}-u_{2})+C_{eq}(\dot{u}_{1}-\dot{ u}_{2})\\ f_{u2}=K_{eq}(u_{2}-u_{1})+C_{eq}(\dot{u}_{2}-\dot{u}_{1})\end{array}\right.. \tag{6}\]  
Therefore, the virtual work done by the translational connecting force at the connection is as follows:  
The virtual work of the rotating connecting force can be written as the following:  
\[\begin{cases}\delta W_{M_{1}}=k_{\theta}\Big{(}\frac{3u_{1}}{2x_{1}}-\frac{3u_{2}}{ 3x_{1}}\Big{)}\delta\frac{3u_{1}}{2x_{1}}\\ \delta W_{M_{2}}=k_{\theta}\Big{(}\frac{3u_{2}}{2x_{2}}-\frac{3u_{1}}{3x_{1}} \Big{)}\delta\frac{3u_{2}}{3x_{2}}.\end{cases} \tag{9}\]  
Thus, the total virtual work of the connecting force can be obtained as follows:  
\[\delta W=\delta W_{f_{i1}}+\delta W_{f_{i2}}+\delta W_{M_{1}}+\delta W_{M_{2}}. \tag{10}\]