# Vibration and Reliability Analysis of Non-Uniform Composite Beam under Random Load
## 2 Theoretical Model
### Natures (Nature Frequencies and Mode Shapes) of Non-Uniform Composite Beam  
In this study, a sample non-uniform rectangular composite cantilever beam structure with varied cross section is analyzed and shown in Figure 1. In the middle of the composite beam is a host beam made of aluminum, and composite layers of piezoelectric ceramic material are pasted above and below the host beam. At the left fixed end, the length, width and thickness of the composite beam and the thickness of the host beam are \(L\), \(b_{0}\), \(2H_{0}\), and \(2h_{0}\), respectively. The thicknesses of the host beam and composite layer at \(x\) position are \(2h_{1}(x)\) and \(2h_{2}(x)\). \(x\) is the position variable along the beam length. The width and thickness of the composite beam at \(x\) position are \(b(x)\) and \(2H(x)=2h_{1}(x)+2h_{2}(x)\). \(F(x,t)\) is the random excitation. The mass distribution and cross-section rigidity at \(x\) position of the composite beam are  
\[\rho(x)A(x)=\rho_{1}A_{1}(x)+\rho_{2}A_{2}(x), \tag{1}\]  
\[E(x)I(x)=(2/3)b(x)h_{1}{}^{3}(x)(E_{1}-E_{2})+(2/3)E_{2}b(x)H^{3}(x), \tag{2}\]  
where \(A(x)=2H(x)b(x)\), \(\rho_{1}\), \(A_{1}(x)\), \(E_{1}\), \(\rho_{2}\), \(A_{2}(x)\) and \(E_{2}\) are mass density, cross-sectional area and modulus of elasticity of host beam and composite layer. The equation of free vibration of a non-uniform composite beam without considering damping is given below,  
\[\rho(x)A(x)\frac{\partial^{2}w(x,t)}{\partial t^{2}}+\frac{\partial^{2}}{ \partial x^{2}}\left[E(x)I(x)\frac{\partial^{2}w(x,t)}{\partial x^{2}}\right] =0, \tag{3}\]  
where \(w(x,t)\) is the deflection function and \(t\) is the time. The boundary conditions are defined as:  
\[\left\{ \begin{array}{c}w(x,t)|_{x=0}=0\\ \frac{\partial w(x,t)}{\partial x}\Big{|}_{x=0}=0\end{array} \right.^{,} \tag{4}\]  
\[\left\{ \begin{array}{c}E(x)I(x)\frac{\partial^{2}w(x,t)}{\partial x^{2}} \Big{|}_{x=L}=0\\ \frac{\partial}{\partial x}\Big{[}E(x)I(x)\frac{\partial^{2}w(x,t)}{\partial x ^{2}}\Big{]}\Big{|}_{x=L}=0\end{array} \right.=0. \tag{5}\]  
By using the mode superposition method, \(w(x,t)\) in Equation (2) can be decomposed into two parts,  
\[w(x,t)=\sum_{i=1}^{\infty}W_{i}(x)q_{i}(t), \tag{6}\]  
where \(W_{i}(x)\) is the \(i\)-th mode shape and \(q_{i}(t)\) is the \(i\)-th corresponding generalized coordinate of the free vibration response or external force. For free vibration analysis solving the natural frequencies and mode shape functions, considering the orthogonality of a linear vibration problem, substituting Equation (6) into Equation (3), one ordinary differential equation corresponding to the \(i\)-th mode is obtained as,  
\[ \begin{array}{c}E(x)I(x)\frac{d^{4}W_{i}(x)}{dx^{4}}+2\frac{d}{dx}[E(x)I(x)] \frac{d^{3}W_{i}(x)}{dx^{3}}+\frac{d^{2}}{dx^{2}}[E(x)I(x)]\frac{d^{2}W_{i}(x )}{dx^{2}}-\\ \omega_{i}^{2}\rho(x)A(x)W_{i}(x)=0,\end{array}  \tag{7}\]  
Figure 1: Main and right views of a non-uniform composite beam. where \(\omega_{i}\) is the \(i\)-th natural frequency. In order to solve the above equation, ADM is applied. The operator form of the Equation (7) is rewritten as,  
\[L_{x}[W_{i}(x)]=\frac{-2\frac{d}{dx}[E(x)I(x)]W_{i}^{\prime\prime\prime}(x)}{E(x )I(x)}-\frac{\frac{d^{2}}{dx^{2}}[E(x)I(x)]W_{i}^{\prime\prime}(x)}{E(x)I(x)}+ \omega_{i}^{2}\frac{\rho(x)A(x)W_{i}(x)}{E(x)I(x)}, \tag{8}\]  
where \(L_{x}\) is the fourth order differential operator. \(L_{x}\)\({}^{-1}\) is applied on the both sides of Equation (8) at the same time, where \(L_{x}\)\({}^{-1}\) is the fourth-order integral operator. \[ \begin{split} W_{i}(x)&=C_{1}+C_{2}x+C_{3}\frac{x^{ 2}}{2!}+C_{4}\frac{x^{3}}{3!}\\ &\quad-L_{x}^{-1}\Bigg{[}\frac{2\frac{d}{dx}[E(x)I(x)]W_{i}^{ \prime\prime}(x)}{E(x)I(x)}+\frac{\frac{d^{2}}{dx^{2}}[E(x)I(x)]W_{i}^{\prime \prime}(x)}{E(x)I(x)}-\omega_{l}^{2}\frac{\rho(x)A(x)W_{i}(x)}{E(x)I(x)}\Bigg{]},\end{split}  \tag{9}\]  
where \(C_{1}\) to \(C_{4}\) are constants that can be determined by boundary conditions, the detailed progress defining \(C_{1}\)-\(C_{4}\) can be found in [36; 37; 38].