# Nonlinear Vibration Analysis of a Beam with a Breathing Crack
## 5 Analysis of a Beam with a Breathing Crack
After substituting Equations (19) and (20) into Equation (16), the following hierarchy of equations can be obtained for \(\varepsilon^{0}\) and \(\varepsilon^{1}\), respectively:  
\[D_{0}^{2}\eta_{0}+\mathbf{K}_{0001}\eta_{0}=0 \tag{22}\]  
\[D_{0}^{2}\ddot{\eta}_{1}+\mathbf{K}_{0001}\eta_{1}=-2D_{0}D_{1}\eta_{0}+ \mathbf{F}_{0}-C_{0}D_{0}\eta_{0}+\,\sum\limits_{n=1}^{N}\mathbf{K}_{002}\eta _{0}\vec{k}_{n}\cos(n\alpha t+\varphi_{n}) \tag{23}\]  
Then, the solution to Equation (22) can be written as  
\[\eta_{0r}=A_{r}(T_{1})e^{i\omega_{r}T_{0}}+\overline{A}_{r}(T_{1})e^{-i\omega_ {r}T_{0}} \tag{24}\]  
where \(A_{r}(T_{1})\) is a complex valued amplitude of the \(r\)-th order of the cracked beam, and \(\overline{A}_{r}(T_{1})\) indicates a complex conjugate of \(A_{r}(T_{1})\); \(\omega_{r}\) is the \(r\)-th order natural frequency, which can be determined by the mass matrix \(\mathbf{M}\) the stiffness matrix of the uncracked beam \(\mathbf{K}_{01}\), and the stiffness matrix \(\mathbf{K}_{02}\) of Equation (12). Substituting Equation (24) into Equation (23) yields  
\[ \begin{array}{ll}D_{0}^{2}\eta_{r1}+\omega_{r}^{2}\eta_{r1}=&-2j\omega_{r}(A ^{\prime}{}_{r}+c_{0r}A_{r})e^{i\omega_{r}T_{0}}+\,\sum\limits_{n=1}^{N}\, \sum\limits_{p=1}^{q}\,\frac{\vec{k}_{n}}{2}(\mathbf{K}_{002})_{p,r}A_{r}e^{i \omega_{r}T_{0}}e^{i(n\omega T_{0}+\varphi_{n})}+\\ &\sum\limits_{n=1}^{N}\,\sum\limits_{p=1}^{q}\,\frac{\vec{k}_{n}}{2}( \mathbf{K}_{002})_{p,r}\overline{A}_{r}e^{-i\omega_{r}T_{0}}e^{i(n\omega T_{0 }+\varphi_{n})}+(\mathbf{F}_{0})_{r}e^{i\omega T_{0}}+cc.\end{array}  \tag{25}\]  
where the prime represents a time derivative of slow timescale \(T_{1}\), and \(cc\). represents the complex conjugate of the proceeding terms. According to the definition by [PERSON] et al. [39] and [PERSON] et al. [40, 41], the frequency of each term in the first-order approximation solution of a nonlinear system is defined as the frequent factor. Therefore, the corresponding frequent factors of Equation (25) are as follows:  
\[\omega,\,n\omega\pm\omega_{r}\,(r=1,\,2,\,\ldots,\,q;\,n=1,\,2,\,\ldots,\,N),\]  
It is seen that the steady state responses consist of several sub- and super-harmonic components, whose frequencies related to the natural frequencies are \(n\omega\pm\omega_{r}\). Based on the theory of nonlinear resonances [44], the cracked beam may be at resonance under the following conditions:  
(1) If \(\omega\approx\omega_{r}\), primary resonance will occur. (2) If \(\delta\omega-\omega_{r}\approx\omega_{r}\) or \(\omega\approx 1/3\)\(\omega_{r}\), third-order super-harmonic resonance will occur. (3) If \(4\omega-\omega_{r}\approx\omega_{r}\) or \(\omega\approx 1/2\omega_{r}\), second-order super-harmonic resonance will occur. (4) If \(\omega-\omega_{r}\approx\omega_{r}\) or \(\omega\approx 2\)\(\omega_{r}\), \(1/2\)th-order sub-harmonic resonance will occur. (5) If \(n\alpha\pm\omega_{r}\approx\omega_{i}\,(r\
eq i)\), combination resonance will occur. (6) If two types of the resonance occur simultaneously, multiple resonance will occur. As shown above, the natural frequency \(\omega_{r}\) is determined by both the structural parameters and the crack parameters. Apparently, the frequent factors are related to structural parameters, the crack parameters and the exciting force. This indicates that, given an exciting force, the nonlinear vibration of the cracked beam is influenced not only by the parameters of the uncracked beam but also by the crack parameters.