# Frequency Contour-Strip Method for Characterization of Damage in Structures under Noisy Conditions
## 2 Transverse Vibration of Timoshenko Beam Bearing a Crack
### Cracked Beam  
A beam with a crack can be regarded as two intact beam segments linked by the crack. Figure 1 shows a beam with a through-width single-sided crack specified by the dimensionless location \(x=X/L\) and the dimensionless depth \(y=d/H\), respectively. This crack connects two undamaged beam segments for each beam segment, the transverse vibration can be described in terms of Equations (2a) and (2b) as  
\[0\leq v\leq x: W_{1}(v)=A_{1}\cosh\lambda_{1}v+A_{2}\text{sinh}\lambda_{1}v+A_{3} \cos\lambda_{2}v+A_{4}\sin\lambda_{2}v \tag{4}\] \[v_{1}(v)=A_{2}m_{1}\cosh\lambda_{1}v+A_{1}m_{1}\text{sinh}\lambda_ {1}v-A_{4}m_{2}\cos\lambda_{2}v+A_{3}m_{2}\sin\lambda_{2}v\] \[x\leq v\leq 1: W_{2}(v)=B_{1}\cosh\lambda_{1}v+B_{2}\text{sinh}\lambda_{1}v+B_{3} \cos\lambda_{2}v+B_{4}\sin\lambda_{2}v\] \[\psi_{2}(v)=B_{2}m_{1}\cosh\lambda_{1}v+B_{1}m_{1}\text{sinh} \lambda_{1}v-B_{4}m_{2}\cos\lambda_{2}v+B_{3}m_{2}\sin\lambda_{2}v\]  
where \(W_{1}(v)\) and \(W_{2}(v)\) are the transverse deflections of two intact beam segments at the location \(v\), respectively. \(\Psi_{1}(v)\) and \(\Psi_{2}(v)\) are the rotation angles of two intact beam segments at the location \(v\), respectively. \(A_{i}\) and \(B_{i}\) (_i_ = 1-4) are constants associated with the boundary and continuity conditions at the crack location, which are as follows:  
\[W_{1}\big{|}_{v=0}=0,\Psi_{1}\big{|}_{v=0}=0,\Psi_{2}^{\prime}\big{|}_{v=1}=0, \big{(}W_{2}^{\prime}-\Psi_{2}\big{)}\big{|}_{v=1}=0 \tag{5}\]  
\[W_{1}\big{|}_{v=x}=W_{2}\big{|}_{v=x},\big{(}\Psi_{2}-\Psi_{1}\big{)}\big{|}_{v =x}=\theta\Psi_{2}^{\prime}\big{|}_{v=x},\Psi_{1}^{\prime}\big{|}_{v=x}=\Psi_ {2}^{\prime}\big{|}_{v=x}, \tag{6}\]  
where \(\theta(y)\) is the dimensionless crack-sectional flexibility [42]:  
\[\theta(y)=6\pi y^{2}f(y)\big{(}H/L\big{)}\] (7a) where \[f(y)\] is a damage correction factor [43], which can be expressed as \[f(y)=\frac{0.923+0.199\big{(}1-\sin\big{(}\frac{\pi}{2}y\big{)}\big{)}^{4}}{\cos (\frac{\pi}{2}y)}\sqrt{\frac{\tan\big{(}\frac{\pi}{2}y\big{)}}{\frac{\pi}{2}y}} \tag{7b}\]  
From Equations (4) to (6), the frequency characteristic equation depicting the transverse vibration of the cracked beam can be derived as [44]  
\[p(m_{1},m_{2},\lambda_{1},\lambda_{2},x)\theta(y)+q(m_{1},m_{2},\lambda_{1}, \lambda_{2})=0\] (8a) in which \[v_{1}(v)=A_{2}m_{1}\cosh\lambda_{1}v+A_{1}m_{1}\text{sinh}\lambda_ {1}v-A_{4}m_{2}\cos\lambda_{2}v+A_{3}m_{2}\sin\lambda_{2}v\]0 and \[v_{1}(v)=A_{2}m_{1}\cosh\lambda_{1}v+A_{1}m_{1}\text{sinh}\lambda_ {1}v-A_{4}m_{2}\cos\lambda_{2}v+A_{3}m_{2}\sin\lambda_{2}v\]1 are given as \[v_{1}(v)=A_{2}m_{1}\cosh\lambda_{1}v+A_{1}m_{1}\text{sinh}\lambda_ {1}v-A_{4}m_{2}\cos\lambda_{2}v+A_{3}m_{2}\sin\lambda_{2}v\]2 \[v_{1}(v)=A_{2}m_{1}\cosh\lambda_{1}v+A_{1}m_{1}\text{sinh}\lambda_ {1}v-A_{4}m_{2}\cos\lambda_{2}v+A_{3}m_{2}\sin\lambda_{2}v\]3 \[v_{1}(v)=A_{2}m_{1}\cosh\lambda_{1}v+A_{1}m_{1}\text{sinh}\lambda_ {1}v-A_{4}m_{2}\cos\lambda_{2}v+A_{3}m_{2}\sin\lambda_{2}v\]4 \[v_{1}(v)=A_{2}m_{1}\cosh\lambda_{1}v+A_{1}m_{1}\text{sinh}\lambda_ {1}v-A_{4}m_{2}\cos\lambda_{2}v+A_{3}m_{2}\sin\lambda_{2}v\]5 \[v_{1}(v)=A_{2}m_{1}\cosh\lambda_{1}v+A_{1}m_{1}\text{sinh}\lambda_ {1}v-A_{4}m_{2}\cos\lambda_{2}v+A_{3}m_{2}\sin\lambda_{2}v\]6 \[v_{1}(v)=A_{2}m_{1}\cosh\lambda_{1}v+A_{1}m_{1}\text{sinh}\lambda_ {1}v-A_{4}m_{2}\cos\lambda_{2}v+A_{3}m_{2}\sin\lambda_{2}v\]7 \[v_{1}(v)=A_{2}m_{1}\cosh\lambda_{1}v+A_{1}m_{1}\text{sinh}\lambda_ {1}v-A_{4}m_{2}\cos\lambda_{2}v+A_{3}m_{2}\sin\lambda_{2}v\]8  
\[v_{1}(v)=A_{2}m_{1}\cosh\lambda_{1}v+A_{1}m_{1}\text{sinh}\lambda_ {1}v-A_{4}m_{2}\cos\lambda_{2}v+A_{3}m_{2}\sin\lambda_{2}v\]9 \[x\leq v\leq 1: W_{2}(v)=B_{1}\cosh\lambda_{1}v+B_{2}\text{sinh}\lambda_{1}v+B_{3} \cos\lambda_{2}v+B_{4}\sin\lambda_{2}v\]0 \[x\leq v\leq 1: W_{2}(v)=B_{1}\cosh\lambda_{1}v+B_{2}\text{sinh}\lambda_{1}v+B_{3} \cos\lambda_{2}v+B_{4}\sin\lambda_{2}v\]1  
Equation (8) presents an implicit function of the eigen-frequency, \(\omega\), the dimensionless crack parameters, \(x\) and \(y\).