# Free Vibrations of Tapered Horseshoe Circular Arch with Constant Volume
## 2 Problem Formulation
### Governing Differential Equations  
Figure 4 is a free body diagram of an arch element under a free vibration state. The stress resultants \((N,Q,M)\) due to deformations \((w,v)\) act in the cross-section of this element, and the inertia forces \((P_{r},P_{t},\ T)\) occur on this element having mass \(\rho\) due to accelerations. From the free body diagram, equilibrium equations of \(\sum F_{r}=0\), \(\sum F_{t}=0\), \(\sum M=0\) are obtained as  
\[N^{\prime}+Q+rP_{t}=0 \tag{9}\]  
\[Q^{\prime}-N+rP_{r}=0 \tag{10}\]  
\[\frac{M^{\prime}}{r}-Q-T=0 \tag{11}\]  
where \((^{\prime})=d/d\theta\) and \((^{\prime\prime})=d^{2}/d\theta^{2}\) of the differential operators. The equations of \((N,\,M,\psi\,)\) related to the deformations \((w,v)\) consider the axial deformation due to \(N\), those are given in following equations [22]. \[N=\frac{EA}{r}(v^{\prime}+w)-\frac{M}{r}=\frac{EV}{c_{3}\alpha r^{2}}f^{2}(v^{ \prime}+w)-\frac{M}{r} \tag{12}\]  
Figure 4: Free body diagram of small arch element. \[M=-\frac{EI}{r^{2}}(w^{\prime}+w)=-\frac{c_{2}EV^{2}}{c_{1}^{2}c_{3}^{2}a^{2}r^{4} }f^{4}(w^{\prime\prime}+w) \tag{13}\]  
\[\psi=\frac{1}{r}(w^{\prime}-v) \tag{14}\]  
The inertia loadings \((P_{r},P_{t},T)\) are [23]:  
\[P_{r}=\rho A\omega_{i}^{2}w=\frac{\rho V}{c_{3}\alpha r}\omega_{i}^{2}f^{2}w \tag{15}\]  
\[P_{t}=\rho A\omega_{i}^{2}v=\frac{\rho V}{c_{3}\alpha r}\omega_{i}^{2}f^{2}v \tag{16}\]  
\[T=\rho la_{i}^{2}\psi=\frac{c_{2}\ \rho V^{2}}{c_{1}^{2}c_{3}^{2}a^{2}r^{3}}a_{i}^ {2}f^{4}(w^{\prime}-v) \tag{17}\]  
First derivatives of \((N^{\prime},\ M^{\prime})\) in Equations (12) and (13) are obtained as  
\[Q^{\prime}-N+rP_{r}=0 \tag{10}\]0  
\[Q^{\prime}-N+rP_{r}=0 \tag{10}\]1  
Substituting Equations (17) and (19) into Equation (11) and arranging against \(Q\) gives Equation (20), and then the first derivative \(Q^{\prime}\) is obtained as Equation (21). \[Q^{\prime}-N+rP_{r}=0 \tag{10}\]2  
\[Q^{\prime}-N+rP_{r}=0 \tag{10}\]3  
The following non-dimensional parameters are defined to derive differential equations in the non-dimensional forms. \[Q^{\prime}-N+rP_{r}=0 \tag{10}\]4  
\[Q^{\prime}-N+rP_{r}=0 \tag{10}\]5  
\[Q^{\prime}-N+rP_{r}=0 \tag{10}\]6  
\[Q^{\prime}-N+rP_{r}=0 \tag{10}\]7  
where \((\delta,\lambda)\) are the radial and tangential dimensionless deformations, \(\kappa\) is the volume ratio and \(C_{i}\) is the frequency parameter. By substituting Equations (12), (15) and (21) into Equation (10) and applying dimensionless Equations (22)-(25), the differential Equation (26) can be obtained. Similarly, by substituting Equations (16), (18) and (20) into Equation (9), the differential Equation (24) can be derived. \[ \begin{array}{ll}\delta^{\prime\prime\prime}=-\frac{8f^{\prime}}{f}&(\delta ^{\prime\prime}+\delta^{\prime})-\frac{4}{f}\Big{(}f^{\prime\prime}+\frac{3f^ {\prime 2}}{f}\Big{)}(\delta^{\prime\prime}+\delta)-(2\delta^{\prime\prime}+\delta) \\ &-C_{i}^{2}\Big{\{}(\delta^{\prime\prime}-\lambda^{\prime})+\frac{4f^{\prime }}{f}(\delta^{\prime}-\lambda)\Big{\}}+\frac{c_{2}^{2}c_{3}\alpha}{c_{2}\kappa }\frac{1}{f^{2}}\Big{[}\big{(}C_{i}^{2}-1\big{)}\delta-\lambda^{\prime}\Big{]} \end{array}  \tag{26}\]  
\[\lambda^{\prime\prime}=-\delta^{\prime}+\frac{c_{2}\kappa}{c_{1}^{2}c_{3} \alpha}C_{i}^{2}f^{2}(\delta^{\prime}-\lambda)-\frac{2f^{\prime}}{f}(\lambda^ {\prime}+\delta)-C_{i}^{2}\lambda \tag{27}\]where derivatives of \((f^{\prime},f^{\prime\prime})\) in the above two equations are obtained from Equation (2), or  
\[f^{\prime}=4(m-1)\Big{(}\frac{2\theta}{\alpha^{2}}-\frac{1}{\alpha}\Big{)},\ \ f^{ \prime\prime}=\frac{8(m-1)}{\alpha^{2}} \tag{28}\]  
In Equations (26) and (27), the term in \(\{\cdots\}\) is due to the rotatory inertia couple \(T\). If differential equations ignore \(T\), the term in \(\{\cdots\}\) is simply deleted.