## 5 The Analytical Solutions
Equations (51) and (52) become the normalized differential equations in the time functions \(\mathrm{w_{1}}(\mathrm{t}),\mathrm{w_{3}}(\mathrm{t})\):  
\[\mathrm{w_{1}}^{\bullet\bullet}+2\xi\Omega_{1}\sqrt{\frac{1-\mathrm{C_{1}}\cos (2\varphi_{1})}{1-\mathrm{C_{2}}\cos(2\varphi_{1})}}\mathrm{w_{1}}^{\bullet}+ \Omega_{1}^{2}\frac{1-\mathrm{C_{1}}\cos(2\varphi_{1})}{1-\mathrm{C_{2}}\cos (2\varphi_{1})}\mathrm{w_{1}}=-\Gamma_{0}\mathrm{w_{1}}-\Gamma_{1}\mathrm{w_{1 }}^{3}-\Gamma_{2}\mathrm{w_{1}^{5}}, \tag{53}\]  
\[\mathrm{w_{3}}^{\bullet\bullet}+2\xi\Omega_{3}\sqrt{\frac{1-\mathrm{C_{3}}\cos (2\varphi_{3})}{1-\mathrm{C_{4}}\cos(2\varphi_{3})}}\mathrm{w_{3}}^{\bullet}+ \Omega_{3}^{2}\frac{1-\mathrm{C_{3}}\cos(2\varphi_{3})}{1-\mathrm{C_{4}}\cos (2\varphi_{3})}\mathrm{w_{3}}=-\Gamma_{5}\mathrm{w_{3}}-\Gamma_{3}\mathrm{w_{3 }}^{3}-\Gamma_{4}\mathrm{w_{3}^{5}}, \tag{54}\]  
where the coefficients \(\mathrm{C_{1}},\ \mathrm{C_{2}},\ \mathrm{C_{3}},\ \mathrm{C_{4}}\) are presented in Appendix A, and the damping ratio \(\xi\), is given by the following equation:  
\[\zeta=\frac{\Delta}{\sqrt{4\pi^{2}+\Delta^{2}}}, \tag{55}\]  
The natural frequency of the tulip in bending \(\Omega_{1}\) and the natural frequency of the bowl in bending \(\Omega_{3}\) are given by the following equations:  
\[\Omega_{1}=\sqrt{\frac{3\mathrm{E}}{\mathrm{m_{T}}\mathrm{L}_{\mathrm{TA}}^{3 }}\frac{\mathrm{b_{1}}(1+\mathrm{a_{1}})}{1+\frac{\mathrm{b_{1}}}{\mathrm{L_{ 2}}\mathrm{TA}}(1+\mathrm{a_{1}})}}},\mathrm{a_{1}}=\frac{0.5(\mathrm{J_{1T} }+\mathrm{J_{2T}})}{\mathrm{Sr}\left[\frac{\mathrm{L}_{\mathrm{TA}}^{2}}{12}+ \mathrm{d}_{\mathrm{CT}}^{2}\right]},\mathrm{b_{1}}=\mathrm{S_{T}}\left[\frac{ \mathrm{L}_{\mathrm{TA}}^{2}}{12}+\mathrm{d}_{\mathrm{CT}}^{2}\right], \tag{56}\]  
\[\Omega_{3}=\sqrt{\frac{3\mathrm{E}}{\mathrm{m_{B}}\mathrm{L}_{\mathrm{BA}}^{3 }}\frac{\mathrm{b_{2}}(1+\mathrm{a_{2}})}{1+\frac{\mathrm{b_{2}}}{\mathrm{L_{ 2}}\mathrm{BA}}(1+\mathrm{a_{2}})}},\mathrm{a_{2}}=\frac{0.5(\mathrm{J_{1B}}+ \mathrm{J_{2B}})}{\mathrm{S_{B}}\left[\frac{\mathrm{L}_{\mathrm{BA}}^{2}}{12} +\mathrm{d}_{\mathrm{CB}}^{2}\right]},\mathrm{b_{2}}=\mathrm{S_{B}}\left[\frac{ \mathrm{L}_{\mathrm{BA}}^{2}}{12}+\mathrm{d}_{\mathrm{CB}}^{2}\right]}. \tag{57}\]  
The coefficients of the cubic and quintic terms expressed in Equations (53) and (54) can be mathematically explained by development in a Taylor infinite series induced by the excitation function \(\sinh(\mathrm{x})\) (see Equations (53) and (54)) expressed approximately as  
\[\sinh(\mathrm{x})\approx\mathrm{x}+\frac{\mathrm{x}^{3}}{3!}+\frac{\mathrm{x}^ {5}}{5!}. \tag{58}\]  
In the PPR region, the excitation terms \(\cos(2\varphi_{1}(\mathrm{t}))\), \(\sinh(\omega_{\mathrm{n1}}\mathrm{t})\), and \(\cos(2\varphi_{3}(\mathrm{t}))\) must satisfy the Equations (59) and (60), as mentioned in ([PERSON] and [PERSON], 2016) (p. 425), since this resonance is one of the most important, as mentioned in ([PERSON], 2016; [PERSON] and [PERSON], 2016; [PERSON], 2016):  
\[\eta_{1}\simeq 2\Omega_{1}\simeq\frac{\mathrm{d}\theta_{1}}{\mathrm{dt}}= \frac{2\mathrm{d}\varphi_{1}}{\mathrm{dt}}\simeq\omega_{\mathrm{n1}}, \tag{59}\]  
\[\mathrm{m_{T}}^{\bullet\bullet}\mathrm{w_{1}}+\mathrm{c_{1}}^{\bullet}\mathrm{w_ {1}}++\mathrm{k_{1}}\mathrm{w_{1}}=-\frac{\rho\mathrm{A}\mathrm{L}_{\mathrm{Ms} }^{2}}{\mathrm{n}\pi}\omega_{\mathrm{n1}}^{2}\mathrm{C_{2n}sinh}(\omega_{ \mathrm{n1}}\mathrm{t}),\mathrm{n}=1,2,3,\ldots \tag{51}\]0where \(\eta_{1}\) is the excitation frequency in the PPR region for the tulip, \(\eta_{3}\) is the excitation frequency in the PPR region for the bowl, \(\frac{\mathrm{d}\varphi_{1}}{\mathrm{d}t}\) is the excitation induced in the FBV movement (Equation (53)) by the rigid twisting angle of the tulip concerning the \(X_{1}\) axis, and \(\frac{\mathrm{d}\varphi_{3}}{\mathrm{d}t}\) is the excitation induced in the FBV movement (Equation (54)) by the rigid twisting angle of the the bowl concerning the \(X_{3}\) axis, as given by the following equation:  
\[\mathrm{m_{T}}^{\bullet\bullet}\mathrm{w_{1}}+\mathrm{c_{1}}^{\bullet}\mathrm{w_ {1}}++\mathrm{k_{1}}\mathrm{w_{1}}=-\frac{\rho\mathrm{A}\mathrm{L}_{\mathrm{Ms} }^{2}}{\mathrm{n}\pi}\omega_{\mathrm{n1}}^{2}\mathrm{C_{2n}sinh}(\omega_{ \mathrm{n1}}\mathrm{t}),\mathrm{n}=1,2,3,\ldots \tag{51}\]1  
This considers the nonuniformity of the kinematic isometry of the AD [25; 26].