# Analysis of Dynamic Characteristics of Attached High Rise Risers
## 3 Dynamic Analysis of VNA System under Combined Excitation
### Vibration Response of the VNA System under First Order Frequency Locking
\begin{split} a_{1}&=\int_{0}^{1}\Phi_{1}\left(\xi\right) \Phi_{1}^{{}^{\prime\prime\prime}}(\xi)\mathrm{d}\xi+\left(\gamma+\
u^{2}\right) (\int_{0}^{1}\Phi_{1}(\xi)\Phi^{{}^{\prime\prime}}{}_{1}(\xi)\mathrm{d}\xi)- \beta(\int_{0}^{1}\xi\[PERSON]_{1}(\xi)\[PERSON]^{{}^{\prime\prime}}{}_{1}(\xi)\mathrm{d} \xi)\\ &-\beta(\int_{0}^{1}\[PERSON]_{1}\left(\xi\right)\[PERSON]^{{}^{\prime}}{}_{1 }(\xi)\mathrm{d}\xi)+\kappa_{1}\[PERSON]_{1}{}^{2}(\xi_{a})+\kappa_{2}\[PERSON]_{1}{}^{2} (\xi_{b})\\ a_{2}&=2\sqrt{\alpha}\
u(\int_{0}^{1}\[PERSON]_{1}(\xi) \[PERSON]^{{}^{\prime}}{}_{1}(\xi)\mathrm{d}\xi)+f_{D}+c_{1}\[PERSON]_{1}{}^{2}(\xi_{a} )+c_{2}\[PERSON]_{1}{}^{2}(\xi_{b})\\ a_{3}&=o(\int_{0}^{1}\[PERSON]_{1}{}^{2}(\xi)\[PERSON]^{{}^{ \prime}}{}_{1}{}^{2}(\xi)\mathrm{d}\xi)+\kappa_{3}\[PERSON]_{1}{}^{4}(\xi_{a})+ \kappa_{4}\[PERSON]_{1}{}^{4}(\xi_{b})\\ a_{4}&=G(\int_{0}^{1}\[PERSON]_{1}(\xi)d\xi)f_{a},\ a_{5} =-\frac{D^{2}C_{L0}\rho_{f}}{16L\pi^{2}m_{\Sigma}\xi_{i}^{2}}\\ \end{split} \]  
\[b_{1}=-\lambda_{s}\Omega_{s},\ b_{2}=\int_{0}^{1}\Phi_{1}{}^{4}(\xi)d\xi,\ b_{3}=pa_{1},\ b_{4}=pa_{2},\ b_{5}=-pa_{5},\ b_{6}=pa_{3}\]  
\(f_{a}\) is the acceleration amplitude. The multi-scale method can be used to solve Equations (28) and (29) to obtain the approximate periodic response of the system. Two tuning parameters \(\sigma_{1}\) and \(\sigma_{2}\) are introduced, and the following relationship is given,  
\[\Omega_{g}{}^{2}=\Omega_{1}{}^{2}+\varepsilon^{2}\sigma_{1};\Omega_{s}{}^{2}= \Omega_{1}{}^{2}+\varepsilon^{2}\sigma_{2} \tag{30}\]  
where, \(\Omega_{1}\) is the linear dimensionless resonance frequency of the first mode of the system, \(\varepsilon\) is a small quantity. In order to make the damping and forcing terms effects and nonlinear effects appear in the same perturbation equation and the damping term be a quadratic small quantity, select a small parameter \(\varepsilon\) to label the small quantity in the dynamic system. By sorting them out, the following dynamic equations can be obtained:  
\[ \begin{split}&\ddot{\overline{\eta}}_{1}(\tau)+\Omega_{g}{}^{2} \overline{\eta}_{1}[\tau]=-\varepsilon^{2}\widehat{a}_{1}\overline{\eta}_{1}( \tau)-\varepsilon^{2}\widehat{a}_{2}\dot{\overline{\eta}}_{1}(\tau)-a_{3} \overline{\eta}_{1}{}^{3}(\tau)-a_{5}\varepsilon^{2}\widehat{\Omega}_{1}{}^{2 }\overline{\eta}_{1}(\tau)\\ &-a_{5}\varepsilon^{2}\sigma_{2}\overline{\eta}_{1}(\tau)- \varepsilon^{3}\widehat{a}_{4}\cos(\Omega_{S}\tau)+\varepsilon^{2}\widehat{ \Omega}_{1}{}^{2}\overline{\eta}_{1}(\tau)+\varepsilon^{2}\sigma_{1} \overline{\eta}_{1}(\tau)\\ &\ddot{\overline{\eta}}_{1}(\tau)+\Omega_{g}{}^{2}\overline{\eta} _{1}(\tau)=\varepsilon^{2}(\sigma_{1}-\sigma_{2})\overline{\eta}_{1}(\tau)- \varepsilon^{2}\widehat{b}_{1}\dot{\overline{\eta}}_{1}(\tau)-b_{2}\dot{ \overline{\eta}}_{1}(\tau)\overline{\eta}_{1}{}^{2}(\tau)\\ &-\varepsilon^{2}\widehat{b}_{3}\overline{\eta}_{1}(\tau)- \varepsilon^{2}\widehat{b}_{4}\dot{\overline{\eta}}_{1}(\tau)-b_{6}\overline{ \eta}_{1}{}^{3}(\tau)-b_{5}\varepsilon^{2}\widehat{\Omega}_{1}{}^{2} \overline{\eta}_{1}(\tau)-b_{5}\varepsilon^{2}\sigma_{2}\overline{\eta}_{1}( \tau)\end{split}  \tag{31}\]  
where, \(a_{i}=\varepsilon^{2}\widehat{a}_{i}(i=1,2)\), \(a_{4}=\varepsilon^{3}\widehat{a}_{4}\), \(b_{i}=\varepsilon^{2}\widehat{b}_{i}(i=1,3,4)\), \(\Omega_{1}{}^{2}=\varepsilon^{2}\widehat{\Omega}_{1}{}^{2}\). Assume the approximate solution form of Equations (31) and (32) as  
\[ \begin{split}&\overline{\eta}_{1}=\varepsilon^{1}\overline{\eta}_{11}(T _{0},T_{2})+\varepsilon^{3}\overline{\eta}_{13}(T_{0},T_{2})\\ &\overline{\eta}_{1}=\varepsilon^{1}\overline{\eta}_{11}(T_{0},T_ {2})+\varepsilon^{3}\overline{\eta}_{13}(T_{0},T_{2})\end{split}  \tag{33}\]  
where \(T_{n}=\varepsilon^{n}\tau,\ (n=1,2,3,\ldots)\). By substituting Equation (33) into Equations (31) and (32) and comparing the coefficients of the same power at both ends of the equation, the following set of partial differential equations can be obtained:  
\[