# 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{array}{l}\sum\limits_{j=1}^{n}h_{i,j}\dot{\overline{\eta}}_{j}(\tau) +\sum\limits_{j=1}^{n}h_{i,j(^{4})}\overline{\eta}_{j}(\tau)+2\sqrt{\alpha} \
u\sum\limits_{j=1}^{n}h_{i,j(^{3})}\dot{\overline{\eta}}_{j}(\tau)+\left( \gamma+\
u^{2}\right)\sum\limits_{j=1}^{n}h_{i,j(^{2})}\overline{\eta}_{j}( \tau)-\beta\sum\limits_{j=1}^{n}h\xi_{i,j(^{2})}\overline{\eta}_{j}(\tau)\\ -\beta\sum\limits_{j=1}^{4}h_{i,j(^{1})}\overline{\eta}_{j}(\tau)+f_{D}\sum \limits_{j=1}^{n}h_{i,j}\dot{\overline{\eta}}_{j}(\tau)+\phi\sum\limits_{l=1}^ {n}\sum\limits_{m=1}^{n}h_{i,l(^{1}),m,s(^{1})}\overline{\eta}_{l}(\tau) \overline{\eta}_{m}(\tau)\overline{\eta}_{s}(\tau)+\kappa_{1}\sum\limits_{j=1}^ {n}H_{i,j}(\xi_{a})\overline{\eta}_{j}(\tau)\\ +c_{1}\sum\limits_{j=1}^{n}H_{i,j}(\xi_{a})\dot{\overline{\eta}}_{j}(\tau)+ \kappa_{3}\sum\limits_{l=1}^{n}\sum\limits_{m=1}^{n}\sum\limits_{s=1}^{n}H_{i, l(m,s}(\xi_{a})\overline{\eta}_{l}(\tau)\overline{\eta}_{m}(\tau)\overline{\eta}_{s}( \tau)+\kappa_{2}\sum\limits_{j=1}^{n}H_{i,j}(\xi_{b})\dot{\overline{\eta}}_{j}( \tau)+c_{2}\sum\limits_{j=1}^{n}H_{i,j}(\xi_{b})\dot{\overline{\eta}}_{j}( \tau)\\ +\kappa_{4}\sum\limits_{l=1}^{n}\sum\limits_{m=1}^{n}\sum\limits_{s=1}^{n}H_{i, l,m,s}(\xi_{b})\overline{\eta}_{l}(\tau)\overline{\eta}_{m}(\tau)\overline{\eta}_{s}( \tau)=f_{L}h_{i,\tau}\overline{\eta}_{r}(\tau)-Gh_{i}\dot{\overline{\eta}}_{b}( \tau),(i=1,2,\cdots,n)\end{array}  \tag{26}\]  
\[ \begin{array}{l}h_{i,j}\dot{\overline{\eta}}_{r}(\tau)+\Omega_{s}{}^{2}h_{i, j}\dot{\overline{\eta}}_{r}(\tau)-\lambda_{s}\Omega_{s}h_{i,j}\dot{\overline{ \eta}}_{r}(\tau)+h_{i,j}\dot{\overline{\eta}}_{r}(\tau)\overline{\eta}_{r}{}^{2 }(\tau)=p(\sum\limits_{j=1}^{n}h_{i,j}\dot{\overline{\eta}}_{j}(\tau)+Gh_{i} \ddot{\overline{\eta}}_{b}(\tau)),(i=1,2,\cdots,n)\end{array}  \tag{27}\]  
\[ \begin{array}{l}\text{where }h_{i,j},\ h_{i,j(^{4})},\ h_{i,j(^{1})},\ h_{i,j(^{2})},\ h _{\xi_{i,j(^{2})}},\ h_{i,j(^{6})}(\xi_{a}),\ h_{i,l(^{1}),m,s(^{1})},H_{i,j}( \xi_{b}),\ H_{i,l,m,s}(\xi_{a}),\ H_{i,l,m,s}(\xi_{b}),\ H_{i,l,m,s}(\xi_{b}), \ H_{i,r},\ h_{i},\ h_{i,r}{}^{3}\text{ as shown in (A3) in Appendix A}.\end{array} \]  
Through modal truncation analysis, it is easy to know that under frequency locking conditions, studying first-order main resonance is sufficient to predict the vibration response of VNA system pipelines. Taking \(I=1\) from Equations (26) and (27) obtained by the Galerkin discretization method, the following ordinary differential equation under first-order frequency locking can be obtained:  
\[\dot{\overline{\eta}}_{1}(\tau)+a_{1}\overline{\eta}_{1}(\tau)+a_{2}\dot{ \overline{\eta}}_{1}(\tau)+a_{3}\overline{\eta}_{1}{}^{3}(\tau)+a_{5}\Omega_{s}{}^{2} \overline{\eta}_{1}(\tau)+a_{4}\cos(\Omega_{g}\tau)=0 \tag{28}\]  
\[ \begin{array}{l}\ddot{\overline{\eta}}_{1}(\tau)+\Omega_{s}{}^{2}\overline{ \eta}_{1}(\tau)+b_{1}\dot{\overline{\eta}}_{1}(\tau)+b_{2}\dot{\overline{\eta}}_{1}( \tau)\overline{\eta}_{1}{}^{2}(\tau)\\ +b_{3}\overline{\eta}_{1}(\tau)+b_{4}\dot{\overline{\eta}}_{1}(\tau)+b_{6}\dot{ \overline{\eta}}_{1}{}^{3}(\tau)+b_{5}\Omega_{s}{}^{2}\overline{\eta}_{1}( \tau)=0\end{array}  \tag{29}\]where,  
\[