# 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}\widehat{a_{4}}\sin(\theta_{1})-\widehat{a_{2}}\Omega_{g}u_ {1}+\sin(\phi)\Big{(}\widehat{\Omega_{1}}^{-2}{}_{2}s_{12}+a_{5}\sigma_{2}u_{ 2}\Big{)}=0\\ 4\cos(\phi)\widehat{b_{4}}\Omega_{g}u_{1}+\sin(\phi)\Big{(}4\widehat{b_{3}}u_ {1}+3b_{6}{u_{1}}^{3}\Big{)}+4\widehat{b_{1}}\Omega_{g}u_{2}+b_{2}\Omega_{g}{ u_{2}}^{3}=0\\ 4\cos(\theta_{1})\widehat{a_{4}}+4\widehat{a_{1}}u_{1}-4\widehat{\Omega_{1}}^{ -2}{}_{1}u_{1}-4\sigma_{1}u_{1}+3a_{3}{u_{1}}^{3}+\cos(\phi)\Big{(}4\widehat{ \Omega_{1}}^{-2}{}_{2}s_{12}+4a_{5}\sigma_{2}u_{2}\Big{)}=0\\ {-4\widehat{b_{4}}\sin(\phi)\Omega_{g}u_{1}+\cos(\phi)\Big{(}4\widehat{b_{3}}u_ {1}+3b_{6}{u_{1}}^{3}\Big{)}+4\widehat{\Omega_{1}}^{-2}{}_{2}b_{5}u_{2}-4 \sigma_{1}u_{2}+4\sigma_{2}u_{2}+4b_{5}\sigma_{2}u_{2}=0}\end{array}  \tag{39}\]  
Eliminate \(\phi\), \(\theta_{1}\), then two bifurcation equations for amplitude \(u_{1}\) and \(u_{2}\) can be obtained:  
\[\begin{array}{l}-16\widehat{b_{3}}^{2}{}_{1}{}^{-2}16\widehat{b_{4}}^{2} \Omega_{g}^{2}{u_{1}}^{-2}24\widehat{b_{3}}{b_{6}}{u_{1}}^{4}-9b_{6}{}^{2}{u_{1 }}^{6}+16\widehat{\Omega_{1}}^{-4}{}_{1}b_{5}^{2}{u_{2}}^{2}-32\widehat{\Omega_{1 }}^{-2}b_{5}\sigma_{1}{u_{2}}^{2}+\\ 16\sigma_{1}^{2}{u_{2}}^{2}+32\widehat{\Omega_{1}}^{-2}b_{5}\sigma_{2}{u_{2}}^{2 }+32\widehat{\Omega_{1}}^{-2}b_{5}^{2}\sigma_{2}u_{2}u_{2}^{-32}2\widehat{\
u_{ 1}}^{-2}32b_{5}\sigma_{1}{\sigma_{2}}{u_{2}}^{2}+16\sigma_{2}^{2}{u_{2}}^{2}\\ +32b_{5}\sigma_{2}^{2}u_{2}+16\widehat{b_{5}}^{2}\sigma_{2}^{2}u_{2}^{2}+16 \widehat{b_{5}}^{2}\sigma_{2}^{2}u_{2}^{2}+8\widehat{b_{3}}b_{2}\Omega_{g}^{2}u _{2}^{2}{u_{2}}^{4}+b_{2}^{2}\Omega_{g}^{2}u_{2}^{6}=0\\ (-4\widehat{a_{1}}u_{1}+4\widehat{\Omega_{1}}^{-2}{}_{1}u_{1}+4\sigma_{1}u_{1}-3a_ {3}{u_{1}}^{3}+(16\Big{(}\widehat{\Omega_{1}}^{-2}{}_{1}a_{5}u_{2}+a_{5}\sigma_{2 }u_{2}\Big{)}(4\widehat{b_{3}}\widehat{\Omega_{1}}^{-2}b_{5}u_{2}-4\widehat{b_{3}} \sigma_{1}u_{2}+4\widehat{b_{3}}\sigma_{2}u_{2}\\ +4\widehat{b_{3}}b_{5}\sigma_{2}u_{2}+4\widehat{b_{1}}\widehat{b_{4}}\Omega_{g}^{2} u_{2}+3\widehat{\Omega_{1}}^{-2}b_{5}b_{6}\sigma_{1}{u_{1}}^{2}u_{2}-3b_{6}\sigma_{1}u_{1} {u_{2}}+3b_{6}\sigma_{2}{u_{1}}^{2}u_{2}+3b_{5}b_{6}\sigma_{2}{u_{1}}^{2}u_{2} +\widehat{b_{4}}b_{2}\Omega_{g}^{2}u_{3}))/\\ \Big{(}u_{1}\Big{(}16\widehat{b_{3}}^{2}+16\widehat{b_{4}}^{2}\Omega_{g}^{2}+24 \widehat{b_{3}}b_{6}{u_{1}}^{2}+9b_{6}{}^{2}{u_{1}}^{4}\Big{)}\Big{)})^{2}+16( \widehat{a_{2}}\Omega_{g}u_{1}-\big{(}\big{(}\widehat{\Omega_{1}}^{-2}a_{5}u_{2} +a_{5}\sigma_{2}u_{2}\Big{)}\\ (-16\widehat{b_{1}}\widehat{b_{3}}\Omega_{g}\Omega_{g^{\prime}}u_{1}u_{2}+16 \widehat{b_{4}}\widehat{\Omega_{1}}^{-2}b_{5}\Omega_{g}u_{1}u_It can be seen that the linear and nonlinear stiffness of high and low position elastic constraints affect the vortex-induced vibration response of the VNA system. In order to explore the impact of constraint stiffness parameters on the response of the VNA system bearing the combined action of vortex-induced vibration and foundation excitation in the first-order frequency locking circumstance, this section divides the constraint types into seven types, namely,  
* Linear stiffness \(10^{4}\) N/m, nonlinear stiffness \(10^{3}\) N/m\({}^{3}\);
* Linear stiffness \(10^{5}\) N/m, nonlinear stiffness \(10^{3}\) N/m\({}^{3}\);
* Linear stiffness \(10^{3}\) N/m, nonlinear stiffness \(10^{4}\) N/m\({}^{3}\);
* Linear stiffness \(10^{3}\) N/m, nonlinear stiffness \(10^{5}\) N/m\({}^{3}\);
* Linear stiffness \(10^{4}\) N/m, nonlinear stiffness \(10^{4}\) N/m\({}^{3}\);
* Linear stiffness \(10^{5}\) N/m, nonlinear stiffness \(10^{5}\) N/m\({}^{3}\). The high and low constraint stiffnesses and damping are consistent. Applying the multiscale method to solve the bifurcation Equations (40) and (41) of the VNA system under the first-order frequency locking condition, the amplitude frequency response of the system can be obtained as shown in Figure 7, where the abscissa is the dimensionless basic excitation frequency and \(\eta_{A}\) is the dimensionless vibration amplitude of the top of the VNA system pipeline.