# Analysis of Dynamic Characteristics of Attached High Rise Risers
## 3 Dynamic Analysis of VNA System under Combined Excitation
### Mathematical Model
Considering the effect of foundation excitation, Equation (17) for simulating vortex-induced lift can be further written as  
\[\frac{\partial^{2}q}{\partial t^{2}}+\lambda_{s}\omega_{s}\Big{(}q^{2}-1\Big{)} \frac{\partial q}{\partial t}+\omega_{s}{}^{2}q=\frac{P}{D}(\frac{\partial^{2} w_{rel}(x,t)}{\partial t^{2}}+\ddot{w}_{b}) \tag{18}\]  
In addition, according to literature [45], the expression of vortex-induced force is  
\[f(x,t)=\frac{C_{L0}\rho_{f}V_{out}^{2}D}{4}q(x,t)-\frac{C_{D}\rho_{f}V_{out}D}{ 2}\dot{w}_{rel} \tag{19}\]  
\(C_{L_{0}}\) and \(C_{D}\) are respectively the stable lift coefficient and the average section drag coefficient, which are taken as 0.3 and 1.2 for ease of calculation. For the convenience of analysis and calculation, the following dimensionless parameters are introduced:  
\(\eta=w/L\); \(\xi=x/L\); \(\tau=t/\Big{(}L^{2}\sqrt{m_{\Sigma}/(EI)}\Big{)}\); \(\
u=V_{in}L\sqrt{m_{f}/(EI)}\); \(o=\frac{At^{2}}{2T}\);  
\(c_{0}=c_{s}L^{2}/\sqrt{m_{\Sigma}EI}\); \(c_{1,2}=c_{epdm}L^{2}/\sqrt{m_{\Sigma}EI}\); \(\alpha=m_{f}/m_{\Sigma}\); \(\beta=m_{s}L^{3}/(EI)\); \(\gamma=(m_{t}g+m_{s}gL)L^{2}/(EI)\);  
\(\kappa_{1}=K_{1}L^{3}/(EI)\); \(\kappa_{2}=K_{2}L^{3}/(EI)\); \(\kappa_{3}=K_{3}L^{5}/(EI)\); \(\kappa_{4}=K_{4}L^{5}/(EI)\);  
\(\Omega_{s}=\omega_{s}L^{2}\sqrt{m_{\Sigma}/(EI)}\); \(\Omega_{in}=\omega_{in}L^{2}\sqrt{m_{\Sigma}/(EI)}\); \(f_{D}=\frac{C_{D}\rho_{f}DV_{out}L^{2}}{2\sqrt{m_{\Sigma}EI}}\); \(f_{L}=\frac{C_{L0}\rho_{f}DV_{out}L^{2}L^{3}}{4 EI}\); \(p=\frac{PI}{D}\)  
The dimensionless dynamic equation of the exhaust duct can be obtained as follows:  
\[ \begin{array}{l}\frac{\partial^{2}\eta_{rel}}{\partial\xi^{2}}+\frac{ \partial^{2}\eta_{rel}}{\partial\xi^{2}}+\
u^{2}\frac{\partial^{2}\eta_{rel}} {\partial\xi^{2}}+2\
u\sqrt{\alpha^{2}\frac{\partial^{2}\eta_{rel}}{\partial \xi^{2}}}-\beta\frac{\partial\eta_{rel}}{\partial\xi^{2}}+\gamma\frac{ \partial^{2}\eta_{rel}}{\partial\xi^{2}}-\beta\xi\frac{\partial^{2}\eta_{rel}} {\partial\xi^{2}}\\ -\Big{[}o\int_{0}^{1}\Big{(}\frac{\partial\eta_{rel}}{\partial\xi}\Big{)}^{2} \mathrm{d}\xi\Big{]}\frac{\partial^{2}\eta_{rel}}{\partial\xi^{2}}+\kappa_{1} \eta_{rel}\delta(\xi-\xi_{a})+\kappa_{2}\eta_{rel}\delta(\xi-\xi_{b})+\kappa_{3 }\eta_{rel}^{3}\delta(\xi-\xi_{a})\\ +\kappa_{4}\eta_{rel}^{3}\delta(\xi-\xi_{b})+c_{1}\frac{\partial\eta_{rel}}{ \partial\tau}\delta(\xi-\xi_{a})+c_{2}\frac{\partial\eta_{rel}}{\partial\tau} \delta(\xi-\xi_{a})=f(\xi,\tau)-G\tilde{\eta}_{b}\end{array}  \tag{20}\]  
\(\eta_{rel}\) is the dimensionless relative displacement, \(G=m_{\Sigma}L^{3}/(EI)\), \(c_{1,2}=c_{epdm}L^{2}/\sqrt{m_{\Sigma}EI}\), \(c_{epdm}\) is the damping coefficient of the rubber gasket, and the value in this chapter is 0.3 N/(m/s). \(f(\xi,\tau)\) is the dimensionless vortex-induced force. For ease of expression, this chapter will use \(\eta\) instead of \(\eta_{rel}\). Equation (5-5) can be dimensionless and expressed as:  
\[\frac{\partial^{2}q}{\partial\tau^{2}}+\lambda_{s}\Omega_{s}\Big{(}q^{2}-1 \Big{)}\frac{\partial q}{\partial\tau}+\Omega_{s}{}^{2}q=p(\frac{\partial^{2} \eta}{\partial\tau^{2}}+G\tilde{\eta}_{b}) \tag{21}\]