# Analysis of Dynamic Characteristics of Attached High Rise Risers
## 3 Dynamic Analysis of VNA System under Combined Excitation
### Mathematical Model  
This chapter considers a VNA system with one end fixed and vertically placed. The fluid flow rate in the duct is shown in Figure 6. The pipeline is subjected to transverse external flow at a velocity of \(V_{out}\), and external displacement-based excitation recorded as \(w_{b}\). It should be noted that this article only considers the occurrence of transverse flow vibrations in the pipeline system, that is, perpendicular to the direction of the external flow velocity. Refer to the literature [19; 40; 41; 42; 43; 44] for details; this assumption has been proven to be feasible. At the same time, the damping characteristics of EPDM rubber gaskets are considered in this chapter. Figure 5: Modal shapes of VNA system with different outer diameter function. (**a**) First mode shapes; (**b**) Second order mode shapes; (**c**) Third order mode shapes; (**d**) Fourth order mode shapes. Due to the pipe's large length-diameter ratio, the Euler-Bernoulli beam theory can be used for the pipe structure. Here we propose two assumptions: one is that the fluid inside and outside the pipeline is incompressible and flows at a constant rate, and the other is that the damping of the VNA system itself is not considered. According to research by [PERSON] et al., the transverse absolute displacement of a beam, that is, the transverse displacement relative to a fixed inertial system, can be written as  
\[w(x,t)=w_{b}(x,t)+w_{rel}(x,t) \tag{14}\]  
where, \(w_{rel}(x,t)\) is the transverse displacement relative to the fixed end, \(w_{b}(x,t)\) is the base displacement, i.e.,  
\[w_{b}(x,t)=\delta_{1}(x)g_{L}(t)+\delta_{2}(x)h_{L}(t) \tag{15}\]  
In the formula, \(\delta_{1}(x)\) and \(\delta_{2}(x)\) are respectively the influence functions of foundation transverse displacement and small rotation on beam transverse displacement. Simply set \(\delta_{1}(x)=1\), \(\delta_{2}(x)=x\) for a cantilever beam. The dynamic equations of VNA pipeline subjected to both foundation excitation and vortex-induced vibration are as follows:  
\[ \begin{array}{l}EI\frac{\partial^{2}w_{rel}}{\partial x^{4}}-\frac{\partial }{\partial x}\Big{(}T(x)\frac{\partial w_{rel}}{\partial x}\Big{)}+m_{f}V_{in} ^{2}\frac{\partial^{2}w_{rel}}{\partial x^{2}}+2m_{f}V_{in}\frac{\partial^{2 }w_{rel}}{\partial x\partial t}+m_{\sum}\frac{\partial^{2}w_{rel}}{\partial t ^{2}}\\ -[\frac{EA}{2L}\int_{0}^{L}\Big{(}\frac{\partial w_{rel}}{\partial x}\Big{)}^{ 2}dx]\frac{\partial^{2}w_{rel}}{\partial x^{2}}+K_{1}w_{rel}\delta(x-l_{a})+K_ {2}w_{rel}\delta(x-l_{b})+K_{3}w_{rel}{}^{3}\delta(x-l_{a})\\ +K_{4}w_{rel}{}^{3}\delta(x-l_{b})+c_{pdm}\frac{\partial w_{rel}}{\partial t} \delta(x-l_{a})+c_{pdm}\frac{\partial w_{rel}}{\partial t}\delta(x-l_{b})=f( x,t)-m_{\sum}\bar{w}_{b}\end{array}  \tag{16}\]  
In this paper, the empirical model of vortex wake oscillator proposed by [PERSON] et al. is used, and the equation for simulating vortex-induced force is expressed as follows,  
\[\frac{\partial^{2}q}{\partial t^{2}}+\lambda_{s}\omega_{s}\Big{(}q^{2}-1 \Big{)}\frac{\partial q}{\partial t}+\omega_{s}{}^{2}q=\frac{P}{D}\frac{ \partial^{2}w(x,t)}{\partial t^{2}} \tag{17}\]  
where \(q(x,t)\) is the equivalent lift coefficient used to describe the wake behavior, \(\omega_{s}\) is the vortex shedding frequency, \(\omega_{s}=2\pi S_{t}V_{out}/D\), \(\lambda_{s}\) and P are empirical parameters. Figure 6: VNA pipeline model excited by vortex and foundation simultaneously.