# Analysis of Dynamic Characteristics of Attached High Rise Risers
## 2 Natural Vibration Characteristics Analysis  
### Theoretical Model  
The VNA system mainly includes horizontal and vertical pipes, steel structure platforms, fixed supports, guide supports, flow meters, lightning protection equipment, and other components. The specific structural model is shown in Figure 1a. Vertical pipes and horizontal pipes are decoupled by hoses connecting them. The specific layout is shown in Figure 1b. The fixed support and guide support make the vertical section of the VNA system a multi-point constrained riser model. The fixed support restricts all degrees of freedom at the bottom of the riser, and the guide support only releases degrees of freedom of translation and rotation along the axis of the riser. For the reason that there are many components in the VNA system and the contact relationship between each component is complex, it is necessary to simplify the structural model: (1) It is assumed that the steel structure platform and various instruments and equipment are located at the top of the riser. (2) The fixed support is simplified by the fixed constraint located at the bottom of the riser. (3) The guide bracket is simplified into two, which are located at different positions on the pipe body. The simplified physical model is shown in Figure 2.  
Figure 1: VNA system structure model. (**a**) Schematic diagram of general assembly structure; (**b**) Layout of Hualong-1. Figure 2: Simplified mechanical model of the riser. Based on [PERSON] longitudinal and transverse bending beam theory [38; 39], the free vibration control equation of riser is as follows:  
\[ \begin{array}{l}\frac{\partial^{2}}{\partial x^{2}}\Big{(}E(x)I(x)\frac{ \partial^{2}w(x,t)}{\partial x^{2}}\Big{)}-\frac{\partial}{\partial x}\Big{(} T(x)\frac{\partial w(x,t)}{\partial x}\Big{)}+m(x)\frac{\partial^{2}w(x,t)}{ \partial t^{2}}\\ +K_{1}w(x,t)\delta(x-l_{a})+K_{2}w(x,t)\delta(x-l_{b})=0\end{array}  \tag{1}\]  
where \(E(x)I(x)\) is the bending stiffness, \(T(x)\) is the effective axial force of the riser, \(m(x)\) is the effective mass per unit length of the riser, \(K_{1}\) is the stiffness of the low-position guide bracket, the distance from the bottom is \(l_{a}\), \(K_{2}\) is the stiffness of the high- position guide bracket, and the distance from the bottom is \(l_{b}\). \(T(x)=-m_{t}g-\int_{x}^{L}m(x)g\mathrm{d}x\), \(m_{t}\) is the top concentrated mass, and the riser length is \(L\). Use the method of separating variables to solve Equation (1), make  
\[w(x,t)=\phi(x)\eta(t) \tag{2}\]  
Substitute it into the Equation (1),  
\[ \begin{array}{l}\frac{EI}{m}\frac{1}{\phi(x)}\frac{d^{4}\phi(x)}{dx^{4}}- \delta\frac{1}{\phi(x)}\frac{d\phi(x)}{dx}+\frac{m_{t}g+mg(L-x)}{m}\frac{1}{ \phi(x)}\frac{d^{2}\phi(x)}{dx^{2}}\\ +\frac{K_{1}\phi(x-l_{a})+K_{2}\phi(x-l_{b})}{m}=-\frac{1}{\eta(t)}\frac{d^{2} \eta(t)}{dt^{2}}\end{array}  \tag{3}\]  
Owing to the variable coefficient \(x\) on the left side of the Equation (4) and the spring constraints, the vibration mode function cannot be expressed as an elementary function, and the analytical solution cannot be obtained. It can only be solved by numerical methods. The dimensionless variables are defined as follows:  
\[ \begin{array}{l}\overline{x}=\frac{x}{L},\ \overline{\phi}(\overline{x})= \frac{\phi(x)}{L},\ s(\overline{x})=\frac{E(x)I(x)}{E(x_{0})I(x_{0})},\ n( \overline{x})=\frac{T(x)L^{2}}{E(x_{0})I(x_{0})},\\ q(\overline{x})=\frac{m(x)}{m(x_{0})},\ \lambda^{4}=\frac{m(x_{0})\omega^{2}L^{4}}{E(x_{0})I(x_{0})} \end{array} \]  
the control equation becomes  
\[ \begin{array}{l}\frac{d^{2}}{d\overline{x}^{2}}\Big{(}s(\overline{x})\frac {d^{2}\overline{\phi}(x)}{d\overline{x}^{2}}\Big{)}-\frac{d}{d\overline{x}} \bigg{(}n(\overline{x})\frac{d\overline{\phi}(x)}{d\overline{x}}\bigg{)}-q( \overline{x})\lambda^{4}\overline{\phi}(x)=0\end{array}  \tag{4}\]  
Due to the elastic constraints in the beam, the vibration mode of the cantilever beam will be significantly affected in cases of large stiffness.