# Analysis of Dynamic Characteristics of Attached High Rise Risers
## 2 Natural Vibration Characteristics Analysis
### DTM Analysis  
The differential transformation of \(\overline{\phi}(\overline{x})\) is defined as  
\[\Phi(k)=\left.\frac{1}{k!}\left(\frac{d^{k}\overline{\phi}(\overline{x})}{d \overline{x}^{k}}\right)\right|_{\overline{x}=\overline{x}_{0}},0\leq\overline {\pi}\leq 1 \tag{5}\]  
where \(\Phi(k)\) is the \(k\)-order differential function of \(\overline{\phi}(\overline{x})\) after \(\overline{x}=\overline{x}_{0}\) transformation, called T function. The inverse differential transformation of \(\Phi(k)\) is defined as  
\[\overline{\phi}(\overline{x})=\sum_{k=0}^{\infty}(\overline{x}-\overline{x}_{ 0})^{k}\Phi(k) \tag{6}\]Substitute (5) into (6), then obtain the Taylor series expansion of \(\overline{\phi}(\overline{x})\) at \(\overline{x}=\overline{x}_{0}\)  
\[\overline{\phi}(\overline{x})=\sum\limits_{k=0}^{\infty}\frac{\left(\overline{ x}-\overline{x}_{0}\right)^{k}}{k!}\left.\left(\frac{d^{k}\overline{\phi}( \overline{x})}{d\overline{x}^{k}}\right)\right|_{\overline{x}=\overline{x}_{0}} \tag{7}\]  
The basic operation rules of differential transformation, such as linear properties, product properties, differential properties, etc., can be found in Appendix B. Using the differential transformation method and the basic operational properties of DTM, ordinary differential Equation (5) can be transformed into the following algebraic equation  
\[ \begin{array}{l}\sum\limits_{r=0}^{k}S(k-r)(r+1)(r+2)(r+3)(r+4)\Phi(r+4)\\ \\ +2\sum\limits_{r=0}^{k}(k-r+1)S(k-r+1)(r+1)(r+2)(r+3)\Phi(r+3)\\ \\ +\sum\limits_{r=0}^{k}(k-r+1)(k-r+2)S(k-r+2)(r+1)(r+2)\Phi(r+2)-\sum\limits_{r= 0}^{k}N(k-r)(r+1)(r+2)\Phi(r+2)\\ \\ -\sum\limits_{r=0}^{k}(k-r+1)N(k-r+1)(r+1)\Phi(r+1)=\sum\limits_{r=0}^{k} \lambda^{4}Q(k-r)\Phi(r)\end{array}  \tag{8}\]  
where the corresponding primitive functions of \(\Phi(k)\), \(S(k)\), \(N(k)\) and \(Q(k)\) are \(\overline{\phi}(\overline{x})\), \(s(\overline{x})\), \(n(\overline{x})\) and \(q(\overline{x})\). Substituting (9) into boundary condition (A1), can get the differential transformation expression of boundary conditions as (A2) in Appendix A. Equation (9) is a recurrence relation formula about the unknown quantity \(\Phi(0),\Phi(1)\), \(\Phi(2),\Phi(3)\). According to the three vibration modes of segmented beam, assume  
\[ \begin{array}{l}\[PERSON]_{1}(0)=\mathrm{X}_{10},\[PERSON]_{1}(1)=\mathrm{X}_{11}, \[PERSON]_{1}(2)=\mathrm{X}_{12},\[PERSON]_{1}(3)=\mathrm{X}_{13},\\ \[PERSON]_{2}(0)=\mathrm{X}_{20},\[PERSON]_{2}(1)=\mathrm{X}_{21},\[PERSON]_{2}(2)=\mathrm{X }_{22},\[PERSON]_{2}(3)=\mathrm{X}_{23},\\ \[PERSON]_{3}(0)=\mathrm{X}_{30},\[PERSON]_{3}(1)=\mathrm{X}_{31},\[PERSON]_{3}(2)=\mathrm{X }_{32},\[PERSON]_{3}(3)=\mathrm{X}_{33}.\end{array} \]  
Furthermore, by using the recurrence relation formula, various types of boundary conditions in (A2) can be transformed into a set of algebraic equations about these 12 unknown quantities, written in the form of matrix as (10). \[ \begin{array}{l}\left[A\right]_{12\times 12}\{\mathrm{X}_{10},\mathrm{X}_{11}, \mathrm{X}_{12},\mathrm{X}_{13},\mathrm{X}_{20},\mathrm{X}_{21},\mathrm{X}_{2 2},\mathrm{X}_{23},\mathrm{X}_{30},\mathrm{X}_{31},\mathrm{X}_{32},\mathrm{X}_ {33}\}^{T}=\left\{0\right\}_{12\times 1}\end{array}  \tag{9}\]  
The condition that the above equation has nontrivial solution needs to be satisfied  
\[Det(\left[A\right]_{12\times 12})=0 \tag{10}\]  
Equation (11) is a higher-order algebraic equation about \(\lambda\), which can be solved by numerical rooting method. According to \(\omega=\frac{\lambda^{2}}{L^{2}}\sqrt{\frac{E(x_{0})I(x_{0})}{m(x_{0})}}\), the vibration frequency of each order of the high riser pipe can be obtained, then as well the vibration mode of the structure. Firstly, assume \(\mathrm{X}_{10}=1\), and substitute the obtained \(\lambda\) into Formula (10), then \(\mathrm{X}_{11}\sim\mathrm{X}_{33}\) will all be obtained. Furthermore, use recursion Formula (9) to obtain \(\Phi_{1}(k)\sim\Phi_{3}(k)\) successively.