## 3 Basic Theory
### Vortex-Induced Force Coefficients  
Assume that the vibration in the CF direction is composed of periodic vibrations with \(n\) different frequencies, the VIV displacement at node \(x\) can be expressed as  
\[y(x,t)=\sum\limits_{i=1}^{n}y_{0i}(x)\sin(\omega_{i}t+\theta_{i}) \tag{13}\]  
where \(\omega_{i}\) is the \(i\)th frequency of vibration in the CF direction; \(y_{0i}(x)\) and \(\theta_{i}\) are the displacement amplitude and phase angle at frequency \(\omega_{i}\). The VIV velocity \(\dot{y}(x,t)\) and acceleration \(\ddot{y}(x,t)\) can be obtained by differentiating Equation (13):  
\[ \begin{split}\dot{y}(x,t)=\sum\limits_{i=1}^{n}y_{0i}(x)\omega_{ i}cos(\omega_{i}t+\theta_{i})\\ \ddot{y}(x,t)=-\sum\limits_{i=1}^{n}y_{0i}(x)\omega_{i}^{2}\sin (\omega_{i}t+\theta_{i})\end{split}  \tag{14}\]  
The vortex-induced force at node \(x\), \(f_{CF}(x,t)\), can be written as  
\[f_{CF}(x,t)=\sum\limits_{i=1}^{n}f_{0i}(x)\sin(\omega_{i}t+\theta_{i}+\varphi _{i}) \tag{15}\]  
where \(f_{0i}(x)\) is the amplitude at frequency \(\omega_{i}\), and \(\varphi_{i}\) is the phase angle between the vortex-induced force and the displacement at frequency \(\omega_{i}\). Expanding the right side of Equation (15),  
\[ \begin{split} f_{CF}(x,t)&=\sum\limits_{i=1}^{n}f_ {CF}(\omega_{i},x,t)\\ &=\sum\limits_{i=1}^{n}\begin{array}{c}f_{0i}(x)\cos(\omega_{ i}t+\theta_{i})\sin(\varphi_{i})\\ -(-f_{0i}(x)\sin(\omega_{i}t+\theta_{i})\cos(\varphi_{i}))\end{array}\end{split}  \tag{16}\]  
Under a single frequency, the two terms in Equation (16) are excitation force in phase with the velocity and added mass forces in phase with the acceleration, respectively [15]; then, the vortex-induced forces of bare pipe modules, buoyancy modules and straked modules under a single frequency can be expressed as:\[f_{CF}(x,t)=f_{0i}(x)\cos(\omega_{i}t+\theta_{i})\sin(\varphi_{i})-(-f_{0i}(x) \sin(\omega_{i}t+\theta_{i})\cos(\varphi_{i})) \tag{17}\] \[\left\{ \begin{array}{l}-\frac{1}{4}\rho\pi T_{bare}{}^{2}{}^{1}C_{a, bare}(x)\ddot{y}(x,t)+\frac{1}{2}D_{bare}|pU^{2}C_{e,bare}(x)\frac{\ddot{y}(x,t)}{y_{0}(x) \omega}\quad(\text{on bare pipe})\\ -\frac{1}{4}\rho\pi T_{buoyancy}{}^{2}{}^{1}C_{a,buoyancy}(x)\ddot{y}(x,t)+ \frac{1}{2}D_{buoyancy}|pU^{2}C_{e,buoyancy}(x)\frac{\ddot{y}(x,t)}{y_{0}(x) \omega}(\text{on buoyancy element})\\ -\frac{1}{4}\rho\pi T_{stake}{}^{2}{}^{1}C_{a,stake}(x)\ddot{y}(x,t)+\frac{1}{2 }D_{stake}|pU^{2}C_{e,stake}(x)\frac{\ddot{y}(x,t)}{y_{0}(x)\omega}(\text{on }\text{ stracked pipe})\end{array} \right.\]  
where \(C_{e,bare}(x),C_{e,buoyancy}(x),C_{e,stake}(x)\) are the excitation coefficients at node x of the bare segment, buoyancy element and stracked element, respectively; \(C_{a,bare}(x),C_{a,buoyancy}(x)\) and \(C_{a,stake}(x)\) are the added-mass coefficients of the bare pipe, buoyancy element and stracked element, respectively; \(\rho\) is the fluid density; \(U\) is the fluid velocity; \(D_{bare}\), \(D_{buoyancy}\) and \(D_{stake}\) are the diameters of the bare pipe, buoyancy element and stracked element, respectively; and \(y_{0i}(x)\omega\) is the amplitude of VIV velocity at node \(x\). Substituting Equation (17) into Equation (16), the vortex-induced forces under these multi-frequencies can be expressed as the sum of vortex-induced forces under each single frequency:  
\[f_{CF}(x,t)= \tag{18}\] \[\left\{