## 2 Theoretical Background  
The mathematical model for VIV is briefly presented in this section. The equation of motion for a flexible prism with a length L undergoing vortex-induced vibrations may be expressed as:  
\[M_{r}\ddot{Y}_{r}(t)+2\omega_{r}\zeta_{r}M_{r}\dot{Y}_{r}(t)+\omega_{r}^{2}M_{ r}Y_{r}(t)=\int\limits_{0}^{L}\phi_{r}(x)P(t)dx \tag{1}\]  
where \(Y_{r}\) represents the modal displacement; \(M_{r}\), \(\omega_{r}\), and \(\zeta_{r}\) represent the effective mass, the natural circular frequency, and the damping ratio of the \(r\)th mode of vibration, respectively; and \(P(t)\) is the per unit length owing to vortex shedding. According to the single harmonic oscillator vortex-induced vibration mathematical model [1], the vortex shedding lift force approximates a simple harmonic excitation expressed as:  
\[P(t)=\frac{1}{2}\rho\mathcal{U}^{2}D\widetilde{C}_{L}\sin(\omega t+\psi) \tag{2}\]where \(\rho\) is the air density, \(U\) is the incoming wind speed, \(D\) is the cross-wind dimension of a cross-section, \(\widetilde{C}_{L}\) is the fluctuating lift coefficient of vortex shedding, and \(\omega\) is the vortex shedding frequency, which is close to the natural frequency when a lock-in phenomenon occurs. Substitution of Equation (2) into Equation (1) yields:  
\[M_{r}\ddot{Y}_{r}(t)+2\omega_{r}\zeta_{r}M_{r}\dot{Y}_{r}(t)+\omega_{r}^{2}M_{ r}Y_{r}(t)=\int\limits_{0}^{L}\varphi_{r}(x)\frac{1}{2}\rho U^{2}D\widetilde{C}_{L} \sin(\omega t+\psi)dx \tag{3}\]  
When the prism resonates under the excitation of vortex shedding, the vortex shedding frequency \(\omega\) is very close to the prism's natural circular frequency \(\omega_{r}\). The solution for the amplitude of modal displacement \(Y_{r}\) can be expressed as:  
\[Y_{r,max}=\frac{\frac{1}{2}\rho U^{2}DR\widetilde{C}_{L}\int\limits_{0}^{L} \varphi_{r}(x)dx}{M_{r}\omega_{r}^{2}}\frac{1}{2\zeta_{r}} \tag{4}\]  
where the factor \(R\) allows the imperfect correlation of vortex shedding to force along the length of the prism. It can be referenced as bridge deck [23]. Several studies have been conducted with a focus on the spanwise correlation effects of vortex-induced forces in separate smooth and turbulent flow conditions [24; 25; 26]; the main consensus can be drawn as the greater the vibration amplitude, the more the correlation length, while the quantitative results are varying between amplitude, cross section, Reynolds number, etc. The non-dimensional vibration amplitude can be rewritten as:  
\[ \begin{split}\widetilde{Y}_{3D}=\varphi_{r,\max}\frac{Y_{r,\max}} {D}=&\varphi_{r,\max}\frac{\frac{1}{2}\rho U^{2}R\widetilde{C}_{ L}\int\limits_{0}^{L}\varphi_{r}(x)dx}{m\int\limits_{0}^{L}\varphi_{r}^{2}(x)dx \cdot\left(2\pi f_{r}\right)^{2}}\cdot\frac{1}{2\zeta_{r}}\\ =&\frac{1}{4\pi}\cdot\frac{1}{S_{C}}\cdot\frac{1} {St^{2}}\cdot R\cdot\widetilde{C}_{L}\cdot\frac{\varphi_{r,\max}\int\limits_{0 }^{L}\varphi_{r}(x)dx}{\int\limits_{0}^{L}\varphi_{r}^{2}(x)dx}\end{split}  \tag{5}\]  
where \(m\) is the equivalent mass of per unit of length of the prism, \(S_{C}\) is the mass-damping parameter (\(Sc\) number), \(S_{C}=4\pi m\zeta_{r}\rho D^{2}\), \(St\) is the Strouhal number of the prism's cross-section, and \(St=f_{r}D/U\), while the last term on right hand side of Equation (5) represents the modal shape correction factor for the vibration amplitude. Equation (5) is one of the formulae used in Euro Code to estimate the VIV's maximum amplitude [4; 27]. It can be observed from Equation (5) that both the structural and aerodynamic parameters include the Scruton and Strouhal numbers, fluctuating lift coefficient, spanwise correlation, and modal shape correction that affect the maximum VIV amplitude. As mentioned before, it is common to use the section model wind tunnel test to predict the maximum VIV amplitudes for actual structures. However, the longitudinal deformed modal shape and imperfect spanwise correlation of vortex shedding forces are not considered in the sectional model test. This may lead to unreliable results.