Semi-Analytic Solutions for the Bending-Bending-Torsion Coupled Forced Vibrations of a Rotating Wind Turbine Blade by Means of Green’s Functions

Abstract

In recent years, wind power has received continuous attention as a renewable energy source in the context of carbon neutrality. Blades in wind turbines with elongated structures are susceptible to damage due to aeroelastic instability. In this chapter, the basic solution of the system is derived by Laplace transformation and Green’s function method, and then the system of the second category of Fredholm integral equations about steady-state forced vibration of blades can be derived according to the principle of superposition. The second type of Fredholm integral equation system is discretized numerically, and finally, a semi-analytical solution for the bending-bending-torsion coupled forced vibration of rotating wind turbine blades is obtained. The validity of the solution presented in this chapter is verified by comparing the first-order flag mode and first-order lead/lag mode with previously obtained results. The results show that the inflow ratio at the hub and the angular velocity of rotation have a significant influence on the blade’s flag displacement and lead/lag displacement.

Appendix 1

Some parameters used for Eqs. (3)–(5) are listed here:

$$ {\displaystyle \begin{array}{c}{a}_1=E{I}_{zy}/E{I}_y,{a}_2=m{\omega}^2/E{I}_y,{a}_3=-2m\Omega {\beta}_{\textrm{p}}\omega /E{I}_y,{a}_4=E{I}_{zy}/E{I}_z,{a}_5=-2m\Omega {\beta}_{\textrm{p}}\omega /E{I}_z,\\ {}{a}_6{=}\left(m{\omega}^2{-}m{\Omega}^2\right)/E{I}_z,{a}_7{=}-m{\Omega}^2\left[\left({k}_{m2}^2{-}{k}_{m1}^2\right)\left({\cos}^2\theta {-}{\sin}^2\theta \right){+}\left({k}_{m2}^2{+}{k}_{m1}^2\right){\omega}^2/{\Omega}^2\right]/\left(\textrm{GJ}\right),\\ {}{b}_1=1/E{I}_y,{b}_2=1/E{I}_z,{b}_3=1/\left(\textrm{GJ}\right),{A}_1=-{\beta}_{\textrm{p}}\rho ac{\Omega}^2/2,{A}_2=\cos \theta \rho ac{\Omega}^2/2,\\ {}{A}_3=\left[c/2+\left(c/4\right)\cos \theta -{e}_{\textrm{A}}\right]\rho ac{\Omega}^2/2,{A}_4=\left(\sin \theta \cos \theta \rho a{c}^2\right)/8,{A}_5=-\left(\cos \theta \rho a{c}^2\right)/8,\\ {}{B}_1=\left(-\lambda R\cos \theta \rho ac{\Omega}^2\right)/2,{B}_2=-\left({\sin}^2\theta \rho a{c}^2\right)/8,{B}_3=-\left(\sin \theta \rho a{c}^2\right)/8,\\ {}{A}_4^{\prime }={A}_4{\omega}^2,{A}_5^{\prime }={A}_5{\omega}^2,{B}_2^{\prime }={B}_2{\omega}^2,{B}_3^{\prime }={B}_3{\omega}^2,\end{array}} $$

$$ {\displaystyle \begin{array}{c}{f}_1(x)=\left(\rho ac{\Omega}^2/2\right)\left\{\left(1-{\beta}_{\textrm{p}}^2\right){x}^2\sin \theta -\lambda Rx\cos \theta +\left[c/2+\left(c/4\right)\cos \theta -{e}_{\textrm{A}}\right]{\beta}_{\textrm{p}}x\right\},\\ {}{p}_1(x)=\left(\rho ac{\Omega}^2/2\right)\left\{{\lambda}^2{R}^2\cos \theta -\lambda \left(1-{\beta}_{\textrm{p}}^2\right) Rx\sin \theta -\left({C}_{\textrm{DO}}/a\right){x}^2\right\},\\ {}{\textrm{Non}}_1(x)={A}_2{x}^2{\int}_0^LQ\left({x}_0\right){G}_{\Phi}\left(x,{x}_0\right) d x,\\ {}{\textrm{Non}}_2(x)={B}_1x{\int}_0^LQ\left({x}_0\right){G}_{\Phi}\left(x,{x}_0\right)d{x}_0,\\ {}{f}_w(x)={\int}_0^L\left[{f}_1\left(\xi \right)+{\textrm{Non}}_1\left(\xi \right)\right]{G}_{W1}\left(x,\xi \right) d\xi +{\int}_0^L\left[{p}_1\left(\xi \right)+{\textrm{Non}}_2\left(\xi \right)\right]{G}_{W2}\left(x,\xi \right) d\xi, \\ {}{f}_v(x)={\int}_0^L\left[{f}_1\left(\xi \right)+{\textrm{Non}}_1\left(\xi \right)\right]{G}_{V1}\left(x,\xi \right) d\xi +{\int}_0^L\left[{p}_1\left(\xi \right)+{\textrm{Non}}_2\left(\xi \right)\right]{G}_{V2}\left(x,\xi \right) d\xi \end{array}} $$