悬索主共振响应的多尺度解与同伦分析解

The solutions of primary resonance responses of suspended cables via the multiple scales method and homotopy analysis method

利用哈密顿变分原理,引入拟静态假设,建立了悬索面内非线性运动方程,并采用Galerkin方法对其进行离散。接着运用多尺度法和同伦分析法得到了悬索前两阶模态主共振响应的近似解。为验证这两种分析方法的适用性,同时采用龙格-库塔法对方程直接进行了数值积分。数值计算结果表明,随着悬索垂跨比以及振幅的增加,由多尺度法与同伦分析法得到的幅频响应曲线存在明显的定性与定量的差别,而同伦分析法结果与数值法的结果更加接近。最后比较了两种分析方法得到的位移场与索力时程响应曲线。

By applying the Hamilton＇s principle and quasi-static assumption,the equation of motion of the suspended cable was obtained by using the Galerkin procedure. Then, the multiple scales method and homotopy analysis method were applied to obtain the approximate series solutions of the primary resonance response of the suspended cable for the cases of the first two modes. Moreover, in order to verify the accuracy of the approximations,the Runge-Kutta method was also introduced. The numerical results show that.as the increasing of the sag-to-span ratio and the response amplitudes of the suspended cable, there are significant qualitative and quantitative differences in the frequency amplitude curves obtained by the multiple scales method and homotopy analysis method, and the results obtained by the homotopy analysis method are in good agreement with the numerical results. Finally, the displacement fields and the time response curves of the axial tension force are compared and analyzed.