悬索面内1/3次谐波共振瞬时相频特性

考虑悬索垂度和几何非线性，研究垂跨比改变对悬索1/3次谐波共振瞬时相频特性的影响。基于悬索面内分布激励下的运动常微分方程，采用多尺度法进行摄动求解，得到悬索面内1/3次谐波共振二阶近似解，再利用希尔伯特变换得到响应与激励的瞬时相位差及响应与线性项的瞬时相位差，通过数值算例探讨不同垂跨比及调谐参数下瞬时相位差的变化规律。结果表明:垂跨比改变对悬索响应与激励的瞬时相位差影响较大，其主要方式是改变二阶近似解中的高阶项及漂移项成分。随着垂跨比及调谐参数的减小，高阶项及漂移项成分的影响在逐渐增大。同时在垂跨比处于特定的范围内时，其微小的变化可能会引起瞬时相频特性的巨大变化，在这其中起主导作用的是响应中的漂移项成分。

Transient Phase-frequency Characteristics of In-plane 1/3 Harmonic Resonances in Suspended Cables

The research aimed to study the influence of different sag-to-span ratios on the transient phase-frequency characteristics of the 1/3 harmonic resonance of the suspended cables considering the sag and geometric nonlinearity of the suspended cables. On the basis of the ordinary differential equation for the motion of a suspended cable under distributed excitation in the plane, the multiple scales method was used to obtain the second order approximate solution of the 1/3 harmonic resonance in the suspended cable plane. Then, the Hilbert transform is used to obtain the transient phase difference between the response and the excitation, as well as the transient phase difference between the response and the linear term, and the variation of transient phase difference under different sag-to-span ratios and detuning parameters were investigated by using numerical examples. The results show that the variation of the sag-to-span ratios has a great effect on the transient phase difference between the response and the excitation, which is mainly by changing the higher order term and the drift term in the second order approximate solution. With the decrease of sag-to-span ratio and detuning parameters, the influence of higher order term and drift term increases gradually. In a specific range, a small change of the sag-to-span ratio may cause a great change in the transient phase-frequency characteristics, in which the drift term in the response plays a leading role.