温度变化对悬索非线性内共振响应特性影响

Influences of temperature on suspended cables′nonlinear vibration characteristics considering internal resonances

悬索是一类典型的同时包含平方和立方非线性的柔性结构,其模态间极易发生各种形式的内共振。以悬索同时发生主共振和2∶1内共振为例,探究温度变化对悬索非线性内共振响应特性的影响。通过引入张力改变系数,建立了均匀温度场中悬索面内非线性运动微分方程;利用Galerkin法和多尺度法分别得到激励作用在高阶和低阶模态时,系统极坐标和直角坐标形式的平均方程;通过绘制共振响应时的激励响应幅值曲线、幅频响应曲线、动态解、时程曲线、相位图、频率谱以及庞加莱截面,定性和定量地描述了温度变化影响下的内共振响应特性。数值算例表明:频率会明显改变悬索模态频率,影响系统内共振响应,温度上升时,内共振更容易发生在Irvine参数较小的悬索;无论激励直接作用在高阶还是低阶模态,共振响应幅值随温度上升而增加,反之则减小;直接激发的模态响应幅值与因内共振激发的响应幅值受温度变化影响的敏感程度存在明显区别;温度变化对动态分岔(霍普和倍周期分岔)影响要比对静态分岔(鞍结点和叉形分岔)明显得多;动态分岔随着温度上升,向更小激励幅值和频率方向移动;系统的动态解和周期运动与温度变化密切相关,受温度影响,系统可能呈现出截然不同的周期运动。此外对比理论分析解和直接数值积分解,结果表明两者吻合较好。

As a classical flexible structure with quadratic and cubic nonlinearities,the suspended cables′multi-type internal resonances are very easily excited.The paper aims at investigating influences of temperature on nonlinear dynamic behaviors of suspended cables considering simultaneous primary and two-to-one internal resonances.A kinematically condensed nonlinear model of a suspended cable considering temperature effects was introduced for the in-plane displacements by introducing a tension variation factor.Applying the Galerkin method and following the multiple scales procedure as to a weakly nonlinear system,both the polar and Cartesian forms of modulation equations were obtained when the higher/lower mode was excited directly.A complete dynamic scenario of vibration behaviors with thermal effects was presented quantitatively and qualitatively,e.g.,force/frequency-response curves,periodic dynamic solutions,time displacement curves,phase plane diagrams,frequency spectrum,as well as Poincare sections.Numerical examples show that:the mode frequencies were affected by temperature changes significantly,and then the internal resonance would be changed induced by thermal effects.With an increase in temperature,the suspended cable with a smaller Irvine parameter would exhibit internal resonant responses.No matter the higher or the lower order was excited directly,the influence of temperature on the direct and internal excited response amplitudes were different.Temperature effects on the dynamic bifurcations(Hopf and periodic-doubling bifurcations)are much more apparent than the ones on the static bifurcations(saddle-node and pitck-fork bifurcations).The dynamic solutions as well as periodic motions are closely related to thermal effects,and significant different periodic motions were observed due to temperature changes.Besides,a good agreement between the perturbation solutions and the direct numerical integration ones could be observed.