Determination of the regions of dynamic instability has been an important issue for elastic structures. Under the extreme climate, the external load acting on structures is becoming more and more complicated, which can induce dynamic instability of elastic structures. In this study, we explore the dynamic instability and response characteristics of simply supported beams under multi-harmonic parametric excitation. A numerical approach for determining the instability regions under multi-harmonic parametric excitation is developed here by examining the eigenvalues of characteristic exponents of the monodromy matrix based on the Floquet theorem, and the fourth-order Runge–Kutta method is used to calculate the dynamic responses. The accuracy of the method is verified by the comparison with classical approximate boundary formulas of dynamic instability regions. The numerical results reveal that Bolotin’s approximate formulas are only applicable to the low-order instability regions with a small value of the excitation parameter of simple parametric resonance. Multi-harmonic parametric excitation can significantly change the dynamic instability regions, it may cause parametric resonance on beams for longitudinal complex periodic loads. The influence of frequency and number of multiply harmonics on the parametrically excited vibration of the beam is explored. High-order harmonics with low-frequency have positive effects on the stable response characteristics for multi-harmonic parametric excitation. This paper provides a new perspective for the vibration suppression of parametric excitation. The developed procedure can be used for multi-degree-of-freedom (MDOF) systems under complex excitation (e.g. tsunami waves and strong winds).

中文翻译：

---

多谐波参量激励简支梁的动态稳定性

动态不稳定区域的确定一直是弹性结构的一个重要问题。在极端气候条件下，作用在结构上的外荷载越来越复杂，会引起弹性结构的动力失稳。在这项研究中，我们探讨了多谐波参数激励下简支梁的动态不稳定性和响应特性。基于Floquet定理，通过检验单调矩阵特征指数的特征值，开发了一种确定多谐参量激励下不稳定区域的数值方法，并使用四阶龙格-库塔法计算动态响应. 通过与经典动态不稳定区域近似边界公式的比较，验证了该方法的准确性。数值结果表明，Bolotin的近似公式只适用于简单参量共振激发参数值较小的低阶不稳定区域。多谐参量激励可以显着改变动态不稳定区域，在纵向复杂周期载荷下可能引起梁的参量共振。探讨了频率和多次谐波数对梁参激振动的影响。低频高次谐波对多谐波参量激励的稳定响应特性有积极影响。本文为参数激励的振动抑制提供了一个新的视角。所开发的程序可用于复杂激励（例如海啸和强风）下的多自由度（MDOF）系统。