重力场对欧拉屈曲梁非线性吸振器分岔特征的影响研究

Effects of gravity field on bifurcation characteristics of Euler buckling beam nonlinear vibration absorber

以欧拉屈曲梁构成的非线性吸振器为研究对象,针对不同的应用环境,分别建立有无重力条件对应非线性动力学模型并重点研究重力场对非线性系统分岔特征的影响。利用复变量-平均法推导出非线性系统的慢变方程,进而得到其对应的鞍结(saddle-node, SN)分岔以及霍普夫(Hopf)分岔边界。通过对比有无重力条件下的分岔边界发现,重力场使分岔边界受参数影响范围变大;进一步,在有重力场条件下,针对失谐参数和激励幅值对非线性系统分岔特征的影响展开讨论;最后,考虑重力场条件,针对欧拉屈曲梁非线性吸振器部分关键设计参数对系统分岔特征的影响进行了分析。计算结果发现:满足一定条件,SN分岔和Hopf分岔会有共存的情形;而且,频响幅值随激励幅值的变大而产生两个分支,随激励幅值继续增大,将使两个分支重合但多解区间并未消失且增大;欧拉屈曲梁长度和斜置倾角对分岔特性有较大影响,且变化规律相似,随着参数增大,SN分岔与Hopf分岔边界减小。

Here, taking a nonlinear vibration absorber composed of Euler buckling beam as the study object, aiming at different application environments, the corresponding nonlinear dynamic models with and without gravity conditions were established, respectively, and effects of gravity field on bifurcation characteristics of nonlinear system were emphatically studied. The complex variable-average method was used to derive slowly varying dynamic equations of nonlinear system, and then the corresponding saddle-node(SN) bifurcation and Hopf bifurcation boundaries were obtained.By comparing bifurcation boundaries with and without gravity, it was shown that gravity field makes bifurcation boundaries more affected by parameters.Furthermore, under the condition of gravity field, effects of detuning parameters and excitation amplitude on bifurcation characteristics of nonlinear system were discussed. Finally, considering gravity field conditions, effects of some key design parameters of Euler buckling beam nonlinear absorber on bifurcation characteristics of nonlinear system were analyzed.The results showed that under satisfying certain conditions, SN bifurcation and Hopf bifurcation can coexist;frequency response amplitude produces two branches with increase in excitation amplitude, with continuous increase in excitation amplitude, the two branches can coincide but multi-solution interval can’t disappear yet enlarge;length and inclined angle of Euler buckling beam affect bifurcation characteristics more largely, and their variation lawsare similar;with increase in parameters, both SN bifurcation boundary and Hopf bifurcation one decrease.