Page 103 - 《振动工程学报》2025年第9期

第 38 卷第 9 期 振 动 工 程 学 报 Vol. 38 No. 9
2025 年 9 月 Journal of Vibration Engineering Sept. 2025

含 常 数 激 励 的 分 数 阶 非 线 性 隔 振 系 统 幅 频 特 性

及 周 期 运 动 多 样 性 研 究

屈鸣鹤 ， 丁旺才 ， 谭亦秋 ， 周丹峰 ， 王连春 ， 李 杰 1
1
1
1
2
1
（1. 国防科技大学智能科学学院，湖南 长沙 410073； 2. 兰州交通大学机电工程学院，甘肃 兰州 730070）
摘要：针对具有非线性和黏弹性的隔振系统，采用分数阶非线性 Zener 模型对其本构关系进行表征，探究常数激励与简谐激励
联合作用下系统骨架线及幅频响应的变化规律，重点分析常数激励对隔振系统动力学行为的影响。将分数阶项等效成三角
函数的形式，采用谐波平衡法求解系统稳态响应，并结合多种数值方法对解析结果进行比较，总结了不同参数对幅频响应多
态解共存频带范围的影响规律，数值模拟系统在联合激励作用下的动力学行为。研究结果表明，系统在常数激励和简谐激励
联合作用下幅频响应解存在五解共存区，系统出现软、硬特性共存现象，骨架线先向左侧倾斜后向右侧倾斜，也出现多解共存
现象。数值模拟过程中还发现，联合激励作用下系统周期运动和混沌转迁过程存在多种分岔类型。受常数激励影响，系统在
联合激励作用下的周期运动多样性与系统仅在简谐激励单独作用下的动力学行为有明显差异，并结合 Lyapunov 指数总结了
联合激励作用下系统周期运动的转迁规律。
关键词： 分数阶微分；非线性 Zener 模型；常数激励；转迁规律
中图分类号：O322 文献标志码：A DOI：10.16385/j.cnki.issn.1004-4523.202309052

Amplitude-frequency characteristics and diversity of periodic motions
of fractional-order nonlinear vibration isolation system with constant excitation

1
1
1
2
1
QU Minghe ，DING Wangcai ，TAN Yiqiu ，ZHOU Danfeng ，WANG Lianchun ，LI Jie 1
（1.College of Intelligent Science and Technology，National University of Defense Technology，Changsha 410073，China；
2.School of Mechatronic Engineering，Lanzhou Jiaotong University，Lanzhou 730070，China）
Abstract：The fractional nonlinear Zener model is used to describe the nonlinear and viscoelastic constitutive relation of the vibration isolation
system. The variation law of system amplitude-frequency response and backbone under the combined action of constant excitation and harmonic
excitation is discussed，and the influence of constant excitation on the dynamic behavior of vibration isolation system is discussed significantly.
The fractional-order derivative term is made equivalent to a term in the form of trigonometric function，the steady-state response of the system
is solved by harmonic balance method，and the results are compared with a variety of other methods. The influences of different parameters on
the coexistence frequency band range of the amplitude-frequency response multi-state solution are summarized，and the dynamic behaviors of
the system under the combined excitation are obtained by using numerically simulation. The results show that there are five solutions co-
existence region in the amplitude-frequency response solution under the combined effect of constant excitation and harmonic excitation，and
the system shows a phenomenon of coexistence of softening characteristic and hardening characteristic，and the backbone of the amplitude-
frequency curve is tilted firstly to the left and then to the right. Additionally，it is found that the periodic motion and chaos coexist in the system
under the combined excitation， and the transition laws of the polymorphic coexistence region and its adjacent regions are summarized
explicitly. Affected by constant excitation， the diversity of periodic motion of the system under the combined excitation is significantly
different from the dynamic behavior under the action of simple harmonic excitation alone，and the transition rules of periodic motion of the
system under the action of combined excitation are summarized based on the Lyapunov exponent.
Keywords：fractional-order derivative；nonlinear Zener model；constant excitation；transition law

黏弹性材料被广泛应用于航空器的辅助动力装 中的悬架缓冲块等。橡胶由于其特殊结构可以表现
置隔振器、轨道车辆二系悬挂的橡胶堆、汽车底盘 出黏弹性而被广泛应用，但只有变形较小时才可近

收稿日期：2023-09-17；修订日期：2024-03-08
基金项目：国家自然科学基金重点项目（52332011）；国家自然科学基金资助项目（11962013）；国家重点研发计划子任务资
助项目（2016YFB1200601）；湖南省研究生科研创新项目（CX20240124）；设备综合研究项目（4201400013313）