超声激励下微悬臂梁非线性动力学及其混沌控制

It is well known that nonlinear dynamic characteristics of microcantilevers under ultrasonic excitation have a very important influence on the performances and applications of atomic force acoustic microscopy (AFAM). The overall aims of the research project are to explore the strongly nonlinear vibration mechanism and its control of microcantilever under ultrasonic excitation are still scarce. First, through the establishment of a reasonable mathematical model combined with numerical simulation results, the project intends to study the characteristics of nonlinear contact resonance spectrum of microcantilevers under large-amplitude ultrasonic excitation. Moreover, we will analyse the differences of resonance spectra characteristics of the DMT and JKR contact models. Then, the existence, stability and bifurcation characteristics of subharmonic periodic orbits of microcantilever's vibro-impact motion will be studied by analytical subharmonic Melnikov method and numerical method. By using phase plane method, Poincare map, power spectrum and Lyapunov exponents, the nonlinear dynamic behavior of n/m subharmonic motions and chaos are to be characterized. Finally, after analyzing the routes to chaos, we will investigate the feasibility of delayed feedback method and piecewise control strategy on the implementation of chaotic control of mirocantilever as a typical non-smooth dynamical system. The above research will afford a preliminary theoretical basis for strong nonlinear contact resonance spectroscopy and sub-harmonic mode application in AFAM.

众所周知，超声激励下微悬臂梁非线性动力学特性对原子力声学显微镜(AFAM)的性能及其应用具有非常重要的影响。本项目旨在探讨超声激励下微悬臂梁强非线性振动机理及其控制方法。首先，拟通过建立合理的数学模型并结合数值仿真结果，对大振幅超声激励下微悬臂梁强非线性接触共振谱特性进行研究，分析DMT和JKR接触模型共振谱特性差异。其次，运用次谐Melnikov解析方法和数值方法分析微悬臂梁次谐波周期碰撞振动轨道的存在性、稳定性及分岔特性，并利用相平面图、Poincare映射、功率谱和Lyapunov 指数等研究n/m次谐波周期运动和混沌的非线性动力学行为。最后，我们在分析微悬臂梁振动通向混沌的途径及其特征的基础上，进而探索利用分区控制和延迟反馈方法对微悬臂梁非光滑动力学系统的混沌运动实施有效控制。上述研究将为强非线性接触共振谱和次谐波工作模式在AFAM中的应用奠定初步的理论基础。

超声激励下微悬臂梁非线性动力学特性对原子力声学显微镜(AFAM)的性能具有非常重要的影响。本项目主要研究超声激励下微悬臂梁非线性振动现象，并利用相平面图、Poincare 映射、功率谱和Lyapunov 指数等对微悬臂梁周期运动和混沌运动进行表征。我们首先建立了超声激励下板梁结构振动的多自由度动力学模型，通过数值求解并结合已有的实验结果阐明了超声激励下板梁非线性振动机理，为AFAM微悬臂梁振动分析奠定理论基础。其次，利用Euler–Bernoulli梁理论和DMT针尖—样品作用力模型建立了试样激励下轻敲模式的AFAM系统的动力学方程。采用数值方法对AFAM系统微悬臂梁非线性幅频响应特性进行了研究，分析了微悬臂梁非线性共振曲线频移和高频跳变现象。通过应用微分方程数值求解与非线性动力学分析方法对AFAM微悬臂梁的超谐波、次谐波和准次谐波振动特性进行了分析。最后，阐明了超声激励下微悬臂梁混沌振动机理，分析了微悬臂梁混沌振动存在条件、特征及通向混沌的途径以及控制方法。在项目研究期间，在SCI和EI源刊物上发表论文9篇。