Duffing振子在水工结构动力响应信号检测应用中的一种混沌判据

A chaotic criterion for the detection of dynamic response signals of hydraulic structures using duffing oscillators

【目的】Duffing振子是一种典型的非线性动力系统,混沌状态的判断是利用Duffing振子进行水工结构动力响应信号检测的基础准则。Duffing振子系统进入混沌状态时,系统状态时序图和相图会变得更为无序和复杂,可以对系统状态进行直观反映,但直接将二者作为系统混沌状态的判断依据,存在较为模糊和主观性较大的问题。为了对Duffing振子系统状态时序图和相图的复杂程度进行定量描述,以便更准确地利用二者对Duffing振子系统混沌状态的判断,【方法】提出了一种利用系统内状态量的时序图和相图盒维数的指数组合的指标,将该指标是否发生突变作为利用Duffing振子进行信号检测时的混沌状态的判断依据。【结果】所提出的Duffing振子混沌状态识别指标在Duffing方程的仿真分析、正弦波仿真信号检测和悬臂梁锤击动力响应信号检测试验中分别表现出22.611%、17.514%和30.975%的相对变化率。同时振子对应的李雅普诺夫指数也由负转正,但Duffing方程的仿真分析结果显示李雅普诺夫指数作为识别指标更为敏感,在振子系统的时序图和相图未明显表现出混沌状态时,其已经由负转正。【结论】由试验结果可知:可以根据所提出识别指标是否发生突变进行识别振子混沌状态,达到信号检测的目的。同时,所提出的混沌状态识别指标以时序图和相图的复杂程度直接作为判断标准,与其他指标相比更为直接和合理。悬臂梁动力响应信号检测试验初步验证了所选取的识别指标应用到具体工作中的可行性,有望应用于结构损伤识别,共振发现等工作中。

[Objective]Duffing oscillator is a typical nonlinear dynamic system,and the judgment of chaotic state is the basic criterion for detecting the dynamic response signal of hydraulic structures using Duffing oscillator.When the Duffing oscillator system enters a chaotic state,the system state time series diagram and phase diagram will become more disordered and complex,which can intuitively reflect the system state.However,directly using the two as the basis for judging the chaotic state of the system has a relatively fuzzy and subjective problem.To quantitatively describe the complexity of the state time series diagram and phase diagram of the Duffing oscillator system,and to more accurately use them to judge the chaotic state of the Duffing oscillator system,[Methods]this paper proposes an indicator that combines the box dimension of the state variables’time series diagram and the phase diagram.Whether the indicator undergoes a sudden change is used as the basis for judging the chaotic state of the Duffing oscillator system when using the Duffing oscillator for signal detection.[Results]The chaotic state identification index of Duffing oscillator proposed in this article showed relative change rates of 22.611%,17.514%,and 30.975%in the simulation analysis of Duffing equation,sine wave simulation signal detection,and cantilever beam hammer dynamic response signal detection experiments,respectively.At the same time,the Lyapunov exponent corresponding to the oscillator also changes from negative to positive,but the simulation analysis result of the Duffing equation show that the Lyapunov exponent is more sensitive as a recognition index.When the time series and phase diagrams of the oscillator system do not show obvious chaotic states,it has already changed from negative to positive.[Conclusion]From the experimental result,it can be concluded that the chaotic state of the oscillator can be identified based on whether the identification index proposed in this article has undergone a sudden change,achieving the purpose of signal detection.Meanwhile,the chaotic state recognition index proposed in this article uses the complexity of time series and phase diagrams as the judgment criteria directly,which is more direct and reasonable compared to other indicators.The cantilever beam dynamic response signal detection experiment preliminarily verified the feasibility of applying the identification indicators selected in this article to specific work.It is expected to be applied in structural damage identification,resonance detection,and other work.