矩形薄板非线性热、磁弹性振动与混沌研究
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摘要
薄板是工程结构中常见的元件之一,而且它们通常都是在机械场、电磁场和温度场等多场作用环境下工作,其动力特性对系统的结构安全有重要影响。因此,对磁弹性薄板多场耦合作用动力学问题开展研究具有重要的理论意义与实用价值。本文研究了矩形薄板在电磁场、外载荷以及温度场等多场共同作用下的非线性弹性振动和分岔与混沌特性分析。在板壳与磁弹性力学理论的基础上,建立了矩形磁弹性薄板非线性系统动力学模型,对其非线性自由振动、强迫振动和分岔与混沌等问题进行了深入系统的研究,主要包括以下内容:
导出了矩形磁弹性薄板非线性自由振动系统动力学模型及其动力学微分方程式,以四边简单支撑的矩形薄板为例,对其进行了自由振动模态分析。用多尺度法求出了非线性自由振动系统时域响应近似解析解,用四阶Runge-Kutta法编程微分方程进行数值求解计算,绘制系统的位移时间响应曲线和相图,讨论了机电参数对系统响应的影响。
导出了矩形磁弹性薄板非线性强迫振动系统动力学模型及其动力学微分方程式,以四边简单支撑的矩形薄板为例,用多尺度法推导了非线性系统强迫振动时域响应的一次近似解析解,计算并分析了外加激励频率远离和接近派生系统固有频率时系统的主共振、超谐波和亚谐波稳态响应。讨论了机电参数对振动系统频域响应的影响规律。
推导出不同支撑情况下矩形磁弹性薄板在电磁场和机械场耦合作用的非线性振动方程,运用Melnikov函数方法推导系统发生混沌的条件,用四阶Runge-Kutta法编辑程序对系统进行数值求解,并绘制系统分岔图、相平面轨迹图、波形图以及庞伽莱截面图和Lyapunov指数图,讨论机电参数对系统运动特性的影响。
考虑温度场的影响,推导出在横向稳恒磁场和载荷共同作用下不同边界条件下矩形薄板的非线性磁弹性耦合振动方程。用Melnikov函数法给出该非线性动力系统smale马蹄变换意义下出现混沌运动的判据,用四阶Runge-Kutta法编程数值求解系统振动方程,绘制系统的分岔图、Lyapunov指数图、相应位移波形图、相平面轨迹图、庞伽莱截面图。分析了温度场与机电参量对系统运动状态的影响。
Thin plate is one kind of common project components, and they are usually installated in coupled fields, such as the mechanical field, electromagnetic field and temperature field and so on, the dynamic properties of the system have a significant impact on the structural safety. Therefore, study of many fields coupled dynamics of magneto-elastic thin plate is of great theoretical and practical significance. In this paper, the nonlinear elastic vibration and bifurcation and chaos or the rectangular thin plates in the electromagnetic fields, mechanical force and temperature field are studied. Based on the theory of sheet mechanics and magnetoelastic mechanics, the nonlinear system dynamics model of the rectangular Magnetoelastic thin plate is founded, the nonlinear free vibration, forced vibration and bifurcation and chaos of the thin plates are investigated and analyzed. Main results of the paper are as below:
Nonlinear free vibration system dynamics model and dynamics differential equations of rectangular magnetoelastic thin plate are derived. Taken four-side simply supported rectangular plate as an example, the mode of free vibration for the thin plate is analysized. Approximative analytic solutions in time-domain of nonlinear free vibration are achieved by multiscale mechtod. Using the four-order Runge-Kutta method, the numerical solutions of the differential equations are got, and displacement curves and phase diagram of the system are drawn. The influences of the system response of electro-mechanical parameters are discussed.
Nonlinear forced vibration system dynamics model and dynamics differential equations of rectangular magnetoelastic thin plate are reduced. Taken four-side simply supported rectangular plate as an example, one order approximative analytic solutions in time-domain of nonlinear forced vibration are achieved by multiscale mechtod. Stable time responses of the main resonance, superharmonic and subharmonic responses are computed and analyzed when excitation frequency is far from and near to natural frequencies of generating system. The influences of electromechanical parameters on the frequency responses of forced vibration system are discussed.
The nonlinear vibration equations of thin rectangular plate coupled with electromagnetic fields and mechanical loads under different boundary conditions are obtained. The chaotic criterion of this system was got by Melnikov function method, and the vibration equation of the system was solved using the four-order Runge-Kutta method numerical method. And in the specific examples, the bifurcation diagram, the Lyapunov exponent diagram, the wave diagram of displacement, phase diagram and Poincare map are derived. The influences of magnetic parameter and mechanic loads on the vibration of the system are analyzed.
Considering the influence of temperature field, the vibration equations of the rectangular thin plate under the action of mechanic field and steady transverse magnetic field are derived. By Melnikov function method, the chaos condition and judging criterion of the system under the condition of Smale horseshoe map are given. The vibration equations are solved numerically using the four-order Runge-Kutta method. By some examples, the bifurcation diagram, the Lyapunov exponents diagram, the displacement wave diagram, the phase diagram and the Poincare section diagram of the system are obtained. The influences of temperature field, electromagnetic field and mechanic loads on system vibration properties are analyzed.
导出了矩形磁弹性薄板非线性自由振动系统动力学模型及其动力学微分方程式,以四边简单支撑的矩形薄板为例,对其进行了自由振动模态分析。用多尺度法求出了非线性自由振动系统时域响应近似解析解,用四阶Runge-Kutta法编程微分方程进行数值求解计算,绘制系统的位移时间响应曲线和相图,讨论了机电参数对系统响应的影响。
导出了矩形磁弹性薄板非线性强迫振动系统动力学模型及其动力学微分方程式,以四边简单支撑的矩形薄板为例,用多尺度法推导了非线性系统强迫振动时域响应的一次近似解析解,计算并分析了外加激励频率远离和接近派生系统固有频率时系统的主共振、超谐波和亚谐波稳态响应。讨论了机电参数对振动系统频域响应的影响规律。
推导出不同支撑情况下矩形磁弹性薄板在电磁场和机械场耦合作用的非线性振动方程,运用Melnikov函数方法推导系统发生混沌的条件,用四阶Runge-Kutta法编辑程序对系统进行数值求解,并绘制系统分岔图、相平面轨迹图、波形图以及庞伽莱截面图和Lyapunov指数图,讨论机电参数对系统运动特性的影响。
考虑温度场的影响,推导出在横向稳恒磁场和载荷共同作用下不同边界条件下矩形薄板的非线性磁弹性耦合振动方程。用Melnikov函数法给出该非线性动力系统smale马蹄变换意义下出现混沌运动的判据,用四阶Runge-Kutta法编程数值求解系统振动方程,绘制系统的分岔图、Lyapunov指数图、相应位移波形图、相平面轨迹图、庞伽莱截面图。分析了温度场与机电参量对系统运动状态的影响。
Thin plate is one kind of common project components, and they are usually installated in coupled fields, such as the mechanical field, electromagnetic field and temperature field and so on, the dynamic properties of the system have a significant impact on the structural safety. Therefore, study of many fields coupled dynamics of magneto-elastic thin plate is of great theoretical and practical significance. In this paper, the nonlinear elastic vibration and bifurcation and chaos or the rectangular thin plates in the electromagnetic fields, mechanical force and temperature field are studied. Based on the theory of sheet mechanics and magnetoelastic mechanics, the nonlinear system dynamics model of the rectangular Magnetoelastic thin plate is founded, the nonlinear free vibration, forced vibration and bifurcation and chaos of the thin plates are investigated and analyzed. Main results of the paper are as below:
Nonlinear free vibration system dynamics model and dynamics differential equations of rectangular magnetoelastic thin plate are derived. Taken four-side simply supported rectangular plate as an example, the mode of free vibration for the thin plate is analysized. Approximative analytic solutions in time-domain of nonlinear free vibration are achieved by multiscale mechtod. Using the four-order Runge-Kutta method, the numerical solutions of the differential equations are got, and displacement curves and phase diagram of the system are drawn. The influences of the system response of electro-mechanical parameters are discussed.
Nonlinear forced vibration system dynamics model and dynamics differential equations of rectangular magnetoelastic thin plate are reduced. Taken four-side simply supported rectangular plate as an example, one order approximative analytic solutions in time-domain of nonlinear forced vibration are achieved by multiscale mechtod. Stable time responses of the main resonance, superharmonic and subharmonic responses are computed and analyzed when excitation frequency is far from and near to natural frequencies of generating system. The influences of electromechanical parameters on the frequency responses of forced vibration system are discussed.
The nonlinear vibration equations of thin rectangular plate coupled with electromagnetic fields and mechanical loads under different boundary conditions are obtained. The chaotic criterion of this system was got by Melnikov function method, and the vibration equation of the system was solved using the four-order Runge-Kutta method numerical method. And in the specific examples, the bifurcation diagram, the Lyapunov exponent diagram, the wave diagram of displacement, phase diagram and Poincare map are derived. The influences of magnetic parameter and mechanic loads on the vibration of the system are analyzed.
Considering the influence of temperature field, the vibration equations of the rectangular thin plate under the action of mechanic field and steady transverse magnetic field are derived. By Melnikov function method, the chaos condition and judging criterion of the system under the condition of Smale horseshoe map are given. The vibration equations are solved numerically using the four-order Runge-Kutta method. By some examples, the bifurcation diagram, the Lyapunov exponents diagram, the displacement wave diagram, the phase diagram and the Poincare section diagram of the system are obtained. The influences of temperature field, electromagnetic field and mechanic loads on system vibration properties are analyzed.
引文
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