热—机载荷作用下大挠度梁的动力稳定性分析
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摘要
非线性动力系统的分岔和混沌行为是非线性动力学研究的重要内容之一。非线性参数激励振动系统是最一般的、最具代表性的一类非线性动力学系统。在这类系统中存在着极其复杂且丰富的动力学行为,如分岔、分形、混沌等特性。本文针对具有非线性弹性地基的梁在承受热冲击载荷与横向周期机械载荷共同作用下参数激励系统大幅振动的稳定性,包括共振、分岔和混沌等问题开展了理论研究。具体内容和成果分别为:
1.在考虑几何非线性、线性阻尼的情况下,基于Euler-Bernoulli梁理论,利用Hamilton原理建立了具有非线性弹性地基上的梁在热冲击载荷与横向周期载荷共同作用下的几何非线性动力学控制方程。利用Galerkin变分原理将控制方程离散成关于时间的二阶非线性常微分方程:Mathieu-Duffing方程。为了抑制梁的大幅共振,对系统采用了非线性速度反馈控制。
2.在均匀升温载荷小于梁的临界屈曲载荷的情况下,利用多尺度法得到梁主共振、主参数共振响应的一次近似解,获得了振幅和相位慢变的一个常微分方程组,在此基础上分析系统的稳态响应、稳定区域和失稳临界条件。通过数值计算分析了阻尼、地基刚度、热冲击载荷幅值和频率、周期激振载荷幅值和频率、控制参数等对系统主共振和主参数共振响应特性的影响。数值模拟验证了多尺度分析的有效性及速度反馈控制对主参数共振控制的有效性。考察了系统发生3倍超谐波共振和1/3亚谐波共振的可能性及其存在条件,
3.在均匀升温载荷大于梁静态热屈曲临界温度载荷的情况下,求出系统的同、异宿轨道,利用Melnikov方法预测了系统可能发生Smale马蹄变换意义下的混沌运动的临界条件。数值模拟了在屈曲前后梁的分岔历程,综合使用Poincare截面、功率谱密度、Lyapunov指数、分形维数、时间历程图和相图等多种混沌分析手段研究了系统的混沌运动状态,指出了系统发生分岔和混沌等动力学行为的条件和规律。
Behaviors of bifurcation and chaos in nonlinear dynamic systems are key problems in the study of nonlinear dynamics. Nonlinear dynamic systems with parametrical excitations are the common but typical nonlinear dynamic systems which including very complex and abundant dynamic behaviors, such as bifurcation, fractal and chaos. In this thesis, theoretical investigation on the behaviors of dynamic stability, such as bifurcation, fractal and chaos of beams resting on nonlinear elastic foundation and subjected to both thermal shock and periodical mechanical excitation, has been presented. The main content and results are as following:
1. Based on Euler-Bernoulli beam theory , by considering the effects of geometric nonlinearity and linear external damp, geometrical nonlinear dynamic governing equation for simply supported beam resting on nonlinear elastic foundation subjected to both transverse harmonic mechanical load and thermal shock were derived according to Hamilton's theory. Galerkin Discretization method is applied to truncate the nonlinear partial differential governing equation into a second-order nonlinear ordinary differential equation: Mathieu-Duffing equation. In order to decrease the primary parametric resonance of the beam with large amplitude, a nonlinear velocity feedback control is adopted.
2. For the case that the uniform temperature rise less than the critical buckling load, the method of Multiple Scales is used to derive a first-order ordinary-differential equations that govern the time slowly variation of the amplitude and phase of the response of the primary resonance and primary parametric resonance of the beam to obtain the first approximate solution of the system, the steady state responses ,the stability area and the critical value of instability are analyzed. By using numerically computing , the effects of the parameters, such as the viscous damp, the elastic foundation stiffness, the amplitude and frequency of thermal shock as well as the external excitation on the response of the primary resonance and principal parametric resonance of the system. The validity of Multiple Scales and the control law based on cubic nonlinear velocity feedback on the response of the principal parametric resonance is verified by numerical simulations. Furthermore, the possibility and conditions of which the sub-harmonic response of 1/3 order and super-harmonic of 3 order exist are examined.
3. At the conditions that the uniform temperature rise is great than the critical buckling load of the beam, the homoclinic and heteroclinic orbits is derived. The Melnikov technique is used to obtain the critical value of the amplitude of thermal and mechanical load at or over which the Smale horseshoes chaos will take place. Numerical simulation is used to obtain the evolution of the bifurcation process of the response. By numerical simulation and incorporating with many analytical techniques for analysis of chaotic response, such as Poincare Map, power spectrum, Lyapunov exponents, fractal dimension, time history and phase trajectory, the paths of bifurcation of the beam is plotted and the chaotic movement is studied. The conditions and regularities of the evolution of bifurcation and chaotic movement in the system are indicated.
1.在考虑几何非线性、线性阻尼的情况下,基于Euler-Bernoulli梁理论,利用Hamilton原理建立了具有非线性弹性地基上的梁在热冲击载荷与横向周期载荷共同作用下的几何非线性动力学控制方程。利用Galerkin变分原理将控制方程离散成关于时间的二阶非线性常微分方程:Mathieu-Duffing方程。为了抑制梁的大幅共振,对系统采用了非线性速度反馈控制。
2.在均匀升温载荷小于梁的临界屈曲载荷的情况下,利用多尺度法得到梁主共振、主参数共振响应的一次近似解,获得了振幅和相位慢变的一个常微分方程组,在此基础上分析系统的稳态响应、稳定区域和失稳临界条件。通过数值计算分析了阻尼、地基刚度、热冲击载荷幅值和频率、周期激振载荷幅值和频率、控制参数等对系统主共振和主参数共振响应特性的影响。数值模拟验证了多尺度分析的有效性及速度反馈控制对主参数共振控制的有效性。考察了系统发生3倍超谐波共振和1/3亚谐波共振的可能性及其存在条件,
3.在均匀升温载荷大于梁静态热屈曲临界温度载荷的情况下,求出系统的同、异宿轨道,利用Melnikov方法预测了系统可能发生Smale马蹄变换意义下的混沌运动的临界条件。数值模拟了在屈曲前后梁的分岔历程,综合使用Poincare截面、功率谱密度、Lyapunov指数、分形维数、时间历程图和相图等多种混沌分析手段研究了系统的混沌运动状态,指出了系统发生分岔和混沌等动力学行为的条件和规律。
Behaviors of bifurcation and chaos in nonlinear dynamic systems are key problems in the study of nonlinear dynamics. Nonlinear dynamic systems with parametrical excitations are the common but typical nonlinear dynamic systems which including very complex and abundant dynamic behaviors, such as bifurcation, fractal and chaos. In this thesis, theoretical investigation on the behaviors of dynamic stability, such as bifurcation, fractal and chaos of beams resting on nonlinear elastic foundation and subjected to both thermal shock and periodical mechanical excitation, has been presented. The main content and results are as following:
1. Based on Euler-Bernoulli beam theory , by considering the effects of geometric nonlinearity and linear external damp, geometrical nonlinear dynamic governing equation for simply supported beam resting on nonlinear elastic foundation subjected to both transverse harmonic mechanical load and thermal shock were derived according to Hamilton's theory. Galerkin Discretization method is applied to truncate the nonlinear partial differential governing equation into a second-order nonlinear ordinary differential equation: Mathieu-Duffing equation. In order to decrease the primary parametric resonance of the beam with large amplitude, a nonlinear velocity feedback control is adopted.
2. For the case that the uniform temperature rise less than the critical buckling load, the method of Multiple Scales is used to derive a first-order ordinary-differential equations that govern the time slowly variation of the amplitude and phase of the response of the primary resonance and primary parametric resonance of the beam to obtain the first approximate solution of the system, the steady state responses ,the stability area and the critical value of instability are analyzed. By using numerically computing , the effects of the parameters, such as the viscous damp, the elastic foundation stiffness, the amplitude and frequency of thermal shock as well as the external excitation on the response of the primary resonance and principal parametric resonance of the system. The validity of Multiple Scales and the control law based on cubic nonlinear velocity feedback on the response of the principal parametric resonance is verified by numerical simulations. Furthermore, the possibility and conditions of which the sub-harmonic response of 1/3 order and super-harmonic of 3 order exist are examined.
3. At the conditions that the uniform temperature rise is great than the critical buckling load of the beam, the homoclinic and heteroclinic orbits is derived. The Melnikov technique is used to obtain the critical value of the amplitude of thermal and mechanical load at or over which the Smale horseshoes chaos will take place. Numerical simulation is used to obtain the evolution of the bifurcation process of the response. By numerical simulation and incorporating with many analytical techniques for analysis of chaotic response, such as Poincare Map, power spectrum, Lyapunov exponents, fractal dimension, time history and phase trajectory, the paths of bifurcation of the beam is plotted and the chaotic movement is studied. The conditions and regularities of the evolution of bifurcation and chaotic movement in the system are indicated.
引文
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