【作者】 穆晓平；

【导师】 张义民；

【作者基本信息】 吉林大学 ， 固体力学， 2004， 硕士

【摘要】 非线性随机振动的研究是当前力学研究中一个重要而又令人注目的研究方向。在工程中，几乎所有机械（结构）系统都在某种程度上呈现出非线性。许多实际工程中的问题，仅从线性和确定性观点来考虑，不足以反映系统的实际工作情况，采用非线性模型是出于以下两个方面的需要，一是对于某些特定的问题希望得到更准确的定量分析；二是对于某些本质非线性现象，线性模型甚至无法作出合理的解释和正确的定性判断，因此对有些工程实际的动力学问题应该处理为非线性随机模型来加以研究。滞回系统是一种典型的非线性随机振动系统。许多工程结构由于大幅变形而进入非弹性状态，由于屈服而呈现出滞回状态，在周期运动中导致正向运动与反向运动时的恢复力—位移曲线形成滞回环。其效应主要表现为刚度的减少和能量耗散的增加，呈现出复杂的非线形。特别是系统的恢复力不仅依赖于该瞬时的位移，而且取决于该瞬时的速度方向，使得动力方程中出现了多值函数，从而给系统的响应分析带来了极大的困难。对滞回系统的研究主要集中在以下三个方面：一是滞回非线性模型的确定。人们曾根据不同的工程实际的需要提出多种描述滞回恢复力特性的模型，如双线性模型、分布弹塑性元件模型、辅助微分方程模型如Bouc-Wen模型及其它模型，但是每种模型都有一定的局限性。目前对滞回系统的研究应用较广的是Bouc-Wen微分型模型。近几年来，又发展了非对称模型；二是滞回系统在随机激励下响应的求解。人们应用这些模型给出了受随机激励的滞回系统的随机响应的研究，大体上可分为等效线性化方法、一般等效线性化方法、功率平均法、FPK方程法、标准随机平均法和能量包线随机平均法等。应用以上这些方法，依据不同的需要，采用相应的方法便可以解决单自由度和多自由度的滞回系统在随机激励下的响应问题；三是滞回系统的参数辨识问题。一般的只有弄清了结构系统的载荷和位移的滞回非线性关系，才可以进行诸如能量耗散、永久变形、结构响应与退化等后续分析。因此，从已知结构系统的响应中辨识出上述关系是十分有用的。辨识问题主要有两个方面的内容，一是模型结构的确定，二是模型参数的估计。对于滞回系统，较常见的是先根据经验假设一种模型形式，然后用不同的方法对其参数进行识别。除了上述的参数识别手段外，对于滞回系统，非参数辩识的方法一样有效。 <WP=58>以上这些研究只给出了受随机激励的滞回系统的随机响应，而没有给出由于滞回环本身的随机性而引起的非线形随机振动系统的随机响应。但是由于材料性态的离散性，工程机械中的半连续介质尤为突出，必然导致随机滞回系统的非线性随机振动模型。大量的工程实验得出的结论也证明滞回环存在着明显的随机离散性。因此，有必要对滞回环本身的随机性而引起非线性随机振动系统的响应进行研究，从而使得对滞回系统的研究更为完善。从Bouc-Wen模型在不同参数下的滞回环曲线中可以看出，适当的选取该模型的参数，便可以得到具有不同能耗，渐软或渐硬的滞回力模型，以满足不同的工程实际的需要。该模型实质上指出恢复力的增大与状态有关，与加载卸载过程有关，这与材料的本构关系相似。并且它易于与系统的运动微分方程相结合，组成扩阶的系统，便于求解。在Bouc-Wen滞回模型的基础上，根据求解非线性结构动力学的二阶矩法，假设该模型各随机参数之间是不相关的，推导出了求解系统响应的零阶方程，一阶方程和二阶方程，从而得到了由于滞回环本身的随机性而引起的多自由度非线性滞回系统的随机响应的有效的数值方法。应用此方法，就该模型的参数所存在的三种情形各举了一个数值算例，编制相应的程序，得到了系统响应的均值曲线，标准差曲线和协方差曲线。为了验证该方法的正确性，采用了Monte-Carlo数值模拟法，选取了1000个随机点对结构系统的响应进行模拟分析。由于自然界中大量的随机现象均近似服从正态分布，因此模拟过程中假设各随机参数服从方差系数为0.005的正态分布，也得到了系统响应均值曲线，标准差曲线和协方差曲线。应用二阶矩法求解系统响应所得的计算结果与Monte-Carlo数值模拟方法的结果进行比较，非常地吻合，从而验证了运用二阶矩法推导出的数值方法求解滞回系统随机响应的正确性和可靠性，进而解决了由滞回环本身的随机性而引起的多自由度非线形随机滞回系统的振动响应问题。更多还原

【Abstract】 The research on nonlinear vibration is an attracting and important study field in the research of mechanics recently. Almost all systems exhibit nonlinear in some degree. It is not effective for many actual engineer questions if solved only from the point of linear and determination views. Based on two reasons ,we adopt nonlinear models: first ,furthermore exact quantitative analysis is expected to find for the certain problems ;second ,linear models can not even make with reasonable explains and right qualitative analysis for the essence nonlinear phenomena ,So it is very necessary to study the nonlinear random vibration systems. Hysteretic system is one of the most typical nonlinear systems. Many engineer structural systems under dynamic loading usually exhibit hysteretic behavior, especially when the response becomes inelastic. A hysteretic loop produces between hysteretic restoring force and displacement under periodic movement. The effect exhibits the change of stiffness and energy. Because the restoring force of a hysteretic system depends not only on the instantaneous displacement， but also on its past history， analytical modeling and solution of such a system under random excitation has been an interesting and challenging subject. The hereditary nature of the restoring force of an inelastic system indicates that the force can on longer be described by an algebraic function of the instantaneous displacement and velocity. The research on hysteretic system concentrated on three aspects: Firstly: Hysteretic restoring force models. People have developed many different hysteretic restoring force models according to different engineer structural system needs in recent years, for example: Bilinear model，Ramberg-Osgood model，Differential equation models，Bilinear and multilinear model and Biaxial interaction model，etc.But every kind of model has definite limitation。At the present time Bouc-Wen differential coefficient model is used abroad in the research on hysteretic system。Recently，dissymmetrical model is developed。Secondly: The response of hysteretic system under random excitation. According to the hysteretic restoring force models People have solved the response of such hysteretic system under random excitation. The main methods include in K-B approximation for bilinear systems, Semi empirical method for elastic-plastic systems，Equivalent linearization method for smooth <WP=60>systems，and Markov vector and Fokker-Plank equation method, etc. Applying above these methods，it has soved the response of single-degree-of-freedom or multi-degree-of-freedom hysteretic system under random excitation according todifferent requires .Thirdly: Hysteretic system parameter identification. Generally, we can analysis of the response of system and energy dissipation and degrading after we know the hysteretic nonlinear relation of displacement and load of system. A simple technique based on a least square error minimization has been developed for the smooth hysteretic models. Other methods such as those based on an extended Kalman filter method have been applied to the smooth hysteretic restoring force successfully. To hysteretic system ,it is familiar that one model is firstly supposed basing on experience and such the parameters are recognized using vary means.Except above-mentioned measures,the methods of non-parameter recognizing are effective。The research that mentioned above solves the response of stochastic hysteretic systems under random excitation, but people do not solve the random response of nonlinear stochastic vibration system that caused by the hysteretic loop that is random itself. But the semi-continuous medium in engineer mechanism stands out especially, because of the discrete character of material, which leads to the nonlinear random vibration model of hysteretic system. A lot of engineer experiments have already proved the random discrete character that lies in hysteretic loop itself. So it is very necessary to study the random vibration of nonlinear systems that caused by hysteretic loop.I更多还原