碰振/滞迟随机系统的响应与可靠性研究
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摘要
随机系统动力学行为的研究,由于其重要的科学意义和工程价值,一直以来受到科研工作者的广泛关注,在传统随机系统的响应、稳定性和可靠性方面,已经建立了较完整的分析方法。传统系统的保守和耗散因素具有分离形式,而弹塑性、滞迟及粘弹性等材料在系统中引入了保守与耗散耦合因素,该耦合因素致使传统的分析手段失效。本文针对两类典型的具保守与耗散耦合因素的系统,即耗散碰振系统和滞迟系统,研究耦合因素的等效处理方法,并分别讨论其随机响应与系统可靠性。
耗散碰振系统考虑挡板的能量耗散效应,接触力同时包含保守与耗散成分通过修正的Hertz接触模型和Wu-Yu冲击模型描述。对于弱激励、弱能量的耗散碰振系统,采用能量耗散平衡法和随机平均法分析之。接触力的保守部分通过势能导数表示,而耗散部分通过一周内的能量耗散平衡替换为依赖于系统能量的等效阻尼力,进而可建立相应的能量包线随机平均法并得出系统响应统计。考虑到上述分析方法对激励和能量耗散的严格限制,并注意到耗散碰振系统方程的特点,进一步采用等效非线性化方法研究无上述限制的碰振系统随机响应。以无耗散碰振系统为基准,通过引进保守因素的修正系数以及状态依赖的等效阻尼系数描述接触力的耗散效应,从而选取了具有精确平稳解的等效非线性系统类。修正系数及待定函数通过原系统和等效非线性系统之差的均方值最小准则确定。状态依赖的等效阻尼系数取为关于系统能量多项式的形式,将泛函极值问题简化为函数极值问题。从而由等效非线性系统的各统计量近似原耗散碰振系统的响应统计。
滞迟系统可由众多模型描述,本文局限于有重要应用价值的Duhem滞迟系统和Preisach滞迟系统,研究其在随机激励作用下的首次穿越问题。类似于对耗散接触力的处理,滞迟力所包含的保守成分和耗散成分分别由势能导数和具能量依赖阻尼系数的阻尼力近似,从而得到近似的保守与耗散因素不耦合的非线性随机系统。应用能量包线随机平均法可导出关于系统能量的平均Ito随机微分方程,进而得到关于系统可靠性函数的后向Kolmogrov方程,求解该方程可确定系统条件可靠性函数和首次穿越时间的条件概率密度函数。
Mechanical and structural systems under stochastic dynamic loadings are always encountered in engineering applications and attract extensive attention of researchers. Stochastic responses, stability and reliability of traditional stochastic systems, in which conservative components and dissipative components are separated, have been studied sufficiently and many effective traditional analytical methods have been proposed. Unfortunately, some materials such as elastic-plastic, hysteresis and viscoelastic. have the coupling conservation and dissipation components. Stochastic systems with coupling conservation and dissipation components are different from the ordinary systems and traditional analysis methods are not suitable. In the present dissertation, responses and reliability of two typical stochastic systems with coupling conservation and dissipation components, i.e., dissipative vibro-impact system and hysteretic system, are studied.
In consideration of the energy dissipative effect, the contact forces are described by modified Hertzian contact model and Wu-Yu impact model, respectively. In case of weak excitations and weak energy dissipation, the vibro-impact systems are researched by energy dissipation balance technique and stochastic averaging method. The contact force can be approximately by conservative component and dissipative component. The equivalent conservative one is represented by the gradient of total potential energy. While the dissipative component is substituted by equivalent quasi-linear damping with energy-dependent damping, which satisfies that energy dissipation due to the inelastic impact in one cycle is equal to the energy dissipation by the quasi-linear damping during the same period. Then, by adopting the stochastic averaging method, stochastic responses can be obtained. Due to the restriction of weak damping and weak excitations, stochastic averaging method is not always effective. Equivalent nonlinear technique is adopted for strong damping and excitations. Based on the vibro-impact system without dissipation, introducing the conservative component correction fact and state-dependent damping coefficient, one class of equivalent nonlinear systems with exact stationary solution is selected. The correction factor and undetermined damping coefficient are determined through minimizing the mean-square difference. By expressing the state-dependent damping coefficient as polynomial functions of system energy, the functional extremum problem is transformed into function extremum problem and finally the responses of the original vibro-impact system are approximately obtained through the equivalent nonlinear system.
Many hysteretic models have been developed to describe the hysteretic behavior. Due to the important application of Duhem and Preisach hysteretic models, the first-passage problem of hysteretic systems described by Duhem and Preisach hysteretic models under stochastic dynamics are discussed. Similarly to dealing with inelastic contact force, the conservative component of the hysteretic force is expressed by the gradient of the total potential energy and dissipative component is substituted by quasi-linear damping with energy-dependent damping coefficient. Thus equivalent nonlinear stochastic system with separated conservation and dissipation components is obtained, and then by adopting stochastic averaging method, the averaged Ito stochastic differential equation with respect to system energy is dervied. The establishing and solving of the associated backward Kolmogorov equation yield the reliability function and probability density of first-passage time.
耗散碰振系统考虑挡板的能量耗散效应,接触力同时包含保守与耗散成分通过修正的Hertz接触模型和Wu-Yu冲击模型描述。对于弱激励、弱能量的耗散碰振系统,采用能量耗散平衡法和随机平均法分析之。接触力的保守部分通过势能导数表示,而耗散部分通过一周内的能量耗散平衡替换为依赖于系统能量的等效阻尼力,进而可建立相应的能量包线随机平均法并得出系统响应统计。考虑到上述分析方法对激励和能量耗散的严格限制,并注意到耗散碰振系统方程的特点,进一步采用等效非线性化方法研究无上述限制的碰振系统随机响应。以无耗散碰振系统为基准,通过引进保守因素的修正系数以及状态依赖的等效阻尼系数描述接触力的耗散效应,从而选取了具有精确平稳解的等效非线性系统类。修正系数及待定函数通过原系统和等效非线性系统之差的均方值最小准则确定。状态依赖的等效阻尼系数取为关于系统能量多项式的形式,将泛函极值问题简化为函数极值问题。从而由等效非线性系统的各统计量近似原耗散碰振系统的响应统计。
滞迟系统可由众多模型描述,本文局限于有重要应用价值的Duhem滞迟系统和Preisach滞迟系统,研究其在随机激励作用下的首次穿越问题。类似于对耗散接触力的处理,滞迟力所包含的保守成分和耗散成分分别由势能导数和具能量依赖阻尼系数的阻尼力近似,从而得到近似的保守与耗散因素不耦合的非线性随机系统。应用能量包线随机平均法可导出关于系统能量的平均Ito随机微分方程,进而得到关于系统可靠性函数的后向Kolmogrov方程,求解该方程可确定系统条件可靠性函数和首次穿越时间的条件概率密度函数。
Mechanical and structural systems under stochastic dynamic loadings are always encountered in engineering applications and attract extensive attention of researchers. Stochastic responses, stability and reliability of traditional stochastic systems, in which conservative components and dissipative components are separated, have been studied sufficiently and many effective traditional analytical methods have been proposed. Unfortunately, some materials such as elastic-plastic, hysteresis and viscoelastic. have the coupling conservation and dissipation components. Stochastic systems with coupling conservation and dissipation components are different from the ordinary systems and traditional analysis methods are not suitable. In the present dissertation, responses and reliability of two typical stochastic systems with coupling conservation and dissipation components, i.e., dissipative vibro-impact system and hysteretic system, are studied.
In consideration of the energy dissipative effect, the contact forces are described by modified Hertzian contact model and Wu-Yu impact model, respectively. In case of weak excitations and weak energy dissipation, the vibro-impact systems are researched by energy dissipation balance technique and stochastic averaging method. The contact force can be approximately by conservative component and dissipative component. The equivalent conservative one is represented by the gradient of total potential energy. While the dissipative component is substituted by equivalent quasi-linear damping with energy-dependent damping, which satisfies that energy dissipation due to the inelastic impact in one cycle is equal to the energy dissipation by the quasi-linear damping during the same period. Then, by adopting the stochastic averaging method, stochastic responses can be obtained. Due to the restriction of weak damping and weak excitations, stochastic averaging method is not always effective. Equivalent nonlinear technique is adopted for strong damping and excitations. Based on the vibro-impact system without dissipation, introducing the conservative component correction fact and state-dependent damping coefficient, one class of equivalent nonlinear systems with exact stationary solution is selected. The correction factor and undetermined damping coefficient are determined through minimizing the mean-square difference. By expressing the state-dependent damping coefficient as polynomial functions of system energy, the functional extremum problem is transformed into function extremum problem and finally the responses of the original vibro-impact system are approximately obtained through the equivalent nonlinear system.
Many hysteretic models have been developed to describe the hysteretic behavior. Due to the important application of Duhem and Preisach hysteretic models, the first-passage problem of hysteretic systems described by Duhem and Preisach hysteretic models under stochastic dynamics are discussed. Similarly to dealing with inelastic contact force, the conservative component of the hysteretic force is expressed by the gradient of the total potential energy and dissipative component is substituted by quasi-linear damping with energy-dependent damping coefficient. Thus equivalent nonlinear stochastic system with separated conservation and dissipation components is obtained, and then by adopting stochastic averaging method, the averaged Ito stochastic differential equation with respect to system energy is dervied. The establishing and solving of the associated backward Kolmogorov equation yield the reliability function and probability density of first-passage time.
引文
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