结构振动响应预示的能量方法研究
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摘要
振动响应预示一直是航天等工程领域的重要课题。现代社会的发展对航天器等各种工程对象提出了更多的功能要求，使得工程结构的复杂程度越来越高，有限元等传统方法的计算量因此而急剧增大，结构复杂性的提高也令动力学参数的精确确定变得更加困难。计算量过大和参数的不确定性已经成为结构振动响应预示领域面临的两个主要问题。大量的研究工作致力于这两方面问题的解决。能量流方法以能量密度控制方程为基础，取平均化的振动能量为变量，极大地减少了计算量，并且给出了较为精细的结果，逐渐成为响应预示方法研究的新热点。本文利用平均化方法，求解了能量密度控制方程的解析解，并研究了能量流方法在不确定性结构振动预示领域的应用。主要内容概括如下：
首先，介绍了能量密度控制方程的数值解法。进一步从结构的位移解出发，将结构的边界效应考虑为能量反射，基于结构的位移Green核，推导了能量密度控制方程的解析解，与数值解进行了对比验证，并求解了飞轮扰动激励下卫星结构的能量流响应。对二维结构的边界处理时，提出利用能量平均自由程的概念进行描述，通过统计的思想将结构的二维边界转化为一维问题。针对随机白噪声激励下的结构，利用能量流方法通过频域积分进行了求解。
其次，针对随机参数结构，推导了用参数标准差表示的能量响应期望值和标准差公式，得到了给定置信概率时的响应区间。对确定性结构的能量密度控制方程进行摄动分析，通过方程的阶数匹配，得到了随机参数结构的能量密度控制方程组。对Gauss随机参数结构进行摄动，利用概率积分建立了随机参数标准差表示的能量流表达式。进一步利用能量流方法推导了随机结构的能量响应方差和与确定性响应的相对偏差，研究了随机参数对结构振动能量响应的影响。
对参数随机分布模型未知的结构，基于熵最大原理进行了结构振动能量预示的非参数建模。根据结构参数的不确定性程度，将计算方法分为随机摄动法和直接概率法。随机摄动法采用小参数摄动展开，对摄动部分以随机参数的熵最大为依据，直接概率法则对随机参数直接代入熵表达式取最大值，从而得到两种方法下的概率密度。进一步利用能量流法求解了随机结构的能量响应值，通过与确定性结构响应的比较，给出了结构能量密度响应的偏差变化特征，表明在复杂结构的高频振动时需要考虑结构参数的随机特性。
最后，对非概率描述的不确定性参数结构进行了能量流求解。利用复杂结构动力学建模的模糊结构描述，将复杂结构分为可以精确建模的主结构和难以精确建模的附加模糊结构，主结构部分采用能量流方法进行描述，附加模糊结构对主结构的影响采用弹簧振子系统进行建模，得到了模糊结构影响下的结构能量流表达式，研究了模糊结构对结构能量响应的大小及传播速度的影响。对已知参数上下界的结构，采用区间分析法进行结构能量流分析建模。利用一阶Taylor级数展开，得到了区间参数结构能量流的响应区间，并且通过与概率方法进行仿真对比，发现区间分析方法能够给出更小的响应区间。
Prediction of vibration response is an important subject in aerospace and otherfields. Recent development of society demands more functions of spacecrafts andthis makes engineering structures more and more complex and dynamicalparameters are more difficult to obtain. Meanwhile, increasing complexity bringsmore computation cost for FEM and other traditional method. Computation cost andparameter uncertainty are two main problems in prediction of structural vibration.Plenty of work focuses on the two problems. Based on energy density governingequations, energy flow method uses averaged energy as a variable to reducecomputation cost dramatically and provide more precise results. Energy flowmethod is now a popular solution for response prediction. This work providesanalytical solutions for energy density governing equations. Response prediction foruncertainty structures is also studied using energy flow method. Detailed works are:Numerical solutions of energy density governing equations are introduced. Energycharacteristic of a satellite structure under excitation of a flying wheel is studied byEnergy Finite Element Method. Taking boundary effect as energy reflection,analytical solutions of energy density governing equations are derived withdisplacement. And the analytical solutions are validated by numerical solutions.Energy mean free path is introduced to describe boundary effect of two-dimensionalstructures. Frequency integral is used to obtain energy response under randomexcitation.
Further, for structures with probabilistic parameters, the expectation andstandard variance of energy density are expressed by the parameter standardvariance. The response interval is obtained under certain confidence probability.Introducing perturbation technique, new control equation sets of energy density foruncertain structures are obtained. For structures with Gauss parameters, probabilityand perturbation techniques are combined to derive the energy density expressedwith standard variances of the uncertain parameters. Response variances of theenergy density and the relative variance comparing the deterministic structures areobtained.
For structures whose random model is unknown, maximum entropy principle isused to prediction energy response of vibrating structures using non-parametermodel. Random perturbation method and direct probability method are provided.According to the maximum entropy, deployment is carried with random perturbationand direct probability utilizes the formulation directly. Probability densities of theparameters are derived with the two methods and the energy responses are obtainedwith energy flow method. The mean responses and the variance to the normal responses are provided. The results indicate randomness must be considered for highfrequency vibration of complex structures.
Finally, for structures with non-probabilistic parameters, fuzzy structuresmethod is introduce into energy flow method. Fuzzy structure method describescomplex structures with master structure that can be modeled precisely and fuzzystructures that lack of modeling information. The master structure is described byenergy flow method and the fuzzy structures are modeled by spring and masssystems. Energy flow of structures is obtained and simulations show that fuzzystructures influence the wave velocity and energy response through wave number.For prediction of structures with interval parameters, Taylor deployment is used toobtain the energy interval. Comparison with probability method shows that intervalmethod provides smaller response interval.
首先，介绍了能量密度控制方程的数值解法。进一步从结构的位移解出发，将结构的边界效应考虑为能量反射，基于结构的位移Green核，推导了能量密度控制方程的解析解，与数值解进行了对比验证，并求解了飞轮扰动激励下卫星结构的能量流响应。对二维结构的边界处理时，提出利用能量平均自由程的概念进行描述，通过统计的思想将结构的二维边界转化为一维问题。针对随机白噪声激励下的结构，利用能量流方法通过频域积分进行了求解。
其次，针对随机参数结构，推导了用参数标准差表示的能量响应期望值和标准差公式，得到了给定置信概率时的响应区间。对确定性结构的能量密度控制方程进行摄动分析，通过方程的阶数匹配，得到了随机参数结构的能量密度控制方程组。对Gauss随机参数结构进行摄动，利用概率积分建立了随机参数标准差表示的能量流表达式。进一步利用能量流方法推导了随机结构的能量响应方差和与确定性响应的相对偏差，研究了随机参数对结构振动能量响应的影响。
对参数随机分布模型未知的结构，基于熵最大原理进行了结构振动能量预示的非参数建模。根据结构参数的不确定性程度，将计算方法分为随机摄动法和直接概率法。随机摄动法采用小参数摄动展开，对摄动部分以随机参数的熵最大为依据，直接概率法则对随机参数直接代入熵表达式取最大值，从而得到两种方法下的概率密度。进一步利用能量流法求解了随机结构的能量响应值，通过与确定性结构响应的比较，给出了结构能量密度响应的偏差变化特征，表明在复杂结构的高频振动时需要考虑结构参数的随机特性。
最后，对非概率描述的不确定性参数结构进行了能量流求解。利用复杂结构动力学建模的模糊结构描述，将复杂结构分为可以精确建模的主结构和难以精确建模的附加模糊结构，主结构部分采用能量流方法进行描述，附加模糊结构对主结构的影响采用弹簧振子系统进行建模，得到了模糊结构影响下的结构能量流表达式，研究了模糊结构对结构能量响应的大小及传播速度的影响。对已知参数上下界的结构，采用区间分析法进行结构能量流分析建模。利用一阶Taylor级数展开，得到了区间参数结构能量流的响应区间，并且通过与概率方法进行仿真对比，发现区间分析方法能够给出更小的响应区间。
Prediction of vibration response is an important subject in aerospace and otherfields. Recent development of society demands more functions of spacecrafts andthis makes engineering structures more and more complex and dynamicalparameters are more difficult to obtain. Meanwhile, increasing complexity bringsmore computation cost for FEM and other traditional method. Computation cost andparameter uncertainty are two main problems in prediction of structural vibration.Plenty of work focuses on the two problems. Based on energy density governingequations, energy flow method uses averaged energy as a variable to reducecomputation cost dramatically and provide more precise results. Energy flowmethod is now a popular solution for response prediction. This work providesanalytical solutions for energy density governing equations. Response prediction foruncertainty structures is also studied using energy flow method. Detailed works are:Numerical solutions of energy density governing equations are introduced. Energycharacteristic of a satellite structure under excitation of a flying wheel is studied byEnergy Finite Element Method. Taking boundary effect as energy reflection,analytical solutions of energy density governing equations are derived withdisplacement. And the analytical solutions are validated by numerical solutions.Energy mean free path is introduced to describe boundary effect of two-dimensionalstructures. Frequency integral is used to obtain energy response under randomexcitation.
Further, for structures with probabilistic parameters, the expectation andstandard variance of energy density are expressed by the parameter standardvariance. The response interval is obtained under certain confidence probability.Introducing perturbation technique, new control equation sets of energy density foruncertain structures are obtained. For structures with Gauss parameters, probabilityand perturbation techniques are combined to derive the energy density expressedwith standard variances of the uncertain parameters. Response variances of theenergy density and the relative variance comparing the deterministic structures areobtained.
For structures whose random model is unknown, maximum entropy principle isused to prediction energy response of vibrating structures using non-parametermodel. Random perturbation method and direct probability method are provided.According to the maximum entropy, deployment is carried with random perturbationand direct probability utilizes the formulation directly. Probability densities of theparameters are derived with the two methods and the energy responses are obtainedwith energy flow method. The mean responses and the variance to the normal responses are provided. The results indicate randomness must be considered for highfrequency vibration of complex structures.
Finally, for structures with non-probabilistic parameters, fuzzy structuresmethod is introduce into energy flow method. Fuzzy structure method describescomplex structures with master structure that can be modeled precisely and fuzzystructures that lack of modeling information. The master structure is described byenergy flow method and the fuzzy structures are modeled by spring and masssystems. Energy flow of structures is obtained and simulations show that fuzzystructures influence the wave velocity and energy response through wave number.For prediction of structures with interval parameters, Taylor deployment is used toobtain the energy interval. Comparison with probability method shows that intervalmethod provides smaller response interval.
引文
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