不确定多体系统动力学分析及可靠性预测
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摘要
传统的多体系统模型是建立在确定性基础上，即把分析工程中各种因素作为确定性物理量来进行处理，认为它们是精确确定的或可以精确测量的。但由于不确定性因素是客观存在的、不可避免的，在实际工程中存在大量的误差和不确定因素，基于确定性假设的模型并不能有效的描述多体系统的动力学行为，甚至可能导致前后矛盾或有悖于工程实际的结果。随着现代机构日益向着高速化、轻量化、精密化的方向发展，对不确定性多体系统进行动力学分析及可靠性预测有着重要的理论和现实意义。本文围绕工程中客观存在的不确定性问题，对含随机参数的刚柔耦合、含动力刚化、多输入输出、含间隙铰、含间隙润滑铰的多体系统动力学问题进行分析并给出了一般解算方法，进一步对多体系统的可靠性问题进行了研究。通过工程实例进行仿真计算，验证了方法的正确性和有效性，并得到了诸多有益的结论。主要内容如下：
1.基于Lagrange方程建立了含随机参数的多体系统的动力学模型，利用广义坐标分离法将随机微分代数方程转化为纯随机微分方程，利用Newmark逐步积分法进行数值解算。应用随机因子法求解系统随机响应的数字特征，获得统计意义下的解。以旋转杆滑块系统为例，考虑系统中载荷、物理和几何参数的随机性，验证了文中方法的正确性和有效性。计算结果表明，部分随机参数的分散性对多体系统动力响应的影响不可忽略，利用随机参数的动力学模型将能客观地反映出系统的动力学行为。
2.在柔性梁的纵向变形位移中计及耦合变形量的条件下，采用Lagrange方程与假设模态法建立了旋转柔性梁系统的刚柔耦合动力学模型，考虑系统物理参数和几何参数的随机性，基于随机响应面法对其动力响应进行随机性分析并对其运动功能的可靠性进行预测。通过算例验证了文中模型和方法的合理性和有效性。研究结果表明，三阶随机响应面法具有良好的计算精度，且效率更高。
3.将Lagrange方程和假设模态法相结合建立了考虑摩擦的双连杆柔性机械臂的动力学模型。考虑随机因素的影响，将随机因子法的处理方式引入随机响应面法中，提出一种处理多输入随机参数的双连杆柔性机械臂系统分析方法，并分别建立了系统强度、刚度和运动功能的功能函数，然后对该系统动力响应的随机性及可靠性进行分析。通过算例验证了文中模型和方法的合理性和可行性，同时预测了系统的可靠度，并分析了系统参数的随机性对双连杆柔性机械臂可靠性的影响。
4.以含间隙的曲柄滑块系统为研究对象，分析其在考虑铰间摩擦力及系统参数具有随机性时的动力响应及系统中滑块位移的可靠性问题。利用连续接触模型和修正的Coulomb摩擦模型分别求出间隙处接触力及切向摩擦力，基于Lagrange方程建立曲柄滑块机构的动力学模型。分别利用BP神经网络法和支持向量机法给出了系统随机参数与动力响应之间的近似函数关系式。在此基础上，利用矩法求解系统动力响应的数字特征和系统中滑块位移的可靠度。通过算例，考察了系统物理参数和几何参数的随机性对系统动力响应及可靠性的影响，并验证了所建模型和方法的合理性和可行性。研究结果表明，系统参数的随机性不可忽略，在参数变异系数相同的情况下，间隙的随机性对系统动力响应随机性和系统可靠性的影响较大。
5.研究了在考虑间隙润滑作用下并同时考虑系统参数具有随机性时曲柄滑块机构的可靠性问题。利用连续接触模型和流体动压润滑理论分别求出铰间隙处的接触力及润滑作用力，基于Lagrange方法建立曲柄滑块系统的动力学模型。为了克服因为参数选择不准确而使支持向量机回归的预测精度难以达到目标精度这一缺陷，通过遗传算法对支持向量机回归预测模型的各项参数进行寻优处理，获得最优参数值，有效提高了分析的精度，为解决支持向量机的参数选择问题提供了一条有效途径。然后利用该方法讨论了曲柄滑块机构中润滑铰间反作用力的可靠性问题。通过算例，验证了该方法的可行性和有效性。
The traditional dynamic model of multi-body system is based on the certainty,namely, all kinds of physical factors are considered as certain datum in the process ofanalysis, and they are accurate or can be accurately measured. But there are plenty oferror and uncertain factors in actual engineering because uncertainty is objective andinevitable. The dynamic model based on the accurate assumption cannot effectivelydiscrible the dynamic action of multi-body system, and may lead to contradictory results.With the development of modern mechanisms increasingly towards high-speed,lightweight and high precision, the dynamic analysis and prediction on reliability ofmulti-body with uncertainty have important theoretical significance and researchworthiness. Concertrated on the uncertain phenomenon existing in engineeringproblems, multibody systems including rigid-flexible coupling, dynamic stiffeningterms, multiple stochastic inputs and outputs, clearance joint and lubricate joint arepresented and their solutions are proposed. Furthermore the reliability of them is studied,the engineering examples are simulated according to the above theories and theeffectiveness and validity of the theories are verified by the results as well as somebenefit conclusions are obtained. The main research works can be described as follows:
1. Dynamic analysis of multibody systems with probabilistic parameters waspresented. Dynamic modeling of multibody systems was obtained by Lagrange’smethod. The Probabilistic Differential Algebraic equations were transformed into pureProbabilistic Differential equations by Generalized Coordinate Partitioning method. TheNewmark step by step integration method was used to calculate the results. Using themethod of random factor method, the numerical characteristics of responses of thesystem were derived, and the results were expressed in statistic. As an illustratingexample, dynamic modeling of a rotating bar and sliding block system considering theprobabilistic of load, geometric and physical parameters is presented and the accuracyand efficiency of the method are verified. The results illustrate the probabilisticparameters affect the dynamic response of the multibody system and the dynamicmodeling with probabilistic parameters can objectively reflect the dynamic behavior ofthe objective systems.
2. Based on the Lagrange’s equations and the assumed mode method, therigid-flexible coupling dynamic model of a rotating flexible beam which took the coupling term of the deformation in the expression of longitudinal deformation wasstudied. Then considering the physical and geometrical parameters under randomness,the randomness analysis of dynamic responses was developed and the motion functionreliability was forecasted by stochastic response surface method. The rationality andefficiency of the modeling and the method presented were verified by an example. Theresults demonstrate third order stochastic response surface method has good precisionwith acceptable time consumption.
3. The Lagrange dynamic differential equations and the assumed mode methodwere combined to establish the modeling of a two-link flexible robot manipulatorconsidering friction in the joints. Considering the effect of stochastic factors, therandom factor method was embedded into the stochastic response surface method toimprove the analysis method of two-link flexible robot manipulator with multiplestochastic inputs. The system performance functions of strength, stiffness and themotion function were developed. Then, the dynamic responses and the reliability of thesystem were analysised. The rationality and feasibility of the modeling and the methodpresented were verified by an example, and the reliability of the system was predicted.Some expressions of random parameters about the two-link fexible robot manipulatorare derived.
4. The dynamic responses and the reliability of output displacement for slider-cranksystem with clearance considering of the friction forces and the randomness of systemparameters were developed. The continuous contact force approach and a modifiedCoulomb`s friction model were used to evaluate the contact force and the friction of theintra-joint respectively. The dynamic model of mechanism was set up based onLagrange method. The approximate functional relationship between the system randomparameters and the dynamic responses was given by using the BP neural network andthe support vector machine method, respectively. The numerical characteristics of thesystem dynamic responses and the reliability of the output displacement of the sliderwere solved by moment method. The effects of the system physical parameters andgeometric parameters on the system dynamic response and the reliability were inspected.In addition, the results illustrate that the randomness of the system parameters can’t beignored, and the randomness of the clearance has greatest effect on the randomness ofthe responses and the reliability of the system under the same coefficient of variationconditions.
5. The reliability of the slider-crank mechanism considering realistic jointcharacteristic, namely, joint with clearance and lubrication, and the randomness of the system parameters is presented. The continuous approach and the hydrodynamic theoryare used to evaluate the contact force for the case of joints modeled as a contact pairwith dry contact and the force generated by lubrication action respectively. The systemdynamic model was set up based on Lagrange’s equation. The prediction accurary ofSupport Vector Machine Regression is difficult to reach the target accurary because theselection of parameters isn’t accurate. In order to overcome this limitation, theparameters are pretreated through Genetic Algorithm to get the optimum parametersvalues. The precision of analysis on the system is effectively improved, which providesan effectively way to solve the problem of parameters selection in Support VectorMachine Regression. Then the reliability of the reaction force developed in thelubricated revolute joint of the crank-slider mechanism is discussed by the method.Finally, take an example to verify the feasibility and effectiveness of the proposedmethod.
1.基于Lagrange方程建立了含随机参数的多体系统的动力学模型，利用广义坐标分离法将随机微分代数方程转化为纯随机微分方程，利用Newmark逐步积分法进行数值解算。应用随机因子法求解系统随机响应的数字特征，获得统计意义下的解。以旋转杆滑块系统为例，考虑系统中载荷、物理和几何参数的随机性，验证了文中方法的正确性和有效性。计算结果表明，部分随机参数的分散性对多体系统动力响应的影响不可忽略，利用随机参数的动力学模型将能客观地反映出系统的动力学行为。
2.在柔性梁的纵向变形位移中计及耦合变形量的条件下，采用Lagrange方程与假设模态法建立了旋转柔性梁系统的刚柔耦合动力学模型，考虑系统物理参数和几何参数的随机性，基于随机响应面法对其动力响应进行随机性分析并对其运动功能的可靠性进行预测。通过算例验证了文中模型和方法的合理性和有效性。研究结果表明，三阶随机响应面法具有良好的计算精度，且效率更高。
3.将Lagrange方程和假设模态法相结合建立了考虑摩擦的双连杆柔性机械臂的动力学模型。考虑随机因素的影响，将随机因子法的处理方式引入随机响应面法中，提出一种处理多输入随机参数的双连杆柔性机械臂系统分析方法，并分别建立了系统强度、刚度和运动功能的功能函数，然后对该系统动力响应的随机性及可靠性进行分析。通过算例验证了文中模型和方法的合理性和可行性，同时预测了系统的可靠度，并分析了系统参数的随机性对双连杆柔性机械臂可靠性的影响。
4.以含间隙的曲柄滑块系统为研究对象，分析其在考虑铰间摩擦力及系统参数具有随机性时的动力响应及系统中滑块位移的可靠性问题。利用连续接触模型和修正的Coulomb摩擦模型分别求出间隙处接触力及切向摩擦力，基于Lagrange方程建立曲柄滑块机构的动力学模型。分别利用BP神经网络法和支持向量机法给出了系统随机参数与动力响应之间的近似函数关系式。在此基础上，利用矩法求解系统动力响应的数字特征和系统中滑块位移的可靠度。通过算例，考察了系统物理参数和几何参数的随机性对系统动力响应及可靠性的影响，并验证了所建模型和方法的合理性和可行性。研究结果表明，系统参数的随机性不可忽略，在参数变异系数相同的情况下，间隙的随机性对系统动力响应随机性和系统可靠性的影响较大。
5.研究了在考虑间隙润滑作用下并同时考虑系统参数具有随机性时曲柄滑块机构的可靠性问题。利用连续接触模型和流体动压润滑理论分别求出铰间隙处的接触力及润滑作用力，基于Lagrange方法建立曲柄滑块系统的动力学模型。为了克服因为参数选择不准确而使支持向量机回归的预测精度难以达到目标精度这一缺陷，通过遗传算法对支持向量机回归预测模型的各项参数进行寻优处理，获得最优参数值，有效提高了分析的精度，为解决支持向量机的参数选择问题提供了一条有效途径。然后利用该方法讨论了曲柄滑块机构中润滑铰间反作用力的可靠性问题。通过算例，验证了该方法的可行性和有效性。
The traditional dynamic model of multi-body system is based on the certainty,namely, all kinds of physical factors are considered as certain datum in the process ofanalysis, and they are accurate or can be accurately measured. But there are plenty oferror and uncertain factors in actual engineering because uncertainty is objective andinevitable. The dynamic model based on the accurate assumption cannot effectivelydiscrible the dynamic action of multi-body system, and may lead to contradictory results.With the development of modern mechanisms increasingly towards high-speed,lightweight and high precision, the dynamic analysis and prediction on reliability ofmulti-body with uncertainty have important theoretical significance and researchworthiness. Concertrated on the uncertain phenomenon existing in engineeringproblems, multibody systems including rigid-flexible coupling, dynamic stiffeningterms, multiple stochastic inputs and outputs, clearance joint and lubricate joint arepresented and their solutions are proposed. Furthermore the reliability of them is studied,the engineering examples are simulated according to the above theories and theeffectiveness and validity of the theories are verified by the results as well as somebenefit conclusions are obtained. The main research works can be described as follows:
1. Dynamic analysis of multibody systems with probabilistic parameters waspresented. Dynamic modeling of multibody systems was obtained by Lagrange’smethod. The Probabilistic Differential Algebraic equations were transformed into pureProbabilistic Differential equations by Generalized Coordinate Partitioning method. TheNewmark step by step integration method was used to calculate the results. Using themethod of random factor method, the numerical characteristics of responses of thesystem were derived, and the results were expressed in statistic. As an illustratingexample, dynamic modeling of a rotating bar and sliding block system considering theprobabilistic of load, geometric and physical parameters is presented and the accuracyand efficiency of the method are verified. The results illustrate the probabilisticparameters affect the dynamic response of the multibody system and the dynamicmodeling with probabilistic parameters can objectively reflect the dynamic behavior ofthe objective systems.
2. Based on the Lagrange’s equations and the assumed mode method, therigid-flexible coupling dynamic model of a rotating flexible beam which took the coupling term of the deformation in the expression of longitudinal deformation wasstudied. Then considering the physical and geometrical parameters under randomness,the randomness analysis of dynamic responses was developed and the motion functionreliability was forecasted by stochastic response surface method. The rationality andefficiency of the modeling and the method presented were verified by an example. Theresults demonstrate third order stochastic response surface method has good precisionwith acceptable time consumption.
3. The Lagrange dynamic differential equations and the assumed mode methodwere combined to establish the modeling of a two-link flexible robot manipulatorconsidering friction in the joints. Considering the effect of stochastic factors, therandom factor method was embedded into the stochastic response surface method toimprove the analysis method of two-link flexible robot manipulator with multiplestochastic inputs. The system performance functions of strength, stiffness and themotion function were developed. Then, the dynamic responses and the reliability of thesystem were analysised. The rationality and feasibility of the modeling and the methodpresented were verified by an example, and the reliability of the system was predicted.Some expressions of random parameters about the two-link fexible robot manipulatorare derived.
4. The dynamic responses and the reliability of output displacement for slider-cranksystem with clearance considering of the friction forces and the randomness of systemparameters were developed. The continuous contact force approach and a modifiedCoulomb`s friction model were used to evaluate the contact force and the friction of theintra-joint respectively. The dynamic model of mechanism was set up based onLagrange method. The approximate functional relationship between the system randomparameters and the dynamic responses was given by using the BP neural network andthe support vector machine method, respectively. The numerical characteristics of thesystem dynamic responses and the reliability of the output displacement of the sliderwere solved by moment method. The effects of the system physical parameters andgeometric parameters on the system dynamic response and the reliability were inspected.In addition, the results illustrate that the randomness of the system parameters can’t beignored, and the randomness of the clearance has greatest effect on the randomness ofthe responses and the reliability of the system under the same coefficient of variationconditions.
5. The reliability of the slider-crank mechanism considering realistic jointcharacteristic, namely, joint with clearance and lubrication, and the randomness of the system parameters is presented. The continuous approach and the hydrodynamic theoryare used to evaluate the contact force for the case of joints modeled as a contact pairwith dry contact and the force generated by lubrication action respectively. The systemdynamic model was set up based on Lagrange’s equation. The prediction accurary ofSupport Vector Machine Regression is difficult to reach the target accurary because theselection of parameters isn’t accurate. In order to overcome this limitation, theparameters are pretreated through Genetic Algorithm to get the optimum parametersvalues. The precision of analysis on the system is effectively improved, which providesan effectively way to solve the problem of parameters selection in Support VectorMachine Regression. Then the reliability of the reaction force developed in thelubricated revolute joint of the crank-slider mechanism is discussed by the method.Finally, take an example to verify the feasibility and effectiveness of the proposedmethod.
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