摩擦连接的结构非线性动力学研究
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摘要
作为一种重要的紧固式机械连接形式，摩擦连接大量存在于各类飞行器结构以及其它工业产品中。实际上这类连接并不完全紧固，在振动环境下，连接结合面上常常存在着滑移运动，这会对连接子结构间的载荷传递和整体结构的动力学特性产生重要影响。本文所关注的正是在摩擦连接处滑移运动影响下结构的非线性振动问题，主要研究了包含描述摩擦连接的Iwan模型的一系列振动系统，包括振子系统、连续梁以及一类高维自激振动系统的非线性振动问题的求解方法。本论文主要包括以下几方面的内容：
当连接模型用离散Iwan模型描述时，相应的振动系统存在分段线性迟滞特点。针对此类非线性系统，提出了系统振动响应的解析求解方法。该方法是在获得线性阶段内运动解析式的基础上，根据位移和速度的连续性条件，依次将振动过程中所历经的线性阶段的参数代入解析式从而获得系统在整个时域上的解。对于含离散Iwan模型的振子系统，通过变量代换，便可将任意线性阶段的运动方程转化为可解析求解的形式。对于含离散Iwan模型描述的迟滞边界的连续梁系统，通过位移变换，便将该分段线性迟滞边界转化为可解析求解的线性边界问题。算例计算结果表明，随着激励量级的增大，在摩擦面滑移影响下，幅频响应表现出共振峰左漂的和阻尼强化等非线性特征，并对悬臂梁模型实施了振动台正弦扫频试验，验证了模型描述以及计算结果的合理性。
含无穷数量Jenkins单元的连续Iwan模型，更适合于描述结构摩擦连接面处的微滑移运动。在推导出模型非线性恢复力计算式的基础上，求解了含连续Iwan模型的干摩擦振子系统的非线性振动问题。针对振幅衰减的自由振动，提出了对加/卸载各自对应的半个谐波运动过程进行独立求解的分析方法，并基于谐波平衡法获得了系统响应的解析解；对于系统微滑移下的受迫振动，通过谐波平衡方程求得了系统稳态运动的幅频关系式以及系统阻尼与振幅的非线性关系。对算例的计算结果表明，解析解所得结果与数值解吻合较好，从而验证了方法的有效性。
研究了一端存在微滑移的固支梁系统在基础简谐激励下的非线性振动问题。当滑移较小时，将边界处迟滞约束力写成含有小参数的形式。考虑到由Iwan模型描述的迟滞边界条件中显含位移幅值项，将位移项连同其幅值一并展开成小参数的幂级数形式，通过多尺度法，获得了系统主共振下二阶近似解及幅频关系，并基于Lyapunov线性化稳定性理论得到了系统响应的稳定边界。仿真结果显示，当激励幅值、粘性阻尼和约束刚度等参数取值在一定范围内，幅频曲线均会出现不稳定的多值区域。当滑移较大时，对系统方程和边界条件进行了重构，通过一阶谐波解获得了系统的幅频关系，仿真结果表明，由于摩擦阻尼显著影响，幅频曲线中并未出现不稳定域。
最后，将Iwan模型引入到一类高维自激振动系统的迟滞非线性环节的描述中。在采用高阶谐波平衡法进行求解系统极限环振荡问题时，针对谐波项数增加而使得谐波平衡方程组的规模成倍扩充从而造成求解困难的问题，提出了在数值求解高阶谐波平衡方程组时，以一阶谐波解所得结果作为对应项初值，高阶谐波项系数初值赋零的初值选取策略。该策略在对微滑移下带外挂的二元机翼系统的极限环振荡问题的求解中得到了验证。
As an important form of mechanical joints for fastening, frictional joints arewidely used in the structures of vehicles and other industrial products. In fact, thistype of connections are not fully tightened, for slipping often take place on thecontact interface during vibration of the structure, which may have an importantimpact on loads transfer between substructures as well as the mechanicalcharacteristics of the composed whole structures. In this thesis, the nonlinearvibration of the structures under the influence of the slipping on the interface offrictional joints is concerned. The solution methods for nonlinear vibration of aseries of vibration systems, including oscillators, beams and a high-dimensionalself-excited vibration system with Iwan model, which can be used for friction jointsmodeling, will be deeply studied in this thesis. The main contents of the thesis are asfollows:
When the joint is modeled by a discrete Iwan model, the correspondingvibration systems shows piecewise linear hysteresis nonlinear characteristics. Forsuch nonlinear systems, we proposed an analytical method to calculate the vibrationresponse of the system. In the method, when the general analytical expression of thesolution for each linear phase is obtained, by the continuity of displacement and ve-locity, substituting the parameters of each linear phase into the analytical expressionsequentially during the vibration process, then the solution of entire time domain isconstituted. For an oscillator with a discrete Iwan model, by variable substitution,the equation of motion for arbitrary linear phase can be transformed into the formwhich can be solved analytically. For an continuious beam with an piecewise linearhysteresis boundary modeled by a discrete Iwan model, by displacement conversion,the piecewise linear hysteresis boundary problem is then converted into linearboundary problem which can be solved by linear vibration theory. The results ofnumerical examples show that, as the excitation magnitude increases, slipping onthe friction surface would take place, which makes the peaks of ampli-tude-frequency curves left drifted as well as damping enhancement. Finally, The ra-tionality of frictional joint modeling and the calculation results is verified by a sinesweep vibration test of the beam model in a vibration shaker.
The continuous Iwan model, which contains infinite number of Jenkinselements, is more appropriate to depict the microslip on the frictional interfaces ofstructures. Based on the expressions of restoring force of the continous Iwan model,we solved the nonlinear vibration problems of a dry friction oscillator with continuous Iwan model. For the free vibration problem, the half-harmonic motioncorresponding to each monotone loading/unloading movement independently isanalyzed, then based on the harmonic balance method, the analytical solution of thesystem response is obtained. For the forced vibration of the oscillator undermicrosilp, on the basis of the harmonic balance, the amplitude-frequency relation-ship as well as the nonlinear relationship of damping and the vibration amplitude areboth obtained. Simulation results show that the numerical results agree well withthose of analytical solutions, which verified the validity of the both methods.
The nonlinear vibration of a clamped-clamped beam with microslip at one endunder harmonic base excitation is also studied. When the slip is small, the hysteresisforce at the nonlinear boundary can be rewritten in a form containing a smallparameter. Taking into account that hysteresis boundary condition which describedIwan model containing displacement amplitude items, we expand both thedisplacement and its amplitude into a power series of the small parameter. Then, bythe method of multiple scales, the second-order approximate solution and theamplitude-frequency relationship of system's primary resonance are obtaind. Andbased on Lyapunov linearity stability theory, the stability boundary of the responseof the system is derived. Simulation results show that an unstable multi-valuedregion would arise in the frequency response curve when the excitation amplitude,viscous damping and joint stiffness are in a specific range. When the slip is large,the governing equation and the nonliear boundary condition are reconstructed, thenamplitude-frequency relationship can be obtained by the first harmonic balancemethod. The simulation results show that there are no unstable region in theamplitude-frequency for the significant influence of the friction damping.
In the final chapter of the thesis, the Iwan model is introduced for modeling thehysteresis nonlinearity in a class of high-dimensional self-excited system. Whensolving the high-dimensional nonlinear differential equations by the higher orderharmonic balance method, as the number harmonic terms increases, the dimensionsof the harmonic balance equations wound increase manyfold, which makes it toughto solve the equtions directly. For this problem, an strategy for init valuesdetermination is proposed. In the strategy, the results of the first order harmonicsolution are considered as the initial values of the corresponding coefficients of thehigh order harmonic solutions, and the initial value of the higher order harmonicscoefficients are assigned to zero. This strategy is verified by seccessfully solving ofthe limit cycle oscillations of an airfoil with an external store under microslip.
当连接模型用离散Iwan模型描述时，相应的振动系统存在分段线性迟滞特点。针对此类非线性系统，提出了系统振动响应的解析求解方法。该方法是在获得线性阶段内运动解析式的基础上，根据位移和速度的连续性条件，依次将振动过程中所历经的线性阶段的参数代入解析式从而获得系统在整个时域上的解。对于含离散Iwan模型的振子系统，通过变量代换，便可将任意线性阶段的运动方程转化为可解析求解的形式。对于含离散Iwan模型描述的迟滞边界的连续梁系统，通过位移变换，便将该分段线性迟滞边界转化为可解析求解的线性边界问题。算例计算结果表明，随着激励量级的增大，在摩擦面滑移影响下，幅频响应表现出共振峰左漂的和阻尼强化等非线性特征，并对悬臂梁模型实施了振动台正弦扫频试验，验证了模型描述以及计算结果的合理性。
含无穷数量Jenkins单元的连续Iwan模型，更适合于描述结构摩擦连接面处的微滑移运动。在推导出模型非线性恢复力计算式的基础上，求解了含连续Iwan模型的干摩擦振子系统的非线性振动问题。针对振幅衰减的自由振动，提出了对加/卸载各自对应的半个谐波运动过程进行独立求解的分析方法，并基于谐波平衡法获得了系统响应的解析解；对于系统微滑移下的受迫振动，通过谐波平衡方程求得了系统稳态运动的幅频关系式以及系统阻尼与振幅的非线性关系。对算例的计算结果表明，解析解所得结果与数值解吻合较好，从而验证了方法的有效性。
研究了一端存在微滑移的固支梁系统在基础简谐激励下的非线性振动问题。当滑移较小时，将边界处迟滞约束力写成含有小参数的形式。考虑到由Iwan模型描述的迟滞边界条件中显含位移幅值项，将位移项连同其幅值一并展开成小参数的幂级数形式，通过多尺度法，获得了系统主共振下二阶近似解及幅频关系，并基于Lyapunov线性化稳定性理论得到了系统响应的稳定边界。仿真结果显示，当激励幅值、粘性阻尼和约束刚度等参数取值在一定范围内，幅频曲线均会出现不稳定的多值区域。当滑移较大时，对系统方程和边界条件进行了重构，通过一阶谐波解获得了系统的幅频关系，仿真结果表明，由于摩擦阻尼显著影响，幅频曲线中并未出现不稳定域。
最后，将Iwan模型引入到一类高维自激振动系统的迟滞非线性环节的描述中。在采用高阶谐波平衡法进行求解系统极限环振荡问题时，针对谐波项数增加而使得谐波平衡方程组的规模成倍扩充从而造成求解困难的问题，提出了在数值求解高阶谐波平衡方程组时，以一阶谐波解所得结果作为对应项初值，高阶谐波项系数初值赋零的初值选取策略。该策略在对微滑移下带外挂的二元机翼系统的极限环振荡问题的求解中得到了验证。
As an important form of mechanical joints for fastening, frictional joints arewidely used in the structures of vehicles and other industrial products. In fact, thistype of connections are not fully tightened, for slipping often take place on thecontact interface during vibration of the structure, which may have an importantimpact on loads transfer between substructures as well as the mechanicalcharacteristics of the composed whole structures. In this thesis, the nonlinearvibration of the structures under the influence of the slipping on the interface offrictional joints is concerned. The solution methods for nonlinear vibration of aseries of vibration systems, including oscillators, beams and a high-dimensionalself-excited vibration system with Iwan model, which can be used for friction jointsmodeling, will be deeply studied in this thesis. The main contents of the thesis are asfollows:
When the joint is modeled by a discrete Iwan model, the correspondingvibration systems shows piecewise linear hysteresis nonlinear characteristics. Forsuch nonlinear systems, we proposed an analytical method to calculate the vibrationresponse of the system. In the method, when the general analytical expression of thesolution for each linear phase is obtained, by the continuity of displacement and ve-locity, substituting the parameters of each linear phase into the analytical expressionsequentially during the vibration process, then the solution of entire time domain isconstituted. For an oscillator with a discrete Iwan model, by variable substitution,the equation of motion for arbitrary linear phase can be transformed into the formwhich can be solved analytically. For an continuious beam with an piecewise linearhysteresis boundary modeled by a discrete Iwan model, by displacement conversion,the piecewise linear hysteresis boundary problem is then converted into linearboundary problem which can be solved by linear vibration theory. The results ofnumerical examples show that, as the excitation magnitude increases, slipping onthe friction surface would take place, which makes the peaks of ampli-tude-frequency curves left drifted as well as damping enhancement. Finally, The ra-tionality of frictional joint modeling and the calculation results is verified by a sinesweep vibration test of the beam model in a vibration shaker.
The continuous Iwan model, which contains infinite number of Jenkinselements, is more appropriate to depict the microslip on the frictional interfaces ofstructures. Based on the expressions of restoring force of the continous Iwan model,we solved the nonlinear vibration problems of a dry friction oscillator with continuous Iwan model. For the free vibration problem, the half-harmonic motioncorresponding to each monotone loading/unloading movement independently isanalyzed, then based on the harmonic balance method, the analytical solution of thesystem response is obtained. For the forced vibration of the oscillator undermicrosilp, on the basis of the harmonic balance, the amplitude-frequency relation-ship as well as the nonlinear relationship of damping and the vibration amplitude areboth obtained. Simulation results show that the numerical results agree well withthose of analytical solutions, which verified the validity of the both methods.
The nonlinear vibration of a clamped-clamped beam with microslip at one endunder harmonic base excitation is also studied. When the slip is small, the hysteresisforce at the nonlinear boundary can be rewritten in a form containing a smallparameter. Taking into account that hysteresis boundary condition which describedIwan model containing displacement amplitude items, we expand both thedisplacement and its amplitude into a power series of the small parameter. Then, bythe method of multiple scales, the second-order approximate solution and theamplitude-frequency relationship of system's primary resonance are obtaind. Andbased on Lyapunov linearity stability theory, the stability boundary of the responseof the system is derived. Simulation results show that an unstable multi-valuedregion would arise in the frequency response curve when the excitation amplitude,viscous damping and joint stiffness are in a specific range. When the slip is large,the governing equation and the nonliear boundary condition are reconstructed, thenamplitude-frequency relationship can be obtained by the first harmonic balancemethod. The simulation results show that there are no unstable region in theamplitude-frequency for the significant influence of the friction damping.
In the final chapter of the thesis, the Iwan model is introduced for modeling thehysteresis nonlinearity in a class of high-dimensional self-excited system. Whensolving the high-dimensional nonlinear differential equations by the higher orderharmonic balance method, as the number harmonic terms increases, the dimensionsof the harmonic balance equations wound increase manyfold, which makes it toughto solve the equtions directly. For this problem, an strategy for init valuesdetermination is proposed. In the strategy, the results of the first order harmonicsolution are considered as the initial values of the corresponding coefficients of thehigh order harmonic solutions, and the initial value of the higher order harmonicscoefficients are assigned to zero. This strategy is verified by seccessfully solving ofthe limit cycle oscillations of an airfoil with an external store under microslip.
引文
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