轴向运动梁动力学及控制研究
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摘要
轴向运动梁是一种重要的工程元件简化模型，如硬式空中加油管、平推式架桥、快速升起的野战雷达天线、高速飞行的火箭和导弹等，都可以将其简化为轴向运动梁。对轴向运动梁的动力学及振动控制研究具有重要的理论意义和工程应用价值。
本文针对两种轴向运动梁动力学模型进行了较深入研究。一种为轴向运动悬臂梁模型，对其横向振动动力学建模、振动控制方法和试验等进行了研究；另一种为轴向运动两端自由梁模型，分析了其横向振动的动力学建模和稳定性，并研究了热效应对梁的动力学特性的影响以及金属热防护系统(MTPS)动力学建模等问题。论文的具体内容如下：
（1）分析了端部集中质量对轴向运动悬臂梁振动模态的影响。运用D'Alembert原理建立了轴向运动悬臂梁的横向振动的动力学模型，导出了基于伽辽金法的瞬时线性化近似求解方程组。通过数值计算和试验研究了两种满足不同正交性条件的悬臂梁的振型对系统响应求解结果的影响。结果表明，在使用伽辽金法对自由端带有集中质量的轴向运动悬臂梁响应求解的过程中，当端部集中质量较小时，使用均匀悬臂梁的振型函数代替自由端含有集中质量悬臂梁的振型函数进行求解，造成的误差较小。
（2）对轴向运动悬臂梁横向振动的力控制方法和子系统吸振方法进行了研究。采用Hamilton原理得到了跨内含有主动控制力或端部带主动振子的轴向运动悬臂梁的振动方程和边界条件，并且运用修正的伽辽金法得到了求解系统响应的近似方程。运用线性二次型调节器（LQR）方法设计了主动振子和主动力控制器，采用加权系数法选择Q，R矩阵。用数值计算仿真了两种控制方案的效果。结果表明，采用主动振子或主动力都能够有效地控制轴向运动悬臂梁的横向振动。在相同的初始条件下，主动力控制的效果比主动振子要好，但主动振子的物理可实现性要优于主动力。
（3）设计制造了轴向运动悬臂梁试验平台，通过该平台研究了轴向运动悬臂梁的阻尼与边界条件等效建模方法。利用模态叠加法得到了轴向运动梁的横向振动方程，通过实测梁在不同长度下的第一阶固有频率，调整理论计算模型中的悬臂长度来达到对悬臂边界条件的修正。研究发现，梁的长度修正量与梁的实际悬臂长度无明确相关性。使用对数衰减率方法识别出多个长度下梁的第一阶衰减系数，通过数值拟合，建立了衰减系数与梁长度的关系。结果表明，修正后模型的横向振动响应计算结果与实测结果吻合较好，验证了模型修正的有效性。
（4）设计制造了非接触激振装置，并分别运用速度反馈控制方法、线性二次型高斯模型（LQG）控制理论和H∞控制理论设计了三种控制器。在试验平台上对轴向运动悬臂梁横向振动控制进行了较深入研究。结果表明，当施加控制后，梁的横向振动可更快速衰减，验证了控制方法的有效性，且试验数据与数值仿真结果吻合也较好。
（5）以高超声飞行导弹为背景，研究轴向运动两端自由梁的横向振动特性。运用Hamilton原理建立了梁截面连续变化，任意位置带有附加质量的轴向运动两端自由梁横向振动动力学模型，并且运用修正的伽辽金法得到了求解系统响应的近似方程。分别研究了轴向运动效应，质量耗损和温度效应对梁的动力学特性影响。通过Lyapunov稳定性准则得到系统的指数一致稳定判据，详细研究了集中质量的放置位置和截面形状对梁稳定性的影响。结果表明，具有正值的单位长度质量密度斜率的系统更加稳定，合适放置集中质量可以增加结构的阻尼，并且轴向运动和热效应会降低结构的刚度使得结构固有频率下降。
（6）研究了轴向运动两端自由梁在热冲击作用下的动力学响应及控制。采用Hamilton原理建立了受热冲击和舵面控制力作用的轴向运动两端自由梁的方程，运用伽辽金法对系统响应进行近似求解。运用LQR法设计了最优控制器，采用加权系数法选择Q，R矩阵。采用数值方法研究了轴向速度为2马赫时，模型在受到热冲击下的动力学响应及控制。
（7）研究了外部带金属热防护系统的梁式结构的动力学建模，以及热模态计算的简化方法。给出了蜂窝结构面外的等效热传导系数、比热容、密度和弹性系数的表达式，建立了包括外层蜂窝、中间防热层、内层蜂窝、应变隔离层和承力结构的完整模型。研究了外表面受高温加热情况下，结构的热力学响应及弯曲模态。提出了带金属热防护系统的结构弯曲热模态计算的简化方法，实现了带金属热防护系统的结构弯曲热模态的快速计算，并且验证了温度对结构模态有较大影响。
Axially moving beams are very important mechanism. Mechanisms, such as rigid in-flightrefuelling pipe, bridge of the bridge laying truck, rapid raising military radar antenna, flying rocketand missile, can all be modeled as axially moving beams. The studies on the dynamics and vibrationcontrol of axially moving beams are very valuable.
Two dynamic configurations of axially moving beams are investigated in the article. One is anaxially moving cantilever beam, where the dynamic modeling and vibration control of the transverseoscillation of the beam are studied. The other is an axially moving free-free beam, where the dynamicmodeling and the stability of the beam are analyzed, the thermal effect on the dynamics of the beamand the dynamics of the metallic thermal protection system (MTPS) are studied as well. The mainpoints of the concrete content are as follows.
(1) The influence of the end mass on the modes of the axially moving cantilever beam isinvestigated. Firstly, the transverse vibration equation of an axially moving cantilever beam with tipmass is formulated by the D’Alembert principle. The instant linearized equations are set up based on伽辽金法’s method subsequently. At last, two modal shapes that satisfy different orthogonallyconditions are studied in the vbrational calculation of the axially moving cantilever beam. It is foundthat the modal shapes of the uniform beam can substitute the ones of the beam with light lumpedmass in the伽辽金法’method.
(2) Based on the theoretical analysis, the control of the transverse vibration of an axially movingcantilever beam is studied, and the control configurations for the active vibrator and the active forceare presented. Firstly, the transverse vibration equation and boundary conditions are formulated by theHamilton principle. Then, the instant linearized equations are set up based on伽辽金法’s method forthe approximation solution. The controllers for the active vibrator and the active force are designedbased on linear quadratic regulator (LQR) method subsequently, where the optimal Q and R matricesare selected by the weighting coefficient method. At last, the two control methods are simulated bynumerical examples. The results show that the two control methods can all suppress the transversevibration effectively. The control result of the active force method is better than that of the activevibrator method, but the active vibrator is more realizable than the active force in practice.
(3) An experimental platform is designed and presented for the axially moving cantilever beam,and the damping and boundary condition of the axially moving beam are updated by experiments.The beam which slides through a prismatic joint is considered as an axially moving cantilever beam, the transverse vibration equation is given, and the discretized equations of motion of the translatingbeam with time-dependent coefficients is derived by mode superposition method. Firstly, the fixedboundary condition is modified by the adjustment of the cantilever length of the theoretical beammodel with the measured results of its first order natural frequencies under various cantilever lengths.It is found that the correction lengths to the model beam have no explicit relationship with thecantilever lengths. Secondly, the first order decay coefficients are identified by LogarithmicDecrement method and it is find that the decay coefficients of the beam decrease with the increase ofthe cantilever length. At last, the calculated responses by the modified model are fit well with theexperimental results. It verifies the effectiveness of the proposed model modification method.
(4) A noncontact exciting equipment is designed and put forward. Based on the experimentalplatform and the exciting equipment, vibration control of the axially moving cantilever beam isstudied, where the velocity feedback control method, the linear quadratic Gaussian (LQG) controlmethod and the H∞control method are studied, respectively. According to the control methods,numerical simulation and actual experiments are implemented for the beam with constant length andvarying length, respectively. And the results show that the control methods are effective and thesimulations fit well with the experiments.
(5) Frequency characteristics and stability of the axially moving free-free beam with high velocitywhich simulates a filying missile is investigated. Firstly, the transverse vibration equation of theaxially moving free-free beam is derived by Hamilton’s principle, where the lumped mass attached atarbitrary position of the beam is taken into account, and the cross section is consecutive variation. Theinstant linearized equations are set up based on伽辽金法’s method for the approximation solutionsubsequently. Secondly, the influence of the axially movment, the changing of the mass density, andthe temperature on the natural frequencies of the beam are investigated, respectively. At last, thesufficient conditions for uniform stability and uniform exponential stability of the beam areestablished via Lyapunov stability criteria. It is found that the damping can be increased if the lumpedmass is placed suitably, the slope of the mass per unit length of the beam influences system stabilitysignificantly, and the axially movment and temperature can decrease the stiffness of the beam whichinduces the vibration frequencies descending.
(6) The response and control of the transverse vibration of an axially moving free-free beam withhigh velocity subjected to the thermal shock is studied. Considered the thermal shock and the controlforce, the transverse vibration equation and boundary conditions are formulated by the Hamiltonprinciple. The displacement responses are subsequently simulated, where the axially velocity is2Ma.Then the controller is designed based on LQR method, and the displacement response for the controlmethod is simulated by a numerical example. The result shows that the control method can suppress the transverse vibration effectively.
(7) Dynamic modeling of the beam with MTPS is investigated, and the simplified method forcalculation of thermal modes is studied. Firstly, the effective thermal conductivities, specific heatcapacity, density and elastic coefficient of metallic honeycombs are presented. Then a completemodel including outer honeycombs, middle thermal protection layer, inner honeycombs, isolation andload-carrying structure are modeling, and the thermodynamical response and the bending modes aresubsequently studied. At last, the simplified model for bending modes calculation is investigated, andthe results show that it is effective in the fast calculation of the modes of the structure with MTPS.Also it’s verified that the influence of the high temperature on structure’s modes is notable.
本文针对两种轴向运动梁动力学模型进行了较深入研究。一种为轴向运动悬臂梁模型，对其横向振动动力学建模、振动控制方法和试验等进行了研究；另一种为轴向运动两端自由梁模型，分析了其横向振动的动力学建模和稳定性，并研究了热效应对梁的动力学特性的影响以及金属热防护系统(MTPS)动力学建模等问题。论文的具体内容如下：
（1）分析了端部集中质量对轴向运动悬臂梁振动模态的影响。运用D'Alembert原理建立了轴向运动悬臂梁的横向振动的动力学模型，导出了基于伽辽金法的瞬时线性化近似求解方程组。通过数值计算和试验研究了两种满足不同正交性条件的悬臂梁的振型对系统响应求解结果的影响。结果表明，在使用伽辽金法对自由端带有集中质量的轴向运动悬臂梁响应求解的过程中，当端部集中质量较小时，使用均匀悬臂梁的振型函数代替自由端含有集中质量悬臂梁的振型函数进行求解，造成的误差较小。
（2）对轴向运动悬臂梁横向振动的力控制方法和子系统吸振方法进行了研究。采用Hamilton原理得到了跨内含有主动控制力或端部带主动振子的轴向运动悬臂梁的振动方程和边界条件，并且运用修正的伽辽金法得到了求解系统响应的近似方程。运用线性二次型调节器（LQR）方法设计了主动振子和主动力控制器，采用加权系数法选择Q，R矩阵。用数值计算仿真了两种控制方案的效果。结果表明，采用主动振子或主动力都能够有效地控制轴向运动悬臂梁的横向振动。在相同的初始条件下，主动力控制的效果比主动振子要好，但主动振子的物理可实现性要优于主动力。
（3）设计制造了轴向运动悬臂梁试验平台，通过该平台研究了轴向运动悬臂梁的阻尼与边界条件等效建模方法。利用模态叠加法得到了轴向运动梁的横向振动方程，通过实测梁在不同长度下的第一阶固有频率，调整理论计算模型中的悬臂长度来达到对悬臂边界条件的修正。研究发现，梁的长度修正量与梁的实际悬臂长度无明确相关性。使用对数衰减率方法识别出多个长度下梁的第一阶衰减系数，通过数值拟合，建立了衰减系数与梁长度的关系。结果表明，修正后模型的横向振动响应计算结果与实测结果吻合较好，验证了模型修正的有效性。
（4）设计制造了非接触激振装置，并分别运用速度反馈控制方法、线性二次型高斯模型（LQG）控制理论和H∞控制理论设计了三种控制器。在试验平台上对轴向运动悬臂梁横向振动控制进行了较深入研究。结果表明，当施加控制后，梁的横向振动可更快速衰减，验证了控制方法的有效性，且试验数据与数值仿真结果吻合也较好。
（5）以高超声飞行导弹为背景，研究轴向运动两端自由梁的横向振动特性。运用Hamilton原理建立了梁截面连续变化，任意位置带有附加质量的轴向运动两端自由梁横向振动动力学模型，并且运用修正的伽辽金法得到了求解系统响应的近似方程。分别研究了轴向运动效应，质量耗损和温度效应对梁的动力学特性影响。通过Lyapunov稳定性准则得到系统的指数一致稳定判据，详细研究了集中质量的放置位置和截面形状对梁稳定性的影响。结果表明，具有正值的单位长度质量密度斜率的系统更加稳定，合适放置集中质量可以增加结构的阻尼，并且轴向运动和热效应会降低结构的刚度使得结构固有频率下降。
（6）研究了轴向运动两端自由梁在热冲击作用下的动力学响应及控制。采用Hamilton原理建立了受热冲击和舵面控制力作用的轴向运动两端自由梁的方程，运用伽辽金法对系统响应进行近似求解。运用LQR法设计了最优控制器，采用加权系数法选择Q，R矩阵。采用数值方法研究了轴向速度为2马赫时，模型在受到热冲击下的动力学响应及控制。
（7）研究了外部带金属热防护系统的梁式结构的动力学建模，以及热模态计算的简化方法。给出了蜂窝结构面外的等效热传导系数、比热容、密度和弹性系数的表达式，建立了包括外层蜂窝、中间防热层、内层蜂窝、应变隔离层和承力结构的完整模型。研究了外表面受高温加热情况下，结构的热力学响应及弯曲模态。提出了带金属热防护系统的结构弯曲热模态计算的简化方法，实现了带金属热防护系统的结构弯曲热模态的快速计算，并且验证了温度对结构模态有较大影响。
Axially moving beams are very important mechanism. Mechanisms, such as rigid in-flightrefuelling pipe, bridge of the bridge laying truck, rapid raising military radar antenna, flying rocketand missile, can all be modeled as axially moving beams. The studies on the dynamics and vibrationcontrol of axially moving beams are very valuable.
Two dynamic configurations of axially moving beams are investigated in the article. One is anaxially moving cantilever beam, where the dynamic modeling and vibration control of the transverseoscillation of the beam are studied. The other is an axially moving free-free beam, where the dynamicmodeling and the stability of the beam are analyzed, the thermal effect on the dynamics of the beamand the dynamics of the metallic thermal protection system (MTPS) are studied as well. The mainpoints of the concrete content are as follows.
(1) The influence of the end mass on the modes of the axially moving cantilever beam isinvestigated. Firstly, the transverse vibration equation of an axially moving cantilever beam with tipmass is formulated by the D’Alembert principle. The instant linearized equations are set up based on伽辽金法’s method subsequently. At last, two modal shapes that satisfy different orthogonallyconditions are studied in the vbrational calculation of the axially moving cantilever beam. It is foundthat the modal shapes of the uniform beam can substitute the ones of the beam with light lumpedmass in the伽辽金法’method.
(2) Based on the theoretical analysis, the control of the transverse vibration of an axially movingcantilever beam is studied, and the control configurations for the active vibrator and the active forceare presented. Firstly, the transverse vibration equation and boundary conditions are formulated by theHamilton principle. Then, the instant linearized equations are set up based on伽辽金法’s method forthe approximation solution. The controllers for the active vibrator and the active force are designedbased on linear quadratic regulator (LQR) method subsequently, where the optimal Q and R matricesare selected by the weighting coefficient method. At last, the two control methods are simulated bynumerical examples. The results show that the two control methods can all suppress the transversevibration effectively. The control result of the active force method is better than that of the activevibrator method, but the active vibrator is more realizable than the active force in practice.
(3) An experimental platform is designed and presented for the axially moving cantilever beam,and the damping and boundary condition of the axially moving beam are updated by experiments.The beam which slides through a prismatic joint is considered as an axially moving cantilever beam, the transverse vibration equation is given, and the discretized equations of motion of the translatingbeam with time-dependent coefficients is derived by mode superposition method. Firstly, the fixedboundary condition is modified by the adjustment of the cantilever length of the theoretical beammodel with the measured results of its first order natural frequencies under various cantilever lengths.It is found that the correction lengths to the model beam have no explicit relationship with thecantilever lengths. Secondly, the first order decay coefficients are identified by LogarithmicDecrement method and it is find that the decay coefficients of the beam decrease with the increase ofthe cantilever length. At last, the calculated responses by the modified model are fit well with theexperimental results. It verifies the effectiveness of the proposed model modification method.
(4) A noncontact exciting equipment is designed and put forward. Based on the experimentalplatform and the exciting equipment, vibration control of the axially moving cantilever beam isstudied, where the velocity feedback control method, the linear quadratic Gaussian (LQG) controlmethod and the H∞control method are studied, respectively. According to the control methods,numerical simulation and actual experiments are implemented for the beam with constant length andvarying length, respectively. And the results show that the control methods are effective and thesimulations fit well with the experiments.
(5) Frequency characteristics and stability of the axially moving free-free beam with high velocitywhich simulates a filying missile is investigated. Firstly, the transverse vibration equation of theaxially moving free-free beam is derived by Hamilton’s principle, where the lumped mass attached atarbitrary position of the beam is taken into account, and the cross section is consecutive variation. Theinstant linearized equations are set up based on伽辽金法’s method for the approximation solutionsubsequently. Secondly, the influence of the axially movment, the changing of the mass density, andthe temperature on the natural frequencies of the beam are investigated, respectively. At last, thesufficient conditions for uniform stability and uniform exponential stability of the beam areestablished via Lyapunov stability criteria. It is found that the damping can be increased if the lumpedmass is placed suitably, the slope of the mass per unit length of the beam influences system stabilitysignificantly, and the axially movment and temperature can decrease the stiffness of the beam whichinduces the vibration frequencies descending.
(6) The response and control of the transverse vibration of an axially moving free-free beam withhigh velocity subjected to the thermal shock is studied. Considered the thermal shock and the controlforce, the transverse vibration equation and boundary conditions are formulated by the Hamiltonprinciple. The displacement responses are subsequently simulated, where the axially velocity is2Ma.Then the controller is designed based on LQR method, and the displacement response for the controlmethod is simulated by a numerical example. The result shows that the control method can suppress the transverse vibration effectively.
(7) Dynamic modeling of the beam with MTPS is investigated, and the simplified method forcalculation of thermal modes is studied. Firstly, the effective thermal conductivities, specific heatcapacity, density and elastic coefficient of metallic honeycombs are presented. Then a completemodel including outer honeycombs, middle thermal protection layer, inner honeycombs, isolation andload-carrying structure are modeling, and the thermodynamical response and the bending modes aresubsequently studied. At last, the simplified model for bending modes calculation is investigated, andthe results show that it is effective in the fast calculation of the modes of the structure with MTPS.Also it’s verified that the influence of the high temperature on structure’s modes is notable.
引文
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