热机电耦合压电智能薄板结构不确定性及其H_∞振动控制研究
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摘要
本文以‘热机电耦合压电智能薄板结构'作为研究对象,对其不确定性和鲁棒H_∞振动控制进行了研究,主要工作包含如下内容:
1、以每层材料各不相同的层叠板为研究对象,在热传导、对流以及热辐射边界条件共同作用下,采用4节点矩形单元,利用伽辽金法建立了该结构瞬态温度场的有限元微分方程,进而推导出了温度场对任意设计参数的灵敏度表达式,发现了灵敏度表达式中存在的公共项具有惯性效应。
2、对于线性瞬态温度场,当所有结构参数均具有随机性时,为了避开矩阵指数的随机性分析难题,通过方程解耦、特征对随机性分析以及一次二阶矩法,给出了瞬态随机温度场数字特征的求解过程。以三层复合板结构为例进行了求解,并将所得结果与Monte Carlo数值模拟法的结果进行比较,证明了求解过程的实用性。
3、对于非线性瞬态温度场,当所有结构参数均具有随机性时,为了节省随机温度场Monte Carlo数值模拟的时间,在引入随机因子后,提出了一类关于热传导矩阵和热容矩阵的近似处理方法,并给出了误差的定量计算方法,即通过灵敏度和一阶泰勒展开线性化的方法去近似求解由于近似处理所造成的温度场数字特征的误差。算例表明大量的Monte Carlo数值模拟时间得以节省,且所得的近似误差具有一定的精度。
4、将3中获得的思路运用到了平面桁架结构的随机性分析中,对结构的质量矩阵和刚度矩阵的随机性提出了两种近似处理方法,通过对误差的定性分析表明第一种方法能获得结构动力特性均方根上限,第二种方法能获得结构动力特性随机性的近似解,这两种方法都能显著节俭Monte Carlo数值模拟法的计算量。算例表明所给方法不但节省了数值模拟的时间,且具有较好的精度。
5、对于3和4中提及的随机因子,这里首次从数学证明的角度对其进行了论述:将随机物理参数结构系统的质量矩阵和刚度矩阵均以随机因子的齐次形式表示,通过对QR变换法求解随机矩阵特征值过程的论述,证明了系统中各阶固有频率具有相同的随机因子,并导出了求解固有频率随机变量概率密度的公式;在此基础上,论述了系统正则振型矩阵中各元素亦具有相同的随机因子,并导出了求解正则振型随机因子概率密度的公式。算例表明了所得结论的正确性。
6、对于非线性瞬态温度场,为了避开对复杂分布进行假设检验时面临的统计量寻找和分位数确定等经典难题,通过综合运用差分法、Monte Carlo数值模拟法、Q-Q图法、Box-Cox幂变换法,正态假设检验以及参数区间估计,提出了一整套近似求解温度响应主体区间的方法和表达式,并通过算例表明所提出的分析和求解策略的有效性与合理性。
7、针对热机电耦合压电智能薄板结构,提出了一种新型的包含4个位移节点、2个电势节点和8个温度节点的混合有限元模型,并基于虚功原理推倒了热机电耦合有限元微分方程,通过与已有文献的比较,表明所得有限元方程的正确性;同时也给出了这类有限元模型具体的编程实现方法。以智能悬臂薄板为例,对位移响应、温度响应和输出电压进行了数值仿真,凸现了在小挠度变形下,变化幅值较大的温度对输出电压具有重大影响。
8、对于7中所得有限元微分方程,当所有参数皆具有随机性时,通过各响应量对各个随机参数灵敏度的解析求解和一次二阶矩法,给出了依次求解结构固有频率、固有振型、温变、位移以及输出电压等随机响应数字特征的方法和步骤。并通过算例表明处理方法可行且有较好精度,但求解过程十分费时费力。
9、针对8中存在的求解困难问题,研究了综合应用差分法和基于灵敏度求解的矩法对结构位移和温度响应的数字特征进行了求解:算例表明改进后的方法运算快捷且具有一定精度。
10、为了实现对小区间参数不确定性智能薄板结构的鲁棒H_∞振动控制,这里先研究了‘如何由原小区间参数不确定性系统的不确定性求解平衡降阶后系统的不确定性'这一难题。针对平衡降阶过程,提出了一种近似求解方法,即:通过正交变换获得实对称矩阵的特征对,在求解矩阵指数灵敏度和实对称矩阵特征对灵敏度的基础上,获得了平衡变换矩阵的灵敏度,在探讨了排序矩阵的选取后,进而利用一阶泰勒展开式和区间运算法则求得了降阶后的不确定性系统。在对原不确定性系统和降阶后的不确定性系统分别施加相同的输入后,通过二者输出区间的比较对所提方法的正确性进行了验证。
11、针对10中获得的不确定性系统,提出了一类维数较低的不确定性矩阵的分解形式:利用H_∞控制理论对降阶后的增广被控对象进行了鲁棒控制器求解,并给出了该控制器对原不确定性系统的鲁棒性验证过程。算例表明10中的研究成果起到了连接降阶与鲁棒控制的桥梁作用。
12、为了实现对小区间参数不确定性智能薄板结构的鲁棒H_∞振动控制,在完成10和11针对一般系统的研究内容后,还需研究确定性智能薄板结构的振动控制问题。为此,这里利用平衡降阶法对其进行了降阶处理;论述了增广被控对象的构建以及利用H_∞控制理论对振动进行抑制的过程。并以悬臂薄板为例,其控制效果表明所给处理过程的合理性,也反映出该类结构在同时进行弯曲抑制和拉伸抑制时,压电致动层所具有的控制作用与干扰作用的双面性。
13、通过综合运用10～12中的研究成果,实现了对小区间参数不确定性智能薄板结构的鲁棒H_∞振动控制。
The paper studies on robust control of uncertain piezothermoelasticity intelligent laminated Plates by using H_∞control law, the main works finished in this paper include the following:
1. For composite plates with materials of each layer different, considering the heat conduction, convection and radiation boundary conditions, a 4-node rectangle element model is adopted and its temperature differential equations are deduced by means of Galerkin method, at the same time, the general formulation about the sensitivity of the temperature with respect to each design parameter is deduced, and the common part in the general formulation having an obvious inertial effect is founded in computer simulation.
2. For linear transient temperature field, when all parameters are random, on the basis of decoupling method, randomicity analysis of eigenvalue and eigenvector, and second-moment method, the numeric characteristics of the temperature field are obtained. A 3-layer plates was taken as an example, the numerical results are compared with those obtained by Monte Carlo method, by which the practicability of the presented method is verified.
3. For nonlinear transient temperature field, when all parameters are random, in order to decrease the operating time of Monte Carlo method, this paper introduce random factor and presents an approximate treatment of heat conduction matrix and heat capacity matrix, at the same time, the errors of temperature's numerical characteristics caused by that approximate treatment method are computed approximately by computing sensitivity and the linearization technique of first order Taylor series expansion. Computer simulation shows the time is saved largely, and the errors computed approximately are precision in a certain extent.
4. The method presented in 3 is used in 2-D truss structure, two methods to deal with approximatively the randomicities of the mass matrix and stiffness matrix are presented, the first method can obtain an upper limit of standard deviation of dynamic characteristic, the second method can obtain an approximation of it, both of them can greatly decrease the works of Monte Carlo numerical simulation. Finally, the frugality and precision of the two methods proposed in this paper are proved by computer simulation.
5. For random factor mentioned in 3 and 4, although it has been used for many years, it has not been proven form the angle of mathematics, here, it is proven for the first time, namely, after random variables are described by random factor method, when all random factors can be extracted from mass matrix and stiffness matrix, through analyzing the QR method process of solving random matrix's eigenvalue, the natural frequencies' random factors are proved equally, based on it, all elements of the normal model shape matrix having the same random factor are proven too. In computer simulation, all conclusions are proven by Monte Carlo method.
6. For nonlinear transient temperature field, when all parameters are random, this paper presents a method to obtain the approximate solution of the temperature's main interval by using synthetically the following techniques: difference method, Monte Carlo method, Quantile-quantile plot, Box-Cox transformation, hypothesis testing and parameter interval estimation. In computer simulation, the temperature's main interval obtained is compared with lots of temperature's random samples, which shows that the method presented is reasonable.
7. For piezothermoelasticity intelligent thin plate, a cube finite element model including 4 displacement nodes, 2 electric potential nodes and 8 temperature nodes is presented, the displacement field is defined by means of plane shell element model, its electric potential field and temperature field are all defined by means of linear interpolation; the finite element equations are deduced by using virtual work principle, its correctness is proved by compared with other papers; finally, an intelligent cantilever plate is taken as an example , its responses of temperature field, displacement field and output voltages are simulated, which show that the temperature having big change amplitude value has large affection on output voltages.
8. For element finite differential equations in 7, when all parameters are random, through the analytic solutions of sensitivities with respect to random parameters and first order second moment method, the numerical characteristics of natural frequencies, mode shapes, temperature field, displacement field and output voltages are solved in turn; finally , an intelligent cantilever plate is taken as an example , the numerical results are compared with those of Monte Carlo Method, the results show that the computing process presented has good precision.
9. For the problem of computing speed slowness existed in 8, the response's numerical characteristics are computed by using synthetically deference method and moment method, the computer simulation shows that the computing process presented is practicability and has great efficiency.
10. In order to realize the robust control of uncertain piezothermoelasticity intelligent laminated plates with small interval parameters, it is studied firstly how to derive the reduced system's uncertainty from the original uncertain system with small interval parameters. On the basis of sensitivities of matrix exponential and the sensitivities of real symmetric matrixes' eigenvalue and eigenvector in orthogonal transform, the sensitivities of state transformation matrix are derived, then the reduced system's uncertainty are obtained by the first order Taylor series expansion and interval mathematics, in computer simulation, a same input is applied respectively on the original system with parametric uncertainty and the reduced system with obtained uncertainty, their output's intervals are almost the same shows the method presented is corrected.
11. For the reduced uncertain system obtained in 10, a low dimension method is presented to decompose uncertain matrix; the robust controller of the reduced generalized plants is solved by using H_∞control theory, and a process is given to verify the robustness of the controller on original uncertain system. A simple example's computer simulation shows the giving method is feasible.
12. In order to realize the robust control of uncertain piezothermoelasticity intelligent laminated plate with small interval parameters, except the works in 10 and 11, it is also need to be studied beforehand that the vibration control of the certain piezothermoelasticity intelligent laminated plate. Here, without treating of thermal radiation, the linear system is reduced by using Balancing reduction; it is discussed how to derive the generalized plants and using H_∞control theory to restrain vibration; an intelligent cantilever plate is taken as an example , its control effect shows that the method presented is valid; simultaneity, the example shows for this kind of structure, when both the bending displacement and tensile deformation are wanted to be controlled, the actuator has the two faces of control action and interference action in control process.
13. By using synthetically the wors presented in 10-12, the robust H_∞control of uncertain piezothermoelasticity intelligent laminated plate with small interval parameters is realized.
1、以每层材料各不相同的层叠板为研究对象,在热传导、对流以及热辐射边界条件共同作用下,采用4节点矩形单元,利用伽辽金法建立了该结构瞬态温度场的有限元微分方程,进而推导出了温度场对任意设计参数的灵敏度表达式,发现了灵敏度表达式中存在的公共项具有惯性效应。
2、对于线性瞬态温度场,当所有结构参数均具有随机性时,为了避开矩阵指数的随机性分析难题,通过方程解耦、特征对随机性分析以及一次二阶矩法,给出了瞬态随机温度场数字特征的求解过程。以三层复合板结构为例进行了求解,并将所得结果与Monte Carlo数值模拟法的结果进行比较,证明了求解过程的实用性。
3、对于非线性瞬态温度场,当所有结构参数均具有随机性时,为了节省随机温度场Monte Carlo数值模拟的时间,在引入随机因子后,提出了一类关于热传导矩阵和热容矩阵的近似处理方法,并给出了误差的定量计算方法,即通过灵敏度和一阶泰勒展开线性化的方法去近似求解由于近似处理所造成的温度场数字特征的误差。算例表明大量的Monte Carlo数值模拟时间得以节省,且所得的近似误差具有一定的精度。
4、将3中获得的思路运用到了平面桁架结构的随机性分析中,对结构的质量矩阵和刚度矩阵的随机性提出了两种近似处理方法,通过对误差的定性分析表明第一种方法能获得结构动力特性均方根上限,第二种方法能获得结构动力特性随机性的近似解,这两种方法都能显著节俭Monte Carlo数值模拟法的计算量。算例表明所给方法不但节省了数值模拟的时间,且具有较好的精度。
5、对于3和4中提及的随机因子,这里首次从数学证明的角度对其进行了论述:将随机物理参数结构系统的质量矩阵和刚度矩阵均以随机因子的齐次形式表示,通过对QR变换法求解随机矩阵特征值过程的论述,证明了系统中各阶固有频率具有相同的随机因子,并导出了求解固有频率随机变量概率密度的公式;在此基础上,论述了系统正则振型矩阵中各元素亦具有相同的随机因子,并导出了求解正则振型随机因子概率密度的公式。算例表明了所得结论的正确性。
6、对于非线性瞬态温度场,为了避开对复杂分布进行假设检验时面临的统计量寻找和分位数确定等经典难题,通过综合运用差分法、Monte Carlo数值模拟法、Q-Q图法、Box-Cox幂变换法,正态假设检验以及参数区间估计,提出了一整套近似求解温度响应主体区间的方法和表达式,并通过算例表明所提出的分析和求解策略的有效性与合理性。
7、针对热机电耦合压电智能薄板结构,提出了一种新型的包含4个位移节点、2个电势节点和8个温度节点的混合有限元模型,并基于虚功原理推倒了热机电耦合有限元微分方程,通过与已有文献的比较,表明所得有限元方程的正确性;同时也给出了这类有限元模型具体的编程实现方法。以智能悬臂薄板为例,对位移响应、温度响应和输出电压进行了数值仿真,凸现了在小挠度变形下,变化幅值较大的温度对输出电压具有重大影响。
8、对于7中所得有限元微分方程,当所有参数皆具有随机性时,通过各响应量对各个随机参数灵敏度的解析求解和一次二阶矩法,给出了依次求解结构固有频率、固有振型、温变、位移以及输出电压等随机响应数字特征的方法和步骤。并通过算例表明处理方法可行且有较好精度,但求解过程十分费时费力。
9、针对8中存在的求解困难问题,研究了综合应用差分法和基于灵敏度求解的矩法对结构位移和温度响应的数字特征进行了求解:算例表明改进后的方法运算快捷且具有一定精度。
10、为了实现对小区间参数不确定性智能薄板结构的鲁棒H_∞振动控制,这里先研究了‘如何由原小区间参数不确定性系统的不确定性求解平衡降阶后系统的不确定性'这一难题。针对平衡降阶过程,提出了一种近似求解方法,即:通过正交变换获得实对称矩阵的特征对,在求解矩阵指数灵敏度和实对称矩阵特征对灵敏度的基础上,获得了平衡变换矩阵的灵敏度,在探讨了排序矩阵的选取后,进而利用一阶泰勒展开式和区间运算法则求得了降阶后的不确定性系统。在对原不确定性系统和降阶后的不确定性系统分别施加相同的输入后,通过二者输出区间的比较对所提方法的正确性进行了验证。
11、针对10中获得的不确定性系统,提出了一类维数较低的不确定性矩阵的分解形式:利用H_∞控制理论对降阶后的增广被控对象进行了鲁棒控制器求解,并给出了该控制器对原不确定性系统的鲁棒性验证过程。算例表明10中的研究成果起到了连接降阶与鲁棒控制的桥梁作用。
12、为了实现对小区间参数不确定性智能薄板结构的鲁棒H_∞振动控制,在完成10和11针对一般系统的研究内容后,还需研究确定性智能薄板结构的振动控制问题。为此,这里利用平衡降阶法对其进行了降阶处理;论述了增广被控对象的构建以及利用H_∞控制理论对振动进行抑制的过程。并以悬臂薄板为例,其控制效果表明所给处理过程的合理性,也反映出该类结构在同时进行弯曲抑制和拉伸抑制时,压电致动层所具有的控制作用与干扰作用的双面性。
13、通过综合运用10～12中的研究成果,实现了对小区间参数不确定性智能薄板结构的鲁棒H_∞振动控制。
The paper studies on robust control of uncertain piezothermoelasticity intelligent laminated Plates by using H_∞control law, the main works finished in this paper include the following:
1. For composite plates with materials of each layer different, considering the heat conduction, convection and radiation boundary conditions, a 4-node rectangle element model is adopted and its temperature differential equations are deduced by means of Galerkin method, at the same time, the general formulation about the sensitivity of the temperature with respect to each design parameter is deduced, and the common part in the general formulation having an obvious inertial effect is founded in computer simulation.
2. For linear transient temperature field, when all parameters are random, on the basis of decoupling method, randomicity analysis of eigenvalue and eigenvector, and second-moment method, the numeric characteristics of the temperature field are obtained. A 3-layer plates was taken as an example, the numerical results are compared with those obtained by Monte Carlo method, by which the practicability of the presented method is verified.
3. For nonlinear transient temperature field, when all parameters are random, in order to decrease the operating time of Monte Carlo method, this paper introduce random factor and presents an approximate treatment of heat conduction matrix and heat capacity matrix, at the same time, the errors of temperature's numerical characteristics caused by that approximate treatment method are computed approximately by computing sensitivity and the linearization technique of first order Taylor series expansion. Computer simulation shows the time is saved largely, and the errors computed approximately are precision in a certain extent.
4. The method presented in 3 is used in 2-D truss structure, two methods to deal with approximatively the randomicities of the mass matrix and stiffness matrix are presented, the first method can obtain an upper limit of standard deviation of dynamic characteristic, the second method can obtain an approximation of it, both of them can greatly decrease the works of Monte Carlo numerical simulation. Finally, the frugality and precision of the two methods proposed in this paper are proved by computer simulation.
5. For random factor mentioned in 3 and 4, although it has been used for many years, it has not been proven form the angle of mathematics, here, it is proven for the first time, namely, after random variables are described by random factor method, when all random factors can be extracted from mass matrix and stiffness matrix, through analyzing the QR method process of solving random matrix's eigenvalue, the natural frequencies' random factors are proved equally, based on it, all elements of the normal model shape matrix having the same random factor are proven too. In computer simulation, all conclusions are proven by Monte Carlo method.
6. For nonlinear transient temperature field, when all parameters are random, this paper presents a method to obtain the approximate solution of the temperature's main interval by using synthetically the following techniques: difference method, Monte Carlo method, Quantile-quantile plot, Box-Cox transformation, hypothesis testing and parameter interval estimation. In computer simulation, the temperature's main interval obtained is compared with lots of temperature's random samples, which shows that the method presented is reasonable.
7. For piezothermoelasticity intelligent thin plate, a cube finite element model including 4 displacement nodes, 2 electric potential nodes and 8 temperature nodes is presented, the displacement field is defined by means of plane shell element model, its electric potential field and temperature field are all defined by means of linear interpolation; the finite element equations are deduced by using virtual work principle, its correctness is proved by compared with other papers; finally, an intelligent cantilever plate is taken as an example , its responses of temperature field, displacement field and output voltages are simulated, which show that the temperature having big change amplitude value has large affection on output voltages.
8. For element finite differential equations in 7, when all parameters are random, through the analytic solutions of sensitivities with respect to random parameters and first order second moment method, the numerical characteristics of natural frequencies, mode shapes, temperature field, displacement field and output voltages are solved in turn; finally , an intelligent cantilever plate is taken as an example , the numerical results are compared with those of Monte Carlo Method, the results show that the computing process presented has good precision.
9. For the problem of computing speed slowness existed in 8, the response's numerical characteristics are computed by using synthetically deference method and moment method, the computer simulation shows that the computing process presented is practicability and has great efficiency.
10. In order to realize the robust control of uncertain piezothermoelasticity intelligent laminated plates with small interval parameters, it is studied firstly how to derive the reduced system's uncertainty from the original uncertain system with small interval parameters. On the basis of sensitivities of matrix exponential and the sensitivities of real symmetric matrixes' eigenvalue and eigenvector in orthogonal transform, the sensitivities of state transformation matrix are derived, then the reduced system's uncertainty are obtained by the first order Taylor series expansion and interval mathematics, in computer simulation, a same input is applied respectively on the original system with parametric uncertainty and the reduced system with obtained uncertainty, their output's intervals are almost the same shows the method presented is corrected.
11. For the reduced uncertain system obtained in 10, a low dimension method is presented to decompose uncertain matrix; the robust controller of the reduced generalized plants is solved by using H_∞control theory, and a process is given to verify the robustness of the controller on original uncertain system. A simple example's computer simulation shows the giving method is feasible.
12. In order to realize the robust control of uncertain piezothermoelasticity intelligent laminated plate with small interval parameters, except the works in 10 and 11, it is also need to be studied beforehand that the vibration control of the certain piezothermoelasticity intelligent laminated plate. Here, without treating of thermal radiation, the linear system is reduced by using Balancing reduction; it is discussed how to derive the generalized plants and using H_∞control theory to restrain vibration; an intelligent cantilever plate is taken as an example , its control effect shows that the method presented is valid; simultaneity, the example shows for this kind of structure, when both the bending displacement and tensile deformation are wanted to be controlled, the actuator has the two faces of control action and interference action in control process.
13. By using synthetically the wors presented in 10-12, the robust H_∞control of uncertain piezothermoelasticity intelligent laminated plate with small interval parameters is realized.
引文
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