多参数结构动态二阶灵敏度及重分析研究
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摘要
特征灵敏度是指利用结构的模态参数(特征值和特征向量)与结构参数的关系计算结构参数对振动系统影响的敏感程度。充分掌握结构特征值和特征向量的灵敏度信息对增强系统的稳定性,提高结构动态优化设计效率,缩短设计周期具有重要意义。
自1968年首次提出特征向量灵敏度的计算概念以来,结构特征灵敏度一直是人们热衷研究的领域。但由于特征值与特征向量都是关于设计参数的隐函数,无法计算其导数矩阵,从而不能用直接求导法对多参数结构特征灵敏度问题进行研究。现有的求解方法一般是在简单求导法的基础上结合模态展开法或Nelson法,但这些方法都要涉及一系列的方程运算,其算法步骤繁琐且计算量大。
本文在矩阵摄动理论的基础上提出一种新的计算特征值和特征向量灵敏度的方法——摄动灵敏度法。首先,将结构的系统增量矩阵(刚度矩阵、质量矩阵)作为设计参数的隐函数进行Taylor展开,得到系统增量关于设计参数的函数关系,然后根据特征值与特征向量的一阶、二阶摄动理论,推导出多参数结构特征值和特征向量的一阶、二阶摄动灵敏度和摄动灵敏度矩阵。此外,本文还对复模态特征灵敏度问题进行了研究,给出多参数结构复特征值与右、左特征向量的一阶、二阶摄动灵敏度和摄动灵敏度矩阵。
摄动灵敏度法通过在建立模型的过程中引入设计变量,使分析结果具有较明确的物理意义,提高了理论模态和试验模态的相关程度。它所提供的一阶、二阶摄动灵敏度矩阵是特征值和特征向量关于多参数的一阶、二阶导数矩阵的有效近似,解决了梯度阵和Hessian阵不能用求导法直接计算的问题,为结构重分析提供有力帮助。
本文通过对大型有限元分析软件I-DEAS的二次开发,成功地将该方法应用于此软件平台,解决了多参数结构特征灵敏度的工程计算问题,充分体现了该方法对结构设计的理论指导作用,展示了该方法的有效性与优越性。
Design sensitivity analysis of eigenvalues and eigenvectors will reveal how the changes in some design parameters in the system affect the dynamic characteristics of the structure. The designer can use this information directly in an interactive computer-aided design process as a valuable guide; it will shorten the time and improve work efficiency. It has been used in a variety of engineering disciplines ranging from automatic control theory to analysis of large-scale physiological systems. Some of the areas where sensitivity analysis has been applied include: system identification; development of insensitive control system; the approximation of system response to system parameters change; assessment of design changes on system performance; analysis of structural systems with random parameters. Recent applications have been made in vibration diagnosis on structural fault, the optimization of structural analysis, robust control and inverse problem of eigenvalue. The design sensitivity analysis of eigenvalues and eigenvectors play an essential role.
From 1960, many researchers have made investigations on the sensitivity of eigenpairs. The two major methods for investigate the sensitivity of eigenvector are proposed by Fox and Nelson. In 1968, Fox and Kapoor gave exact expressions for the first-order derivative of the eigenvalues and eigenvectors with respect to any design variables. In 1976, Nelson proposed an effective method to calculate sensitivity of eigenvectors which requires only the eigenvalue and eigenvector under consideration. The method represented eigenvector derivatives by a sum of the homogeneous solution and particular solution. In modal method the total information of eigenvectors is often used, however, for large-scale dynamic analysis it is impractical to obtain all modes. Therefore, some other methods to modify it are inevitable. In this paper, improved modal method, high accurate modal superposition method and Neumann series expansion method are introduced.
However, in many engineering problems such as the system with proportional or nonproportional damping, coupling between structure and control system, the system matrices are not real symmetric and may be complex asymmetric. In this case, the real mode theory is limited for dynamic modification and the complex mode is needed. After introducing the state vector and the state transformation matrix, the vibration equation of the linear systems is doubled. Similarly, the mode method and Nelson method will change their forms. Sondipon Adhikari’s method is consistent with the notion of traditional modal analysis methods. By finding the relationship of complex eigenvector, it avoids using state space approach and reduced the dimension from 2n to n . Najeh Guedria extended Nelson’s method into complex mode. He integrated the sensitivity of eigenvalue and eigenvector for only one parameter in one term and solved them in a series of equations simultaneity.
Generally, it is difficult to express the eigenvalues and eigenvectors as explicit functions of design variables. To carry out the calculations of the gradient and Hessian matrix by using the above methods are impossible. It is desirable to transfer the direct differential approach into the perturbation method.
It is well known that the matrix perturbation theory is concerned with how the eigenvalues and eigenvectors will change when small modifications are imposed on the structural parameters. The matrix perturbation method is an extremely useful tool for sensitivity analysis and it has been well developed. In this paper, the methods for computing the sensitivity of eigenvalue and eigenvector for real mode and complex mode are introduced. Based on the second-order Taylor expansion and second-order perturbation theory the estimation of the changes of eigenvalues and eigenvectors are obtained when design parameters changed. It also gave the first-and secong-order sensitivity matrices for eigenpairs.
In the third chapter of this paper, the first derivatives matrices of the mass and stiffness matrices are expressed as the linear functions of changed parameters. Then, based on matrix perturbation theory, the second-order perturbations of eigenvalues and eigenvectors are transformed into the multiple parameters forms, and the second-order perturbation sensitivity matrices of eigenvalues and eigenvectors are developed. Using these formulations, the efficient methods based on the second-order Taylor expansion and second-order perturbation are obtained to estimate the changes of eigenvalues and eigenvectors when design parameters changed. The examples demonstrate the application of the proposed method. From the numerical results it can be seen that the present method is validity and yields to excellent precision.
The fifth chapter extends the above method into complex mode. Similarly, in this case, the matrix perturbation for real modes can not be used and we have to use the matrix perturbation for complex mode. However, we assumed the system is not defective and the eigenvalues are all different. After introduced the state transformation matrix, we can have 2n dimension eigenvalues with left and right eigenvectors. From the perturbation formula and Taylor series, we can have the first-and second-order sensitivity matrix of eigenvalue and right and left eigenvectors.
In the end, we talked about the implement of the methods in computer. I-DEAS software system has extensive and advanced simulation capabilities, powerful calculation and analysis components. The software’s open-platform is an effective tool for implementing the proposed algorithm. Using these secondary platform and compile programs, we investigated the structure dynamic modification and verified the correctness of the method.
自1968年首次提出特征向量灵敏度的计算概念以来,结构特征灵敏度一直是人们热衷研究的领域。但由于特征值与特征向量都是关于设计参数的隐函数,无法计算其导数矩阵,从而不能用直接求导法对多参数结构特征灵敏度问题进行研究。现有的求解方法一般是在简单求导法的基础上结合模态展开法或Nelson法,但这些方法都要涉及一系列的方程运算,其算法步骤繁琐且计算量大。
本文在矩阵摄动理论的基础上提出一种新的计算特征值和特征向量灵敏度的方法——摄动灵敏度法。首先,将结构的系统增量矩阵(刚度矩阵、质量矩阵)作为设计参数的隐函数进行Taylor展开,得到系统增量关于设计参数的函数关系,然后根据特征值与特征向量的一阶、二阶摄动理论,推导出多参数结构特征值和特征向量的一阶、二阶摄动灵敏度和摄动灵敏度矩阵。此外,本文还对复模态特征灵敏度问题进行了研究,给出多参数结构复特征值与右、左特征向量的一阶、二阶摄动灵敏度和摄动灵敏度矩阵。
摄动灵敏度法通过在建立模型的过程中引入设计变量,使分析结果具有较明确的物理意义,提高了理论模态和试验模态的相关程度。它所提供的一阶、二阶摄动灵敏度矩阵是特征值和特征向量关于多参数的一阶、二阶导数矩阵的有效近似,解决了梯度阵和Hessian阵不能用求导法直接计算的问题,为结构重分析提供有力帮助。
本文通过对大型有限元分析软件I-DEAS的二次开发,成功地将该方法应用于此软件平台,解决了多参数结构特征灵敏度的工程计算问题,充分体现了该方法对结构设计的理论指导作用,展示了该方法的有效性与优越性。
Design sensitivity analysis of eigenvalues and eigenvectors will reveal how the changes in some design parameters in the system affect the dynamic characteristics of the structure. The designer can use this information directly in an interactive computer-aided design process as a valuable guide; it will shorten the time and improve work efficiency. It has been used in a variety of engineering disciplines ranging from automatic control theory to analysis of large-scale physiological systems. Some of the areas where sensitivity analysis has been applied include: system identification; development of insensitive control system; the approximation of system response to system parameters change; assessment of design changes on system performance; analysis of structural systems with random parameters. Recent applications have been made in vibration diagnosis on structural fault, the optimization of structural analysis, robust control and inverse problem of eigenvalue. The design sensitivity analysis of eigenvalues and eigenvectors play an essential role.
From 1960, many researchers have made investigations on the sensitivity of eigenpairs. The two major methods for investigate the sensitivity of eigenvector are proposed by Fox and Nelson. In 1968, Fox and Kapoor gave exact expressions for the first-order derivative of the eigenvalues and eigenvectors with respect to any design variables. In 1976, Nelson proposed an effective method to calculate sensitivity of eigenvectors which requires only the eigenvalue and eigenvector under consideration. The method represented eigenvector derivatives by a sum of the homogeneous solution and particular solution. In modal method the total information of eigenvectors is often used, however, for large-scale dynamic analysis it is impractical to obtain all modes. Therefore, some other methods to modify it are inevitable. In this paper, improved modal method, high accurate modal superposition method and Neumann series expansion method are introduced.
However, in many engineering problems such as the system with proportional or nonproportional damping, coupling between structure and control system, the system matrices are not real symmetric and may be complex asymmetric. In this case, the real mode theory is limited for dynamic modification and the complex mode is needed. After introducing the state vector and the state transformation matrix, the vibration equation of the linear systems is doubled. Similarly, the mode method and Nelson method will change their forms. Sondipon Adhikari’s method is consistent with the notion of traditional modal analysis methods. By finding the relationship of complex eigenvector, it avoids using state space approach and reduced the dimension from 2n to n . Najeh Guedria extended Nelson’s method into complex mode. He integrated the sensitivity of eigenvalue and eigenvector for only one parameter in one term and solved them in a series of equations simultaneity.
Generally, it is difficult to express the eigenvalues and eigenvectors as explicit functions of design variables. To carry out the calculations of the gradient and Hessian matrix by using the above methods are impossible. It is desirable to transfer the direct differential approach into the perturbation method.
It is well known that the matrix perturbation theory is concerned with how the eigenvalues and eigenvectors will change when small modifications are imposed on the structural parameters. The matrix perturbation method is an extremely useful tool for sensitivity analysis and it has been well developed. In this paper, the methods for computing the sensitivity of eigenvalue and eigenvector for real mode and complex mode are introduced. Based on the second-order Taylor expansion and second-order perturbation theory the estimation of the changes of eigenvalues and eigenvectors are obtained when design parameters changed. It also gave the first-and secong-order sensitivity matrices for eigenpairs.
In the third chapter of this paper, the first derivatives matrices of the mass and stiffness matrices are expressed as the linear functions of changed parameters. Then, based on matrix perturbation theory, the second-order perturbations of eigenvalues and eigenvectors are transformed into the multiple parameters forms, and the second-order perturbation sensitivity matrices of eigenvalues and eigenvectors are developed. Using these formulations, the efficient methods based on the second-order Taylor expansion and second-order perturbation are obtained to estimate the changes of eigenvalues and eigenvectors when design parameters changed. The examples demonstrate the application of the proposed method. From the numerical results it can be seen that the present method is validity and yields to excellent precision.
The fifth chapter extends the above method into complex mode. Similarly, in this case, the matrix perturbation for real modes can not be used and we have to use the matrix perturbation for complex mode. However, we assumed the system is not defective and the eigenvalues are all different. After introduced the state transformation matrix, we can have 2n dimension eigenvalues with left and right eigenvectors. From the perturbation formula and Taylor series, we can have the first-and second-order sensitivity matrix of eigenvalue and right and left eigenvectors.
In the end, we talked about the implement of the methods in computer. I-DEAS software system has extensive and advanced simulation capabilities, powerful calculation and analysis components. The software’s open-platform is an effective tool for implementing the proposed algorithm. Using these secondary platform and compile programs, we investigated the structure dynamic modification and verified the correctness of the method.
引文
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