一维不定参数结构系统的摄动Riccati传递矩阵方法及其应用
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摘要
本文深入地研究了摄动Riccati传递矩阵方法的理论和应用，在理论方面取得的研究成果包括：
1、 导出了一维不定参数结构系统振动特征问题的二阶摄动计算公式和摄动方程，利用矩阵的奇异值分解方法，成功地使得摄动方程中的特征值摄动变量和特征向量摄动变量完全分离，奠定了求解各阶摄动方程、特别是得到高精度的各阶摄动特征向量的理论基础；
2、 给出了一维不定参数结构系统的孤立特征值和特征向量的二阶摄动计算方法以及重频特征值和特征向量的二阶摄动计算方法，该方法不仅适用于实数特征值问题的摄动分析，而且适用于复数特征值问题的摄动分析；
3、 在进行重频特征向量的一阶摄动计算时并不需要求解二阶或者更高阶的摄动方程，这也是有别于基于有限元的矩阵摄动方法的地方，从而也使得重频特征值问题的摄动分析得到了简化；
4、 由于避免了在求解特征向量的摄动时的模态截断误差问题，用该方法得到的特征值和特征向量的摄动解是高精度的摄动解；
5、 本文给出的摄动分析方法是一个适用于一维及类一维结构系统动力学问题的摄动分析方法。
在应用方面取得的成果包括：
1、 开发出了摄动Riccati传递矩阵法的计算分析程序。该程序可以对杆、梁结构横向弯曲振动的实数、复数的孤立和重频特征值问题进行摄动分析和灵敏度分析，特别是适合于转子动力学系统特征值和特征向量问题的摄动分析和灵敏度分析；
2、 用摄动Riccati传递矩阵方法解决了某电站锅炉给水泵转子的参数识别及动力模型修改的问题，并给出了该种型号的给水泵转子的更准确的力学模型，为进一步的转子动力学分析与设计奠定了可靠基础；
3、 给出了摄动理论在相关领域如随机特征值分析、随机振动响应分析、可靠性分析、灵敏度分析、优化设计以及参数识别中的应用公式。
In this paper, the perturbation RICCATI transfer matrix method was comprehensively investigated. The important developments achieved in theories include:
1. Based on RICCATI transfer matrix method, the calculating formulas and equations for perturbation analysis of one-dimensional structural system with parameter uncertainties are derived. By making singular value decomposition, the unknown perturbing eigenvalues and unknown perturbing eigenvectors in the perturbation equations are separated, consequently, the theoretical basis for solving perturbation equations and for getting perturbation eigenvectors with high precision is established;
2. General methods are presented to calculate the 2nd order perturbations of eigendatas for one-dimensional structural system with parameter uncertainties, the methods are applicable to both single and repeated eigendata problems as well as both real and complex eigendatas;
3. It doesn't needed that solve the 2nd or the other higher order perturbation equations as the 1st order perturbations of eigenvectors are calculated for repeated eigenvalues, and the perturbation analysis of eigenvectors for repeated eigenvalue problems is simplified. This is different from the matrix perturbation method based on the FEM;
4. Because the errors caused by truncating the high modals are avoided, the perturbation solutions of eigenvalues and eigenvectors are high accurate perturbation solutions;
5. The perturbation method presented in this paper is a general method for any one-dimensional and like one-dimensional structural system.
The achievements in applications include:
1. Based on the perturbation RICCATI transfer matrix method, the calculating program are developed. The program can be used to the perturbation analysis and the sensitivity analysis of the real and complex, the single and repeated eigenvalues and eigenvectors for lateral vibration of rod and beam structures, especially suitable to the perturbation analysis and the sensitivity analysis of eigenvalues and eigenvectors for rotordynamic systems;
2. The perturbation RICCATI transfer matrix method was applied to identify the parameters of the rotor for a boiler supply pump, and the accurate dynamic model of the rotor was archived. It is a important basis for advanced dynamic analysis and redesign;
3. The application formulas based on this perturbation theory in some areas such as stochastic eigendatas, stochastic response, dynamic reliability, sensitivity analysis, parameter identification, optimal design are given.
1、 导出了一维不定参数结构系统振动特征问题的二阶摄动计算公式和摄动方程，利用矩阵的奇异值分解方法，成功地使得摄动方程中的特征值摄动变量和特征向量摄动变量完全分离，奠定了求解各阶摄动方程、特别是得到高精度的各阶摄动特征向量的理论基础；
2、 给出了一维不定参数结构系统的孤立特征值和特征向量的二阶摄动计算方法以及重频特征值和特征向量的二阶摄动计算方法，该方法不仅适用于实数特征值问题的摄动分析，而且适用于复数特征值问题的摄动分析；
3、 在进行重频特征向量的一阶摄动计算时并不需要求解二阶或者更高阶的摄动方程，这也是有别于基于有限元的矩阵摄动方法的地方，从而也使得重频特征值问题的摄动分析得到了简化；
4、 由于避免了在求解特征向量的摄动时的模态截断误差问题，用该方法得到的特征值和特征向量的摄动解是高精度的摄动解；
5、 本文给出的摄动分析方法是一个适用于一维及类一维结构系统动力学问题的摄动分析方法。
在应用方面取得的成果包括：
1、 开发出了摄动Riccati传递矩阵法的计算分析程序。该程序可以对杆、梁结构横向弯曲振动的实数、复数的孤立和重频特征值问题进行摄动分析和灵敏度分析，特别是适合于转子动力学系统特征值和特征向量问题的摄动分析和灵敏度分析；
2、 用摄动Riccati传递矩阵方法解决了某电站锅炉给水泵转子的参数识别及动力模型修改的问题，并给出了该种型号的给水泵转子的更准确的力学模型，为进一步的转子动力学分析与设计奠定了可靠基础；
3、 给出了摄动理论在相关领域如随机特征值分析、随机振动响应分析、可靠性分析、灵敏度分析、优化设计以及参数识别中的应用公式。
In this paper, the perturbation RICCATI transfer matrix method was comprehensively investigated. The important developments achieved in theories include:
1. Based on RICCATI transfer matrix method, the calculating formulas and equations for perturbation analysis of one-dimensional structural system with parameter uncertainties are derived. By making singular value decomposition, the unknown perturbing eigenvalues and unknown perturbing eigenvectors in the perturbation equations are separated, consequently, the theoretical basis for solving perturbation equations and for getting perturbation eigenvectors with high precision is established;
2. General methods are presented to calculate the 2nd order perturbations of eigendatas for one-dimensional structural system with parameter uncertainties, the methods are applicable to both single and repeated eigendata problems as well as both real and complex eigendatas;
3. It doesn't needed that solve the 2nd or the other higher order perturbation equations as the 1st order perturbations of eigenvectors are calculated for repeated eigenvalues, and the perturbation analysis of eigenvectors for repeated eigenvalue problems is simplified. This is different from the matrix perturbation method based on the FEM;
4. Because the errors caused by truncating the high modals are avoided, the perturbation solutions of eigenvalues and eigenvectors are high accurate perturbation solutions;
5. The perturbation method presented in this paper is a general method for any one-dimensional and like one-dimensional structural system.
The achievements in applications include:
1. Based on the perturbation RICCATI transfer matrix method, the calculating program are developed. The program can be used to the perturbation analysis and the sensitivity analysis of the real and complex, the single and repeated eigenvalues and eigenvectors for lateral vibration of rod and beam structures, especially suitable to the perturbation analysis and the sensitivity analysis of eigenvalues and eigenvectors for rotordynamic systems;
2. The perturbation RICCATI transfer matrix method was applied to identify the parameters of the rotor for a boiler supply pump, and the accurate dynamic model of the rotor was archived. It is a important basis for advanced dynamic analysis and redesign;
3. The application formulas based on this perturbation theory in some areas such as stochastic eigendatas, stochastic response, dynamic reliability, sensitivity analysis, parameter identification, optimal design are given.
引文
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