二阶系统的同谱解耦研究
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摘要
二阶系统解耦不仅在线性振动系统方面，而且在量子力学、数量经济学和计算科学等多个领域中起着重要的作用。通过解耦，一个多自由二阶系统被看成多个无关的单自由度子系统。二阶系统解耦不仅为系统响应的计算提供有效方法，而且为系统的定量分析带来极大的便利，因此，研究二阶系统解耦意义重大。
本文旨在提出一种二阶系统的同谱解耦方法，无论系统是否亏损，或首系数矩阵是否奇异，该方法始终有效。本文主要从二阶系统解耦的三个方面来寻求发展与突破：首先，实现亏损二阶阻尼系统的同谱解耦和解耦变换的获取；其次，求解可解耦二阶系统的实值保结构变换；最后，解决奇异二阶系统的解耦问题。本文的主要工作可概括如下：
提出了可解耦二阶系统的一种Jordan标准形构造方法。在质量矩阵非奇异的情况下，对可解耦的二阶系统的Jordan标准形进行研究，根据解耦条件和解耦前后系统同谱的性质，首先将二阶系统的特征值分成三类，然后分别求解每一类特征值所对应的Jordan子块，最后利用直和得到了二阶系统的Jordan标准形。这种构造方法为本文后面的工作提供了基础。
提出了基于相位同步原理的一种亏损二阶系统同谱解耦方法。针对可解耦的亏损二阶系统，首先利用相位同步原理使每一个阻尼模态的相位角同步，从而使得非经典阻尼系统被转换为经典阻尼系统。其次，通过经典模态分析将经典阻尼系统转换成解耦系统。再次，基于相位同步原理得到解耦变换，并且证明该解耦变换是原始系统与解耦系统友矩阵的一个相似变换。最后，提出解耦变换的一种构造方法。该方法具有广泛的适用性，它不仅对所有可解耦亏损二阶系统有效，而且同样适用于非亏损二阶系统。数值试验证明了该方法的有效性。
在质量矩阵非奇异的情况，提出可解耦二阶系统的一种实值保结构变换求解方法。对于二阶系统的解耦问题，如何求解保结构变换一直是个难点。目前，虽然从理论上证明对所有的非亏损二阶系统在Lancaster结构下都存在保结构变换，但是很难实现实值保结构变换求解。针对可解耦的二阶系统，首先，利用Jordan三元组理论和标准三元组理论证明了保结构变换满足的一种形式。其次，根据可解耦二阶系统的Jordan标准形给出其中心化子。最后，利用置换矩阵和一系列构造得到了保结构变换，并且该变换是实值的。这种实值保结构变换求解方法适用于所有的可解耦二阶系统，无论是否亏损。数值试验证明了该方法的有效性。
针对可解耦的奇异二阶系统，提出其一种同谱实对角系统的构造方法和一种基于谱变换的同谱解耦方法。一方面，为了构造同谱的解耦系统，提出了利用原始系统的有限特征值和无限特征值来构造解耦系统的三个参数矩阵的方法；另一方面，针对奇异二阶系统的同谱解耦问题，提出了利用谱变换将原始奇异二阶系统转换为非奇异二阶系统的方法，然后可以利用已有方法来解耦，最后将该系统还原，从而实现奇异二阶系统解耦。数值试验证明了该方法的有效性。
本文的研究不仅是非奇异二阶系统解耦理论的一个发展与完善，也是奇异二阶系统解耦理论的一个突破。
Quadratic system decoupling plays a fundamental role not only in linear vibrations butalso in diverse areas such as quantum mechanics, mathematical economics, andcomputational science. Upon decoupling, a quadratic system can be regarded as a series ofindependent single-degree-of-freedom quadratic subsystems. This not only provides anefficient means of evaluating the system response but also greatly facilitates qualitativeanalysis, so it is necessary to research quadratic system decoupling.
The purpose of this dissertation is to propose a new isospectral decoupling method ofquadratic system, no matter defective or not and singular leading coefficient or not, thismethod is still effective. This paper will mainly seek development and breakthrough ofquadratic system decoupling from three aspects. Firstly, realizing the isospectral decouplingof defective quadratic system and decoupling transformations are obtained; Secondly,solving real-valued structure preserving transformation of diagonalizable quadratic system;Thirdly, resolving the decoupling problem of quadratic system with singular leadingcoefficient. Major results presented in this dissertation are summarized in the followingstatements:
A construction method of Jordan canonical form is proposed for diagonalizablequadratic system. With mass matrix nonsingular, the Jordan canonical form ofdiagonalizable quadratic system are researched, according to the characteristics ofdecoupling conditions and the decoupled system is isospectral as original system, theeigenvalues are classified into three categories, then solve the corresponding Jordansub-matrices of each kind of eigenvalues, respectively. Finally, the Jordan canonical form ofquadratic system is obtained by using direct sum. The construction method providefoundation for further work.
An isospectral decoupling method of defective quadratic system based on phasesynchronization is proposed. For diagonalizable defective quadratic system, Firstly, bysynchronizing the phase angles in each damped mode, a non-classically damped system canbe transformed into one with classical damping. Secondly, the classically damped system isconverted into the decoupled system by classical modal analysis. Thirdly, the decoupling transformations are obtained based on based on phase synchronization of decoupling, and itis proved to be a similar transformation of the companion between decoupled system andoriginal system. Finally, a construction method of decoupling transformation is given. Thismethod has the extensive applicability, and it is not only effective to all defective quadraticsystem, but also validity to non-defective quadratic system. numerical experiment ispresented to prove the availability of the method.
With mass matrix nonsingular, a solution method of real-valued structure preservingtransformation is given for diagonalizable quadratic system. The solution of structurepreserving transformation is always a nodus for quadratic system decoupling, At present，although it has been proved in theory that the structure preserving transformation esist forall the non-defective quadratic system based on Lancaster structure，it is difficult to obtainreal-valued structure preserving transformation. For diagonalizable quadratic system, firstof all, by using theory of Jordan triple and standard triple, a of structure preservingtransformation is proved. Then, the centralizer for the Jordan canonical form is given.Finally, the structure preserving transformation are obtained by using permutation matrixand a series of contruction, in addition, it is real-valued. This method apply to alldiagonalizable quadratic system, no matter defective or not. numerical experiment ispresented to prove the availability of the method.
A construction method of isospectral real diagonal system and an isospectraldecoupling method based on spectral transformation are proposed for diagonalizablequadratic system with singular leading coefficient. For one thing, to construct isospectraldecoupled system, a method which construct three parameters of the decoupled system byusing finite eigenvalues and infinite eigenvalues of original system is given. For anotherthing, spectral transformation is introduced to transform singular system into nonsingularsystem, so it can use nonsingular quadratic system decoupling to construct and restoredecoupled system, thus realize the decoupling of singular quadratic system. Finallynumerical experiment is presented to prove the availability of the method.
The study in this paper is not only a development and improvement for the quadratic system decoupling with nonsingular leading coefficient, but also a breakthrough forsingular quadratic system decoupling.
本文旨在提出一种二阶系统的同谱解耦方法，无论系统是否亏损，或首系数矩阵是否奇异，该方法始终有效。本文主要从二阶系统解耦的三个方面来寻求发展与突破：首先，实现亏损二阶阻尼系统的同谱解耦和解耦变换的获取；其次，求解可解耦二阶系统的实值保结构变换；最后，解决奇异二阶系统的解耦问题。本文的主要工作可概括如下：
提出了可解耦二阶系统的一种Jordan标准形构造方法。在质量矩阵非奇异的情况下，对可解耦的二阶系统的Jordan标准形进行研究，根据解耦条件和解耦前后系统同谱的性质，首先将二阶系统的特征值分成三类，然后分别求解每一类特征值所对应的Jordan子块，最后利用直和得到了二阶系统的Jordan标准形。这种构造方法为本文后面的工作提供了基础。
提出了基于相位同步原理的一种亏损二阶系统同谱解耦方法。针对可解耦的亏损二阶系统，首先利用相位同步原理使每一个阻尼模态的相位角同步，从而使得非经典阻尼系统被转换为经典阻尼系统。其次，通过经典模态分析将经典阻尼系统转换成解耦系统。再次，基于相位同步原理得到解耦变换，并且证明该解耦变换是原始系统与解耦系统友矩阵的一个相似变换。最后，提出解耦变换的一种构造方法。该方法具有广泛的适用性，它不仅对所有可解耦亏损二阶系统有效，而且同样适用于非亏损二阶系统。数值试验证明了该方法的有效性。
在质量矩阵非奇异的情况，提出可解耦二阶系统的一种实值保结构变换求解方法。对于二阶系统的解耦问题，如何求解保结构变换一直是个难点。目前，虽然从理论上证明对所有的非亏损二阶系统在Lancaster结构下都存在保结构变换，但是很难实现实值保结构变换求解。针对可解耦的二阶系统，首先，利用Jordan三元组理论和标准三元组理论证明了保结构变换满足的一种形式。其次，根据可解耦二阶系统的Jordan标准形给出其中心化子。最后，利用置换矩阵和一系列构造得到了保结构变换，并且该变换是实值的。这种实值保结构变换求解方法适用于所有的可解耦二阶系统，无论是否亏损。数值试验证明了该方法的有效性。
针对可解耦的奇异二阶系统，提出其一种同谱实对角系统的构造方法和一种基于谱变换的同谱解耦方法。一方面，为了构造同谱的解耦系统，提出了利用原始系统的有限特征值和无限特征值来构造解耦系统的三个参数矩阵的方法；另一方面，针对奇异二阶系统的同谱解耦问题，提出了利用谱变换将原始奇异二阶系统转换为非奇异二阶系统的方法，然后可以利用已有方法来解耦，最后将该系统还原，从而实现奇异二阶系统解耦。数值试验证明了该方法的有效性。
本文的研究不仅是非奇异二阶系统解耦理论的一个发展与完善，也是奇异二阶系统解耦理论的一个突破。
Quadratic system decoupling plays a fundamental role not only in linear vibrations butalso in diverse areas such as quantum mechanics, mathematical economics, andcomputational science. Upon decoupling, a quadratic system can be regarded as a series ofindependent single-degree-of-freedom quadratic subsystems. This not only provides anefficient means of evaluating the system response but also greatly facilitates qualitativeanalysis, so it is necessary to research quadratic system decoupling.
The purpose of this dissertation is to propose a new isospectral decoupling method ofquadratic system, no matter defective or not and singular leading coefficient or not, thismethod is still effective. This paper will mainly seek development and breakthrough ofquadratic system decoupling from three aspects. Firstly, realizing the isospectral decouplingof defective quadratic system and decoupling transformations are obtained; Secondly,solving real-valued structure preserving transformation of diagonalizable quadratic system;Thirdly, resolving the decoupling problem of quadratic system with singular leadingcoefficient. Major results presented in this dissertation are summarized in the followingstatements:
A construction method of Jordan canonical form is proposed for diagonalizablequadratic system. With mass matrix nonsingular, the Jordan canonical form ofdiagonalizable quadratic system are researched, according to the characteristics ofdecoupling conditions and the decoupled system is isospectral as original system, theeigenvalues are classified into three categories, then solve the corresponding Jordansub-matrices of each kind of eigenvalues, respectively. Finally, the Jordan canonical form ofquadratic system is obtained by using direct sum. The construction method providefoundation for further work.
An isospectral decoupling method of defective quadratic system based on phasesynchronization is proposed. For diagonalizable defective quadratic system, Firstly, bysynchronizing the phase angles in each damped mode, a non-classically damped system canbe transformed into one with classical damping. Secondly, the classically damped system isconverted into the decoupled system by classical modal analysis. Thirdly, the decoupling transformations are obtained based on based on phase synchronization of decoupling, and itis proved to be a similar transformation of the companion between decoupled system andoriginal system. Finally, a construction method of decoupling transformation is given. Thismethod has the extensive applicability, and it is not only effective to all defective quadraticsystem, but also validity to non-defective quadratic system. numerical experiment ispresented to prove the availability of the method.
With mass matrix nonsingular, a solution method of real-valued structure preservingtransformation is given for diagonalizable quadratic system. The solution of structurepreserving transformation is always a nodus for quadratic system decoupling, At present，although it has been proved in theory that the structure preserving transformation esist forall the non-defective quadratic system based on Lancaster structure，it is difficult to obtainreal-valued structure preserving transformation. For diagonalizable quadratic system, firstof all, by using theory of Jordan triple and standard triple, a of structure preservingtransformation is proved. Then, the centralizer for the Jordan canonical form is given.Finally, the structure preserving transformation are obtained by using permutation matrixand a series of contruction, in addition, it is real-valued. This method apply to alldiagonalizable quadratic system, no matter defective or not. numerical experiment ispresented to prove the availability of the method.
A construction method of isospectral real diagonal system and an isospectraldecoupling method based on spectral transformation are proposed for diagonalizablequadratic system with singular leading coefficient. For one thing, to construct isospectraldecoupled system, a method which construct three parameters of the decoupled system byusing finite eigenvalues and infinite eigenvalues of original system is given. For anotherthing, spectral transformation is introduced to transform singular system into nonsingularsystem, so it can use nonsingular quadratic system decoupling to construct and restoredecoupled system, thus realize the decoupling of singular quadratic system. Finallynumerical experiment is presented to prove the availability of the method.
The study in this paper is not only a development and improvement for the quadratic system decoupling with nonsingular leading coefficient, but also a breakthrough forsingular quadratic system decoupling.
引文
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