Research On Isospectral Decoupling Of Quadratic System

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.