Research On Inverse Spectrum Of Structured Matrix Based On Vibration System

One or more elastic vibration systems were associated with the design and storage of weapons and ammunition. In theory,The system can often be described by a mathematical model of which the parameters represent important physical quantities.In the course of constructing mathematical model,The vibrational spectra obtained from the tests were used. Through theoretical analysis, can predict the effect of the vibration behavior, and compared with the experimental results, and then update the model to make it more accurate expression of the vibration process.Discussion on the inverse eigenvalue problem,was the process of establishing such a matrix model、analysis, correction and solution,It relies on the spectral information of the corresponding system, the basic structure of the system constraints, the necessary simplification and mathematical theory tools.Based on the above considerations, this paper focuses on the inverse eigenvalue problem of structured matrix.In the introduction,the development course of the Vibration and vibration inverse problem,the inverse eigenvalue problem, and the inverse spectrum of nonnegative matries was described,and the research status at home and abroad was introduced.By analyzing the structural characteristics of mass spring system in Vibration Engineering, the structural characteristics of the system were analyzed, the source of the six main problems of the inverse problem of the Jacobi matrix eigenvalue problem was found,and the correlation and the change of these problems were discussed. Applying Cauchy’s alternating theorem, an interlacing property of invers eigenvalues of the nonnegative quasi-symmetric tridiagonal plus arrow form matrices was exploited, Describes the characteristics of the minimal and maximal eigenvalues of the leading principal submatrix of the matrix, Necessary and sufficient conditions for the solution and constructive algorithm of the problem were obtained under the condition of spectral constraints.Applying the Perron-Frobenius theorem to characterize the spectral constraint, discussed the positive SDIEP, and the sufficient conditions for the solution of the problem were found by combining the spectral decomposition theorem of the simple matrix and the properties of the Soules matrix,and the algorithm of constructing SDIEP matrix was given. On the other hand, the work of this chapter, we elaborate the four methods and discuss the inclusion relations and intersection relations between the general realizability criteria obtaining from the four methods.Next,based on the FFT fast algorithm and spectral decomposition theory,and properties of generalized circulant matrices, we discuss the inverse spectral problem for a class of nonnegative r?circulant matrices and give the necessary and sufficient conditions of solvability for this problem and constructive algorithms. Furthermore, we give an extention to nonnegative symmetric circulant matrices on this basis, then the necessary and sufficient conditions of solvability for this problem and constructive algorithms were also given. Eventually, we suggest the nonnegative symmetric circulant matrices were an s.d.s. realizable.In this paper, the spectrally arbitrary of sign patterns was also discussed, one was a class of spectrally arbitrary ray pattern,via Nilpotent-Jacobian method of promotion,in addition,we study the spectrally arbitrary complex sign patterns using two-colored digraph,and show a necessary condition for a spectrally arbitrary complex sign pattern and introduce a minimal spectrally arbitrary complex sign pattern all of whose superpatterns were also spectrally arbitrary for n ?2.In the end,the summerary of the paper was given and the further research work was put forward.The conclusions provide theoretical analysis and support for the inverse design of weapon system, Vibration control in ammunition highway transportation, structure analysis, mechanics and so on.