Research On Some Problems On Defective And Near Defective Systems

Structural modification of systems is an important research issue and has a wide range ofapplications such as vibration suppression, system design and control. Changes of structuresare often necessary to satisfy predetermined demands in various design and optimizationproblems, the response of the numerous modified structures need to be evaluated repeatedly.In the structural dynamic optimization, the multiple repeated analyses are ones of the mostcostly computations. The need for efficient and accurate reanalysis technique in modernstructural design is crucial because the design becomes more complex and large. Until now,many researches of structural vibration reanalysis are concerned on non-defective systems. Inproblems such as those of dynamic and symmetric structures, however, the correspondingmatrices can have repeated eigenvalues. Very often indeed, the geometric multiplicity of theeigenvalues is less than the algebraic multiplicity, and so the system has an incomplete set ofeigenvectors, insufficient to form a base for the state space. Systems of this type are called thedefective systems. The defective systems can be encountered in actual engineering problems,such as flutters of airplane and missile wings or long blades of turbines which thecorresponding matrices are defective.Many engineering optimization problems, for example, model updating or structuraldamage detections, lead to a sensitivity analysis of eigenproblems. As a result, the study ofthe sensitivity of eigensolutions due to variations in the system parameters has been animportant research area. A dynamic model can be far from the assumed prototype. Since thereis usually a variation, such as a mistuned parameter or a irregularity geometrical. For thesereasons, sensitivity analysis is meaningful to perform a theoretical study and give a guide for engineering practice. There are two main difficulties in computing the eigenvectorsderivatives. One of the main difficulties is how to change the irreversible state ofcharacteristic matrix. The other difficulty is how to establish a uniform efficient method forcomputing the eigenvectors derivatives.The grant of the projects from the National Natural Science Foundation of China “FastOptimization of Cross-Sectional Parameters for Simplified Car Body Multi-Elements FrameStructure Based on Reanalysis Theory”(No.50975121) and Doctoral Program of HigherEducation “The Research of Adaptive Reanalysis Algorithm of the Defective VibrationSystem”(No.20090061110022) and2011Graduate Innovation Fund of Jilin University“Research on Dynamic Modification for Defective Systems Based on CombinedApproximations Method”(No.20111056) and2012Graduate Innovation Fund of JilinUniversity “Research on Relaxation Combined Approximations Method for SensitivityAnalysis”(No.20121097) is gratefully acknowledged for the financial support.The main contents of this paper can be summarized as follows:The Relaxation Combined Approximations (RCA) method is proposed which aims tosolve reanalysis problems of defective systems. A relaxation factor is introduced into the CAmethod based on the general mode theory, and it changes the characteristic matrix from beingsingular to non-singular. For N repeated eigenvalues and general defective systems, thismodification makes possible to the solution of generalized eigenvectors. An additionaladvantage of the method is that the generalized eigenvectors are expressed by a series ofbasic vectors and the dimension of basic vectors is usually much less than the dimension ofeigenvectors, so the computational cost is also reduced. Numerical examples show that theRCA method can lead to exact generalized eigenvectors, and is very easy to be used for thesame kind of these problems.The RCA method gives a solution to the problem of defective vibration systems basedon the advantage of relaxation factor. During system optimization, some originally separatedfrequencies can approach closer and closer. In these cases, if the associated eigenvectors make groups of nearly parallel vectors the system can be classified as near defective. Fromthe view point of mathematics, the close eigenvalues of near defective systems are distinct,but the dynamic characteristic is still defective. By the frequency shift (Shift RelaxationCombined Approximations method), the reanalysis problems of near defective systems withclose eigenvalues can be transformed into one of the defective systems with repeated ones,which is equal to the average of the close ones. The relationship between the relaxation factorand error shows that the accuracy of the algorithm.As the research basis of the reanalysis algorithms, the sensitivity analysis problems ofthe structural modification and structure optimization are discussed for the defective and neardefective systems. The formulae for calculating the sensitivity analysis are derived from theRCA and SRCA approach. Sensitivity algorithms are not only neat format but also accurate,efficient and more applicable to the particular system applicable to an ordinary system.