بررسی تجزیه و تحلیل حساسیت طراحی برای مسائل EIGEN غیرخطی

A general nonlinear eigenproblem is considered in this paper. It is shown that the widely used undamped, viscously or nonviscously damped eigenproblem can be considered as a special case of the more general nonlinear eigenproblem. The existing formulas of the derivatives of eigensolutions of this nonlinear eigenproblem are very complex since the normalizations of undamped eigenproblems are considered in their studies. To simplify the computation of eigensensitivity, a new normalization of the general nonlinear eigenproblem is presented. The new normalization of the general eigenproblem derived here can degenerate to the familiar mass orthogonal relationship of undamped eigenvectors. Based on this normalization, the design sensitivity analysis for the general nonlinear eigenproblem with respect to arbitrary design parameters is studied. Moreover, this method can address the eigenproblem for both repeated and distinct eigenvalues. As it can be seen, under such general normalization the derivatives of eigensolutions can be expressed in a way similar to those of undamped systems. Finally, five numerical examples are provided to illustrate the effectiveness of the derived results. It is shown that the derivatives of eigensolutions can be treated in a unified way for different structural systems (i.e., undamped systems, viscously and nonviscously damped systems, and nonlinear systems).

Design sensitivity analysis of eigenproblems of structural and mechanical systems with respect to structural design parameters has become an integral part of many engineering design methodologies including structural health monitoring, structural reliability, dynamic model updating, structural design optimization, structural dynamic modification, approximate reanalysis techniques and many other applications. Mottershead et al. [1] pointed out that the eigensolution sensitivity method is probably the most successful of the many approaches to the problem of models updating and has developed into a mature technology applied successfully for the correction of enterprise-level finite element models. Although computing eigenvalue sensitivity is straightforward, finding eigenvector derivatives resides several challenges. There are two main difficulties residing in computing the eigenvector derivatives. One of the main difficulties in computation of eigenvector sensitivities is the singularity issue. And the other difficulty is the derivatives of eigenvectors corresponding to the repeated eigenvalues. Eigensensitivity analysis has received much attention over the past four decades. Several methods have been developed for the calculation of the eigensolution sensitivities. For undamped eigensystems, it is well known that the equations of motion for the free vibration can be expressed by equation(1) Ku(s)=sMu(s)Ku(s)=sMu(s) Turn MathJax on where View the MathML sourceK,M∈RN×N are, respectively, the stiffness and mass matrices. The eigenvalues λ i are the roots of the characteristic equation, det[K−sM]=0det[K−sM]=0. The i th eigenvalue can be expressed as View the MathML sourceλi=ωi2 where ωiωi is the ith undamped natural frequency. Fox and Kapoor [2] derived the expressions of design sensitivity analysis of eigenproblem with respect to any design variable for symmetric undamped systems. To simplify the computation of design sensitivity of eigenproblem, Nelson [3] presented an efficient approach to calculate the eigenvector derivatives with distinct eigenvalues for undamped systems by expressing the derivative of each eigenvector as a particular solution and a homogeneous solution. One of the main advantages of Nelson’s method is that it requires the information of only those eigenvectors that are to be differentiated. Sutter et al. [4] presented a comparison of several methods on calculating vibration mode shape derivatives and pointed out that Nelson’s method may be more efficient than the modal method for the reason that the modal method needs all or most of the eigenvectors to determine the eigenvector derivatives. Later, Lee and Jung [5] and [6] proposed an efficient algebraic method with symmetric coefficient matrices for calculating the derivatives of vibration mode shapes of symmetric undamped systems with distinct and repeated eigenvalues, respectively. The main idea of this method is to obtain a non-singular equation for computing the eigensensitivity by assembling the derivatives of eigenproblems and the additional constraints of eigensolution derivatives into a linear system of algebraic equations. Unfortunately, Wu et al. [7] pointed out that the method [6] was not correct because a mistake was made in the derivation of equations on the derivatives of the normalization for the systems with repeated eigenvalues. There are many repeated eigenvalues or nearly equal eigenvalues in typical structural or mechanical systems for certain reasons. Ojalvo [8], Mills-Curran [9], Dailey [10] developed Nelson’s method for solving the derivatives of eigensolutions of the real symmetric eigensystems with repeated eigenvalues. Shaw and Jayasuriya [11] generalized these methods to calculate the derivatives of eigensolutions in the case of repeated eigenvalues with repeated first-order derivatives. Eigensensitivity analysis for viscously damped systems has received much attention over the past two decades. The symmetric eigenproblem with viscous damping is given by the second-order system equation(2) (2sM+sDV+K)u(s)=0(s2M+sDV+K)u(s)=0 Turn MathJax on where DVDV is the viscous damping matrix. Adhikari [12] and [13] presented N-space modal methods to compute the eigenvector sensitivities for non-proportionally symmetric systems with viscous damping. Later, Adhikari and Friswell [14] extended the N-space modal method to asymmetric damped systems and the second-order eigensolution derivatives were also derived. Lee et al. [15] further extended their algebraic method to the case of distinct eigenvalues in terms of second-order symmetric viscously damped systems. Friswell and Adhikari [16] extended Nelson's method to symmetric and asymmetric systems with viscous damping. Choi et al. [17] presented an efficient algebraic method for symmetric and asymmetric viscously damped systems. One of the main advantages is that this method can compute the derivatives of eigensolutions of asymmetric systems without using the left eigenvectors. Guedria et al. [18] developed an algebraic method to calculate the derivatives of eigensolutions for general asymmetric viscously damped systems. Chouchane et al. [19] simply reviewed the algebraic method for symmetric and asymmetric systems and developed their method to the second-order derivatives of eigensolutions. Mirzaeifar et al. [20] presented a new method based on a combination of the algebraic method and the modal method for general asymmetric viscously damped systems. This combined method neither has the complications of the modal method on the computation of the complex left and right eigenvector derivatives nor suffers from the numerical instability issues usually associated with the algebraic method. Later, Xu and Wu [21] derived a new normalization and presented a new method for computing the derivatives of eigensolutions of asymmetric viscously damped eigensystems with distinct and repeated eigenvalues. Recently, Xu et al. [22] derived an efficient algebraic method for the computations of derivatives of eigensolutions of asymmetric damped eigensystems with distinct eigenvalues. More recently, Li et al. [23] suggested a new normalization for the left eigenvectors, from which the left and right eigenvector derivatives can be computed separately and independently for asymmetric eigensystems. Moreover, this method can address the eigenproblem with distinct and repeated eigenvalues and is well-conditioned since the components of coefficient matrices are all of the same order of magnitude.

Increasing the use of viscoelastic damping technology such as composite structural materials, active control and damage tolerant systems in rockets, spacecrafts, shuttles, satellites, ships and automobiles has led to renewed demand for accurate and efficient dynamic analysis of nonlinear eigensystems. This paper describes a general nonlinear eigenproblem. It is shown that the widely used undamped, viscously or nonviscously damped eigenproblem can be considered as a special case of the more general nonlinear eigenproblem. A new normalization of the general nonlinear eigenproblem is derived and can degenerate to the familiar mass orthogonal relationship of undamped eigenvectors. Based on this normalization, the design sensitivity analysis for the general nonlinear eigenproblem with respect to arbitrary design parameters is studied. In the presented paper, for the first time in the literature, this method can accurately address the eigenproblem for nonviscous damped systems with repeated eigenvalues. As it can be observed, based on the general normalization, the derivatives of eigensolutions can be expressed in a way similar to those of undamped systems. Finally, five numerical examples are considered to show the effectiveness of the derived results. The results in three distinct cases: the undamped system, the viscous system and the nonviscous system are shown in good agreement with the results obtained by common existing methods. As can be seen, although the expressions of general system matrices are different, depending on the different structural systems (i.e., undamped systems, viscously and nonviscously damped systems, and nonlinear systems), the derivatives of eigensolutions can be treated in a unified way. The proposed method can be also applied in many issues in dynamics control and other branches of applied mathematics, including dynamic optimization, the theory of oscillation and finite element (FE) method with frequency-dependent system matrices. The study carried out here speeds up the recent developments of composite and smart materials and actively controlled structures for engineering applications and further research in this direction is worth perusing.