A sensitivity‐based approach to solving the inverse eigenvalue problem for linear structures carrying lumped attachments

Abstract

This paper presents an efficient algorithm for designing dynamical systems to exhibit a desired spectrum of eigenvalues. Focusing on combined systems of linear structures carrying various lumped element attachments, we apply the assumed-modes method and the implicit function theorem to derive analytical expressions for eigenvalue sensitivities, which are used to efficiently determine the minimal set of structural modifications needed to achieve a set of desired eigenvalues. The proposed algorithm employs an adaptive step size, performs significantly better than existing approaches, and can be easily applied to a broad range of structures. Convergence properties and limitations on achievable eigenvalues are also discussed, and a number of case studies demonstrating the performance of the algorithm in a wide variety of different applications are also included. KEYWORDSinverse problem, structural design, structural dynamics, eigenvalue sensitivity Int J Numer Methods Eng. 2019;120:537-566.wileyonlinelibrary.com/journal/nme

A modified polynomial-based approach to obtaining the eigenvalues of a uniform Euler–Bernoulli beam carrying any number of attachments

“…Tests were executed on an Intel i5-10400 CPU running at 2.90 GHz. Then, the first natural frequencies (for undamped systems) or eigenvalues (for damped systems) of several case studies previously validated in the literature (Cha, 2002(Cha, , 2005Lin and Tsai, 2007;Cha and Hu, 2017;Dawson and Cha, 2019) were computed using the proposed scheme. All of them correspond to simply supported Euler-Bernoulli uniform beams with different lumped attachment configurations as denoted by ID in Table 1.…”

“…This procedure allows to easily solve the inverse problem (e.g. Mir Hosseini and Baddour (2014); Dawson and Cha (2019)) with numerous engineering applications in order to either impose or avoid certain frequencies of the vibrational system by modifying the characteristics of the combined system. In this way, Figure 5 shows the graphical solution for the second and third (Figure 5 1 and 2 in Mir Hosseini and Baddour (2014).…”

A direct solution method for generalized stochastic eigenvalue problems of structures with random parameters

“…Estimating the eigenvalues or natural frequencies of a given structure precisely is fundamental and essential for structural analysis and design, 1,2 which provides a wealth of information related to its dynamic characteristics 3 . Therefore, the eigenvalues obtained by solving the generalized stochastic eigenvalue problems are considered as significant criteria of analysis and design in engineering structures 4 …”