242 T. Kalayc{og˘lu and H.N. Özgüven substituting internal nonlinear forces by their corresponding multi-harmonic describing functions. Huang [18] also proposed a method for predicting dynamic responses of a complex structural assembly with nonlinearity at joints. In this paper, applicability and validity of the nonlinear structural modification/coupling method proposed recently by the authors [19] are demonstrated by applying it on a test structure and by comparing the experimentally measured nonlinear FRFs with those calculated using the method. The method proposed is for structural modification analysis problems where a linear system can be modified with a nonlinear subsystem for the following four cases: structural modification without additional degrees of freedom (DOFs), structural coupling with linear elements and structural coupling with nonlinear elements. In the previous work, the applicability of the method is demonstrated for all those cases by using several theoretical case studies. In this work, the proposed approach is applied to an experimental test system in which two linear structures are coupled by a nonlinear element. 23.2 Theory The structural modification method proposed by Özgüven [12] has recently been extended to nonlinear systems [19, 20]. The formulation was given for two different cases: structural modification without adding new DOFs to the original system, and structural modification which increases the total DOF of the original system (this formulation is also applicable to structural coupling problems). The resultant equations for the second case are given as follows [12]: H ba H ca D ŒI Œ0 Œ0 Œ0 C ŒHbb Œ0 Œ0 ŒI : ŒZ 1 ŒHba Œ0 (23.1) H bb H bc H cb H cc D ŒI Œ0 Œ0 Œ0 C ŒHbb Œ0 Œ0 ŒI : ŒZ 1 ŒHbb Œ0 Œ0 ŒI (23.2) [ ] [ ] [ ] 0 Haa Haa Hab Z = - * Hba * Hca * (23.3) [ ] [ ] [ ] [ ] 0 I Z = - Hab * Hbb * Hcb * Hbc * Hcc * Hac Hab * (23.4) where [H] and [H*] are the receptance matrices of the original and modified systems, respectively. Here, the subscript a represents the coordinates that belong to the original system only, the subscript b denotes connection coordinates, and the subscript c represents coordinates that belong to modifying system only. [Z] denotes the dynamic stiffness matrix of the modifying structure, which can be written for a linear modification case as follows: ŒZ DŒ K !2 Œ M Cj!Œ C CjŒ D (23.5) where [4K], [4M], [4C] and [4D] represent stiffness, mass, viscous and structural damping matrices of the modifying structure, respectively. For a nonlinear modification [Z] will be expressed as [19, 20]: ŒZ.X/ DŒ K !2 Œ M Cj!Œ C CjŒ D CŒ .X/ (23.6) where [4(X)] is the “nonlinearity matrix” whose elements are functions of unknown response amplitudes and can be written in terms of describing functions for any type of nonlinearity as [21]:
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