Chapter 41 Modal Analysis of Axially Deforming Rods with Isolated LapJoints D. Dane Quinn Almost every modern engineering structure incorporates some form of mechanical interface, a connection between two otherwise separate mechanical structures. Complex machines and structures such as automobiles, bridges, aircraft, rockets, etc. rely heavily on these interfaces; however, high-fidelity numerical analysis of such connected structures is currently extremely difficult and computationally expensive due to the disparate length and time scales of the interface as compared to those characterizing the overall structure. This work describes recent developments in the reduced order modeling of structures with isolated interfaces. In particular, the response of the mechanical system with interfaces can be described in terms of the corresponding monolithic system, with the addition of isolated contributions of the interface that are assumed to be small compared to those that are derived from the monolithic system. As a result, the underlying modal structure of the monolithic system can be used to describe the response of the jointed structure. Further, the contributions from the interface can be described in terms of a physically motivated reduced-order model, so that nonlinear damping and mode coupling are an integral part of the observed response. This approach is applied to study the dynamic response of axially deforming lap joints. The results show that the reducedorder model is capable of reproducing experimentally observed characteristics of the system, including mode coupling and softening effects with increasing excitation. Further, the model easily incorporates both transient and the forced response of the system. Consider a uniform rod of length ` D1 that contains an internal joint located in the interval x 2 .x1; x2/ D.0:45; 0:55/ with a uniform frictional intensity at the interface [4]. Outside of the isolated joint interval the system is described by the equations of motion for the axial vibrations of a linearly elastic rod. However, within the isolated joint region microslip can occur due to the presence of the interface, and the governing equations are no longer linear within this region. The development of reduced order model for an elastic rod with a single interface, together with an expansion in terms of the modes of the monolithic system was developed previously in [3]. The response of the rod u.s; / can be expanded in terms of the spatial functions j.s/ as u.s; / D 1 XjD1 Aj. / j.s/; (41.1) leading to modal equations of the form @2Ai @ 2 . / C i @Ai @ . / C!2 i Ai. / D Ofi. / CıQŒ . / . i.s2/ i.s1// ; (41.2) where !i is modal frequency corresponding to the eigenmode i.s/, i is the ith modal damping coefficient, and Ofi. / is the modal forcing. If QJi .t/ represents the axial tension in the jointed structure at x Dxi and QMi .t/ is the axial tension in the monolithic structure at x Dxi, then this deviatoric force is defined as ıQi.t/ DQJi .t/ QMi .t/: D. Dane Quinn ( ) Department of Mechanical Engineering, The University of Akron, Akron, OH, USA e-mail: quinn@uakron.edu © The Society for Experimental Mechanics, Inc. 2019 G. Kerschen (ed.), Nonlinear Dynamics, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-74280-9_41 375
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