Galerkin–Ivanov transformation for nonsmooth modeling of vibro-impacts in continuous structures

Abstract

This work deals with the modeling of nonsmooth vibro-impact motion of a continuous structure against a rigid distributed obstacle. Galerkin’s approach is used to approximate the solutions of the governing partial differential equations of the structure, which results in a system of ordinary differential equations. When these ordinary differential equations are subjected to unilateral constraints and velocity jump conditions, one must use an event detection algorithm to calculate the time of impact accurately. Event detection in the presence of multiple simultaneous impacts is a computationally demanding task. Ivanov (Ivanov A 1993 “Analytical methods in the theory of vibro-impact systems”. Journal of Applied Mathematics and Mechanics 57(2): pp. 221–236.) proposed a nonsmooth transformation for a vibro-impacting multi-degree-of-freedom system subjected to a single unilateral constraint. This transformation eliminates the unilateral constraints from the problem and, therefore, no event detection is required during numerical integration. This nonsmooth transformation leads to sign function nonlinearities in the equations of motion. However, they can be easily accounted for during numerical integration. Ivanov used his transformation to make analytical calculations for the stability and bifurcations of vibro-impacting motions; however, he did not explore its application for simulating distributed collisions in spatially continuous structures. We adopt Ivanov’s transformation to deal with multiple unilateral constraints in spatially continuous structures. Also, imposing the velocity jump conditions exactly in the modal coordinates is nontrivial and challenging. Therefore, in this work, we use a modal-physical transformation to convert the system from modal to physical coordinates on a spatially discretized grid. We then apply Ivanov’s transformation on the physical system to simulate the vibro-impact motion of the structure. The developed method is demonstrated by modeling the distributed collision of a nonlinear string against a rigid distributed surface. For validation, we compare our results with the well-known penalty approach.

Nonsmooth modal analysis via the boundary element method for one-dimensional bar systems

“…However, we recognize that the systems considered are of limited interest in the industrial sphere even though NSA was initially motivated by aerospace applications [12]. It also has ramifications in areas like music instruments [8], breathers [9] or applied mathematics [20] to cite a few. More generally, vibration analysis is commonly conducted during the design of a mechanical component and it is now recognized that unilaterally contact conditions, when unavoidable, strongly affect the dynamics and cannot be ignored [21].…”

“…Discretization comes into the proposed solution strategy when the Fourier expansions (20) are truncated to a finite number m of harmonics, such that we define u .m/ .1; t/ u.1; t/ and p .m/ .1; t/ p.1; t/. A second level of discretization lies in the computation of the integrals (29).…”