D’Alembert function for exact non-smooth modal analysis of the bar in unilateral contact

• Published in: Nonlinear analysis. Hybrid systems Vol. 43; p. 101115
• Main Authors: Urman, David, Legrand, Mathias, Junca, Stéphane
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.12.2021; Elsevier
• Subjects: D’Alembert function; Engineering Sciences; Mechanics; Method of steps; Neutral Delay Differential Equation; Non-smooth modal analysis; Periodic solutions; Signorini complementarity conditions; Structural mechanics; Unilateral contact; Vibration analysis; Unilateral contact; D’Alembert function; Neutral Delay Differential Equation; Non-smooth modal analysis; Periodic solutions; Method of steps; Vibration analysis; Signorini complementarity conditions; d'Alembert function; unilateral contact; nonsmooth modal analysis; vibration analysis
• ISSN: 1751-570X
• DOI: 10.1016/j.nahs.2021.101115

Non-smooth modal analysis is an extension of modal analysis to non-smooth systems, prone to unilateral contact conditions for instance. The problem of a one-dimensional bar subject to unilateral contact on its boundary has been previously investigated numerically and the corresponding spectrum of vibration could be partially explored. In the present work, the non-smooth modal analysis of the above system is reformulated as a set of functional equations through the use of both d’Alembert solution to the wave equation and the method of steps for Neutral Delay Differential Equations. The system features a strong internal resonance condition and it is established that irrational and rational periods of vibration should be carefully distinguished. For irrational periods, it was previously proven that the displacement field of the non-smooth modes of vibration is characterized with piecewise-linear functions in space and time and such a motion is unique for a prescribed energy. However, for rational periods, which are the subject of this work, new periodic solutions are found analytically. Findings consist of families of iso-periodic solutions with piecewise-smooth displacement fields in space and time and continua of piecewise-smooth periodic solutions of the same energy and frequency.