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

Nonsmooth modal analysis of a varying cross-sectional area bar in unilateral contact

“…In NBM, the FE approximation is obtained by plugging the approximations (11) into the complementarity conditions (3) such that…”

“…To clarify, sticking phases stand in contrast to chattering exhibited by schemes utilizing a Newton impact law with e D 1 [1]. This is a noteworthy property of this scheme since sticking phases occur in the true solution to the Signorini problem [1,11,12,14].…”

“…While the problem of modal analysis of the bar in unilateral contact with a constant cross-section, A 0 .x/ D 0, has been studied before both analytically and numerically [1,11,12,20], the same techniques could not be implemented for the case of A 0 .x/ ¤ 0. Namely, analytical techniques have relied on the exact solution to the wave equation to describe both inactive and active contact phases in a closed-form manner [1,11,20].…”

“…While the problem of modal analysis of the bar in unilateral contact with a constant cross-section, A 0 .x/ D 0, has been studied before both analytically and numerically [1,11,12,20], the same techniques could not be implemented for the case of A 0 .x/ ¤ 0. Namely, analytical techniques have relied on the exact solution to the wave equation to describe both inactive and active contact phases in a closed-form manner [1,11,20]. In contrast, the WFEM, in the case of the bar of uniform area, exhibits properties favoring the existence of periodic solutions such as: energy conservation and preservation of characteristic quantities [13].…”

“…The dynamics of the bar are formulated via the wave equation with Signorini conditions for treatment of unilateral contact conditions [9,10]. While periodic solutions have been obtained analytically for the case of the bar with constant area [1,11], the same analytical method cannot be generalized to the case of the bar in varying area as a closed-form solution to the wave equation with varying coefficients has yet been obtained. Therefore, periodic solutions to the Signorini problem must be obtained numerically.…”