The representation of periodic vibration—impact processes via pulse—phase motion parameters

The Analysis of Non-Newtonian Vibro-Impact Processes in Tube Constructions and Systems with Parallel Impact Pairs

“…In accordance with the method, we can approximate the forces from equation (8) as a singular generalized function. In particular, for T-periodic process under the assumption of a single interaction for the period of movement in each impact pair: (10) where t n is a fixed moment of the beginning of interaction in the n-th impact pair (phase); furthermore, -half time of interaction in this pair determined from the previously obtained formulas [10], [6], [11] (see also below). Representation (9)- (10) is caused by the fact that, first, we find the modes of motion namely of periodic type in a conservative system, and, secondly, the forces of interaction, as in the case of hypothesis of momentary impact, are given by equations of motion containing singular generalized functions.…”

“…The representation of PFG-s is useful in their final form at intervals of periodicity (symmetry) -see [4], [6], [9] and below.…”

“…Following the first works, this approach was later developed in the book [4], while for systems with multiple impact pairs -in [6]. We fix the hypothesis of elastic interaction in each impact pair in form (3) and consider problem (1) [ (2)] using the method of singularisation.…”

“…PFG properties are described in detail in [4], [6]; they represent the response of linear systems to periodic sequence of Dirac δ-functions (Dirac comb):…”

“…The impact force, in this case, is written using a singular generalized function [1], [3], [4], [6]. However, if it is important to take into account either finite duration of impact force or energy losses during the interaction, one should take into account some theories of contact damping based, for example, on the material properties of the interacting bodies, and here it is necessary to abandon assumptions about the momentary nature of the interaction.…”