ANALYTIC STUDY OF DISCRETENESS EFFECTS IN A STRING RESTING ON A PERIODIC ARRAY OF VIBRO-IMPACT SUPPORTS

摘要

We analyze the forced oscillations of an infinite stringrnsupported by an array of vibro-impact supports. The envelopernof the excitation possesses 'slow' and 'fast' scales and is periodicrnwith respect to the 'fast' scale. The 'fast' spatial scale is definedrnby the distance between adjacent nonlinear supports. Torneliminate the singularities from the governing equations ofrnmotion that arise due to the discrete nature of the supports, wernemploy the nonsmooth transformations of the spatial variablernfirst introduced in (Pilipchuk, 1985) and (Pilipchuk, 1988).rnThus, we convert the problem to a set of two nonhomogeneousrnnonlinear boundary value problems which we solve by means ofrnperturbation theory. The boundary conditions of these problemsrnarise from 'smoothness conditions' that are imposed tornguarantee sufficient differentiability of the results. Therntransformed system of equations is simplified using regularrnperturbation and harmonic balancing. Standing solitary wavernsolutions reflecting the discreteness effects inherent in therndiscrete foundation are calculated numerically for the unforcedrnsystem.