Parametric oscillation of the rectangular thin plate on the elastic foundation with two coefficients1 when making mention of the creep of material, has been investigated in earlier publications (see for example [2, 3]). However forced oscillation of the rectangular thin plate, to the author's knowledge, has not been hitherto examined. | T~p chi CO' h9c Journa1 of Mechanics, NGNST of Vietnam T. XVI, 1994, No 2 (13- 19) FORCED OSCILLATION OF THE RECTANGULAR THIN PLATE ON THE ELASTIC FOUNDATION WITH TWO COEFFICIENTS HOANG VAN DA The Technical University of Mining and Geology Hanoi, SRV §0. INTRODUCTION Parametric oscillation of the rectangular thin plate on the elastic foundation with two coefficients1 when making mention of the creep of material, has been investigated in earlier publications (see for example [2, 3]). However; forced oscillation of the rectangular thin plate, to the author's knowledge, has not been hitherto examined. ,This problem is studied here by means of an asymptotic method for high-order systems [1] and boundary value problem [4]. §1. FORMULATION OF THE PROBLEM. THE EQUATION OF MOTION Now, let's determine forced oscillation of a rectangular thin plate, having thickness h, Young's modulus E, specific mass M and lengths of edges b, c, which is supported on four edges and lying on the elastic foundation with two coefficients as shown in Fig. 1. Its motion is loaded by direction force, equally distributed q = q( t) 0 ;1 7 ;; ~ 2 b - E {) =O, The solution of this problem can be found in the form Wo(x, y, t) = Z(x, y)T(t) 14 {) Substituting () into () and () we obtain () () zl zl aByzl 8x =o, a2 z + "a2 z I 2 2 x==O,b y::::O,c =O, aZ --+v-2 2 By2 -o ' x=O,b - Bx 2 = () o. y=O,c It is easy seen that the solution () takes form 00 W0 (x, y, t) = L A,Z,,,(x, y) cos¢,+ Zrs(x, y) • = sm T'll"X S1ry -b- sin-,-, L D,.Z,,,(x, y)e-,,, = (fl,t + .;.,), () where A,. 11 , Dr,, 1/Jrll are positive constants determined from the initial conditions () It is supposed that when e = 0 there exists a periodic solution with frequency flu () and there is a resonance relation () 6 is the detuning coefficient. With these assumptions, we are going to find the partial solution of the boundary value problem (), () .
