Steady-state transverse vibrations of sectional cylindrical anisotropic plates of variable thickness

Abstract

Using the perturbation method we solve the problem of the steady-state transverse vibrations of a plate of variable thickness consisting of circular rings made of different cylindrically anisotropic materials. Numerical studies are carried out for plates consisting of two rings with different laws of variation of thickness. We give the graphs of the distribution of values of the bending moments.2 Figures. Bibliography: 5 titles.The problem of the symmetric bending of a circular orthotropic plate of constant thickness was solved by S. G. Lekhnitskii [1]. N. M. Maslov [2] gives the solution of the problem of free axisymmetric vibrations of a circular structurally orthotropic plate of variable thickness. In the present article we use the approximate method of [3] to solve the problem of the steady-state vibrations of articulated plates of variable thickness whose materials have cylindrical anisotropy. Consider a plate consisting of circular rings S t (j = 1, k) with contours of radius Rm (m = 1, k + 1). The rings are made of different cylindrically anisotropic materials. They are welded or glued together along the corresponding surfaces without any preliminary distention. The plate is loaded with a transverse distributed load of intensity qt pulsing with frequency w.The equation of the steady-state vibrations of a plate of variable thickness in polar coordinates (r, 0) for each of the regions St, taking account of the axial symmetry, has the form [4] 02(rM( S)) t)Or 2 Or --where M~ (j) and M~ j) are the bending moments, Wj is the deflection, and pj and h I are the density of the material and the thickness of the ring S 1. In what follows the subscript j denoting membership in the region S 1 is omitted except where it is necessary. We assume the dependence between the moments and the deflection are given by [2] M~=-Dn2(W" + ~W'); Mo= "Dn2(vW" +n--2r W').(2)Here D = Eh3/12(n 2v 2) is the cylindrical stiffness of the plate, and E and v are the reduced Young's modulus and Poisson coefficient. The parameters n = x/'~" n2 can be found from relations [5]fi l TI'2 where Er, vr, Eo, and vo are the Young's moduli and the Poisson coefficient in the directions r and respectively. Substituting the expressions (2) into Eq. (1), we obtain w(") + "2w" • + (w" + D' r ,,, v + :w"-w'5 +=.2w + = 19\ x z 2 / q A4 + P hA4 2, n2 D n---i-ffD w vv.