Resonant effects of local loads on circular sandwich plates on an elastic foundation

Abstract

The axisymmetric forced vibrations of a circular sandwich plate on an elastic foundation are studied. The plate is subjected to axisymmetric surface and mechanical loads with frequency equal to one of the natural frequencies of the plate. The foundation reaction is described by the Winkler model. To describe the kinematics of an asymmetric sandwich, the hypothesis of broken normal is used. The core is assumed to be light. The analytical solution of the problem is obtained and numerical results are analyzed Keywords: circular sandwich plate, axisymmetric forced vibrations, Winkler elastic foundation, axisymmetric surface, mechanical loadIntroduction. Analytic and numerical results on the vibrations of circular sandwich plates not bonded to an elastic foundation were obtained in [1][2][3]. The nonlinear vibrations of layered plates were studied in [4]. The quasistatic deformation of sandwich structures on an elastic foundation was studied in [5]. The natural vibrations of a sandwich rod bonded to a Winkler foundation were studied in [6].We will consider an asymmetric (throughout the thickness) circular elastic sandwich plate bonded to an elastic foundation and undergoing small axisymmetric transverse vibrations excited by surface and mechanical resonant loads, i.e., loads with frequency equal to one of the natural frequencies of the plate.1. Problem Formulation. We will use a cylindrical coordinate system r, j, z (Fig. 1) fixed to the midsurface of the core. For isotropic face layers of thickness h 1 and h 2 , the Kirchhoff hypotheses are accepted. The incompressible core (h 3 = 2ñ) is light, i.e., the work of the tangential stresses s rz in the tangential direction can be neglected. The deformed normal of the core remains straight, but turns through some additional angle y. The displacements at the boundaries of the layers are continuous. There is a rigid diaphragm at the edge of the plate to prevent the relative movement of the layers. The external vertical load does not depend on the coordinate j: q = q(r, t). The outside surface of the second face layer sustains the reaction q R of the elastic foundation.For symmetry reasons, the tangential displacements of the layers are zero, and the deflection w of the plate, the relative shear y in the core, and the radial displacement u of the coordinate surface do not depend on the coordinate j, i.e., u(r, t), y(r, t), w (r, t). Hereafter these functions are unknown. The thickness and density of the kth layer are denoted by h k and r k , respectively.The relationship between the reaction and the deflection is described by the Winkler model (q w R = k 0 , k 0 is the foundation modulus (modulus of subgrade reaction)).The system of partial differential equations describing the forced transverse vibrations of a circular sandwich plate not bonded to an elastic inertialess foundation and disregarding the reduction and rotary inertia of the normal in the layers is obtained from Lagrange's variational principle taking into account the variation of the work of inertial forces [1,3...

Impact of thermal and ionizing radiation on a circular sandwich plate on an elastic foundation

“…The system of differential equations describing the transverse vibrations of the plate and disregarding the reduction and rotary inertia of the normal in the layers is obtained from Lagrange's variational principle. This system coincides with that presented in [16] and describing the free vibrations of a similar plate on an elastic foundation (1.4) in the absence of external physical fields: where a subscript comma followed by a subscript variable indicates differentiation with respect to that variable; the coefficients a i and differential operators L 2 and L 3 are defined by …”