Acta Mech DOI 10.1007/s00707-013-1018-8 Roshan Lal · Rashmi Rani Mode shapes and frequencies of radially symmetric vibrations of annular sandwich plates of variable thickness Received: 28 June 2013 / Revised: 27 September 2013 © Springer-Verlag Wien 2013 Abstract An analysis and numerical results of radially symmetric vibrations of annular sandwich plates with core of linearly varying thickness are presented. The face sheets are treated as membranes of constant thickness, and the core is assumed to be solid as well as moderately thick. Due to linear thickness variation in the core, the face sheets take the shape of a truncated conical shell and because of this the face sheets membrane forces contribute to the bending and transverse shear of the core of the sandwich plate. Keeping this in view, the equations of motion for such a plate are developed by Hamilton’s energy principle. The frequency equations for three different combinations of boundary conditions, namely clamped at the inner edge and clamped or simply supported or free at the outer edge, are obtained by employing the differential quadrature method. The lowest three roots of these frequency equations have been reported as the frequencies for the first three modes of vibration. The effect of various plate parameters such as taper parameter, thickness of the core at the center, face thickness, and radii ratio on the natural frequencies has been analyzed. Three-dimensional mode shapes for a specified plate for all the three boundary conditions are illustrated. A comparison of results is presented. 1 Introduction Sandwich construction has been used for many years in aerospace industries and aviation as well as in marine and civil engineering applications due to their specific stiffness, light weight, and design versatility besides good damping characteristics and maximum fatigue resistance. In recent years, composite sandwich plates are widely used as structural elements in many other industrial products such as trains, ships, and housing components. In particular, such plates are not only used for aircraft interior components, e.g. overhead bins, floor panels, etc. but also for numerous exterior parts, e.g. radome, rudder, aileron, aerodynamic fairings, etc. [1, 2]. Often, these structural components are tapered for further weight reduction which necessitated to develop an analytic theory for sandwich plates of variable thickness. A sandwich essentially consists of two thin faces sandwiching a light core between them. A very popular type of core is the “honeycomb” core, other types being corrugated sheet, expanded materials such as cellulose acetate, synthetic rubber, and balsa wood. The selection of materials and dimensions of the face and core combination depends on the particular application. A number of researchers have been concerned with the dynamic behavior of sandwich structures. Yu [3–5] studied the vibrations of sandwich plates by considering the flexural theory of isotropic elastic plates. The effects of transverse shear deformation and rotatory inertia for the core as well as the facings were taken into account. In a later paper, Yu [6] simplified the earlier analysis by considering the face sheets as membranes for vibration of rectangular sandwich plates. Torsional vibrations of sandwich circular plates by taking facings as membranes have been investigated by Yu and Koplik [7]. Mirza and Singh [8] analyzed the axisymmetric vibrations of circular sandwich plates by taking bending stresses in the facings but no stresses parallel to the R. Lal · R. Rani (B ) Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247667 UK, India E-mail: rani.msrashmi9@gmail.com