Flexural vibrations of clamped-free rhombic plates with corner stress singularities. Part II: comparison of solutions

Abstract

An ample review of the published literature of skew (rhombic) plate vibrations in a companion Part I paper serves as a background that motivates the need for accurate solutions incorporating stress singularity-based methodologies for analyzing the titled problem. Such an accurate method is presented in this Part II paper. The prime focus here is that the vibration analysis explicitly considers the bending stress singularities that occur in the two opposite, clamped-free corners with obtuse angles of the rhombic plates. The strength of these singularities increases significantly, as the obtuse angles at the clamped-free corners exceeds 95o. A single-field energy-based Ritz procedure is employed with the dynamical energies derived from classical Kirchhoff thin-plate theory. The normal displacement of the rhombic plate is approximated as a hybrid series of (i) admissible and mathematically complete algebraic polynomials, and (ii) corner functions which account for both the kinematic boundary conditions and the bending stress singularities at the obtuse clamped-free corners. It is surmised from extensive convergence studies that the corner functions accelerate the convergence of upper bounds on the exact solutions, and that these functions are required if accurate solutions are to be obtained for highly skewed plates (including very thin ones incorporating shear deformable plate theories). Accurate non-dimensional frequencies and normalized contours of the vibratory transverse displacement are presented for rhombic plates having a large enough skew angle of 750 (i.e., obtuse angle of 1650), so that a significant influence of clamped-free corner stress singularities may be understood. Given the double symmetry axis of the rhombus domain, accurate solutions for a number of isosceles and right triangular plates (depending on the symmetry axis invoked) with various combinations of clamped, free, and sliding edges are also available from the frequency and mode shape data presented. Frequency data obtained from the present analysis are compared with nearly six decades of previously published data obtained using alternative theoretical plate analyses (including shear deformable ones) and classic experimental bench tests, clearly answering the inquiry of whether the stress singularities at the obtuse clamped-free corners of rhombic plates must be explicitly taken into consideration to obtain accurate three to four significant digit upper bounds on exact clamped-free skew (rhombic) plate vibration solutions reported in this work.