Semi-Exact Natural Frequencies for Kirchhoff–Love Plates Using Wave-Based Phase Closure

Abstract

This paper presents semi-exact, closed-form algebraic expressions for the natural frequencies of Kirchhoff–Love plates by analyzing plane waves, their edge reflections, and their phase closure. The semi-exact nature is such that the analysis exactly satisfies plate boundary conditions along each edge when taken in isolation, but not fully when combined, and thus is approximate near a corner. As frequency increases, the expressions become increasingly more accurate. For clamped square plates, closed-form expressions are reported in algebraic form for the first time. These expressions are developed by tracing the path of plane waves as they reflect from edges while accounting for phase changes over a total trip. This change includes phase addition/subtraction due to edge reflections. A natural frequency is identified as a frequency in which three phase changes (in the plate's horizontal, vertical, and path directions) each sum to an integer multiple of 2π, enforcing phase closure along each direction. A solution of the subsequent equations is found in closed form, for multiple boundary conditions, such that highly convenient algebraic expressions result for the plate natural frequencies. The expressions are exact for the case of all sides simply supported, while for other boundary conditions, the expressions are semi-exact. For the practically important and difficult case of a fully clamped plate, the expressions for a square plate yield the first 20 nondimensional natural frequencies to within 0.06% of their exact values.