Exact Equations and Finding the Cutoff Frequencies of Functionally Graded Plates in Free Vibrations

In an explicit form, the frequency equations of cutoff frequencies for free vibrations of functionally graded plates whose elastic moduli depend on the transverse coordinate are considered. Plate faces are stress-free, rigidly clamped, or subjected to combined boundary conditions. It is shown that the corresponding left-hand sides of the equations are entire functions, which can be presented as a particular case of Peano series, and their respective mathematical estimates are given. It is also shown that the method suggested is effective and all integration stages can be easily realized by the present-day program packages of numerical and symbolic computations. The new final method is efficient for calculating the cutoff frequencies. They are verified on numerical examples and by comparing with results of the Wentzel–Kramer–Brillouin (WKB) method allowing one to find the asymptotics of high vibrations