Abstract
To calculate the frequencies of free vibrations is very bothersome, but from the characteristic values concerned with oscillatory solutions the character of oscillation, and insufficiency of many other approximation methods can be shown. If we set order of Legendre function as follows, [numerical formula] we can get the following three dimensional polynomial of λ from the expression of stress function. λ^3+{4+(1-ν^2)Ω^2}λ^2+k(1-Ω^2)λ+k{2+(1+3ν)Ω^2-(1-ν^2)Ω^4}=0 and from compensated solutions λ_4=-{2+2(1+ν)Ω^2} λ can be considered as characteristic value of free vibrations with parameters [numerical formula] 1) λ: plus real number monotoneous damping 2) λ: minus real number harmonic (non-damping) oscillation 3) λ: complex conjugate oscillatory damping It becomes apparent with these values λ_i that to neglect the inertia terms occured by displacements u, v, to approximate as inextentional or extentional vibrations are nonreasonable, and lowest frequency may be assumed as near value of Ω^2=1. Finally, we may find the approximation of solutions, with which we can calculate free vibration frequencies rather easily.
