厚さが半径方向に漸変する円板の振動

概要

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The vibrational equation of plate : [numerical formula] where E=Young's modulus, σ=Poisson's ratio, ρ=density, 2h=thickness varying along the radial direction, w=deflection and t=time, was led to ∇_1^2 [κ^3 (ζ) ∇_1^2u (ζ)]-λ^4κ (ζ) u (ζ)=0,under the substitution w=u (ζ) sin nθsin pt, ζ=r/a (a=outer radius of disc), h=h_0κ (ζ), λ^4=3/2×(1-σ^2) ρp^2 : Eh_0^2/a^4,∇_1^2=d^2/dζ^2+ζ^<-1>d/dζ-n^2ζ^<-2>. Assuming [numerical formula], the general solution for u was determined, and the frequency equation, under generalized boundary conditions, such as, α_<j1>u+α_<j2>u'+α_<j3>u"+α_<j4>u'''=0,(j=1,2,3,4) was established. The frequency equation, here given, contains the Prescott's results as a special case. Finally the authors describes an application to a lens-formed section : h=h_0 (1+αζ^2), concave and convex corresponding to the sign of α, where |α| is very small, resulting λ^4=104^・2+113^・8α, for the fundamental frequency of disc clamped at its circunference, and free at the centre.
一般社団法人日本機械学会の論文
1951-12-20