On the Transverse Vibration of a Square Plate with Four Clamped Edges

3. The problem of calcuating the frequency of the fundamental mode of transverse vibration of a square plate with clamped edges is equivalent to a minimum problem of computing the minimum value of the expression :

V(W)/T(W) ≡ (∫∫__s(⊿W)^2dξdη)/(∫∫__s[W^2 + 2g^2{((∂W)/(∂ξ))^2 + ((∂W)/(∂η))^2}]dξdη) (10)

for all function W (ξ, η) which have continuous derivatives up to the fourth orderin the square S : │ξ│≦π/2, │η│≦π/2 and which also satisfy the clamped edges conditions :

W = 0, (∂W)/(∂n) = 0 (11)

at the bonndary C : ｜ξ｜= π/2, ｜η｜ = π/2 of the square, the double integrals being over the square S and n denoting the normal to the boundary.
In the present paper, we consider the modified minimum problem which may be expressed as follows :
It is required to find the minimum value of the expression :
(V(v))/(T(v)) = (∫∫__s(⊿v)^2dξdη)/(∫∫__s[v^2 + 2g^2{((∂v)/(∂ξ))^2 + ((∂v)/(∂η))^2}]dξdη)
for all functions v (ξ,η) which vanish on the boundary C of the square and satisfy the following boundary m conditions on C :

G_<2∫-1> =∫__o(∂v)/(∂n)g_(2j-1)ds=0 , (j=1,2,...,m)

where ds is a line element along C so that ds=d ξ on η= ± π/2 , and ds = d ξd=dη on ξ = ± π/2 . The double integrals are taken over the square S, while the single integrals are taken along the boundary C and functions g_(2j-1) are taken follows :

g_(2j-1) = C_j cos(2j-1)ξ on η= ± π/2 ,
= C_j cos(2j-1)η on ξ= ± π/2 ,
where the Ci's are certain constants.
Applying to the modified problem the general principle in the calculus of variation, a non decreasing sequence of lower limits for the true minimum value of k^2 = (ρha^4p^2)/(Dπ＾4) is calculated for h/a = 0.1, 0.2 and 0.3. From these sequences we find the Table IV, giving the frequency p of the fundamental mode of transverse vibration of a square plate with clamped edges.
Table IV.
h/a 0.1 0.2 0.3
(ρha^4p^2)/(Dπ＾4) 13.037 12.308 11.260

In conclusion, I wish to thank Prof. Tomotika for his encouragement during this work.