File  
language 
eng

Author 
Hasegawa, Misao

Description  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_(2j1)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_(2j1) are taken follows : g_(2j1) = C_j cos(2j1)ξ on η= ± π/2 , = C_j cos(2j1)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. 
Journal Title 
島根大学論集. 自然科学

Volume  1

Start Page  25

End Page  34

ISSN  04886542

Published Date  19510331

NCID  AN0010814X

Publisher  島根大学

Publisher Aalternative  Shimane University

NII Type 
Departmental Bulletin Paper

OAIPMH Set 
Faculty of Science and Engineering
