where N[ and NK are the number of beam eigenfunctions in the [ and K parametric directions, respectively. If an integer n is assigned at any combination i, j, then the flexural displacement Eq. 10 can be rewritten as: 1 ( , ) N n n n w q w I [ K 7 ¦ q I (11) where N = N[uNK and q is the generalized coordinate vector. Introducing the displacement expansion Eq. 11 in the quadratic functional Eq. 5, and imposing the stationarity of the potential energy [13]: w3 w 0 q (12) yields the approximate natural eigenfrequencies of the plate by solving the algebraic eigenproblem: [ ] ' Mq K K q 0 (13) where: T T T T T ( ) [ ( ) 2(1 )( )] ( ) s h xx xx yy yy xx yy yy xx xy xy s s dS D dS dS U Q Q N 7 7 ' ³ ³ ³ M K K II I I I I I I I I I I II (14) are the mass matrix and stiffness matrix, computed according to Eqs. 8, 6 and 9 respectively. The integrals in Eq. 14 can be computed in the natural coordinates [ and K through Eqs. 3 and 4. In particular, the stiffness matrix K can be computed by means of the following expression: ȉ ˆ ˆ ˆ [ ] [ ] det[ ] s D d d [[ KK [K [ K [ K ³ K L J I I I I I I I I (15) where the elements in the symmetric 5u5 matrix L are given by: 2 2 2 11 11 21 11 21 31 2 2 2 22 12 22 12 22 32 2 2 2 33 13 23 13 23 33 2 2 2 44 14 24 14 24 34 2 2 2 55 15 25 15 25 35 12 21 11 12 21 22 11 22 12 L C C 2CC 2(1)C L C C 2CC 2(1)C L C C 2CC 2(1)C L C C 2CC 2(1)C L C C 2CC 2(1)C LL CCCC (CCCC Q Q Q Q Q Q Q Q Q Q Q 21 31 32 13 31 11 13 21 23 11 23 13 21 31 33 14 41 11 14 21 24 11 24 14 21 31 34 15 51 11 15 21 25 11 25 15 21 31 35 23 32 12 13 22 23 12 23 ) 2(1 )C C L L CC CC (CC CC) 2(1 )CC L L CC CC (CC CC) 2(1 )CC L L CC CC (CC CC) 2(1 )CC L L C C C C (C C Q Q Q Q Q Q Q Q 13 22 32 33 24 42 12 14 22 24 12 24 14 22 32 34 25 52 12 15 22 25 12 25 15 22 32 35 34 43 13 14 23 24 13 24 14 23 33 34 35 53 13 15 23 25 13 C C ) 2(1 )C C L L CC CC (CC CC) 2(1 )CC L L CC CC (CC CC) 2(1 )CC L L CC CC (CC CC) 2(1 )CC L L C C C C (C Q Q Q Q Q Q Q Q 25 15 23 33 35 45 54 14 15 24 25 14 25 15 24 34 35 C CC) 2(1 )CC L L CC CC (CC CC) 2(1 )CC Q Q Q (16) Each element of matrix L is a functions of Poisson’s coefficient and of the elements in matrix C, Eq. 3. 80
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