Example 2. A trapezoid plate is considered, clamped on three edges and free on the longest one, with parameters H = 1, a1 = a2 = 0.5, b1 = b2 = 3/2,and Q = 0.3. In Tab. 2, the dimensionless frequencies O 2 computed using 6 (clamped-clamped) u 6 (clamped-free) eigenfunctions (36 global dofs) are compared with those obtained using the finite element method (14883 dofs). Shapes of modes 3 and 4 are plotted in Fig. 5. Table 2 Example 2, dimensionless frequency parameter O2 ( Q = 0.3, H = 1, a 1 = a2 = 0.5, b1 = b2 = 3/2). Mode Present approach [6u6 dofs] FEM 1 10.715 10.632 2 24.288 24.061 3 35.517 35.283 4 45.028 44.452 5 53.611 53.181 6 70.977 69.981 7 78.671 77.493 8 91.418 90.758 9 101.094 100.001 10 105.546 101.320 1600 elements, 14883 dofs. -1 -0.5 0 0.5 1 0 0.5 1 -0.5 0 0.5 Mode 3 -1 -0.5 0 0.5 1 0 0.5 1 -0.5 0 0.5 Mode 4 Fig. 5 Trapezoid clamped-free plate: modes 3 and 4 Example 3. A triangular plate is considered, clamped on all three edges, with a1 = 1/ 2, a2 = 1/ 6, b1 = 1/ 2, b2 = 1/ 6, and Q = 0.3 (angles D12 = 90°, D3 = 30°, D4 = 60°). In Tab. 3, the dimensionless frequencies O 2 computed using 6u6 (clampedclamped) eigenfunctions (36 global dofs) are compared with those reported in Ref. [15], computed using boundary characteristic orthogonal polynomials in two dimensions as shape functions. Note that the singularity in K = –1 does not affect the results. Shapes of modes 1, 2, 3 and 4 are plotted in Fig. 6. Table 3 Example 3, dimensionless frequency parameter O2 ( Q = 0.3, a 1 = 1/ 2, a2 = 1/ 6, b1 = 1/ 2, b2 = 1/ 6). Mode Ref. [15] Present approach [6u6 dofs] 1 176.58 176.618 2 280.95 280.046 3 380.92 380.058 4 416.53 412.145 5 558.48 532.878 83
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