-0.4 -0.2 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0.5 0 Mode 1 -0.4 -0.2 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0.5 0 -0.5 Mode 2 -0.4 -0.2 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0.5 0 -0.5 Mode 3 -0.4 -0.2 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0.5 0 -0.5 Mode 4 Fig. 6 Triangular clamped plate: modes 1, 2, 3 and 4 Plates with curved edges A plate with curved parabolic edges as represented in Fig. 7 is considered. Since in Eq. 1 the number of boundary points with predefined coordinates should be increased according to the polynomial order of the edges, in this example the quadratic Lagrange function is adopted: 2 2 2 2 ( , ) ( )( ) /4 1,2,3,4 ( , ) ( )(1 )/2 5,7 ( , ) ( )(1 ) /2 6,8 ( , ) (1 )(1 ) 9 i i i i i i i i P i P i P i P i [ K [K [ [ K K [ K K K K [ [ K [ [ [ K [ K [ K (23) where the numbering convention is shown in Fig. 1. Introducing the coordinates of points 1 to 9 in Eq. 23 yields the coordinate mapping: 2 2 ( 1) (3 ) x y [ K K [ ° ® °¯ (24) Example 4. A free plate on all edges is considered, assuming Q = 0.3. In Tab. 4, the dimensionless frequencies computed using 6u6 and 12u12 free-free beam eigenfunctions (with NuN = 144 global dofs) are compared with those obtained using the finite element method (8463 dofs). Shapes of modes 4, 5, 6 and 9 are plotted in Fig. 8. The first three modes are rigid body motions. 84
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