Table 1 Example 1, dimensionless frequency parameter O2 ( Q = 0.3, D = S / 6, a = b = 1, r = 0.3) Present approach Mode 6u6 dofs 8u8 dofs 10u10 dofs 12u12 dofs FEM 1 15.013 14.873 14.814 14.773 14.652 2 16.671 16.397 16.295 16.234 16.012 3 53.532 51.883 51.157 50.841 49.563 4 60.943 57.217 56.872 56.741 55.967 5 62.272 60.392 59.644 59.317 57.900 1715 elements, 5304 nodes, 15912 dofs. 0 0.5 1 1.5 0 0.5 1 -0.5 0 0.5 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 -0.5 0 0.5 Fig. 3 Skew plate on annular support ( Q = 0.3, D = S / 6, a = b = 1, r = 0.3): modes 1 (left) and 3 (right) Trapezoid and triangular plates A trapezoid plate as represented in Fig. 4 is considered. The interpolation functions are linear, as in Eq. 18. Introducing the coordinates of points 1 to 4 yields the coordinate mapping: > @ ^ ` > @ 1 2 1 2 1 2 1 2 1 2 1 2 ( ) ( 2 ) ( ) ( 2 ) /4 ( ) ( ) (1 )/4 x a a a a a a a a y b b b b H [ H [ K [ K ° ® °¯ (21) Letting H o 0 yields the coordinate mapping for a triangle (Fig. 4): > @ > @ 1 2 1 2 1 2 1 2 ( ) ( ) (1 ) /4 ( ) ( ) (1 ) /4 x a a a a y b b b b [ K [ K ° ® °¯ (22) Note that the determinant of the Jacobian matrix becomes singular for K o –1, however not affecting the results in the case of zero displacement in point 1. y x 1 (–H, 0) 2 (0, H) 4 (– a2, b2) 3 (a1, b1) y x 1 (0, 0) {2 4 (– a2, b2) 3 (a1, b1) Fig. 4 Trapezoid (left) and triangular (right) plate 82
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