4 Numerical examples Five example cases are considered, the first one regarding skew plates with non-standard boundary conditions, the second and third ones regarding trapezoid and triangular plates, the fourth one regarding plates with curved parabolic edges, the last one regarding elliptic annular plates. In all cases the dimensionless frequency parameters O2 are computed according to the above described technique. The natural frequencies are then given by: 2 2 h D a O Z U (17) where a is a reference-length parameter. The shape functions are selected in the form of products of beam eigenfunctions: free-free, clamped-clamped and clampedfree [12]. Skew plates A skew plate as represented in Fig. 2 is considered, with b = a and angle D. For a plate with straight edges, the interpolation functions can be defined in the form: ( , ) (1 )(1 )/4 1,2,3,4 i i i P i [ K [[ KK (18) in which [i and Ki are the natural coordinates of the i-th corner, as shown in Fig. 1. Introducing the coordinates of points 1 to 4 in Eq. 18 yields the coordinate mapping: > @ (1 ) sin (1 )/2 cos (1 ) /2 x y [ D K D K ° ® °¯ (19) In this case the determinant of the Jacobian matrix J is constant. y a x b D 4 3 2 1 y a x b D 4 3 2 1 r Fig. 2 Skew plate (left) and skew plate with annular support (right) Example 1. A free skew plate simply supported on an annular constraint as represented in Fig. 2 is considered, with D = S / 6, a = b = 1 and r = 0.3. The equation describing the shape of the support J, expressed in polar form, can be given in natural coordinates using Eq. 19: 2 (cos tan sin ) 2 sin /cos r r [ - D - K - D ® ¯ (20) The frequencies computed using 6u6 to 12u12 free-free beam eigenfunctions (assuming Q = 0.3 and N = 109uD/a3) are reported in Tab. 1, where they are compared with those computed with the finite element method (15912 dofs, using quadratic serendipity plane elements with 8 nodes each). Shapes of modes 1 and 3 are plotted in Fig. 3. 81
RkJQdWJsaXNoZXIy MTMzNzEzMQ==