y a x D 4 3 2 1 b rb ra Fig. 9 Sector or annular plate Sector and annular elliptic plates Circular plates are special cases of elliptic plates and are quite simple to analyze using polar coordinates, since the solution can be expressed in the form of Bessel functions for all nine cases for inner and outer boundary conditions [14]. When compared with the amount of information available for circular plates, studies on the vibration of elliptic plates are limited, especially on elliptic annular plates [15]. The main difficulty in studying elliptic plates is the choice of coordinates: elliptic coordinates can be used with the exact mode shape in the form of Mathieu functions, but they are quite cumbersome to handle [15-16]. In the present study, a general case is considered, consisting of a sector elliptic plate as represented in Fig. 9. Annular elliptic plates, sector circular plates and annular circular plates can be considered as special cases, and they can be solved following the same approach within the proposed technique. In this case the domain mapping can be given in polar coordinates: 2 2 2 2 [(1 ) ] cos( ) [0,1] 1 1 sin ( ) [(1 ) ] sin( ) [0,1] 1 1 sin ( ) a r r x k a r r y k [ DK [ DK [ DK K DK ° ° ® ° ° °¯ (25) where D (0,2 S] is the sector angle, a and b are the lengths of the major and minor semiaxes of the external ellipse, k = a/b is their ratio, aur and bur with r [0,1) are the lengths of the major and minor semiaxes of the internal ellipse. Clearly, D = 2 S yields a circular or elliptic annular plate, D = 2 S and r = 0 yields a circular or elliptic plate and k = 1 yields a circular annular plate. In the latter case Eq. 25 reduces to the simpler form: [(1 ) ]cos( ) [0,1] [(1 ) ]sin( ) [0,1] x a r r y a r r [ DK [ [ DK K ® ¯ (26) If D = 2 S (annular plate) the shape functions along the K coordinate can be selected as terms of the Fourier series: cos(2 ) 0,1,2... ( ) sin(2 ) 1,2,3... i i j j S K I K S K ® ¯ (27) Example 5. An elliptic annular plate ( D = 2 S ) is considered, clamped on the external edge and free on the internal one, with k = 2, a = 1, r = 0.4 and Q = 1/3. In Tab. 8, the dimensionless frequencies O2 computed using 4u11, 7u11 and 4u22 (44, 77 and 88 global dofs) eigenfunctions are compared with those reported in Ref. [15]. Note how the different composition of eigenfunction sets in the cases 7u11 and 4u22 affects the results: the 7u11 set yields better results on modes 2 and 3, while the 4u22 set gives better results on modes 1, 4, 5 and 6. Shapes of modes 1 to 6 are plotted in Fig. 10. 86
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