2 ï ï ï ï ï ï î ï ï ï ï ï ï í ì = ÷ ÷ ø ö ç ç è æ - = - = ÷ ÷ ø ö ç ç è æ - - + = - = þ ý ü î í ì + + + + + - + + + - = = þ ý ü î í ì + + + + + + + = ò + - + 0 0 3 1 2 0 2 0 2 0 1 cos 2 2 1 0 2 2 3 0 2 0 2 2 2 1 2 2 2 2 2 2 2 2 1 2 2 2 2 2 sin 4 ) 3 (1 cos ( ) 1 cos 2 3(4 1) 0 2 (2 1)[(8 8 3) 2(4 2 1) ] 4 ( 1) (4 4 1) 0 (8 8 3) 2(4 2 1)} (4 3)(2 1) 0 q n q q p m l m m l m l q a a E Fa a a a a P x dx x a n F a a E C n n n n n n n n n a E C A n n n n n n a E A n n n n n n n n n n (4) Free Vibration Problems of Spheres Instead of using wave potential approach (Chau [11]), the Helmholtz decomposition theorem will be used here to solve the following "reduced dynamics problem" (Achenbach [12]): ( ) 2 2 2 t V V V ¶ ¶ + ÑÑ× - Ñ´Ñ´ = r r r r m l m (5) 0 0 = = s q s r rr on r a = ; i iV U= , 0 = ¶ ¶ t Vi at t=0 In particular, we can assume the displacement vector as y j r r = Ñ +Ñ ´ V (6) These scalar and vector potentials satisfy 2 2 2 1 2 1 t c ¶ ¶ Ñ = j j 2 2 2 2 2 1 t c ¶ ¶ Ñ = y y r v (7) For our axisymmetric problem, the second of these becomes ( ) ( y ) y = ¶ ¶ y = q y Ñ y - . . 0,0, 1 sin 2 2 2 2 2 2 2 r ie t c r (8) The general solutions for and are: ( ) () f t r P c k A r J n n n n n å ¥ = + - ÷ ÷ ø ö ç ç è æ = 1 2 1 1 2 1 cosq j , ( ) () f t r P c k B r J n n n n n å ¥ = + - ÷ ÷ ø ö ç ç è æ = 1 1 2 2 1 2 1 cosq y (9) Substitution of these solutions to the displacement-strain and stress-strain relations leads to å ¥ = s = m 1 ( , ) ( ) ( ) 2 n n n n n rr A F r k P x f t (10)
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