36 Vibrations of Discretely Layered Structures Using a Continuous Variation Model 387 36.3.1 Frequencies Assuming a vibration formui DSi(x)sin(!t) leads to: @2Si @x2 C !2 c2 i Si D0; i D1;2: : : (36.3) The solutions of which are S1 DB1 cos ! c1 x CB2 sin ! c1 x S2 DB3 cos ! c2 x CB4 sin ! c2 x (36.4) Two sets of boundary conditions are investigated, namely, fixed-fixed and free-fixed. Fixed-fixed conditions and interface continuity gives: S1 D0;x D0I S2 D0;x DL .LDL1 CL2/I S1 DS2;x DL1I (36.5) Whereas, force continuity leads to: E1 @S1 @x DE2 @S2 @x ;x DL1 (36.6) These four conditions lead to a system of algebraic equations inB1, B2, B3 andB4: B1 D0 B3 cos ! c2 L CB4 sin ! c2 L D0 B2 sin ! c1 L1 DB3 cos ! c2 L1 CB4 sin ! c2 L1 E1c2 E2c1 B2 cos ! c1 L1 D B3 sin ! c2 L1 CB4 cos ! c2 L1 (36.7) The natural frequencies are found on setting the determinant of the coefficients to zero. After some lengthy manipulations, one finds: det 2 6 4 0 cos.cr!d/ sin.cr!d/ sin !d 1C˛ cos cr !d 1C˛ sin cr !d 1C˛ mrcr cos !d 1C˛ sin cr !d 1C˛ cos cr !d 1C˛ 3 7 5D0 (36.8) where L2 D˛L1, !d D ! c1 L, cr D c1 c2 andmr D 1 2 , with LD(1C˛)L1 and E1c2 E2c1 Dmrcr. For free-fixed boundary conditions and interface continuity: dS1 dx ˇ ˇ ˇ ˇxD0 D0I S2 D0;x DL .LDL1 CL2/I S1 DS2;x DL1I (36.9) Note that force continuity, Eq. 36.6, still applies. Then the corresponding system of algebraic equations is: B2 D0
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