35 Spectral Element Method for Cable Harnessed Structure 379 where fagD AB C D T and ŒD.!/ D 2 6 6 4 1 0 1 0 0 ˇ 0 ˇ cosˇL sinˇL coshˇL sinhˇL ˇsinˇL ˇcosˇL ˇsinhˇL ˇcoshˇL 3 7 7 5 (35.8) And the transverse shear force and bending moments at the nodal points are given by ffgD 8 ˆ < ˆ : V.0/ M.0/ V.L/ M.L/ 9 > = > ; DEI 2 6 6 4 0 ˇ3 0 ˇ3 ˇ2 0 ˇ2 0 ˇ3 sinˇL ˇ3 cosˇL ˇ3 sinhˇL ˇ3 coshˇL ˇ2 cosˇL ˇ2 sinˇL ˇ2 coshˇL ˇ2 sinhˇL 3 7 7 5 fag (35.9) where ŒF.!/ DEI 2 6 6 4 0 ˇ3 0 ˇ3 ˇ2 0 ˇ2 0 ˇ3 sinˇL ˇ3 cosˇL ˇ3 sinhˇL ˇ3 coshˇL ˇ2 cosˇL ˇ2 sinˇL ˇ2 coshˇL ˇ2 sinhˇL 3 7 7 5 (35.10) From the Eqs. (35.8), (35.9) and (35.10), the relationf andd can be obtained such as ŒS fdgDffg; ŒS DŒF ŒD 1 (35.11) Finally, the spectral element matrix of Euler Bernoulli Beam is given by ŒS DEI 2 6 6 4 S11 S12 S13 S14 S12 S22 S23 S24 S13 S23 S33 S34 S14 S24 S34 S44 3 7 7 5 (35.12) where D 1 1 cos.ˇL/cosh.ˇL/ S11 DS33 D .ˇL/ 3 .cos.ˇL/sinh.ˇL/ Csin.ˇL/cosh.ˇL// S22 DS44 D ˇL3 . cos.ˇL/sinh.ˇL/ Csin.ˇL/cosh.ˇL// S12 D S34 D ˇ 2L3 sin.ˇL/sinh.ˇL/ S13 D .ˇL/ 3 .sin.ˇL/ Csinh.ˇL// S14 D S23 D ˇ 2L3 . cos.ˇL/ Ccosh.ˇL// S24 D ˇL3 . sin.ˇL/ Csinh.ˇL// (35.13) After obtaining the spectral element matrix, we can assemble elements to generate the global spectral matrix. And we apply the boundary conditions to the global system. The global system can be expressed by ŒSg fdggDffgg (35.14) Owing to usage of the exact dynamic stiffness matrix to formulate the spectral element matrix, we can solve exactly the system characteristics with minimum number of element matrix. Especially, single Euler-Bernoulli beam with free-free boundary condition case, we can obtain the exact solution with only one element and no application of boundary condition. Now, we can calculate the natural frequencies by solving the eigenvalue problem for spectral element model given by Sg ˚dg D0 (35.15)
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