380 J. Choi and D.J. Inman Table 35.1 Natural frequecies (Hz) of exact solution, SEM and FEM !FEM !FEM !FEM Mode !exact !spec: n D10 n D30 n D50 1 0 0 0:00001 0:00009 0:0002 2 11:1794 11:1794 11:1798 11:1794 11:1794 3 30:8165 30:8165 30:8242 30:8166 30:8165 4 60:4127 60:4127 60:4683 60:4135 60:4128 5 99:8654 99:8654 100:1052 99:8687 99:8658 6 149:5272 149:1817 149:9387 149:1928 149:1831 7 208:3612 208:3612 210:2857 208:3915 208:3652 8 277:4040 277:4040 281:5237 277:4749 277:4134 Similar to the regular eigenvalue problem, we need to consider that determinant of spectral element matrix [Sg] is zero, det.S.!NAT// D0, to find the natural frequencies. However, the spectral element matrix consists of transcendental functions such as sine, cosine, hyperbolic cosine (cosh), and hyperbolic sine (sinh). We cannot use the linear eigensolver such as ‘eig’ in MATLAB. The several approaches to find the eigenvalues are summarized by Lee [9]. The determinant of global stiffness matrix [Sg] can be simplified as det Sg D E4I4ˇ8 .cos.2ˇL/ cos.ˇL/cosh.ˇL//sin2 .ˇL/sinh2 .ˇL/ 4. 1Ccos.ˇL/cosh.ˇL//3 (35.16) The conventional and spectral beam elements will be compared using a simple example. A free vibration of free-free beam will be analyzed using both types of elements. The material used in the example is aluminum with elastic modulus of 7 1010 N=m2 and density of 2;700kg=m3. The dimensions of beam are1:2192 0:0254 0:003175m. From the characteristic equation of free-free beam [1], cos.ˇL/cosh.ˇL/ D1, the exact natural frequencies are calculated. And the root-finding algorithms [15] are used to find the !n inSEM. Table 35.1 shows the natural frequencies of free-free beam. The SEM results are identical to exact solution. And the FEM results go closer to SEM and exact solution as the number of elements is increased. This tendency also can be notified by means of the receptance versus frequency graph in Fig. 35.1. In Fig. 35.1a, FEM gives 3 good natural frequencies. The number of elements is increased to 10 in Fig 35.1b. And the 5 natural frequencies are almost identical with SEM frequencies. Considering the high frequency range, the more elements are necessary to obtain the good results. 35.3 The SEM for Double Beam with a Free-Free Boundary Condition Most of papers considered the double beam such that two beams are connected with distributed spring connections, not in specific location. In this paper, we can define the exact number and locations of connections. Through this approach, the effect of number and locations of connection can also be identified. Considering the double beam system in Fig. 35.2 each beam is 2Nd.o.f system. And the spring connection exists between ith nodes of beam 1 and beam 2. For beam 1 and beam 2, we can formulate the global stiffness matrix by means of Eq. (35.12). From the relation between nodal displacement and force, the combined global stiffness matrix without spring connection can be shown as Sg;1 0 0 Sg;2 ˚dg;1 ˚dg;2 D ˚fg;1 ˚fg;2 (35.17) where ˚dg;1 D W1;1 1;1 Wi;1 i;1 WN;1 N;1 T ˚dg;2 D W1;2 1;2 Wi;2 i;2 WN;2 N;2 T ˚fg;1 D V1;1 M1;1 Vi;1 Mi;1 VN;1 MN;1 T ˚fg;2 D V1;2 M1;2 Vi;2 Mi;2 VN;2 MN;2 T (35.18)
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