36 Vibrations of Discretely Layered Structures Using a Continuous Variation Model 391 Young’s modulus increase (3 times) Properties Variation 5 4 3 2 1 0 Xd 0 0.2 0.4 0.6 0.8 1 Density increase (1.1 times) Fig. 36.4 Properties transition for a single cell layered rod f1 .xd/ D1C1:9577Heaviside.xd 0:5/ f2 .xd/ D1C0:1439Heaviside.xd 0:5/ (36.21) The logistic function approximations are: f1 .xd/ 1:9789C0:9789tanh.50xd 25/ f2 .xd/ 1:0720C0:0720tanh.50xd 25/ (36.22) Subdividing the cell into ten elements, the FDM gives the results shown in Fig. 36.5, for the first two frequencies and mode shapes of the fixed-fixed case. Increasing the number of elements to 30 produces !d,1 D3.9845, !d,2 D6.7176 (accuracy is increased, as seen via a comparison with values given above). With ten elements the results for the first two frequencies and mode shapes of the free-fixed case are shown in Fig. 36.6. In this case a first-order forward difference scheme was used for the leftmost point calculation. Increasing the number of elements to 30 produces !d,1 D1.8093, !d,2 D5.4604. 36.4.2 Forced Motion Approach Note that, in general, the problem posed by Eq. 36.19 subjected to a specific set of boundary conditions does not have analytic solutions. In principle solutions can be obtained (numerically) by solving an eigenvalue problem. However a problem arises in that MAPLE®’s off-the-shelf solver (the software utilized here) only gives the trivial solution. The strategy employed next is similar to one described in reference [8] and consists of using MAPLE®’s two-point boundary value solver to solve a forced motion problem. In this case an axial non-dimensional forcing functionF*DFsin(!d ) is introduced into Eq. 36.18, which leads to the following form for Eq. 36.19: d dxd f1 .xd/ dS .xd/ dxd Cf2 .xd/!d 2S .xd/ DF (36.23) ThenFis taken to be equal to 1. By varying the frequency!d and observing the mid-span deflection of the rod, resonant frequencies can be found on noting where abrupt changes in the sign of the deflection occurs.
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