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Chris Bickford Introduction to Finite Elements HW #5; Problem 1 MAPLE: Using Maple software, the frequencies for the first six modes of vibration in the beam were calculated. See attached Maple sheets for details. ABAQUS FEA CODE: ABAQUS 6.14 software was used to perform finite element analysis (FEA) to obtain an approximate solution for problem (1). A 3-dimensional eigenvalue analysis was performed using the Lanczos solver. Two models were analyzed, one with linear elements and one with quadratic elements. Both models had 108 elements, Figure 1 shows the mesh and second bending mode shape. The results are shown in Table 1. Both elements gave pretty good approximations for the first and second modes, with the quadratics out-performing the linear elements. However, after the second mode of vibration, the quality of the approximation decreased significantly. Table 1 Summary of FEA Results Exact Soln [Hz] 1st Mode 2nd Mode 3rd Mode 4th Mode 5th Mode 6th Mode 58.902 369.134 1033.586 2025.416 5001.569 6985.662 Linear Elements Freq. [Hz] % Error 58.353 0.93% 349.57 5.30% 919.85 11.00% 1669.8 17.56% 2542.8 49.16% 3493.3 49.99% Quadratic Elements Freq. [Hz] % Error 59.145 -0.41% 354.48 3.97% 932.63 9.77% 1692.7 16.43% 2578.7 48.44% 3547.5 49.22% Figure 1 First Bending Mode