By Farzad Ebrahimi
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667. 426. x d )Q = f (t ) MQ L1 , c 1 Li , c i Ln , cn (19) 3. 1 Free vibration analysis for a free-free beam with a single step The free vibration of the free-free beam studied in Koplow et al. (2006) is restudied using the CEM and the results are compared with those in Koplow et al. Figure 2 shows the geometry of the beam under study. 7 GPa . In the CEM when 350 numbers of c-dofs are used, the first three natural frequencies are converged. 7Hz, respectively. The calculated natural frequencies from the CEM are very close to the experimental values in Koplow et al.
The second term uCT(x,t) is obtained by the multiplication of the analytical mode shapes with a vector of N coefficients c ( also called the c degrees-of-freedom or c-coordinates). N uCT ( x , t ) = ∑ ϕi ( x )c i (t ) (3) i =1 where ϕi (i=1,2,…N) is the analytical shape function of the beam. Different analytical shape functions are used according to the boundary conditions of the beam. Like the FEM, the CEM can be refined using the h-refinement technique by increasing the number of finite elements.
When the test is measured at location A in Figure 2, which are 291Hz, 1165Hz and 1771Hz, respectively. 39%, respectively. This shows the proposed method is accuracte. 2 Free vibration analysis for a cantilever beam with a several steps The cantilever beam studied in Jaworski and Dowell (2008) is restudied to further check the accuracy and effectiveness of the proposed method. Figure 3 shows the dimensions of the beam under study. 6GPa and ρ = 2664 kg / m3 . In the CEM model of the beam, the beam is discretized into one element and 350 terms of c-dofs are used in the calculation.
Advances in Vibration Analysis Research by Farzad Ebrahimi