By Farzad Ebrahimi
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Extra info for Advances in Vibration Analysis Research
Moreover, it can also be refined through the c-refinement method, by increasing the number of shape functions. Here, we apply the c-refinement from the CEM, where the beam needs only to be discretized into one element. This will reduce the total number of degrees-of-freedom in the FEM. , c N (t )]T is the vector of generalized displacements, and N is the number of shape functions used from the classical theory. 2 Vibration analysis for stepped beams without crack Figure 1 shows the sketch of a beam with n steps, the height of the beam d( x ) with n step changes in cross section is expressed as ⎧ d1 0 ≤ x < L1 ⎪ ⎪ d L1 ≤ x ≤ L2 d( x ) = ⎨ 2 # ⎪ ⎪⎩dn Ln − 1 ≤ x ≤ Ln (5) It is assumed that the beam has aligned neutral axis, the flexibility of the beam EI ( x ) can be expressed as ⎧ wd 3 1 ⎪ 0 ≤ x < L1 ⎪ 12 ⎪ wd 3 2 ⎪ L1 ≤ x ≤ L2 EI ( x ) = ⎨ 12 ⎪ # ⎪ ⎪ wd 3 ⎪ n Ln − 1 ≤ x ≤ Ln ⎩ 12 (6) where w is the width of the beam.
Vibration of stepped beams and rectangular plates by an elemental dynamic flexibility method. Journal of Sound and Vibration, Vol. 5, 617–640, ISSN: 0022-460X. R. S. (2009). Dynamic condition assessment of a cracked beam with the composite element model. Mechanical Systems and Signal Processing, Vol. 23, No. 3, 415-431, ISSN: 0888-3270. Michael A. Koplow, Abhijit Bhattacharyya and Brian P. Mann. (2006). Closed form solutions for the dynamic response of Euler–Bernoulli beams with step changes in cross section.
Natural frequencies [Hz] comparison for the stepped beam in Jaworski and Dowell (2008) Note: ω1B , ω2 B are the first and second out-of-plane bending mode frequencies, respectively. ω1c denotes the first in-plane bending mode frequency. CMA represents component modal analysis. 34 Mod e Advances in Vibration Analysis Research No crack Exp. (2002) ed Exp. (2002) d Exp. (2002) ed Exp. 003 Table 2. (2002) dc 1 = 12 mm at Mod e No crack Exp. Propos ed dc 1 = 12 mm at dc 1 = 12 mm at x1 = 595 mm x1 = 595 mm x1 = 595 mm dc 2 = 4 mm at dc 2 = 8 mm at dc 2 = 12 mm at x2 = 800 mm x2 = 800 mm x2 = 800 mm Exp.