By Farzad Ebrahimi

ISBN-10: 9533072091

ISBN-13: 9789533072098

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The beam mass per unit length is ⎧ ρ wd1 0 ≤ x < L1 ⎪ ⎪ ρ wd2 L1 ≤ x ≤ L2 m( x ) = ⎨ # ⎪ ⎪⎩ ρ wdn Ln − 1 ≤ x ≤ Ln (7) where ρ is the mass density of the beam. The elemental stiffness matrix of the stepped beam can be obtained from the following equation Ke = ∫ L 0 ⎡[ kqq ] [ kqc ]⎤ d 2 ST d 2S EI ( x ) 2 dx = ⎢ ⎥ 2 dx dx ⎢⎣[ kcq ] [ kcc ]⎥⎦ (8) where the submatrix [ kqq ] corresponds to the element stiffness matrix from the FEM for the stepped beam; the submatrix [ kqc ] corresponds to the coupling terms of the q-dofs and the c-dofs; submatrix [ kcq ] is a transpose matrix of [ kqc ] , and the submatrix [ kcc ] corresponds to the c-dofs and is a diagonal matrix.

After introducing the boundary conditions, this can be performed by setting the associated degrees-of-freedom in the systematic stiffness matrix K to be a large number, say, 1012 , the governing equation for free vibration of the beam can be expressed as (K − ω 2M )V = 0 (10) where K and M are system stiffness and mass matrices, respectively, ω is the circular frequency, from which and the natural frequencies are identified. The ith normalized mode shapes of the stepped beam can be expressed as 4 N i =1 i =1 Ψ i = ∑ N iVi + ∑ ϕiVi + 4 (11) The equation of motion of the forced vibration of the beam with n steps when expressed in terms of the composite element method is  + CQ + KQ = f (t ) MQ (12) where M and K are the system mass and stiffness matrices, which are the same as those shown in Equation (10), C is the damping matrix which represents a Rayleigh damping model,say, C = a1M + a2 K , a1 and a2 are constants to be determined from two modal damping ratios.

China 1. Introduction Beams are fundamental models for the structural elements of many engineering applications and have been studied extensively. There are many examples of structures that may be modeled with beam-like elements, for instance, long span bridges, tall buildings, and robot arms. The vibration of Euler–Bernoulli beams with one step change in cross-section has been well studied. Jang and Bert (1989) derived the frequency equations for combinations of classical end supports as fourth order determinants equated to zero.

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Advances in Vibration Analysis Research by Farzad Ebrahimi


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