The effects of shear deformation on the free vibration of elastic beams with general boundary conditions

Abstract

Free vibration characteristics of shear deformable elastic beams subjected to different sets of boundary conditions are investigated. The analysis is based on a unified one-dimensional shear deformation beam theory. The governing equations of the elastic beams are obtained by means of Hamilton's principle. Four different boundary conditions are considered. The natural frequencies and mode shapes are obtained by applying the dynamic stiffness method, where the elements of the exact dynamic stiffness matrix are derived by using the analytical solutions of the governing equations of the beam in free vibration. The numerical results for the particular beams with different slenderness ratios are presented and compared with those available in the literature.

Free and forced vibration analysis of coupled conical–cylindrical shells with arbitrary boundary conditions

“…It can be proved that the requirement for the selection of the auxiliary polynomial function is guaranteed, since it is easy to verify that ς 0 (11) and (21,22) together with the admissible functions defined in Eqs. (23)-(34) into Eq.…”

“…Furthermore, the general boundary conditions are often encountered in practical engineering applications compared to the classical boundary conditions since the support types of practical structures are always complicated and variable in nature. Li [21] proposed originally the modified Fourier series solution for the vibration analysis of beams with general elastic restraints. The flexural displacements are expressed by an improved Fourier series, which is composed a standard Fourier series and an auxiliary polynomial function introduced to remove all the relevant discontinuities at the boundaries.…”

Vibrational energy flow analysis of coupled cylindrical shell-plate structure with general boundary and coupling conditions

“…The transverse velocity response for the partition and the radial velocity response for the shell are shown in Figure 3(a) and (b), respectively. In fact, the curves in Figure 3 also indicate the transfer mobility responses according to equation (26). Although there are about 70 natural frequencies in the frequency range from 0 to 1000 Hz, only a few of them are excited to form the resonant peaks.…”

“…To satisfy the general boundary conditions, an improved Fourier series method, which was previously proposed for studying the vibrations of generally supported beams 26 and plates, 27,28 is extended to model the coupled cylindrical shell-plate structure. The in-plane displacements (u p and v p ) and the transverse displacement (w p ) of the plate are expressed in the corresponding local coordinates as follows.…”

Analysis of Nonlinear Vibration of Euler-Bernoulli Beams Subjected to Compressive Axial Forcevia the Equivalent Linearization Method with a Weighted Averaging

“…The Homotopy Analysis method was used by Pirbodaghi et al to investigate nonlinear vibration behaviour of Euler-Bernoulli beams subjected to axial loads [15]. Based onHe's Variational Approach and Laplace Iteration method, Bagheri et al analyzed nonlinear responses of a clamped-clamped buckled beam [16]. Effect of shear deformation on free vibration of elastic beams with general boundary conditions was investigated by Li and Hua [17].…”