Nonlinear dynamics of an inclined beam subjected to a moving load

Abstract

In this paper, the nonlinear dynamic response of an inclined pinned-pinned beam with a constant cross section, finite length subjected to a concentrated vertical force traveling with a constant velocity is investigated. The study is focused on the mode summation method and also on frequency analysis of the governing PDEs equations of motion. Furthermore, the steady-state response is studied by applying the multiple scales method. The nonlinear response of the beam is obtained by solving two coupled nonlinear PDEs governing equations of planar motion for both longitudinal and transverse oscillations of the beam. The dynamic magnification factor and normalized time histories of mid-pint of the beam are obtained for various load velocity ratios and A. Mamandi is a Ph.D. Student. the outcome results have been illustrated and compared to the results with those obtained from traditional linear solution. The appropriate parametric study considering the effects of the linear viscous damping, the velocity of the traveling load, beam inclination angle under zero or nonzero axial load are carried out to capture the influence of the effect of large deflections caused by stretching effects due to the beam's immovable ends. It was seen that quadratic nonlinearity renders the softening effect on the dynamic response of the beam under the act of traveling load. Also in the case where the object leaves the inclined beam, its planar motion path is derived and the targeting accuracy is investigated and compared with those from the rigid solution assumption. Moreover, the stability analysis of steady-state response for the modes equations having quadratic nonlinearity was carried out and it was observed from the frequency response curves that for the considered parameters in the case of internal-external primary resonance, both saturation phenomenon and jump phenomenon can be predicted for the longitudinal excitation.Keywords Nonlinear vibrations · Inclined beam · Mode summation method · Frequency response analysis · Multiple scales method · Quadratic nonlinearity 278 A. Mamandi et al.

Nonlinear Dynamic Analysis of an Inclined Timoshenko Beam Subjected to a Moving Mass/Force with Beam’s Weight Included

“…Suppose that point p 1 is the position of the impact point predicted by the rigid type of analysis and point p 2 the same point predicted by the nonlinear analysis under beam's inclination angle of θ rigid and beam's end slope of θ endslope (see Fig. 10) [25]. Table 2 • , and keeping all other parameters the same, the difference in target position with respect to the rigid beam assumption considering both weightless beam and the self-weight beam models are 125.38 m and 182.76 m, respectively which have about 0.83% error and 1.21% error, respectively (see Fig.…”

“…The linear analysis for beams has been studied in [24] using Newmark-β method to solve the discrete equations of motion for thin and thick beam theories neglecting the longitudinal displacement. The nonlinear dynamic analysis of an inclined Euler-Bernoulli beam with finite length subjected to a concentrated moving force has been studied in [25]. The nonlinear coupled PDEs of motion were solved by using mode summation method.…”

An explicit closed-form solution for transverse and longitudinal vibration of beam with multi-directional elastic constraints under an arbitrary moving load

“…Dmitriev [12] presented the early analytical solution on the longitudinal vibration of the beam subjected to a constant axial force and fixed axially in one end. Mamandi et al [13] studied analytically not only the transverse vibration, but also the longitudinal vibration of inclined beam subjected to a moving load. The effects of the transverse or axial elastic constraint on the longitudinal vibration were also considered in the researches of Nguyen and Duhamel [14], Wu [15], Toth and Ruge [16], but the means adopted in their works are the finite element method, not the analytical solution.…”

Vibration analysis of a beam with moving support subjected to a moving mass travelling with constant and variable speed

“…The chaotic behavior was investigated by utilizing the Poincare´maps, Fourier spectra, and Lyapunov exponents. The nonlinear dynamic response of an inclined beam subjected to a concentrated vertical force traveling with a constant velocity was investigated in [3]. Two nonlinear coupled equations of longitudinal and transverse motion of the beam were solved using the multiple scales method.…”