Nonlinear transverse vibrations of a slightly curved beam carrying a concentrated mass

Abstract

In this study, a slightly curved Euler Bernoulli beam carrying a concentrated mass was handled. The beam was resting on an elastic foundation and simply supported at both ends. Effects of the concentrated mass on nonlinear vibrations were investigated. Sinusoidal and parabolic type functions were used as curvature functions. Equations of motion have cubic nonlinearities because of elongations during vibrations. Damping and harmonic excitation terms were added to the equations of motion. Method of multiple scales, a perturbation technique, was used for solving integro-differential equation analytically. Natural frequencies were calculated exactly for different mass ratios, mass locations, curvature functions, and linear elastic foundation coefficients. Amplitude-phase modulation equations were found by considering primary resonance case. Effects of nonlinear terms on natural frequencies were calculated. Frequency-amplitude and frequency-response graphs were plotted. Finally effects of concentrated mass and chosen curvature function on nonlinear vibrations were investigated.

Natural frequencies of nonlinear vibration of axially moving beams

“…Naturally, (3) and (4) need condition (13) only. In the calculations, the spatial step h = 0.01 (L = 100) and the temporal step τ = 0.000025.…”

“…The transverse displacement of the beam center for (3) and (4) will be numerically solved from the two nonlinear transverse equations (10) and (11) under the boundary conditions (6) and the initial conditions (13). Based on numerically solutions of the (9), (10), and (11), the differences among the nonlinear models can be investigated via the first two natural frequencies of equations by the FFT.…”

“…Tang, Chen, and Yang calculated the natural frequencies of axially moving Timoshenko beams on different supports [12]. Özkaya et al [13] calculated natural frequencies for a slightly curved beam carrying a concentrated mass. All of above literatures, the natural frequency of axially moving beams was calculated from linear governing equation of transverse vibration.…”

A unified approach for nonlinear vibration analysis of curved structures using non-uniform rational B-spline representation

“…The nonlinear vibration of curved beams, carrying a concentrated mass, employing the multiple time scales method, was investigated and the effects of the concentrated mass and curvature shape were examined in detail [14].…”

Nonlinear Transverse Vibrations of a Slightly Curved Beam resting on Multiple Springs

“…By assuming the sinusoidal rising function for the initial curvature of the beam, Erdogan et al studied nonlinear vibrations of curved beams carrying a concentrated mass and multiple concentrated masses. 29,30 Xiuchang et al proposed a wave approach to investigate the wave propagation in the structural waveguides with curved beam components. 31 In the recent years, some researchers focused on the continua resting on partially supported elastic foundation/multiple springs.…”