The nonlinear vibration of a travelling beam subjected to principal parametric resonance in presence of internal resonance is investigated. The beam velocity is assumed to be comprised of a constant mean value along with a harmonically varying component. The stretching of neutral axis introduces geometric cubic nonlinearity in the equation of motion of the beam. The natural frequency of second mode is approximately three times that of first mode; a three-to-one internal resonance is possible. The method of multiple scales (MMS) is directly applied to the governing nonlinear equations and the associated boundary conditions. The nonlinear steady state response along with the stability and bifurcation of the beam is investigated. The system exhibits pitchfork, Hopf, and saddle node bifurcations under different control parameters. The dynamic solutions in the periodic, quasiperiodic, and chaotic forms are captured with the help of time history, phase portraits, and Poincare maps showing the influence of internal resonance.
In this paper, analytical and numerical approach is applied to find the steady-state and dynamic behaviors of an axially accelerating viscoelastic beam subject to two-frequency parametric excitation in presence of internal resonance. Direct method of multiple scales is employed to solve the cubic nonlinear integropartial differential equation. As a result, the governing equation of motion is reduced to a set of nonlinear first-order partial differential equations. These equations are solved through continuation algorithm approach to find the frequency and amplitude response curves and their stability and bifurcation. The system reveals the presence of Hopf, saddle node, and pitchfork bifurcations. The dynamic bevavior of the system is estimated through phase portraits, time traces, Poincare maps, and FFT power spectra obtained via direct time integration. The evolution of maximum Lyapunov exponent reveals the system parameter where the dynamic response changes from stable periodic to unstable chaotic motion of the system.
Analytical-numerical approach has been adopted to investigate the stability, bifurcation and dynamic behavior (including chaotic behavior) of axially moving viscoelastic beam subjected to parametric excitation resulting from speed variation in the presence of 3:1 internal resonance between the first two modes of vibration. The governing equation of transverse vibration is a nonlinear integro-partial-differential equation with time-dependent coefficients. The direct method of multiple scales is employed to analyze the joint influence of the combination of parametric resonance and internal resonance with the focus on steady state responses. Equilibrium solutions along with their stability and bifurcations are determined by continuation algorithm while direct time integration is used for dynamic behavior for various system parameters. The results are compared with the previous works depicting the principal parametric resonances of the first and second modes. Significant comparative analysis results are reported in the stability and bifurcation of frequency response analysis. The dynamic responses show a range of behavior viz. stable periodic, mixed mode, quasiperiodic and unstable chaotic motion of the system. Numerical results illustrate various typical and interesting nonlinear phenomena of the traveling system which are not found in the existent literature.
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