No abstract

In this paper, the nonlinear planar vibration of a pipe conveying pulsatile fluid subjected to principal parametric resonance in the presence of internal resonance is investigated. The pipe is hinged to two immovable supports at both ends and conveys fluid at a velocity with a harmonically varying component over a constant mean velocity. The geometric cubic nonlinearity in the equation of motion is due to stretching effect of the pipe. The natural frequency of the second mode is approximately three times the natural frequency of the first mode for a range of mean flow velocity, resulting in a three-to-one internal resonance. The analysis is done using the method of multiple scales (MMS) by directly attacking the governing nonlinear integralpartial-differential equations and the associated boundary conditions. The resulting set of first-order ordinary differential equations governing the modulation of amplitude and phase is analyzed numerically for principal parametric resonance of first mode. Stability, bifurcation, and response behavior of the pipe are investigated. The results show new zones of instability due to the L. N. Panda ( ) presence of internal resonance. A wide array of dynamical behavior is observed, illustrating the influence of internal resonance.

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.

No abstract

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.