Narrow Results

This paper focuses on the bifurcation and chaos of an axially accelerating viscoelastic beam in the supercritical regime. For the first time, the nonlinear dynamics of the system under consideration are studied via the high-order Galerkin truncation as well as the differential and integral quadrature method (DQM & IQM). The speed of the axially moving beam is assumed to be comprised of a constant mean value along with harmonic fluctuations. The transverse vibrations of the beam are governed by a nonlinear integro-partial-differential equation, which includes the finite axial support rigidity and the longitudinally varying tension due to the axial acceleration. The Galerkin truncation and the DQM & IQM are, respectively, applied to reduce the equation into a set of ordinary differential equations. Furthermore, the time history of the axially moving beam is numerically solved based on the fourth-order Runge–Kutta time discretization. Based on the numerical solutions, the phase portrait, the bifurcation diagrams and the initial value sensitivity are presented to identify the dynamical behaviors. Based on the nonlinear dynamics, the effects of the truncation terms of the Galerkin method, such as 2-term, 4-term, and 6-term, are studied by comparison with DQM & IQM.

This paper investigates some nonlinear dynamical behaviors about domains of attraction, bifurcations, and chaos in an axially accelerating viscoelastic beam under a time-dependent tension and a time-dependent speed. The axial speed and the axial tension are coupled to each other on the basis of a harmonic variation over constant initial values. The transverse motion of the moving beam is governed by nonlinear integro-partial-differential equations with the rheological model of the Kelvin–Voigt energy dissipation mechanism, in which the material derivative is applied to the viscoelastic constitutive relation. The fourth-order Galerkin truncation is employed to transform the governing equation to a set of nonlinear ordinary differential equations. The nonlinear phenomena of the system are numerically determined by applying the fourth-order Runge–Kutta algorithm. The tristable and bistable domains of attraction on the stable steady state solution with a three-to-one internal resonance are analyzed emphatically by means of the fourth-order Galerkin truncation and the differential quadrature method, respectively. The system parameters on the bifurcation diagrams and the maximum Lyapunov exponent diagram are demonstrated by some numerical results of the displacement and speed of the moving beam. Furthermore, chaotic motion is identified in the forms of time histories, phase-plane portraits, fast Fourier transforms, and Poincaré sections.

The well-known vibration model of axially moving beam is considered. Both axial moving speed and axial force are assumed to vary harmonically. The Method of Multiple Time Scales (a perturbation method) is used. The natural vibrations of beam are considered for different values of beam parameters. Resonances are obtained for seven different conditions. Solvability conditions for each resonance case are found analytically. Effects of transport velocity, axial force, rigidity and damping are discussed. Stability analysis are obtained for principal parametric resonances. Stable and unstable regions are obtained regarding velocity and force effects separately and together.

Principal-internal joint resonance of an axially moving beam with elastic constraints is investigated, where the beam is located in magnetic field excited by current-carrying wires. Based on the magnetoelasticity theory and Hamilton principle, the nonlinear magneto-elastic vibration equations are derived. The displacement function is obtained by the elastic constraint boundary condition, and then the equation is discretized into 2-DOF ordinary differential equations using the Galerkin integral method. The method of multiple scales is employed for obtaining the amplitude–frequency response equations with coupling of the first two vibration modes. Through examples, amplitudes varying with frequency tuning parameter, axial velocity, current intensity, and external excitation force are exhibited. Results indicate that as current intensity increases, electromagnetic damping increases and the amplitude decreases; As external excitation force increases and axial velocity decreases respectively, the amplitude increases and the system changes from single periodic motion to quasi-periodic motion and then to single periodic motion; The lower-order modes are always dominant when the principle-internal resonance occurs.

This study focuses on the steady-state periodic response and the chaotic behavior in the transverse motion of an axially moving viscoelastic tensioned beam with two-frequency excitations. The two-frequency excitations come from the external harmonic excitation and the parametric excitation from harmonic fluctuations of the moving speed. A dynamic model is established to include the finite axial support rigidity, the material derivative in the viscoelastic constitution relation, and the longitudinally varying tension due to the axial acceleration. The derived nonlinear integro-partial-differential equation of motion possesses space-dependent coefficients. Applying the differential quadrature method (DQM) and the integral quadrature method (IQM) to the equation of the transverse motion, a set of nonlinear ordinary differential equations is obtained. Based on the Runge–Kutta time discretization, the time history of the axially moving beam is numerically solved for the case of the primary resonance, the super–harmonic resonance, and the principal parametric resonance. For the first time, the nonlinear dynamics is studied under various relations between the forcing frequency and the parametric frequency, such as equal, multiple, and incommensurable relationships. The stable periodic response and its sensitivity to initial conditions are determined using the bidirectional frequency sweep. Furthermore, chaotic motions are identified using different methods including the Poincaré map, the maximum Lyapunov exponent, the fast Fourier transforms, and the initial value sensitivity. Numerical simulations reveal the characteristics of the periodic, quasiperiodic, and chaotic motion of a nonlinear axially moving beam under two-frequency excitations.

This paper clarified kinematic aspects of motion of axially moving beams undergoing large-amplitude vibration. The kinematics was formulated in the mixed Eulerian–Lagrangian framework. Based on the kinematic analysis, the governing equations of nonlinear vibration were derived from the extended Hamilton principle and the higher-order shear beam theory. The derivation considered the effects of material parameters on the beam deformation. The proposed governing equations were compared with a few previous governing equations. The comparisons show that proposed equations are with higher precision. Besides, the proposed equations can be viewed as the asymptotic governing equations of Lagrange’s equations of motion for large displacement. Finally, the corresponding boundary conditions and the comparison between the presented model equation and classical model equation were provided.