scholarly journals Parametric and Internal Resonances of an Axially Moving Beam with Time-Dependent Velocity

2013 ◽  
Vol 2013 ◽  
pp. 1-18 ◽  
Author(s):  
Bamadev Sahoo ◽  
L. N. Panda ◽  
G. Pohit
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Time History ◽  
Mean Value ◽  
Phase Portraits ◽  
Axially Moving Beam ◽  
Internal Resonances ◽  
Moving Beam ◽  
Axially Moving ◽  
The Stability

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.

The Magneto-Elastic Internal Resonances of Rectangular Conductive Thin Plate With Different Size Ratios

2017 ◽  
Vol 34 (5) ◽  
pp. 711-723
Author(s):  
J. Li ◽  
Y. D. Hu ◽  
Y. N. Wang
Keyword(s):  
Magnetic Field ◽  
Differential Equations ◽  
Thin Plate ◽  
Internal Resonance ◽  
Multiple Scales ◽  
Time History ◽  
Transverse Magnetic ◽  
Phase Portraits ◽  
Internal Resonances ◽  
Basic Equations

AbstractBased on the basic equations of electromagnetic elastic motion and the expression of electromagnetic force, the electromagnetic vibration equation of the rectangular thin plate in transverse magnetic field is obtained. For a rectangular plate with one side fixed and three other sides simply supported, its time variable and space variable are separated by the method of Galerkin, and the two-degree-of-freedom nonlinear Duffing vibration differential equations are proposed. The method of multiple scales is adopted to solve the model equations and obtain four first-order ordinary differential equations governing modulation of the amplitudes and phase angles involved via 1:1 or 1:3 internal resonances with different size ratios. With a numerical example, the time history response diagrams, phase portraits and 3-dimension responses of two order modal amplitudes are respectively captured. And the effects of initial values, thickness and magnetic field intensities on internal resonance characteristics are discussed respectively. The results also present obvious characteristics of typical nonlinear internal resonance in this paper.

Nonlinear Dynamics and Bifurcations of an Axially Moving Beam

1999 ◽  
Vol 122 (1) ◽  
pp. 21-30 ◽  
Author(s):  
F. Pellicano ◽  
F. Vestroni
Keyword(s):  
High Dimensional ◽  
Axially Moving Beam ◽  
Dimensional System ◽  
Global Dynamics ◽  
Dimensional Phase Space ◽  
Moving Beam ◽  
Axially Moving ◽  
The Stability ◽  
The Galerkin Method ◽  
Sample Case

The present paper analyzes the dynamic behavior of a simply supported beam subjected to an axial transport of mass. The Galerkin method is used to discretize the problem: a high dimensional system of ordinary differential equations with linear gyroscopic part and cubic nonlinearities is obtained. The system is studied in the sub and super-critical speed ranges with emphasis on the stability and the global dynamics that exhibits special features after the first bifurcation. A sample case of a physical beam is developed and numerical results are presented concerning the convergence of the series expansion, linear subcritical behavior, bifurcation analysis and stability, and direct simulation of global postcritical dynamics. A homoclinic orbit is found in a high dimensional phase space and its stability and collapse are studied. [S0739-3717(00)00501-8]

Post-Critical Response of an Axially Moving Beam

1999 ◽  
Author(s):  
Francesco Pellicano ◽  
Fabrizio Vestroni
Keyword(s):  
Axially Moving Beam ◽  
Dimensional System ◽  
Global Dynamics ◽  
Wide Range ◽  
Moving Beam ◽  
Speed Range ◽  
Traveling Beam ◽  
Axially Moving ◽  
The Stability ◽  
The Galerkin Method

Abstract In this paper the dynamic response of a simply supported traveling beam, subjected to a pointwise transversal load, is investigated. The motion is described by means of a high dimensional system of ordinary differential equations with linear gyroscopic part and cubic nonlinearities obtained through the Galerkin method. The system is studied in the super-critical speed range with emphasis on the stability and the global dynamics that exhibits special features after the first bifurcation. A sample case of a physical beam is developed and numerical results are presented concerning bifurcation analysis and stability, and direct simulations of global postcritical dynamics. In the supercritical speed range a regular motion around a bifurcated equilibrium position becomes chaotic for particular values of frequency and force. The bifurcation diagram for varying force intensity is shown, it can be noticed that a chaotic motion occurs in a wide range of the forcing parameter, co-existing with a 3T periodic solution in a limited window.

Parametric Vibration and Numerical Validation of Axially Moving Viscoelastic Beams With Internal Resonance, Time and Spatial Dependent Tension, and Tension Dependent Speed

2019 ◽  
Vol 141 (6) ◽  
Author(s):  
You-Qi Tang ◽  
Zhao-Guang Ma ◽  
Shuang Liu ◽  
Lan-Yi Zhang
Keyword(s):  
Boundary Conditions ◽  
Internal Resonance ◽  
Multiple Scales ◽  
Differential Quadrature Method ◽  
Parametric Vibration ◽  
Axially Moving Beam ◽  
Beam Model ◽  
Solvability Conditions ◽  
Axially Moving ◽  
External Forces

Abstract In this paper, the idea of an axially moving time-dependent beam model is briefly introduced. The nonlinear response of an axially moving beam is investigated. The effects of a time and spatial dependent tension depending on the external forces at the boundary and a tension dependent speed are highlighted, which gives a new model to study the parametric vibration of axially moving structures. This paper focuses on simultaneous resonant cases that are the principal parametric resonance of first mode and internal resonance of the first two modes. In general, the method of multiple scales can study nonlinear vibration of axially moving structures with homogeneous boundary conditions. Taking Kelvin viscoelastic constitutive relation into account, the inhomogeneous boundary conditions make the solvability conditions fail, which is also one of the highlights of this paper. In order to resolve this problem, the technique of the modified solvability conditions is employed. The influence of some parameters, such as material’s viscoelastic coefficients, viscous damping coefficients, and the axial tension fluctuation amplitudes, on the steady-state vibration responses is demonstrated by some numerical examples. Furthermore, the approximate analytical results are verified by using the differential quadrature method (DQM).

Modeling and Stability Analysis of an Axially Moving Beam With Frictional Contact

2008 ◽  
Vol 75 (3) ◽  
Author(s):  
Gottfried Spelsberg-Korspeter ◽  
Oleg N. Kirillov ◽  
Peter Hagedorn
Keyword(s):  
Stability Analysis ◽  
Frictional Contact ◽  
Equations Of Motion ◽  
Axially Moving Beam ◽  
Perturbation Techniques ◽  
Stability Boundaries ◽  
Analytical Approximations ◽  
Moving Beam ◽  
Axially Moving ◽  
The Stability

This paper considers a moving beam in frictional contact with pads, making the system susceptible for self-excited vibrations. The equations of motion are derived and a stability analysis is performed using perturbation techniques yielding analytical approximations to the stability boundaries. Special attention is given to the interaction of the beam and the rod equations. The mechanism yielding self-excited vibrations does not only occur in moving beams, but also in other moving continua such as rotating plates, for example.

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