Non-linear dynamics of a slender beam carrying a lumped mass under principal parametric resonance with three-mode interactions

The non-linear response of a base-excited slender beam carrying a lumped mass subjected to principal parametric resonance is investigated. The attached mass and its location are so adjusted that the system exhibits 1:3:5 internal resonances. Method of multiple scales is used to reduce the second-order temporal differential equation to a set of first-order differential equations which is then solved numerically to obtain the steady-state response and stability of the system. The steady-state response thus obtained is compared with those found by single- and two-mode analyses and very significant differences are observed in the bifurcation and stability of the response curves. The effects of external and internal detuning, amplitude of excitation and damping on the non-linear steady state, periodic, quasi-periodic and chaotic responses of the system are investigated. Funnel-shaped chaotic orbits, fractal orbits, cascade of period-doubling, torus doubling and intermittency routes to chaos are observed in this system. A simple illustration is given to control chaos by changing the system parameters.

[1]  K. Soh,et al.  Nonlinear analysis of the forced response of a beam with three mode interaction , 1994, Nonlinear Dynamics.

[2]  A. H. Nayfeh,et al.  The response of two-degree-of-freedom systems with quadratic non-linearities to a parametric excitation , 1983 .

[3]  Matthew Cartmell,et al.  Introduction to Linear, Parametric and Non-Linear Vibrations , 1990 .

[4]  A. Nayfeh,et al.  Nonlinear Analysis of the Lateral Response of Columns to Periodic Loads , 1978 .

[5]  H. Saito,et al.  Parametric vibrations of a horizontal beam with a concentrated mass at one end , 1982 .

[6]  A. H. Nayfeh,et al.  The Non-Linear Response of a Slender Beam Carrying a Lumped Mass to a Principal Parametric Excitation: Theory and Experiment , 1989 .

[7]  J. W. Roberts,et al.  NON-LINEAR VIBRATORY INTERACTIONS IN SYSTEMS OF COUPLED BEAMS , 1986 .

[8]  Hideo Saito,et al.  The Parametric Response of a Horizontal Beam Carrying a Concentrated Mass Under Gravity , 1978 .

[9]  A. Nayfeh,et al.  Applied nonlinear dynamics : analytical, computational, and experimental methods , 1995 .

[10]  R. A. Ibrahim,et al.  Parametric Vibration Part IV: Current Problems (2) , 1978 .

[11]  Ali H. Nayfeh,et al.  Modal Interactions in Dynamical and Structural Systems , 1989 .

[12]  R. A. Ibrahim,et al.  Parametric Vibration: Part Ii: Mechanics of Nonlinear Problems , 1978 .

[13]  Raouf A. Ibrahim,et al.  Parametric Vibration: Part I: Mechanics of Linear Problems , 1978 .

[14]  Santosha K. Dwivedy,et al.  Non-linear dynamics of a slender beam carrying a lumped mass with principal parametric and internal resonances , 1999 .

[15]  A. Nayfeh,et al.  A theoretical and experimental investigation of a three-degree-of-freedom structure , 1994 .

[16]  Ali H. Nayfeh,et al.  Non-linear non-planar parametric responses of an inextensional beam☆ , 1989 .

[17]  A. H. Nayfeh,et al.  The response of multidegree-of-freedom systems with quadratic non-linearities to a harmonic parametric resonance , 1983 .

[18]  W. K. Lee,et al.  Combination Resonances of a Circular Plate With Three-Mode Interaction , 1995 .

[19]  A. Barr,et al.  The resonances of structures with quadratic inertial non-linearity under direct and parametric harmonic excitation , 1987 .

[20]  Raouf A. Ibrahim,et al.  AUTOPARAMETRIC RESONANCE IN A STRUCTURE CONTAINING A LIQUID, PART I: TWO MODE INTERACTION , 1975 .

[21]  A. Ertas,et al.  Dynamics and bifurcations of a coupled column-pendulum oscillator , 1995 .