The Hamiltonian dynamics of a resonantly excited linear spring-mass-damper system coupled to an array of pendulums is investigated in this study under 1:1:1:…:2 internal resonance between the pendulums and the linear oscillator. To study the small-amplitude global dynamics, a Hamiltonian formulation is introduced using generalized coordinates and momenta, and action-angle coordinates. The Hamilton’s equations are averaged to obtain equations for the first-order approximations to free and forced response of the system. Equilibrium solutions of the averaged Hamilton’s equations in action-angle or comoving variables are determined and studied for their stability characteristics. The system with one pendulum is known to be integrable in the absence of damping and external excitation. Exciting the system with even a small harmonic forcing near a saddle point leads to stochastic response, as clearly demonstrated by the Poincaré sections of motion. Poincaré sections are also computed for motions started with initial conditions near center-center, center-saddle and saddle-saddle-type equilibria for systems with two, three and four pendulums. In case of the system with more than one pendulum, even the free undamped dynamics exhibits irregular exchange of energy between the pendulums and the block. The increase in complexity is also demonstrated as the number of pendulums is increased, and when external excitation is present.
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e-mail: ashwinv@purdue.edu
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January 2006
Research Papers
Global Dynamics of an Autoparametric System With Multiple Pendulums
Ashwin Vyas,
Ashwin Vyas
School of Mechanical Engineering, 585 Purdue Mall, Mechanical Engineering Building,
e-mail: ashwinv@purdue.edu
Purdue University
, West Lafayette, IN 47907-2088
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Anil K. Bajaj
Anil K. Bajaj
School of Mechanical Engineering, 585 Purdue Mall, Mechanical Engineering Building,
e-mail: bajaj@purdue.edu
Purdue University
, West Lafayette, IN 47907-2088
Search for other works by this author on:
Ashwin Vyas
School of Mechanical Engineering, 585 Purdue Mall, Mechanical Engineering Building,
Purdue University
, West Lafayette, IN 47907-2088e-mail: ashwinv@purdue.edu
Anil K. Bajaj
School of Mechanical Engineering, 585 Purdue Mall, Mechanical Engineering Building,
Purdue University
, West Lafayette, IN 47907-2088e-mail: bajaj@purdue.edu
J. Comput. Nonlinear Dynam. Jan 2006, 1(1): 35-46 (12 pages)
Published Online: May 18, 2005
Article history
Received:
May 1, 2005
Revised:
May 18, 2005
Citation
Vyas, A., and Bajaj, A. K. (May 18, 2005). "Global Dynamics of an Autoparametric System With Multiple Pendulums." ASME. J. Comput. Nonlinear Dynam. January 2006; 1(1): 35–46. https://doi.org/10.1115/1.1994879
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