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RESEARCH PAPERS

Global Dynamics of an Autoparametric System With Multiple Pendulums

[+] Author and Article Information
Ashwin Vyas

School of Mechanical Engineering, 585 Purdue Mall, Mechanical Engineering Building, Purdue University, West Lafayette, IN 47907-2088ashwinv@purdue.edu

Anil K. Bajaj1

School of Mechanical Engineering, 585 Purdue Mall, Mechanical Engineering Building, Purdue University, West Lafayette, IN 47907-2088bajaj@purdue.edu

1

Corresponding author.

J. Comput. Nonlinear Dynam 1(1), 35-46 (May 18, 2005) (12 pages) doi:10.1115/1.1994879 History: Received May 01, 2005; Revised May 18, 2005

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.

Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 1

A linear single degree-of-freedom (SDOF) system with n attached uncoupled pendulums

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Figure 2

Orbits in the invariant (P1,Q1) plane for the unforced system with one pendulum for different values of internal mistuning (d1). δ=0.0, P00=1: (a) d1=−1.2, (b) d1=−0.2, (c) d1=0.0, (d) d1=0.2, and (e) d1=1.2.

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Figure 3

Region denoted by S in the parameter plane (d1ν1∕P00, d2ν2∕P00) where possible equilibrium solutions given by Eq. 26 exist for different length fraction ratios: (a) ν2∕ν1=1, (b) ν2∕ν1=(5), (c) ν2∕ν1=1∕5

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Figure 4

Trajectories for a two-pendulum system simulated for two different initial conditions and projected on (a) (P1,Q1) plane and (b) (P1,Q2) plane. Dotted curve: initial conditions (0.3,0,0,1.21) and (0.6,0,0,1.21). Solid curve: initial conditions (0.3,π∕2,0,1.21) and (0.6,π∕2,0,1.21). d1=0.0, d2=0.2; ν1=ν2=1∕2; δ=0.0, P00=1.

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Figure 5

Equilibrium solutions (P1,Q1,P2,Q2) for a two-pendulum system as a function of internal mistuning (d1) of pendulum 1. Different symbols represent different types of equilibrium solutions. System parameters: d2=0; ν1=ν2=1∕2; P00=1.

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Figure 6

Eigenvalues of equilibrium solutions for a two-pendulum system having both pendulums in motion at different values of pendulum 1 internal mistuning (d1). Big (small) symbols represent real (imaginary) parts of the eigenvalues. d2=0; ν1=ν2=1∕2; P00=1.

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Figure 7

Equilibrium solutions (P1,Q1,P2,Q2) as a function of internal mistuning (d1) of pendulum 1 for a two-pendulum system. Different symbols represent different types of equilibrium solutions. d2=0; ν1=1∕6, ν2=5∕6; P00=1.

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Figure 8

Eigenvalues of equilibrium solutions for a two-pendulum system having both pendulums in motion at different values of pendulum 1 internal mistuning (d1). Big (small) symbols represent real (imaginary) parts of the eigenvalues. d2=0; ν1=1∕6, ν2=5∕6; P00=1.

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Figure 9

Equilibrium solutions (P1,Q1,P2,Q2) as a function of internal mistuning (d1) of pendulum 1 for a two-pendulum system. Different symbols represent different types of equilibrium solutions. d2=0; ν1=5∕6, ν2=1∕6; P00=1.

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Figure 10

Eigenvalues of equilibrium solutions for a two-pendulum system having both pendulums in motion at different values of pendulum 1 internal mistuning (d1). Big (small) symbols represent real (imaginary) parts of the eigenvalues. d2=0; ν1=5∕6, ν2=1∕6; P00=1.

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Figure 11

Poincare sections at cos2Q0=0.1 for two orbits started with different initial conditions for the externally excited one-pendulum system. Motions started near a center [(P1,Q1)=(0.8,π∕2)] are denoted by “∙”, and those started near a saddle (P1,Q1)=(0.01,0.7) are denoted by “+.” Solid curves show the heteroclinic orbits for the unforced system. d1=0.2, σb=1.5; F̂=2.0; ξ0=ξ1=0. (a) δ=0.02 and (b) δ=0.2.

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Figure 12

Trajectories of pendulum 1 action and angle as a function of time τ for motion started near saddle-saddle for the two-pendulum system. d1=0, d2=0; ν1=ν2=1∕2; δ=0, P00=1. (a) Pendulum 1 action P1. (b) Pendulum 1 angle Q1.

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Figure 13

Response of the system with two identical pendulums when the pendulum motions are started with (different) initial conditions (0.1,0.88,0.1,0.98) near the saddle-saddle (0,π∕4,0,π∕4). d1=0, d2=0; ν1=ν2=1∕2; δ=0, P00=1. (a) Projection of Poincaré sections at cos2Q2=−0.15 on (P1,Q1) plane. Trajectories of actions of (b) pendulum 1 and (c) pendulum 2.

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Figure 14

Projection of Poincaré sections at cos2Q2=−0.15 on (P1,Q1) plane for motions started with three different initial conditions for an unperturbed two-pendulum system: (+ and ×) denote motions initiated near two center-saddles {(0.73,1.57,0,1.11) and (0.59,0,0,0.62)} with initial conditions (0.74,1.67,0.05,1.2) and (0.69,0.1,0.05,0.72), and (o) denotes motion initiated near a center-center (0.68,0,0.31,1.57) with initial conditions (0.78,0.1,0.21,1.67). d1=0.2, d2=0;ν1=5∕6, ν2=1∕6; δ=0, P00=1.

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Figure 15

Trajectories of pendulum 2 action P2 as a function of time τ for a two-pendulum system. d1=0.2, d2=0; ν1=5∕6, ν2=1∕6; δ=0, P00=1. (a) Motion started near the center-saddle (0.73,1.57,0,1.11), (b) motion started near the center-saddle (0.59,0,0,0.62), and (c) motion started near the center-center (0.68,0,0.31,1.57).

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Figure 16

Response of a two-pendulum system with unequal length pendulums for motion started near a saddle-saddle (0,0.69,0,.78) with initial condition (0.1,0.79,0.1,0.89). d1=0.2, d2=0; ν1=5∕6, ν2=1∕6; δ=0, P00=1. (a) (P1,Q1) plane projection of Poincaré section at cos2Q2=−0.15, (b) pendulum 1 action P1 as a function of time τ, and (c) pendulum 2 action P2 as a function of time τ.

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Figure 17

Projection of Poincaré sections at cos2Q2=−0.15 on (P1,Q1) plane for motions started with three different initial conditions for a perturbed two-pendulum system with δ=0.02 and forcing F̂=2: (+ and ×) denote motions initiated near two center-saddles {(0.73,1.57,0,1.11) and (0.59,0,0,0.62)} with initial conditions (0.74,1.67,0.05,1.2) and (0.69,0.1,0.05,0.72), and (o) denotes motion initiated near a center-center (0.68,0,0.31,1.57) with initial conditions (0.78,0.1,0.21,1.67). d1=0.2, d2=0; ν1=5∕6, ν2=1∕6; P00=1.

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Figure 18

Response of the system with three pendulums when the pendulum motions are started near a saddle-saddle-saddle with initial condition (0.1∕3,0.83,0.1∕3,0.81,0.1∕3,0.78). d1=0, d2=0.1, d3=0.2; ν1=ν2=ν3=1∕3; δ=0, P00=1. (a) Projection of Poincaré sections at cos2Q2=−0.15 on (P1,Q1) plane. Trajectories for actions of (b) pendulum 1, (c) pendulum 2, and (d) pendulum 3.

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Figure 19

Response of the system with four pendulums when the pendulum motions are started near a four-saddle with initial condition (0.1∕4,0.83,0.1∕4,0.81,0.1∕4,0.78,0.1∕4,0.76). d1=0, d2=0.1, d3=0.2, d4=0.3; ν1=ν2=ν3=ν4=1∕4; δ=0, P00=1. (a) Projection of Poincaré sections at cos2Q2=−0.15 on (P1,Q1) plane. Trajectories for actions of (b) pendulum 1, (c) pendulum 2, (d) pendulum 3, and (e) pendulum 4.

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