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Research Papers

Synchronization of Slowly Rotating Nonidentically Driven Pendula

[+] Author and Article Information
Tomasz Kapitaniak

e-mail: tomaszka@p.lodz.pl
Division of Dynamics,
Technical University of Lodz,
Stefanowskiego 1/15,
Lodz 90-924, Poland

As the analytical studies of Sec. 3 are valid for the system consisting of n pendula, we introduce such a model.

For the larger values of cx, the values of the phase shifts between pendula are slightly different from the derived values of 0 and 2π, but the behavior of the system is qualitatively the same.

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received March 22, 2013; final manuscript received August 28, 2013; published online October 30, 2013. Assoc. Editor: Stefano Lenci.

J. Comput. Nonlinear Dynam 9(2), 021010 (Oct 30, 2013) (10 pages) Paper No: CND-13-1069; doi: 10.1115/1.4025576 History: Received March 22, 2013; Revised August 28, 2013

We study the synchronization of two rotating pendula mounted on a horizontal beam, which can roll on the parallel surface. The pendula are forced to rotate by different driving torques. It has been shown that after a transient two different types of synchronization between the pendula can be observed. The approximate analytical methods allow us to derive the synchronization conditions and explain the observed types of synchronous configurations. The energy balance in the system allows us to show how the energy is transferred between the pendula via the oscillating beam.

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Figures

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Fig. 1

The model of the considered system

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Fig. 2

Two types of synchronous configurations of identically driven pendula: l1 = l2 = 1.0, cφ1 = cφ2 = 0.03, m1 = m2 = 1.0, mb = 10.0, cx = 3.4247, p01 = 2.0, p11 = 1.0, p02 = 2.0, p12 = 1.0. (a) The time series of the pendula's angular velocities ω1 = ϕ·1(τ), ω2 = ϕ·2(τ) during the state of complete synchronization (ϕ2-ϕ1 = 0), kx = 60.0, (b) time series of the pendula's angular velocities ω1 = ϕ·1(τ),ω2 = ϕ·2(τ) during the state of antiphase synchronization (ϕ2-ϕ1 = 0), kx = 10.0, (c) the pendula's configuration during the complete synchronization, (d) the pendula's configuration during the antiphase synchronization, and (e) basins of attraction of complete and antiphase synchronizations; p01 = 2.0, p11 = 1.0, p02 = 2.0, p12 = 1.0, φ10 = 0, ϕ·10 = 0, ϕ·20 = 0, x0 = 0, and x·0 = 0.

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Fig. 3

Almost complete synchronization in the system with pendula with different driving torques: ϕ·10=2.0, p01 = 2.0, p02 = 1.5, p11 = 1.0. (a) The time series of the pendula's angular velocities ω1 = ϕ·1(τ),ω2 = ϕ·2(τ) during the state of almost complete synchronization (ϕ2-ϕ1≈0), p12 = 0.75, (b) time series of the the pendula's angular velocities ω1 = ϕ·1(τ),ω2 = ϕ·2(τ) at the threshold between almost complete synchronization and quasi-periodic motion, p12 = 0.89, (c) energy balance of the system versus p12, and (d) the example of the energy balance in the case when W2SYN < 0.

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Fig. 4

Bifurcation diagrams: ω1 = ϕ·1,ω2 = ϕ·2 (at the time instances when pendulum 1 passes through the equilibrium position, i.e., when φ1(τ) = 2πn) versus p12: kx = 60.0, p01 = 2.0, p11 = 1.0, p02 = 1.5; φ10 = 0, ϕ·10=0, ϕ·20=0, x0 = 0, x·0=0. (a) The value of p12 is increased from 0.8 to 2.0 and next decreased from 2.0 to 0.8, the intervals of 1/1 almost complete synchronization and intervals of 1/2, 1/3, 2/3, 3/4 synchronizations are indicated, (b) the value of p12 is decreased from 0.8 to 0.1 and interval of 1/1 almost complete synchronization is indicated, and (c) the value of p12 is increased from 0.1 to 0.8, intervals of 3/1, 2/1, and 1/1 synchronizations are indicated.

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Fig. 5

The examples of different types of system behaviors shown in Figs. 4(a)4(e): ϕ·10 = 2.0, kx = 60.0, p01 = 2.0, p11 = 1.0, p02 = 1.5. (a) The time series of the pendula's angular velocities ω1 = ϕ·1(τ),ω2 = ϕ·2(τ) during the state of 3/1 synchronization, p12 = 0.22, φ10 = 0.0, ϕ·10 = 2.0, φ20 = 1.57, ϕ·20 = 2.0, x0 = 0, x·0 = 0, (b) the time series of the pendula's angular velocities ϕ·1(τ),ϕ·2(τ) during the state of almost complete synchronization (ϕ2-ϕ1≈0) 1/1 synchronization (coexisting with 3/1 synchronization), p12 = 0.22, φ10 = 0, ϕ·10 = 2.0, ϕ·10 = 2.36, φ20 = 1.80, ϕ·20 = 2.1, x0 = 0.05, and x·0 = 0.95, (c) time series of the pendula's angular velocities ω1 = ϕ·1(τ),ω2 = ϕ·2(τ) during the state 1/3 synchronization, p12 = 1.9, φ10 = 0, ϕ·10 = 0, φ20 = 1.57, ϕ·20 = 0, x0 = 0.0, x·0 = 0, and (d) Poincare map of quasi-periodic motion, p12 = 0.9, φ10 = 0.0, ϕ·10 = 2.0, φ20 = 0.0, ϕ·20 = 2.0, x0 = 0, and x·0 = 0.

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Fig. 6

Bifurcation diagrams: ω1 = ϕ·1,ω2 = ϕ·2 (at the time instances when pendulum 1 passes through the equilibrium position, i.e., when φ1(τ) = 2πn) versus p12, kx = 60.0, p01 = 2.0, p11 = 1.0, p02 = 0.9. (a) The value of p12 is increased from 0.4 to 2.0, φ10 = 0, ϕ·10 = 2.0, φ20 = 0, ϕ·20 = 0, x0 = 0, x·0 = 0, (b) the value of p12 is decreased from 0.4 to 0.1, φ10 = 0, ϕ·10 = 2.0, φ20 = 0, ϕ·20 = 2.0, x0 = 0, x·0 = 0, and (c) the value of p12 is increased from 0.1 to 0.4, φ10 = 0, ϕ·10 = 0, φ20 = 0, ϕ·20 = 9.0, x0 = 0, x·0 = 0.

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Fig. 7

The examples of different types of coexisting attractors shown in Figs. 6(a)6(c): kx = 60.0, p01 = 2.0, p11 = 1.0, p02 = 0.9, p12 = 0.17. (a) The time series of the pendula's angular velocities ω1 = ϕ·1(τ),ω2 = ϕ·2(τ) during the state of 1/1 almost complete synchronization, p12 = 0.22, φ10 = 0, ϕ·10 = 0, φ20 = 3.14, ϕ·20 = 3.0, x0 = 0, x·0 = 0, (b) time series of the pendula's angular velocities ϕ·1(τ),ϕ·2(τ) during the state of 2/1 synchronization, φ10 = 0, ϕ·10 = 0, φ20 = 3.14, ϕ·20 = 4.0, x0 = 0, x·0 = 0, (c) time series of the pendula's angular velocities ω1 = ϕ·1(τ),ω2 = ϕ·2(τ) during 0/1 state, φ10 = 0, ϕ·10 = 0, φ20 = 1.57, ϕ·20 = 0, x0 = 0, x·0 = 0, and (d) basins of attraction of the attractors of (a)–(c) in the plane ω20 = ϕ·20-ϕ20, φ10 = 0.0, ϕ·10 = 2.0, x0 = 0, x·0 = 0.

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Fig. 8

Type of synchronization for different values of driving torques: map p02 versus p12, initially the driving torques of both pendula are identical p01 = p02 = 2.0, p11 = p12 = 1.0, φ10 = 0, ϕ·10=1.0, φ20 = 0, ϕ·20=1.0, x0 = 0, x·0=0 (in the initial moment both pendula are passing through the equilibrium with the same angular velocity equal to 12ϕ·n1 and after the transient the system reaches 1/1 complete synchronization)

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Fig. 9

Bifurcation diagrams: ω1 = ϕ·1,ω2 = ϕ·2 (at the time instances when pendulum 1 passes through the equilibrium position, i.e., when φ1(τ) = 2πn) versus p12: kx = 10.0, p01 = 2.0, p11 = 1.0, p02 = 1.5; φ10 = 0, ϕ·10 = 0, ϕ·20 = 0, x0 = 0, x·0 = 0. (a) The value of p12 is increased from 0.8 to 2.0 and next decreased from 2.0 to 0.8, the intervals of 1/1 almost complete synchronization and intervals of 1/2, 1/3, synchronizations are indicated, and (b) the value of p12 is decreased from 0.8 to 0.1 and next increased from 0.1 to 0.8, intervals of 3/1, 3/1, 2/1, and 3/2 synchronizations are indicated.

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