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Research Papers

Strongly Nonlinear Subharmonic Resonance and Chaotic Motion of Axially Moving Thin Plate in Magnetic Field

[+] Author and Article Information
Hu Yuda

Key Laboratory of Mechanical Reliability
for Heavy Equipments
and Large Structures of Hebei Province,
Yanshan University,
Qinhuangdao 066004, China
e-mail: huyuda03@163.com

Hu Peng

Shanghai Institute of Special Equipment
Inspection and Technical Research,
Shanghai 200333, China
Laboratory of Mechanical Reliability
for Heavy Equipments
and Large Structures of Hebei Province,
Yanshan University,
Qinhuangdao 066004, China
e-mail: hupenghhu@163.com

Zhang Jinzhi

Key Laboratory of Mechanical Reliability
for Heavy Equipments
and Large Structures of Hebei Province,
Yanshan University,
Qinhuangdao 066004, China
e-mail: zhangjinzhi86@163.com

1Corresponding author.

Manuscript received November 20, 2013; final manuscript received April 19, 2014; published online January 12, 2015. Assoc. Editor: D. Dane Quinn.

J. Comput. Nonlinear Dynam 10(2), 021010 (Mar 01, 2015) (12 pages) Paper No: CND-13-1292; doi: 10.1115/1.4027490 History: Received November 20, 2013; Revised April 19, 2014; Online January 12, 2015

In this paper, the nonlinear vibration and chaotic motion of the axially moving current-conducting thin plate under external harmonic force in magnetic field is studied. Improved multiple-scale method is employed to derive the strongly nonlinear subharmonic resonance bifurcation-response equation of the strip thin plate in transverse magnetic field. By using the singularity theory, the corresponding transition variety and bifurcation, which contain two parameters of the universal unfolding for this nonlinear system, are obtained. Numerical simulations are carried out to plot the bifurcation diagrams, corresponding maximum Lyapunov exponent diagrams, and dynamical response diagrams with respect to the bifurcation parameters such as magnetic induction intensity, axial tension, external load, external excited frequency, and axial speed. The influences of different bifurcation parameters on period motion, period times motion, and chaotic motion behaviors of subharmonic resonance system are analyzed. The results show that the complex dynamic behaviors of resonance system can be controlled by changing the corresponding parameters.

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Figures

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Fig. 1

Axially moving thin plate in magnetic field

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Fig. 2

Transition variety and bifurcation diagram of unfolding parametric plane

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Fig. 3

The bifurcation diagram and the corresponding maximum Lyapunov exponent for B0z. (a) bifurcation diagram and (b) maximum Lyapunov exponent.

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Fig. 4

The period-2 motion of the system exists when B0z = 2T. (a) Poincaré map, (b) time history plot, and (c) phase portrait.

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Fig. 5

The chaotic motion of the system exists when B0z = 3T. (a) Poincaré map, (b) time history plot, and (c) phase portrait.

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Fig. 6

The period-3 motion of the system exists when B0z = 4.5T. (a) Poincaré map, (b) time history plot, and (c) phase portrait.

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Fig. 7

The bifurcation diagram and the corresponding maximum Lyapunov exponent for N0. (a) bifurcation diagram and (b) maximum Lyapunov exponent.

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Fig. 8

The period motion of the system exists when N0 = 200 N/m. (a) Poincaré map, (b) time history plot, and (c) phase portrait.

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Fig. 9

The chaotic motion of the system exists when N0 = 250 N/m. (a) Poincaré map, (b) time history plot, and (c) phase portrait.

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Fig. 10

The period-9 motion of the system exists when N0 = 630 N/m. (a) Poincaré map, (b) time history plot, and (c) phase portrait.

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Fig. 11

The chaotic motion of the system exists when N0 = 678 N/m. (a) Poincaré map, (b) time history plot, and (c) phase portrait.

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Fig. 12

The period-2 motion of the system exists when N0 = 750 N/m. (a) Poincaré map, (b) time history plot, and (c) phase portrait.

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Fig. 13

The bifurcation diagram and the corresponding maximum Lyapunov exponent for P. (a) bifurcation diagram and (b) maximum Lyapunov exponent.

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Fig. 14

The period-3 motion of the system exists when P = 4300 N/m2. (a) Poincaré map, (b) time history plot, and (c) phase portrait.

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Fig. 15

The period-6 motion of the system exists when P = 4450 N/m2. (a) Poincaré map, (b) time history plot, and (c) phase portrait.

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Fig. 16

The period-12 motion of the system exists when P = 4456 N/m2. (a) Poincaré map, (b) time history plot, and (c) phase portrait.

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Fig. 17

The chaotic motion of the system exists when P = 4500 N/m2. (a) Poincaré map, (b) time history plot, and (c) phase portrait.

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Fig. 18

The bifurcation diagram and the corresponding maximum Lyapunov exponent for . (a) bifurcation diagram and (b) maximum Lyapunov exponent.

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Fig. 19

The period motion of the system exists when  = 2.9. (a) Poincaré map, (b) time history plot, and (c) phase portrait.

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Fig. 20

The chaotic motion of the system exists when  = 2.92. (a) Poincaré map, (b) time history plot, and (c) phase portrait.

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Fig. 21

The period-3 motion of the system exists when  = 3.15. (a) Poincaré map, (b) time history plot, and (c) phase portrait.

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Fig. 22

The bifurcation diagram and the corresponding maximum Lyapunov exponent for V0. (a) bifurcation diagram and (b) maximum Lyapunov exponent.

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Fig. 23

The chaotic motion of the system exists when V0 = 56.8 m/s. (a) Poincaré map, (b) time history plot, and (c) phase portrait.

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Fig. 24

The period-3 motion of the system exists when V0 = 56.85 m/s. (a) Poincaré map, (b) time history plot, and (c) phase portrait.

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Fig. 25

The period motion of the system exists when V0 = 57.4 m/s. (a) Poincaré map, (b) time history plot, and (c) phase portrait.

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