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Research Papers

Nonlinear Vibrations and Chaotic Dynamics of the Laminated Composite Piezoelectric Beam

[+] Author and Article Information
Minghui Yao

Beijing Key Laboratory of Nonlinear Vibrations
and Strength of Mechanical Structures,
College of Mechanical Engineering,
Beijing University of Technology,
Beijing 100124, China
e-mail: Merry_mingming@163.com

Wei Zhang

Beijing Key Laboratory of Nonlinear Vibrations
and Strength of Mechanical Structures,
College of Mechanical Engineering,
Beijing University of Technology,
Beijing 100124, China
e-mail: sandyzhang0@yahoo.com

Zhigang Yao

Beijing Industrial Technician College,
Beijing 100023, China
e-mail: yzgmm@aliyun.com

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received March 30, 2014; final manuscript received September 25, 2014; published online November 12, 2014. Assoc. Editor: Ryan L Harne.

J. Vib. Acoust 137(1), 011002 (Feb 01, 2015) (16 pages) Paper No: VIB-14-1107; doi: 10.1115/1.4028710 History: Received March 30, 2014; Revised September 25, 2014; Online November 12, 2014

This paper investigates the complicated dynamics behavior and the evolution law of the nonlinear vibrations of the simply supported laminated composite piezoelectric beam subjected to the axial load and the transverse load. Using the third-order shear deformation theory and the Hamilton's principle, the nonlinear governing equations of motion for the laminated composite piezoelectric beam are derived. Applying the method of multiple scales and Galerkin's approach to the partial differential governing equation, the four-dimensional averaged equation is obtained for the case of principal parametric resonance and 1:9 internal resonance. From the averaged equations obtained, numerical simulation is performed to study nonlinear vibrations of the laminated composite piezoelectric beam. The axial load, the transverse load, and the piezoelectric parameter are selected as the controlling parameters to analyze the law of complicated nonlinear dynamics of the laminated composite piezoelectric beam. Based on the results of numerical simulation, it is found that there exists the complex nonlinear phenomenon in motions of the laminated composite piezoelectric beam. In summary, numerical studies suggest that periodic motions and chaotic motions exist in nonlinear vibrations of the laminated composite piezoelectric beam. In addition, it is observed that the axial load, the transverse load and the piezoelectric parameter have significant influence on the nonlinear dynamical behavior of the beam. We can control the response of the system from chaotic motions to periodic motions by changing these parameters.

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Figures

Grahic Jump Location
Fig. 1

The model of the laminated composite piezoelectric beam and the position of the piezoelectric layers: (a) the simplified model of the laminated composite piezoelectric beam structure and (b) the continuous piezoelectric layers on the laminated composite beam

Grahic Jump Location
Fig. 2

The effect of the axial load on frequency of the system: (a) the frequency ratio with respect to the axial load and (b) the internal resonances relationship of two nature frequencies

Grahic Jump Location
Fig. 3

The bifurcation diagram is obtained for the transverse load q1 = 300 - 800, and initial conditions x10 = 1.24, x20 = -0.23,x30 = 1.32, x40 = -1.27: (a) the bifurcation diagram on the plane (x1,q1) and (b) the bifurcation diagram on the plane (x3,q1)

Grahic Jump Location
Fig. 4

The chaotic motion is obtained when q1 = 315: (a) the phase portrait on the plane (x1,x2); (b) the phase portrait on the plane (x3,x4); (c) the waveform on the plane (t,x1); and (d) the phase portrait in the three-dimensional space (x1,x2,x3)

Grahic Jump Location
Fig. 5

The periodic motion is obtained when q1 = 400: (a) the phase portrait on the plane (x1,x2); (b) the phase portrait on the plane (x3,x4); (c) the waveform on the plane (t,x1); and (d) the phase portrait in the three-dimensional space (x1,x2,x3)

Grahic Jump Location
Fig. 6

The chaotic motion is obtained when q1 = 720: (a) the phase portrait on the plane (x1,x2); (b) the phase portrait on the plane (x3,x4); (c) the waveform on the plane (t,x1); and (d) the phase portrait in the three-dimensional space (x1,x2,x3)

Grahic Jump Location
Fig. 7

The bifurcation diagram is obtained for the axial load p1 = 50-150, the transverse load q1 = 430 and initial conditions x10 = 1.24, x20 = -0.23,x30 = 1.32, x40 = -1.27: (a) the bifurcation diagram on the plane (x1,p1) and (b) the bifurcation diagram on the plane (x3,p1)

Grahic Jump Location
Fig. 8

The periodic motion is obtained when p1 = 79: (a) the phase portrait on the plane (x1,x2); (b) the phase portrait on the plane (x3,x4); (c) the waveform on the plane (t,x1); and (d) the phase portrait in the three-dimensional space (x1,x2,x3)

Grahic Jump Location
Fig. 9

The periodic motion is obtained when p1 = 100: (a) the phase portrait on the plane (x1,x2); (b) the phase portrait on the plane (x3,x4); (c) the waveform on the plane (t,x1); and (d) the phase portrait in the three-dimensional space (x1,x2,x3)

Grahic Jump Location
Fig. 10

The chaotic motion is obtained when p1 = 137: (a) the phase portrait on the plane (x1,x2); (b) the phase portrait on the plane (x3,x4); (c) the waveform on the plane (t,x1); and (d) the phase portrait in the three-dimensional space (x1,x2,x3)

Grahic Jump Location
Fig. 11

The bifurcation diagram is obtained for the piezoelectric parameter G0 = 0-100, the transverse load q1 = 300 and initial conditions x10 = 1.24, x20 = -0.23,x30 = 1.32, x40 = -1.27: (a) the bifurcation diagram on the plane (x1,G0) and (b) the bifurcation diagram on the plane (x3,G0)

Grahic Jump Location
Fig. 12

The chaotic motion is obtained when G0=0: (a) the phase portrait on the plane (x1,x2); (b) the phase portrait on the plane (x3,x4); (c) the waveform on the plane (t,x1); and (d) the phase portrait in the three-dimensional space (x1,x2,x3)

Grahic Jump Location
Fig. 13

The periodic motion is obtained when G0 = 18: (a) the phase portrait on the plane (x1,x2); (b) the phase portrait on the plane (x3,x4); (c) the waveform on the plane (t,x1); and (d) the phase portrait in the three-dimensional space (x1,x2,x3)

Grahic Jump Location
Fig. 14

The periodic motion is obtained when G0 = 90: (a) the phase portrait on the plane (x1,x2); (b) the phase portrait on the plane (x3,x4); (c) the waveform on the plane (t,x1); and (d) the phase portrait in the three-dimensional space (x1,x2,x3)

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