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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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References

Leng, J. S., Asundi, A., and Liu, Y. J., 1999, “Vibration Control of Smart Composite Beams With Embedded Optical Fiber Sensor and ER Fluid,” ASME J. Vib. Acoust., 121(4), pp. 508–509. [CrossRef]
Huang, D., and Sun, B., 2001, “Approximate Analytical Solutions of Smart Composite Mindlin Beams,” J. Sound Vib., 244(3), pp. 379–394. [CrossRef]
Halim, D., and Moheimani, S. O. R., 2001, “Spatial Resonant Control of Flexible Structures-Application to a Piezoelectric Laminate Beam,” IEEE Trans. Control Syst. Technol., 9(1), pp. 37–53. [CrossRef]
Waisman, H., and Abramovich, H., 2002, “Variation of Natural Frequencies of Beams Using the Active Stiffening Effect,” Composites, Part B, 33(6), pp. 415–424. [CrossRef]
Kapuria, S., Ahmed, A., and Dumir, P. C., 2004, “Static and Dynamic Thermo-Electro-Mechanical Analysis of Angle-Ply Hybrid Piezoelectric Beams Using an Efficient Coupled Zigzag Theory,” Compos. Sci. Technol., 64(16), pp. 2463–2475. [CrossRef]
Heuer, R., and Adam, C., 2000, “Piezoelectric Vibrations of Composite Beams With Interlayer Slip,” Acta Mech., 140(3–4), pp. 247–263. [CrossRef]
Kapuria, S., and Alam, N., 2005, “Nonlinear Zigzag Theory for Buckling of Hybrid Piezoelectric Rectangular Beams Under Electrothermomechanical Loads,” J. Eng. Mech., 131(4), pp. 367–376. [CrossRef]
Kapuria, S., Alam, N., and Jain, N. K., 2005, “Two-Dimensional Piezoelasticity and Zigzag Theory Solutions for Vibration of Initially Stressed Hybrid Beams,” ASME J. Vib. Acoust., 127(2), pp. 116–124. [CrossRef]
Marur, S. R., and Kant, T., 2007, “On the Angle Ply Higher-Order Beam Vibrations,” Comput. Mech., 40(1), pp. 25–33. [CrossRef]
Jiang, J. P., and Li, D. X., 2007, “A New Finite Element Model for Piezothermoelastic Composite Beam,” J. Sound Vib., 306(3–5), pp. 849–864. [CrossRef]
Emam, S. A., and Nayfeh, A. H., 2009, “Postbuckling and Free Vibrations of Composite Beams,” Compos. Struct., 88(4), pp. 636–642. [CrossRef]
Fridman, Y., and Abramovich, H., 2008, “Enhanced Structural Behavior of Flexible Laminated Composite Beams,” Compos. Struct., 82(1), pp. 140–154. [CrossRef]
Mahmoodi, S. N., and Jalili, N., 2008, “Coupled Flexural-Torsional Nonlinear Vibrations of Piezoelectrically Actuated Microcantilevers With Application to Friction Force Microscopy,” ASME J. Vib. Acoust., 130(6), p. 061003. [CrossRef]
Kapuria, S., Kumari, P., and Nath, J. K., 2010, “Efficient Modeling of Smart Piezoelectric Composite Laminates: A Review,” Acta Mech., 214(1–2), pp. 31–48. [CrossRef]
Bilgen, O., Erturk, A., and Inman, D. J., 2010, “Analytical and Experimental Characterization of Macro-Fiber Composite Actuated Thin Clamped-Free Unimorph Benders,” ASME J. Vib. Acoust., 132(5), p. 051005. [CrossRef]
Xu, Y. P., and Zhou, D., 2011, “Two-Dimensional Analysis of Simply Supported Piezoelectric Beams With Variable Thickness,” Appl. Math. Modell., 35(9), pp. 4458–4472. [CrossRef]
Schoeftner, J., and Krommer, M., 2012, “Single Point Vibration Control for a Passive Piezoelectric Bernoulli-Euler Beam Subjected to Spatially Varying Harmonic Loads,” Acta Mech., 223(9), pp. 1983–1998. [CrossRef]
Wang, J. J., Shi, Z. F., and Xiang, H. J., 2013, “Electromechanical Analysis of Piezoelectric Beam-Type Transducers With Interlayer Slip,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 60(8), pp. 1768–1776. [CrossRef] [PubMed]
Lee, U., Kim, D., and Park, I., 2013, “Dynamic Modeling and Analysis of the PZT-Bonded Composite Timoshenko Beams: Spectral Element Method,” J. Sound Vib., 332(6), pp. 1585–1609. [CrossRef]
Hajianmaleki, M., and Qatu, M. S., 2013, “Vibrations of Straight and Curved Composite Beams: A Review,” Compos. Struct., 100, pp. 218–232. [CrossRef]
Beheshti-Aval, S. B., and Lezgy-Nazargah, M., 2013, “Coupled Refined Layerwise Theory for Dynamic Free and Forced Response of Piezoelectric Laminated Composite and Sandwich Beams,” Meccanica, 48(6), pp. 1479–1500. [CrossRef]
Khani, S., Tabandeh, N., and Ghomshei, M. M., 2014, “Natural Frequency Analysis of Non-Uniform Smart Beams With Piezoelectric Layers, Using Differential Quadrature Method,” Composites, Part B, 58, pp. 303–311. [CrossRef]
Mareishi, S., Rafiee, M., He, X. Q., and Liew, K. M., 2014, “Nonlinear Free Vibration, Postbuckling and Nonlinear Static Deflection of Piezoelectric Fiber-Reinforced Laminated Composite Beams,” Composites, Part B, 59, pp. 123–132. [CrossRef]
Kapuria, S., Bhattacharyya, M., and Kumar, A. N., 2006, “Assessment of Coupled 1D Models for Hybrid Piezoelectric Layered Functionally Graded Beams,” Compos. Struct., 72(4), pp. 455–468. [CrossRef]
Alibeigloo, A., 2010, “Thermoelasticity Analysis of Functionally Graded Beam With Integrated Surface Piezoelectric Layers,” Compos. Struct., 92(6), pp. 1535–1543. [CrossRef]
Kiani, Y., Rezaei, M., Taheri, S., and Eslami, M. R., 2011, “Thermo-Electrical Buckling of Piezoelectric Functionally Graded Material Timoshenko Beams,” Int. J. Mech. Mater. Des., 7(3), pp. 185–197. [CrossRef]
Panda, S., and Ray, M. C., 2012, “Active Damping of Nonlinear Vibrations of Functionally Graded Laminated Composite Plates Using Vertically/Obliquely Reinforced 1-3 Piezoelectric Composite,” ASME J. Vib. Acoust., 134(2), p. 021016. [CrossRef]
Dash, P., and Singh, B. N., 2012, “Geometrically Nonlinear Free Vibration of Laminated Composite Plate Embedded With Piezoelectric Layers Having Uncertain Material Properties,” ASME J. Vib. Acoust., 134(6), p. 061006. [CrossRef]
Rafiee, M., Yang, J., and Kitipornchai, S., 2013, “Large Amplitude Vibration of Carbon Nanotube Reinforced Functionally Graded Composite Beams With Piezoelectric Layers,” Compos. Struct., 96, pp. 716–725. [CrossRef]
Shegokar, N. L., and Lal, A., 2013, “Stochastic Nonlinear Bending Response of Piezoelectric Functionally Graded Beam Subjected to Thermoelectromechanical Loadings With Random Material Properties,” Compos. Struct., 100, pp. 17–33. [CrossRef]
Komijani, M., Reddy, J. N., and Eslami, M. R., 2014, “Nonlinear Analysis of Microstructure-Dependent Functionally Graded Piezoelectric Material Actuators,” J. Mech. Phys. Solids, 63, pp. 214–227. [CrossRef]
Reddy, J. N., 2004, Mechanics of Laminated Composite Plates and Shells, 1st ed., McGraw-Hill, New York.
Poulin, K. C., and Vaicaitis, R., 2004, “Vibrations of Stiffened Composite Panels With Smart Materials,” ASME J. Vib. Acoust., 126(3), pp. 370–379. [CrossRef]
Topdar, P., Chakraborti, A., and Sheikh, A. H., 2004, “An Efficient Hybrid Plate Model for Analysis and Control of Smart Sandwich Laminates,” Comput. Meth. Appl. Mech. Eng., 193(42–44), pp. 4591–4610. [CrossRef]
Kapuria, S., and Achary, G. G. S., 2004, “An Efficient Higher Order Zigzag Theory for Laminated Plates Subjected to Thermal Loading,” Int. J. Solids Struct., 41(16–17), pp.4661–4684. [CrossRef]
Kapuria, S., and Achary, G. G. S., 2005, “A Coupled Consistent Third-Order Theory for Hybrid Piezoelectric Plates,” Compos. Struct., 70(1), pp. 120–133. [CrossRef]
Moita, J. M. S., Soares, C. M. M., and Soares, C. A. M., 2005, “Active Control of Forced Vibrations in Adaptive Structures Using a Higher-Order Model,” Compos. Struct., 71(3–4), pp. 349–355. [CrossRef]
Vel, S. S., and Baillargeon, B. P., 2005, “Analysis of Static Deformation, Vibration and Active Damping of Cylindrical Composite Shells With Piezoelectric Shear Actuators,” ASME J. Vib. Acoust., 127(4), pp. 395–407. [CrossRef]
Zhu, L. F., Chattopadhyay, A., and Goldberg, R. K., 2006, “Nonlinear Transient Response of Strain Rate Dependent Composite Laminated Plates Using Multiscale Simulation,” Int. J. Solids Struct., 43(9), pp. 2602–2630. [CrossRef]
Varelis, D., and Saravanos, D. A., 2006, “Small-Amplitude Free-Vibration Analysis of Piezoelectric Composite Plates Subject to Large Deflections and Initial Stresses,” ASME J. Vib. Acoust., 128(1), pp. 41–49. [CrossRef]
Nayfeh, A. H., and Mook, D. T., 1979, Nonlinear Oscillations, 1st ed., Wiley-Interscience, New York.

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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