Abstract
The nonlinear equations of motion derived in Part I are used to investigate the response of an inextensional, symmetric angle-ply graphite-epoxy beam to a harmonic base-excitation along the flapwise direction. The equations contain bending-twisting couplings and quadratic and cubic nonlinearities due to curvature and inertia. The analysis focuses on the case of primary resonance of the first flexural-torsional (flapwise-torsional) mode when its frequency is approximately one-half the frequency of the first out-of-plane flexural (chordwide) mode. A combination of the fundamental-matrix method and the method of multiple scales is used to derive four first-order ordinary-differential equations to describe the time variation of the amplitudes and phases of the interacting modes with damping, nonlinearity, and resonances. The eigenvalues of the Jacobian matrix of the modulation equations are used to determine the stability and bifurcations of their constant solutions, and Floquet theory is used to determine the stability and bifurcations of their limit-cycle solutions. Hopf bifurcations, symmetry-breaking bifurcations, period-multiplying sequences, and chaotic solutions of the modulation equations are studied. Chaotic solutions are identified from their frequency spectra, Poincaré sections, and Lyapunov's exponents. The results show that the beam motion may be nonplanar although the input force is planar. Nonplanar responses may be periodic, periodically modulated, or chaotically modulated motions.
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References
Pai, P. F. and Nayfeh, A. H., ‘Three-dimensional nonlinear vibrations of composite beams—I. Equations of motion’,Nonlinear Dynamics 1, 1990, 477–502.
Friedmann, P. P., ‘Recent developments in rotary-wing aeroelasticity’,Journal of Aircraft 14, 1977, 1027–1041.
Crespo da Silva, M. R. M. and Glynn, C. C., ‘Out-of-plane vibrations of a beam including nonlinear inertia and nonlinear curvature effects’,International Journal of Nonlinear Mechanics 13, 1979, 261–270.
Crespo da Silva, M. R. M. and Glynn, C. C., ‘Nonlinear nonplanar resonant oscillations in fixed-free beams with support asymmetry’,International Journal of Solids and Structures,15, 1979, 209–219.
Crespo da Silva, M. R. M. and Zaretzky, C. L., ‘Nonlinear modal coupling in the response of inextensional beams’,The Second Conference on Nonlinear Vibrations, Stability, and Dynamics of Structures and Mechanisms, June 1–3, 1988, Blacksburg, VA.
Ho, C.-H., Scott, R. A., and Eisley, J. G., ‘Nonplanar, nonlinear oscillations of a beam — II. Free motions’,Journal of Sound and Vibration 47, 1976, 333–339.
Ray, J. D. and Bert, C. W., ‘Nonlinear vibrations of a beam with pinned ends’,Journal of Engineering for Industry, Nov. 1969, 997–1004.
Bauchau, O. A. and Hong, C. H., ‘Large displacement analysis of naturally curved and twisted composite beams’,AIAA Journal 25, 1987, 1469–1475.
Kapania, R. K. and Raciti, S., ‘Nonlinear vibrations of unsymmetrically laminated beams’,AIAA Journal 27, 1989, 201–210.
Minguet, P. and Dugundji, J., ‘Experiments and analysis for composite blades under large deflections, Part 1 — Static behavior’,AIAA Journal 28, 1990, 1573–1579.
Crespo da Silva, M. R. M. and Glynn, C. C., ‘Nonlinear flexural-flexural-torsional dynamics of inextensional beams — I. Equations of motion’,Journal of Structural Mechanics 6, 1978, 437–448.
Pai, P. F., ‘Nonlinear flexural-flexural-torsional dynamics of metallic and composite beams’, Ph.D. Dissertation, Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA, April 1990.
Asfar, O. R. and Hussein, A. M., ‘Numerical solution of linear two-point boundary-value problems via the fundamental matrix method’,International Journal for Numerical Methods in Engineering 28, 1989, 1205–1216.
Nayfeh, A. H.,Perturbation Methods, Wiley-Interscience, New York, 1973.
Nayfeh, A. H.,Introduction to Perturbation Techniques, Wiley-Interscience, New York, 1981.
Nayfeh, A. H. and Mook, D. T.,Nonlinear Oscillations, Wiley-Interscience, New York, 1979.
Gerald, C. F.,Applied Numerical Analysis, Addison-Wesley Publishing Co., 1983.
Nayfeh, A. H. and Pai, P. F., ‘Nonlinear nonplanar parametric responses of an inextensional beam’,International Journal of Nonlinear Mechanics 24, 1989, 139–158.
Pai, P. F. and Nayfeh, A. H., ‘Nonlinear nonplanar oscillations of a cantilever beam under lateral base excitations’,International Journal of Nonlinear Mechanics 25, 1990, 455–474.
Chua, L. O. and Lin, P. M.,Computer-Aided Analysis of Electronic Circuits, Prentice-Hall, New Jersey, 1975.
Marsden, J. E. and McCracken, M.,Hopf Bifurcation and Its Application, Springer-Verlag, New York, 1976.
Seydel, R.,From Equilibrium to Chaos — Practical Bifurcation and Stability Analysis, Elsevier, New York, 1988.
Wolf, A., Swift, J. B., Swinney, H. L., and Vastano, J. A., ‘Determining Lyapunov exponents from a time series’,Physica D 16, 1985, 285–317.
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Pai, P.F., Nayfeh, A.H. Three-dimensional nonlinear vibrations of composite beams — II. flapwise excitations. Nonlinear Dyn 2, 1–34 (1991). https://doi.org/10.1007/BF00045053
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DOI: https://doi.org/10.1007/BF00045053