Abstract
We experimentally investigated nonlinear combination resonances in two graphite-epoxy cantilever plates having the configurations (90/30/-30/-30/30/90)s and (-75/75/75/-75/75/-75)s. As a first step, we compared the natural frequencies and modes shapes obtained from the finite-element and experimental-modal analyses. The largest difference in the obtained frequencies for both plates was 6%. Then, we transversely excited the plates and obtained force-response and frequency-response curves, which were used to characterize the plate dynamics. We acquired time-domain data for specific input conditions using an A/D card and used them to generate time traces, power spectra, pseudo-state portraits, and Poincaré maps. The data were obtained with an accelerometer monitoring the excitation and a laser vibrometer monitoring the plate response. We observed the external combination resonance % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabfM6axjabgIKi7kabeM8a3naaBaaaleaacaaIYaaabeaakiab% gUcaRiabeM8a3naaBaaaleaacaaI3aaabeaaaaa!45C9!\[\Omega \approx \omega _2 + \omega _7 \] in the quasi-isotropic plate and the external combination resonance % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabfM6axjabgIKi7kaacIcacaaIXaGaai4laiaaikdacaGGPaGa% aiikaiabeM8a3naaBaaaleaacaaIYaaabeaakiabgUcaRiabeM8a3n% aaBaaaleaacaaI1aaabeaakiaacMcaaaa!4AAD!\[\Omega \approx (1/2)(\omega _2 + \omega _5 )\] and the internal combination resonance % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabfM6axjabgIKi7kabeM8a3naaBaaaleaacaaI4aaabeaakiab% gIKi7kaacIcacaaIXaGaai4laiaaikdacaGGPaGaaiikaiabeM8a3n% aaBaaaleaacaaIYaaabeaakiabgUcaRiabeM8a3naaBaaaleaacaaI% XaGaaG4maaqabaGccaGGPaaaaa!4FDC!\[\Omega \approx \omega _8 \approx (1/2)(\omega _2 + \omega _{13} )\] in the ±75 plate, where the % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabeM8a3naaBaaaleaacaWGPbaabeaaaaa!3F16!\[\omega _i \] are the natural frequencies of the plate and Ω is the excitation frequency. The results show that a low-amplitude high-frequency excitation can produce a high-amplitude low-frequency motion.
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References
Anderson, T. J., ‘Nonlinear vibrations of metallic and composite structures’, Ph.D. Dissertation, Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA, 1993.
Anderson, T. J., Balachandran, B., and Nayfeh, A. H., ‘Observations of nonlinear interactions in a flexible canbilever beam’, AIAA Paper No. 92-2332, in Proceedings of the 33rd AIAA Structures, Structural Dynamics, and Materials Conference, Dallas, TX, 1992.
Bajaj, A. K. and Johnson, J. M., ‘Asymptotic techniques and complex dynamics in wealky non-linear forced mechanical systems’, International Journal of Non-Linear Mechanics 25, 1990, 211–226.
Bennett, J. A., ‘Nonlinear vibration of simply supported angle-ply laminated plates’, AIAA Journal 9, 1971, 1977–2003.
Bert, C. W., ‘Research on dynamics of composite and sandwich plates’, Shock and Vibration Digest 14, 1982, 17–34.
Burton, T. D. and Kolowidth, M., ‘Nonlinear resonances and chaotic motion in a flexible parametrically excited beam’, in Proceedings of the Second Conference on Nonlinear Vibrations, Stability, and Dynamics of Structures and Mechanisms, Blacksburg, VA, 1988.
Chia, C. Y., Nonlinear Analysis of Plates, McGraw-Hill, New York, 1980.
Chia, C. Y. and Prabhakara, M. K., ‘A general mode approach to nonlinear flexural vibrations of laminated rectangular plates’, ASME Journal of Applied Mechanics, 45, 1978, 623–628.
Crespo da Silva, M. R. M., ‘On the whirling of a base excited cantilever beam’, Journal of the Acoustical Society of America 67, 1980, 707–740.
Cusumano, J. P. and Moon, F. C., ‘Low dimensional behavior in chaotic nonplanar motions of a forced elastic rod: Experiment and theory’, in Nonlinear Dynamics in Engineering Systems, IUTAM Symposium, Germany, 1989.
Dugundji, J. and Mukhopadhyay, V., ‘Lateral bending-torsion vibrations of a thin beam under parametric excitation’, ASME Journal of Applied Mechanics 44, 1973, 693–698.
Eslami, H. and Kandil, O. A., ‘Nonlinear forced vibration of orthotropic plates using the method of multiple scales’, AIAA Journal 27, 1988, 955–960.
Gary, C. E., Decha-Umphai, K., and Mei, C., ‘Large deflection, large amplitude vibration and random response of symmetrically laminated rectangular plates’, in AIAA/ASME/ASCE/AHS 25th Structures, Structural Dynamics and Materials Conference, Palm Springs, CA, 1984.
Haddow, A. G. and Hasan, S. M., ‘Nonlinear oscillations of a flexible cantilever: Experimental results’, in Proceedings of the Second Conference on Nonlinear Vibrations, Stability, and Dynamics of Structures and Mechanisms, Blacksburg, VA, 1988.
Hadian, J. and Nayfeh, A. H., ‘Free vibration and buckling of shear-deformable cross-ply laminated plates using the state-space concept’, Computer & Structures 48, 1993, 677–693.
Huang, C. L., ‘Finite amplitude vibration of an orthotropic plate with an isotropic core’, International Journal of Non-linear Mechanics 8, 1973, 445–457.
Kapania, R. K. and Yang, T. Y., ‘Buckling, postbuckling, and nonlinear vibrations of imperfect plates’, AIAA Journal 25, 1987, 1338–1346.
Moon, F. C., Chaotic and Fractal Dynamics, Wiley-Interscience, New York, 1992.
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., ‘Combination resonances in the non-linear response of bowed structures to a harmonic excitation’, Journal of Sound and Vibration 90, 1983, 457–470.
Nayfeh, A. H. and Balachandran, B., ‘Modal interactions in dynamical and structural systems’, ASME Applied Mechanics Reviews 42, 1989, S175-S201.
Nayfeh, A. H. and Balachandran, B., Applied Nonlinear Dynamics, Wiley, New York, 1995.
Nayfeh, A. H., El-Zein, M. S., and Nayfeh, J. F., ‘Nonlinear oscillations of composite plates using perturbation techniques’, in Proceedings of the American Society for Composites, 4th Technical Conference on Composite Material Systems, October 3–5, VPI&SU, Blacksburg, VA, Technomic Publishing, Lancaster, PA, 1989.
Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley-Interscience, New York, 1979.
Nayfeh, J. F., ‘Nonlinear dynamics of composite plates and other physical systems’, Ph.D. Dissertation, Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA, 1990.
Nayfeh, S. A. and Nayfeh, A. H., ‘Energy transfer from high- to low-frequency modes in a flexible structure via modulation’, ASME Journal of Vibration and Acoustics 116, 1994, 203–207.
Popovic, P., Nayfeh, A. H., Oh, K., and Nayfeh, S. A., ‘Experimental investigation of high-frequency to low-frequency modal interactions in a three-beam frame structure’, in 1994 ASME Winter Annual Meeting, Chicago, IL, Nov. 13–18, 1994.
Reddy, J. N., ‘Large amplitude flexural vibrations of layered composite plates with cutouts’, Journal of Sound and Vibration 83, 1982, 1–10.
Reddy, J. N. and Chao, W. C., ‘Large deflection and large-amplitude free vibrations of laminated composite-material plates’, Composite Structures 13, 1981, 341–347.
Reddy, J. N. and Chao, W. C., ‘Nonlinear oscillations of laminated anisotropic rectangular plates’, Journal of Applied Mechanics 49, 1982, 396–402.
Sathyamoorthy, M., ‘Nonlinear vibration of plates—A review’, Shock and Vibration Digest 15, 1983, 3–16.
Schmidt, G. and Tondl, A., Non-Linear Vibrations, Akademie-Verlag, Berlin, 1986.
User Manual for Modal Analysis 9,0, SDRC I-DEAS, Level 2.5, General Electric CAE International, 1985.
Wentz, K. R., Mei, C., and Chiang, C. K., ‘Large amplitude forced vibration response of laminated composite rectangular plates by a finite element method’, in Composite Structures, I. H. Marshal (ed.), Elsevier Applied Science Publishers, 1988, pp. 703–716.
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Oh, K., Nayfeh, A.H. Nonlinear combination resonances in cantilever composite plates. Nonlinear Dyn 11, 143–169 (1996). https://doi.org/10.1007/BF00044999
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DOI: https://doi.org/10.1007/BF00044999