Abstract
The theory of deterministic generalized viscoelastic linear and nonlinear 1-D oscillators is formulated and evaluated. Examples of viscoelastic Duffing, Mathieu, Rayleigh, Roberts and van der Pol oscillators and pendulum responses are investigated. Material behavior as well as additional effects of structural damping on oscillator performance are also considered. Computational protocols are developed and their results are discussed to determine the influence of viscoelastic and structural (Coulomb friction) damping on oscillator motion. Illustrative examples show that the inclusion of linear or nonlinear viscoelastic material properties significantly affects oscillator responses as related to amplitudes, phase shifts and energy loses when compared to equivalent elastic ones.
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References
Andronov, A. A., Vitt, A. A., and Khaikin, S. E., Theory of Oscillators,Dover, New York, 1966.
Hale, J. K., Oscillations in Nonlinear Systems,Dover, New York, 1963.
Nayfeh, A. H. and Balachandran, B., Applied Nonlinear Dynamics: Analytical, Computational and Experimental Methods,Wiley, New York, 1995.
Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations,Wiley, New York, 1979.
Bergman, L. A., Hilton, H. H., and Tsao, T. C., 'The effect of viscoelasticity on the performance of reaction mass actuators', in Proceedings Damping'93 Conference, WL-TR-93-3105, Wright Patterson AFB, OH, 1993, Vol. 2, pp. GAB–1–GAB–8.
Yi, S., 'Finite element analysis of viscoelastic structures and analytical determination of optimum viscoelastic materials', Ph.D. Dissertation, University of Illinois, Urbana, Illinois, 1992.
Yi, S., 'Thermoviscoelastic of delamination onset and free edge response in epoxy matrix composite laminates', AIAA Journal 32, 1993, 2320–2328.
Yi, S. and Hilton, H. H., 'Dynamic finite element analysis of viscoelastic composite plates in the time domain', International Journal for Numerical Methods in Engineering 37, 1994, 4081–4096.
Hilton, H. H., Yi, S., and Ruijun, T., 'Large deflections of linearly elastic and viscoelastic columns with follower loads', in Recent Advances in Engineering Science, Proceedings of 31st Annual Technical Conference Society of Engineering Science, D. H. Allen and D. C. Lagoudas (eds.), Texas A&M University, College Station, Texas, 1994, pp. 18–19.
Baker, C. T. H., The Numerical Treatment of Integral Equations,Clarendon Press, New York, 1977.
Baker, C. T. H. and Miller, G, F. (eds.), Treatment of Integral Equations by Numerical Methods, Academic Press, New York,1982.
Goldberg, M. (ed.), Solution Methods for Integral Equations-Theory and Applications, Plenum Press, New York, 1979.
Hairer, E., 'Extended Volterra-Runge-Kutta methods', in Treatment of Integral Equations by Numerical Methods,C.T.H. Baker and G. F. Miller (eds.), Academic Press, New York, 1982, pp. 221–231.
Rauber, T. and Rünger, G., 'Parallel implementations of iterated Runge-Kutta methods', The International Journal of Supercomputer Applications and High Performance Computing 10, 1996, 62–90.
Hilton, H. H., 'An introduction to viscoelastic analysis', in Engineering Design for Plastics,E. Baer (ed.), Reinhold, New York, 1964, pp. 199–276.
Christensen, R. M., Theory of Viscoelasticity-an Introduction, 2nd ed., Academic Press, New York, 1981.
Lubich, C., 'On the stability of linear multistep methods for Volterra integral equations of the second kind', in Treatment of Integral Equations by Numerical Methods,C.T.H. Baker and G. F. Miller (eds.), Academic Press, New York, 1982, pp. 199–211.
Keller, H. D., Numerical Methods for Two-Point Boundary Value Problems,Dover, New York, 1992.
Hilton, H. H., Hsu, J., and Kirby, J. S., 'Linear viscoelastic analysis with random material properties', Journal of Probabilistic Engineering Mechanics6, 57–69.
Hilton, H. H. and Feigen, M., 'Minimum weight analysis based on structural reliability', Journal of the Aero/Space Sciences 27, 1960, 641–652.
Hilton, H. H. and Ariaratnam, S. T., 'Invariant anisotropic large deformation deterministic and stochastic combined load failure criteria', International Journal of Solids and Structures 31, 1994, 3285–3293.
Bisplinghoff, R. L., Ashley, H., and Halfman, R., 1955, Aeroelasticity, Addison-Wesley, Cambridge, Massachusetts, 1995.
Hilton, H. H., 'Viscoelastic and structural damping analysis', in Proceedings on Damping'91, Air Force Technical Report WL-TR-91-3078 III, 1991, pp. ICB 1–15.
Hilton, H. H. and Vail, C. F., 'Bending-torsion flutter of linear viscoelastic wings including structural damping', in Proceedings AIAA/ASME/ASCE/AHS/ASC 34th Structures, Structural Dynamics and Materials Conference, AIAA Paper 93-1475,Vol. 3, 1993, pp. 1461–1481.
Eringen, C. A., Nonlinear Theory of Continuous Media, McGraw-Hill, New York, 1962.
Hilton, H. H. and Yi, S., 'Creep divergence of nonlinear viscoelastic lifting surfaces with piezoelectric control', in Proceedings of the Second International Conference on Nonlinear Problems in Aviation and Aerospace,S. Sivasundaram (ed.), Vol. 1,European Conference Publications, Cambridge, UK, 1999, pp. 271–280.
Zak, A. R., 'Structural analysis of realistic solid propellant materials', Journal of Spacecraft and Rockets5, 1967, 270–275.
Taylor, R. L., Pister, K. L., and Goudreau, G. L., Thermomechanical analysis of viscoelastic solids', International Journal for Numerical Methods in Engineering2, 1970, 45–59.
Atkinson, K. E., The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, New York, 1997.
Srivastava, H. M. and Buschma, R. G., 1992, Theory and Applications of Convolution Integral Equations,Kluwer, New York, 1992.
Beldica, C. E. and Hilton, H. H., 'Analytical simulations of optimum anisotropic linear viscoelastic damping properties', Journal of Reinforced Plastics and Composites 18, 1999, 1658–1676.
Hilton, H. H. and Yi, S., 1992, 'Analytical formulation of optimum material properties for viscoelastic damping', Journal of Smart Materials and Structures1, 1992, 113–122.
Bagley, R. L. and Torvik, P. J., 'Fractional calculus-a different approach to the finite element analysis of viscoelastically damped structures', AIAA Journal 21, 1983, 741–748.
Bagley, R. L. and Torvik, P. J., 'A theoretical basis for the application of fractional calculus to viscoelasticity', SIAM Journal on Applied Mathematics 27, 1983, 201–210.
Rogers, L., 'Operators and fractional derivatives for viscoelastic constitutive equations', Journal of Rheology 27, 1983, 351–360.
Hilton, H. H., 'Anisotropic viscoelastic fractional derivative material characterization', in AAE Technical Report TR UILU ENG 95-0507, University of Illinois at Urbana-Champaign, Illinois, 1995.
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Hilton, H.H., Yi, S. Generalized Viscoelastic 1-DOF Deterministic Nonlinear Oscillators. Nonlinear Dynamics 36, 281–298 (2004). https://doi.org/10.1023/B:NODY.0000045520.93189.fe
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DOI: https://doi.org/10.1023/B:NODY.0000045520.93189.fe