Energy dissipation in a nonlinearly damped Duffing oscillator. (English) Zbl 0994.37009

Summary: We study the effect of including a nonlinear damping term proportional to the power of the velocity in the dynamics of a double-well Duffing oscillator. In particular, we focus our attention in understanding how the energy dissipates over a cycle and along the time, by the use of different tools of analysis. Analytical and numerical results for different damping terms are shown, and the presence of a discontinuity and an inversion of behavior depending on the initial energy are discussed. An averaged power loss in a period is defined, showing similar characteristics as the energy dissipation over a cycle, although no discontinuity is present. The discontinuity gap which appears for the energy dissipation at a certain value of the initial energy decreases as the power of the damping term increases and an associated scaling law is found.


37B55 Topological dynamics of nonautonomous systems
37C55 Periodic and quasi-periodic flows and diffeomorphisms
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[1] B. Feeny, A. Guran, Friction as a nonlinearity in dynamics: a historical review, in: A. Guran (Ed.), Nonlinear Dynamics, World Scientific, Singapore, 1997. · Zbl 0931.74004
[2] B.N.J. Persson, Sliding Friction: Physical Principles and Applications, Springer, Berlin, 1998. · Zbl 0966.74001
[3] Röder, J.; Bishop, A.R.; Holian, B.L.; Hammerbereg, J.E.; Mikulla, R.P., Dry friction: modelling and energy flow, Physica D, 142, 306-316, (2000) · Zbl 0961.74044
[4] S.G. Kelly, Fundamentals of Mechanical Vibrations, McGraw-Hill, New York, 1993.
[5] L.N. Virgin, Introduction to Experimental Nonlinear Dynamics, Cambridge University Press, Cambridge, 2000.
[6] Bikdash, M.; Balachandran, B.; Nayfeh, A., Melnikov analysis for a ship with a general roll-damping model, Nonlinear dynam., 6, 101-124, (1994)
[7] Falzarano, J.M.; Shaw, S.W.; Troesch, A.W., Application of global methods for analyzing dynamical systems to ship rolling motion and capsizing, Int. J. bifurcat. chaos, 2, 101-1154, (1992) · Zbl 0900.76064
[8] Chan, H.S.Y.; Xu, Z.; Huang, W.L., Estimation of nonlinear damping coefficients from large-amplitude ship rolling motions, Appl. Ocean res., 17, 217-224, (1995)
[9] N. De Mestre, The Mathematics of Projectiles in Sport, Cambridge University Press, Cambridge, 1990. · Zbl 0746.70001
[10] Huilgol, R.R.; Christie, J.R.; Panizza, M.P., The motion of a mass hanging from an overhead crane, Chaos, solitons fract., 5, 1619-1631, (1995)
[11] Ravindra, B.; Mallik, A.K., Stability analysis of a non-linearly damped Duffing oscillator, J. sound vib., 171, 708-716, (1994) · Zbl 1064.70512
[12] Ravindra, B.; Mallik, A.K., Role of nonlinear dissipation in soft Duffing oscillators, Phys. rev. E, 49, 4950-4954, (1994)
[13] Ravindra, B.; Mallik, A.K., Chaotic response of a harmonically excited mass on an isolator with non-linear stiffness and damping characteristics, J. sound vib., 182, 345-353, (1995) · Zbl 1237.70088
[14] Sanjuán, M.A.F., The effect of nonlinear damping on the universal escape oscillator, Int. J. bifurcat. chaos, 9, 735-744, (1999) · Zbl 0977.34041
[15] Trueba, J.L.; Rams, J.; Sanjuán, M.A.F., Analytical estimates of the effect of nonlinear damping in some nonlinear oscillators, Int. J. bifurcat. chaos, 10, 2257-2267, (2000) · Zbl 0999.70021
[16] Pfenniger, D.; Norman, C., Dissipation in barred galaxies: the growth of bulges and central mass concentrations, The astrophys. J., 363, 391-410, (1990)
[17] J. Binney, S. Tremaine, Galactic Dynamics, Princeton University Press, Princeton, 1987.
[18] D. Lawden, Elliptic Functions and Applications, Springer, Berlin, 1989. · Zbl 0689.33001
[19] L. M. Milne-Thomson, in: A. Abramowitz, I. Stegun (Eds.), Handbook of Mathematical Functions, Dover, New York, 1970. · Zbl 0236.65003
[20] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C, Cambridge University Press, Cambridge, 1992. · Zbl 0778.65003
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