Abstract
Oscillators with a non-negative real-power restoring force and quadratic damping are considered in this paper. The equation of motion is transformed into a linear first-order differential equation for the kinetic energy. The expressions for the energy-displacement function are derived as well as the closed form exact solutions for the relationship between subsequent amplitudes. They are expressed in terms of incomplete Gamma functions. On the basis of these results, expressions for the phase trajectories and the loci of maximal velocities are obtained. It is also demonstrated that the time difference between two consecutive relative maxima and minima of the displacement can both increase and decrease with time.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979)
Ruzicka, J.E., Derby, T.E.: Influence of Damping in Vibration Isolation. T.S.a.V.I. Centre, US Department of Defense, Washington (1971)
Kovacic, I., Brennan, M.J., Waters, T.P.: A study of a non-linear vibration isolator with quasi-zero stiffness characteristic. J. Sound Vib. 315, 700–711 (2008)
Gatti, G., Kovacic, I., Brennan, M.J.: On the response of a harmonically excited two degree-of-freedom system consisting of linear and non-linear quasi-zero stiffness oscillators. J. Sound Vib. 329, 823–1835 (2010)
Alabudzev, P., Gritchin, A., Kim, L., Migirenko, G., Chon, V., Stepanov, P.: Vibration Protecting and Measuring Systems with Quasi-Zero Stiffness. Hemisphere, New York (1989)
Mickens, R.E.: Truly Nonlinear Oscillations: Harmonic Balance, Parametric Expansions, Iteration, and Averaging Methods. World Scientific, Singapore (2010)
Kovacic, I., Rakaric, Z.: Oscillators with a fractional-order restoring force: higher-order approximations for motion via a modified Ritz method. Commun. Non-Linear Sci. Numer. Simul. 15, 2651–2658 (2010)
Cveticanin, L., Zukovic, M.: Melnikov’s criteria and chaos in systems with fractional order deflection. J. Sound Vib. 326, 768–779 (2009)
Bogoliubov, N.N., Mitropolski, Y.A.: Asymptotic Methods in the Theory of Non-Linear Vibrations. Nauka, Moscow (1974) (in Russian)
Yuste, S.B.: Quasi-pure-cubic oscillators studied using a Krylov-Bogoliubov method. J. Sound Vib. 158, 267–275 (1992)
Dasarathy, B.V.: Analysis of a class of non-linear systems. J. Sound Vib. 11, 139–144 (1970)
Jordan, D.W., Smith, P.: Nonlinear Ordinary Differential Equations: An Introduction to Dynamical Systems. Oxford University Press, Oxford (1999)
Andronov, A.A., Vitt, A.A., Hajkin, S.E.: Theory of Vibrations. Nauka, Moscow (1981) (in Russian)
Klotter, K.: Free oscillations of systems having quadratic damping and arbitrary restoring forces. J. Appl. Mech. 22, 493–499 (1955)
Cvetićanin, L.: Oscillator with strong quadratic damping force. Publ. Inst. Math. Nouvelle Sér. 85, 119–130 (2009)
Aslam Chaudhry, M., Zubair, M.S.: A Class of Incomplete Gamma Functions with Applications. Chapman & Hall/CRC, Boca Raton (2002)
http://functions.wolfram.com/GammaBetaErf/Gamma2/03/01/01/0002/. Accessed 9 June 2010
Cveticanin, L.: Oscillator with fraction order restoring force. J. Sound Vib. 320, 1064–1077 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kovacic, I., Rakaric, Z. Study of oscillators with a non-negative real-power restoring force and quadratic damping. Nonlinear Dyn 64, 293–304 (2011). https://doi.org/10.1007/s11071-010-9861-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-010-9861-9