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Free Vibrations

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Strong Nonlinear Oscillators

Part of the book series: Mathematical Engineering ((MATHENGIN))

Abstract

In this chapter the free vibration of the oscillator with pure nonlinearity (2.1) and small additional forces is investigated.

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References

  • Askey, R. A., & Roy, R. (2010). Gamma function. In F. W. J. Olver, D. W. Lozier, R. F. Boisvert, & C. W. Clark (Eds.), NIST handbook of mathematical functions (pp. 135–147). New York: Cambridge University Press.

    Google Scholar 

  • Bagheri, M., Poot, M., Li, M., Pernice, W. P. H., & Tang, H. X. (2011). Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation. Nature Nanotechnology, 6(11), 726–732.

    Article  Google Scholar 

  • Belendez, A., Pascual, C., Gallego, S., Ortufio, M., & Neipp, C. (2007). Application of a modified He’s homotopy perturbation method to obtain higher-order approximations of an \(x^{1/3}\) force nonlinear oscillator. Physica Letters A, 371, 421–426.

    Article  MATH  Google Scholar 

  • Belhaq, M., & Lakrad, F. (2000a). On the elliptic harmonic balance method for mixed parity non-linear oscillators. Journal of Sound and Vibration, 233, 935–937.

    Google Scholar 

  • Belhaq, M., & Lakrad, F. (2000b). The elliptic multiple scales method for a class of autonomous strongly non-linear oscillators. Journal of Sound and Vibration, 234, 547–553.

    Google Scholar 

  • Belhaq, M., & Lakrad, F. (2000c). Prediction of homoclinic bifurcation: The elliptic averaging method. Chaos, Solitons and Fractals, 11, 2251–2258.

    Google Scholar 

  • Bogolubov, N. N., & Mitropolski, J. A. (1974). Asimptoticheskie metodi v teorii nelinejnih kolebanij. Moscow: Nauka.

    Google Scholar 

  • Bhushan, A., Inamdar, M. M., & Pawaskar, D. N. (2013). Dynamic analysis of a double-sided actuated MEMS oscillator using second-order averaging. Lecture Notes in Engineering and Computer Science, 3 LNECS (pp. 1640–1645).

    Google Scholar 

  • Borkje, K., Nunnenkamp, A., Teufel, J. D., & Girvin, S. M. (2013). Signatures of nonlinear cavity optomechanics in the weak coupling regime. Physical Review Letters, 111(5), 053603.

    Article  Google Scholar 

  • Brennecke, F., Ritter, S., Donner, T., & Esslinger, T. (2008). Cavity optomechanics with a Bose–Einstein condensate. Science, 322, 235–238.

    Article  Google Scholar 

  • Byrd, P. F. (1954). Handbook of elliptic integrals for engineers and physicists. Berlin: Springer.

    Book  MATH  Google Scholar 

  • Byrd, P. F., & Friedman, D. (1954). Handbook of elliptic integrals for engineers and physicists. Berlin: Springer.

    Book  MATH  Google Scholar 

  • Chang, D. E., Regal, C. A., Papp, S. B., Wilsonb, D. J., Ye, J., Painter, O., et al. (2010). Cavoty opto-mechanics using an optically levitated nanosphere. Proceedings of the National Academy of Sciences of the United States of America, 107(3), 1005–1010.

    Article  Google Scholar 

  • Chen, C. H., Yang, X. M., & Cheung, Y. K. (1998). Periodic solutions of strongly quadratic nonlinear oscillators by the elliptic perturbation method. Journal of Sound and Vibration, 212, 771–780.

    Article  MathSciNet  MATH  Google Scholar 

  • Cheung, Y. K., Chen, S. H., & Lau, S. L. (1991). A modified Lindstedt–Poincaré method for certain strongly non-linear oscillators. International Journal of Non-Linear Mechanics, 26, 367–378.

    Article  MathSciNet  MATH  Google Scholar 

  • Coppola, V. T., & Rand, R. H. (1990). Averaging using elliptic functions: approximation of limit cycles. Acta Mechanica, 81, 125–142.

    Article  MathSciNet  MATH  Google Scholar 

  • Cveticanin, L. (2000). Vibrations in a parametrically excited system. Journal of Sound and Vibration, 229, 245–271.

    Article  Google Scholar 

  • Cveticanin, L. (2006). Homotopy-perturbation method for pure non-linear differential equation. Chaos, Solitons and Fractals, 30, 1221–1230.

    Article  MATH  Google Scholar 

  • Cveticanin, L. (2009). Oscillator with fraction order restoring force. Journal of Sound and Vibration, 320, 1064–1077.

    Article  Google Scholar 

  • Cveticanin, L. (2014). Strongly nonlinear oscillators—Analytical solutions. Berlin: Springer.

    Book  MATH  Google Scholar 

  • Cveticanin, L. (2014). An analytical method for truly nonlinear oscillators. In A. B. Gumel (Ed.), Mathematics of continuous and discrete dynamics systems, Series Contemporary Mathematics, AMS 618 (pp. 229–245).

    Google Scholar 

  • Cveticanin, L., & Pogany, T. (2012). Oscillator with a sum of non-integer order non-linearities. Journal of Applied Mathematics, 2012(649050), 1–20.

    Google Scholar 

  • Cveticanin, l., Kalmi-Yazdi, M., Saadatnia, Z., & Askari, H. (2010). Application of Hamiltonian approach to the generalized nonlinear oscillator with fractional power. International Journal of Nonlinear Sciences and Numerical Simulation, 11(12), 997–1002.

    Google Scholar 

  • Cveticanin, L., Zukovic, M., Mester, Gy, Biro, I., & Sarosi, J. (2016). Oscillators with symmetric and asymmetric quadratic nonlinearity. Acta Mechanica, 227(6), 1727–1742.

    Article  MathSciNet  MATH  Google Scholar 

  • Drogomirecka, H. T. (1997). Integrating a special Ateb-function. Visnik Lvivskogo Universitetu. Serija mehaniko-matematichna, 46, 108–110. (in Ukrainian).

    Google Scholar 

  • Duffing, G. (1918). Erzwungene Schwingungen bei veranderlicher Eigenfrequenz und ihre technische Bedeutung. Braunschweig: Vieweg & Sohn.

    MATH  Google Scholar 

  • Eichenfield, M., Chan, J., Camacho, R. M., Vahal, K. J., & Painter, O. (2009). Optomechanical crystals. Nature, 462, 78–82.

    Article  Google Scholar 

  • He, J. H. (1999). Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 178, 257–262.

    Article  MathSciNet  MATH  Google Scholar 

  • He, J. H. (2000). A new perturbation technique which is also valid for large parameters. Journal of Sound and Vibration, 229, 1257–1263.

    Article  MathSciNet  MATH  Google Scholar 

  • He, J. H. (2001a). Modified Lindstedt–Poincaré methods for some strongly non-linear oscillations Part III: Double series expansion. International Journal of Nonlinear Sciences and Numerical Simulation, 2, 317–320.

    Google Scholar 

  • He, J. H. (2001b). Bookkeeping parameter in perturbation methods. International Journal of Nonlinear Sciences and Numerical Simulation, 2, 257–264.

    Google Scholar 

  • He, J. H. (2002a). Modified Lindstedt–Poincaré methods for some strongly non-linear oscillations Part I: Expansion of a constant. International Journal of Non-Linear Mechanics, 37, 309–314.

    Google Scholar 

  • He, J. H. (2002b). Modified Lindstedt–Poincaré methods for some strongly non-linear oscillations Part II: A new transformation. International Journal of Non-Linear Mechanics, 37, 315–320.

    Google Scholar 

  • He, J. H. (2003a). Determination of limit cycles for strongly nonlinear oscillators. Physical Review Letters, 90, Art. No. 174301.

    Google Scholar 

  • He, J. H. (2003b). Homotopy perturbation method: A new nonlinear analytical technique. Applied Mathematics and Computation, 135, 73–79.

    Google Scholar 

  • He, J. H. (2004). Asymptology by homotopy perturbation method. Applied Mathematics and Computation, 156, 591–596.

    Article  MathSciNet  MATH  Google Scholar 

  • He, J. H. (2010). Haniltonian approach to nonlinear oscillators. Physics Letters A, 374(23), 2312–2314.

    Article  MathSciNet  MATH  Google Scholar 

  • He, Q., & Daqaq, M. F. (2014). Influence of potential function asymmetries on the performance of nonlinear energy harvesters under white noise. Journal of Sound and Vibration, 333, 3479–3489.

    Article  Google Scholar 

  • Hu, H. (2004). A classical perturbation technique which is valid for large parameters. Journal of Sound and Vibration, 269, 409–412.

    Article  MathSciNet  MATH  Google Scholar 

  • Isaar, A., & Scheid, W. (2002). Deformed quantum harmonic oscillator with diffusion and dissipation. Physica A, 310, 364–376.

    Article  MathSciNet  MATH  Google Scholar 

  • Kamke, A. H. (1959). Differentialgleichungen—Losungsmethoden und Losungen. Leipzig: Akademische Verlagsgesellschaft.

    Google Scholar 

  • Krylov, N., & Bogolubov, N. (1943). Introduction to Nonlinear Mechanics. Princeton, NJ: Princeton University Press.

    Google Scholar 

  • Lai, S. K., & Chow, K. W. (2012). Exact solutions for oscillators with quadratic damping and mixed-parity nonlinearity. Physica Scripta, 85(045006), 1–10.

    MATH  Google Scholar 

  • Lakrad, F., & Belhaq, M. (2002). Periodic solutions of strongly non-linear oscillators by the multiple scales method. Journal of Sound and Vibration, 258, 677–700.

    Article  MathSciNet  MATH  Google Scholar 

  • Liao, S. J. (1995). An approximate solution technique not depending on small parameters: A special example. International Journal of Non-Linear Mechanics, 30, 371–380.

    Article  MathSciNet  MATH  Google Scholar 

  • Liao, S. J., & Tan, Y. (2007). A general approach to obtain series solutions of nonlinear differential equations. Studies in Applied Mathematics, 119, 297–355.

    Article  MathSciNet  Google Scholar 

  • Lim, C. W., & Wu, B. S. (2002). A modified Mickens procedure for certain non-linear oscillators. Journal of Sound and Vibration, 257, 202–206.

    Article  MathSciNet  MATH  Google Scholar 

  • Liu, H. M. (2005). Approximate period of nonlinear oscillators with discontinuities by modified Linstedt–Poincaré method. Chaos, Solitons & Fractals, 23, 577–579.

    Article  MATH  Google Scholar 

  • Liu, Y.-C., Hu, Y.-W., Wong, C.-W., & Xiao, Y.-F. (2013). Review of cavity optomechanical cooling. Chinese Physics B, 22(11), 114213.

    Article  Google Scholar 

  • Ludwig, M., Kubala, B., & Marquardt, F. (2008). The optomechanical instability in the quantum regime. New Journal of Physics, 10(095013), 1–12.

    Google Scholar 

  • Margallo, G. J., & Bejarano, J. D. (1987). A generalization of the method of harmonic balance. Journal of Sound and Vibration, 116, 591–595.

    Article  MathSciNet  MATH  Google Scholar 

  • Mickens, R. E. (2001). Oscillations in an x\(^{4/3}\) potential. Journal of Sound and Vibration, 246, 375–378.

    Article  MathSciNet  MATH  Google Scholar 

  • Mickens, R. E. (2002). Analysis of non-linear oscillators having non-polynomial elastic terms. Journal of Sound and Vibration, 255, 789–792.

    Article  MathSciNet  MATH  Google Scholar 

  • Mickens, R. E. (2010). Truly nonlinear oscillations. Singapore: World Scientific.

    Book  MATH  Google Scholar 

  • Navarro, H. A., & Cveticanin, L. (2016). Amplitude–frequency relationship obtained using Hamiltonian approach for oscillators with sum of non-integer order nonlinearities. Applied Mathematics and Computation, 291, 162–171.

    Article  MathSciNet  Google Scholar 

  • Nayfeh, A. H., & Mook, D. T. (1979). Nonlinear oscillations. New York: Wiley.

    MATH  Google Scholar 

  • Peano, V., & Thorwart, M. (2004). Macroscopic quantum effects in a strongly driven nanomechanical resonator. Physical Review B: Condensed Matter and Materials Physics, 70(23), 1–5.

    Article  Google Scholar 

  • Pham, T. T., Lamarque, C. H., & Pernot, S. (2011). Passive control of one degree of freedom nonlinear quadratic oscillator under combination resonance. Communications in Nonlinear Science and Numerical Simulations, 16, 2279–2288.

    Article  MathSciNet  MATH  Google Scholar 

  • Purdy, T. P., Brooks, D. W. C., Botter, T., Brahms, N., Ma, Z.-Y., & Stamper-Kurn, D. M. (2010). Turnable cavity optomechanics with ultracold atoms. Physical Review Letters, 105(13), 133602.

    Article  Google Scholar 

  • Qaisi, M. I. (1996). A power series approach for the study of periodic motion. Journal of Sound and Vibration, 196, 401–406.

    Article  MathSciNet  MATH  Google Scholar 

  • Ramananarivo, S., Godoy-Diana, R., & Thiria, B. (2011). Rather than resonance, flapping wing flyers may play on aerodynamics to improve performance. Proceedings of the National Academy of Sciences of the United States of America, 108(15), 5964–5969.

    Article  Google Scholar 

  • Rosenberg, R. M. (1963). The Ateb(h)-functions and their properties. Quarterly of Applied Mathematics, 21, 37–47.

    Article  MathSciNet  MATH  Google Scholar 

  • Rudowski, J., & Szemplinska-Stupnicka, W. (1987). On an approximate criterion for chaotic motion in a model of a buckled beam. Ingenieur-Archiv, 57, 243–255.

    Article  MATH  Google Scholar 

  • Safavi-Naeini, A. H., Alegre, T. P. M., Chan, J., Chang, D. E., & Painter, O. (2011). Electromagnetically induced transparency and slowlight with optomechanics. Nature, 472(7341), 69–73.

    Article  Google Scholar 

  • Sankey, J. C., Yang, C., Zwickl, B. M., Jayich, A. M., & Harris, J. G. E. (2010). Strong and tunable nonlineaer optomechanical coupling in a low-loss system. Nature Physics, 6(9), 707–712.

    Article  Google Scholar 

  • Sarma, M. S., & Rao, B. N. (1997). A rational harmonic balance approximation for the Duffing equation of mixed parity. Journal of Sound and Vibration, 207, 597–599.

    Article  MathSciNet  MATH  Google Scholar 

  • Sensoy, S., & Huseyin, K. (1998). On the application of IHB technique to the analysis of non-linear oscillations and bifurcations. Journal of Sound and Vibration, 215, 35–46.

    Article  MathSciNet  MATH  Google Scholar 

  • Sharma, A. K., Patidar, R. K., Raghuramaiah, M., Joshi, A. S., Naik, P. A., & Gupta, P. D. (2012). A study on wavelength dependence and dynamic range of the quadratic response of commercial grade light emitting diodes. Optics Communications, 285, 3300–3305.

    Article  Google Scholar 

  • Suzuki, H., Brown, E., & Sterling, R. (2015). Nonlinear dynamics of an optomechanical system with a coherent mechanical pump: Second order sideband generation. Physical Review A—Atomic, Molecular and Optical Physics, 92(3), 033823.

    Article  Google Scholar 

  • Wang, G., Fang, Z., & Wu, F. (2016). Control of two-dimensional electron population in semiconductor quantum well. Physica E: Low-Dimensional System and Nanostructures, 75(12136), 241–245.

    Article  Google Scholar 

  • Wu, B. S., & Lim, C. W. (2004). Large amplitude non-linear oscillations of a general conservative system. International Journal of Non-Linear Mechanics, 39, 859–870.

    Article  MathSciNet  MATH  Google Scholar 

  • Xiong, H., Si, L.-G., Lu, X.-Y., Yang, X., & Wu, Y. (2013). Carrier-envelope phase-dependent effect of high-order sideband generation in ultrafast driven optomechanical system in sideband generation. Optics Letters, 38(3), 353–355.

    Article  Google Scholar 

  • Xiong, H., Si, L.-G., & Lu, X.-Y. (2014). Nanosecond-pulse-controlled higher order side band comb in a GaAs optomechanical disk resonator in the perturbative regime. Annals of Physics, 349, 43–53.

    Article  MATH  Google Scholar 

  • Yuste, S. B., & Bejarano, J. D. (1990). Improvement of a Krylov–Bogoliubov method that use Jacobi elliptic function. Journal of Sound and Vibration, 139, 151–163.

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang, L., & Kong, H. Y. (2014). Self-sustained oscillation and harmonic generation in optomechanical systems with quadratic couplings. Physical Review A, 89(023847), 1–12.

    Google Scholar 

  • Zurkinden, A. S., Ferri, F., Beatty, S., Kofoed, J. P., & Kramer, M. M. (2014). Non-linear numerical modeling and experimental testing of a point absorber wave energy converter. Ocean Engineering, 78, 11–21.

    Article  Google Scholar 

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  1. Faculty of Technical Sciences, University of Novi Sad, Novi Sad, Serbia

    Livija Cveticanin

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  1. Livija Cveticanin
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Correspondence to Livija Cveticanin .

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Cveticanin, L. (2018). Free Vibrations. In: Strong Nonlinear Oscillators. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-58826-1_4

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  • DOI: https://doi.org/10.1007/978-3-319-58826-1_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-58825-4

  • Online ISBN: 978-3-319-58826-1

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