Amplitude-frequency relationship obtained using Hamiltonian approach for oscillators with sum of non-integer order nonlinearities

In this paper the Hamiltonian approximate analytical approach is extended for solving vibrations of conservative oscillators with the sum of integer and/or non-integer order strong nonlinearities. Solution of the nonlinear differential equation is assumed in the form of a trigonometric function with unknown frequency. The frequency equation is obtained based on the hypothesis that the derivative of the Hamiltonian in amplitude of vibration is zero. The accuracy of the approximate solution is treated with two different approaches: comparing the analytical value for period of vibration with the period obtained numerically and developing an error estimation method based on the ratio between the averaged residual function and the total constant energy of system. The procedures given in the paper are applied for two types of examples: an oscillator with a strong nonlinear term and an oscillator where the nonlinearity is of polynomial type.

[1]  Livija Cveticanin,et al.  Oscillator with a Sum of Noninteger-Order Nonlinearities , 2012, J. Appl. Math..

[2]  L. Cveticanin,et al.  Analytical approximations to the solutions for a generalized oscillator with strong nonlinear terms , 2012, Journal of Engineering Mathematics.

[3]  Ji-Huan He.  Homotopy perturbation technique , 1999 .

[4]  Ji-Huan He,et al.  Homotopy perturbation method: a new nonlinear analytical technique , 2003, Appl. Math. Comput..

[5]  Livija Cveticanin,et al.  Oscillator with fraction order restoring force , 2009 .

[6]  L. Cveticanin.  On the truly nonlinear oscillator with positive and negative damping , 2014, Appl. Math. Comput..

[7]  Livija Cveticanin,et al.  Review on Mathematical and Mechanical Models of the Vocal Cord , 2012, J. Appl. Math..

[8]  Livija Cveticanin,et al.  Nonlinear free vibration of systems with inertia and static type cubic nonlinearities: An analytical approach , 2014 .

[9]  Mahmoud Bayat,et al.  On the approximate analytical solution to non-linear oscillation systems , 2013 .

[10]  Teodor M. Atanackovic,et al.  A Least Square Solution and Error Estimate for some Vibration Problems , 1985, 1985.

[11]  R. Mickens,et al.  OSCILLATIONS IN AN x4/3POTENTIAL , 2001 .

[12]  Ji-Huan He,et al.  Hamiltonian approach to nonlinear oscillators , 2010 .

[13]  Alfredo Aranda,et al.  Improved LindstedtPoincar method for the solution of nonlinear problems , 2003, math-ph/0303052.

[14]  Yadollah Farzaneh,et al.  Global Error Minimization method for solving strongly nonlinear oscillator differential equations , 2010, Comput. Math. Appl..

[15]  Ji-Huan He,et al.  Preliminary report on the energy balance for nonlinear oscillations , 2002 .

[16]  Ivana Kovacic,et al.  Asymptotic methods for vibrations of the pure non-integer order oscillator , 2010, Comput. Math. Appl..

[17]  Ronald E. Mickens,et al.  Comments on the method of harmonic balance , 1984 .

[18]  Ivana Kovacic,et al.  A non-simultaneous variational approach for the oscillators with fractional-order power nonlinearities , 2010, Appl. Math. Comput..

[19]  Nicolae Herisanu,et al.  Accurate analytical solutions to oscillators with discontinuities and fractional-power restoring force by means of the optimal homotopy asymptotic method , 2010, Comput. Math. Appl..

[20]  Ji-Huan He.  Variational iteration method – a kind of non-linear analytical technique: some examples , 1999 .

[21]  L. Cvetićanin,et al.  Application of Hamiltonian Approach to the Generalized Nonlinear Oscillator with Fractional Power , 2010 .

[22]  M. K. Yazdi,et al.  Rational variational approaches to strong nonlinear oscillations , 2017 .