Journal of Theoretical
and Applied Mechanics

0, 0, pp. , Warsaw 0

Steady state vibration of the periodically forced and damped pure nonlinear two-degrees-of-freedom oscillator

Cveticanin Livija, Miodrag Zukovic, Dragan Cveticanin
In the paper the pure nonlinear and damped two-mass oscillator excited with the periodical force is considered. The oscillator is modelled with a system of two coupled second order nonlinear and non-homogenous equations. Using the model two problems are investigated: one, the excitation force identification for the known vibrating response of the system, and the second, determination of vibrations of the system excited with the known periodical force. Using the steady-state motion of the nonlinear oscillator the method for identification of the excitation force is developed. For the pure nonlinear oscillator it is obtained that the forcing function has the form of the Ateb function. However, if the excitation force is known the procedure for computing of the steady-state vibration of the system is introduced. The solution corresponds to steady-state vibrations of the free oscillator, but the amplitude and phase are assumed to be time variable. The averaged solutions are obtained for pure nonlinear oscillator with additional linear elastic force and for the Van de r Pol oscillator. Analytically obtained solutions are compared with numerical ones. They are in good agreement.
Keywords: non-harmonic excitation, force identification, steady-state motion, Van der Pol oscillator, Duffing oscillator


Cveticanin L., 2015, A solution procedure based on the Ateb function for a two-degree-of-freedom oscillator, Journal of Sound and Vibration, 346, 298-313.

Cveticanin L., Zukovic M., 2017, Negative effective mass in acoustic metamaterial with nonlinear mass-in-mass subsystems. Communications in Nonlinear Science and Numerical Simulation, 51, 89-104.

Cveticanin L., 2018a, Pure nonlinear oscillator, in Strong Nonlinear Oscillators, Mathematical Engineering, Springer, 9783319588254, 17-49.

Cveticanin L., 2018b, Free vibrations, in Strong Nonlinear Oscillators, Mathematical Engineering, Springer, 9783319588254, 51-117.

Hsu C., 1960, On the application of elliptic functions in non-linear forced oscillations, Quarterly of Applied Mathematics, 17, 393-407.

Ibrahim A., Jaber N., Chandran A., Thirupathi, M., Younis M., 2017, Dynamics of microbeams under multi-frequewncy excitations, Micromachines MDPI, 32, 8, 1-14.

Ilyas S., Ramini A., Arevalo A., Younis M.I., 2015, An experimental and theoretical investigation of a micromirror under mixed-frequency excitation, Journal of Microelectromechanical Systems, 24, 1124-1131.

Jaber N., Ramini A., Hennawi Q., Younis, M.I., Wideband M.I., 2016a, MEMS resonator using multifrequency excitation, Sensors and Actuators A: Physical, 242, 140-145.

Jaber N., Ramini A., Hennawi Q., Younis, 2016b, Multifrequency excitation of a clamped-clamped microbeam: Analytical and experimental investigation, Microsystems and Nanoengineering, 2, 16002.

Jang T.S., Baek H., Choi H.S., Lee S.G., 2011, A new method for measuring nonharmonic periodic excitation forces in nonlinear damped systems, Mechanical Systems and Signal Processing, 25, 2219-2228.

Jiang Y., Zhu H., Li Z., Peng Z., 2016, The nonlinear dynamics response of cracked gear system in a coal cutter taking environmental multi-frequency excitation forces into consideration, Nonlinear Dynamics, 84, 203-222.

Kovacic I., Zukovic M., 2017, Coupled purely nonlinear oscillators: normal modes and exact solutions for free and forced resonances. Nonlinear Dynamics, 87, 713-726.

Rakaric Z., Kovacic I., Cartmell M., 2017, On the design of external excitations in order to make nonlinear oscillators respond as free oscillators of the same or different type, International Journal of Non-Linear Mechanics, 94, 323-333.

Ramini A., Ibrahim A.I., Younis M.I., 2016, Mixed frequency excitation of an electrostatically actuated resonator, Microsystem Technologies, 22, 1967-1974.

Rosenberg R.M., 1966, On non-linear vibration of systems with many degrees of freedom, Advances in Applied Mechanics, 9, 155-242

Vakakis A.F., Blanchard A., 2018, Exact steady states of the periodically forced and damped Duffing oscillator, Journal of Sound and Vibration, 413, 57-65.

Wei S., Han Q., Peng Z., Chu F., 2016, Dynamic analysis of parametrically excited system under uncertainties and multi-frequency excitations, Mechanical Systems and Signal Processing, 72-73, 762-784.