Vibration of the Liénard Oscillator with Quadratic Damping and Constant Excitation

Livija Cveticanin (), Nicolae Herisanu, Gamal Mohamed Ismail and Miodrag Zukovic
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Livija Cveticanin: Faculty of Technical Sciences, University of Novi Sad, 21000 Novi Sad, Serbia
Nicolae Herisanu: Department of Mechanics and Strength of Materials, University Politehnica Timisoara, 300222 Timisoara, Romania
Gamal Mohamed Ismail: Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi Arabia
Miodrag Zukovic: Faculty of Technical Sciences, University of Novi Sad, 21000 Novi Sad, Serbia

Mathematics, 2025, vol. 13, issue 6, 1-15

Abstract: In this paper, the Liénard oscillator with nonlinear deflection, quadratic damping, and constant excitation is considered. In general, there is no analytic solution for the Liénard equation. However, for certain parameter values, the exact analytic solution exists and has the form of the Ateb function. In addition, for the oscillator with weakly perturbed parameters, the approximate analytic solution is obtained. For the considered Liénard equation, independently of parameter values, the first integral is found. The main advantage of the first integral is that after simple analysis and without solving the equation of motion, it gives important data about oscillation: the dependence of vibration on initial conditions and on the variation of the constant of excitation. In addition, by integration of the first integral, the period of vibration follows. The results of the research on the Liénard equation are applied for optimization of the properties of a sieve in the process industry. For the sieve with mass variation, dependent on the displacement function, the influence of excitation force on the system vibration is analyzed, and the optimal value is suggested.

Keywords: Liénard oscillator; mass-variable oscillator; first integral; Ateb function; exact analytic solution (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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