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Exact Closed-Form Solution for the Oscillator with a New Type of Mixed Nonlinear Restitution Force

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  • Livija Cveticanin

    (Faculty of Technical Sciences, University of Novi Sad, 21000 Novi Sad, Serbia
    Doctoral School of Safety and Security Sciences, Obuda University, 1081 Budapest, Hungary)

Abstract

This paper shows an oscillator with a spring made of material where the stress is a function not only of strain but also strain rate. The corresponding restitution force is of strong nonlinear monomial type and is the product of displacement and velocity of any order. The mathematical model of the oscillator is a homogenous strong nonlinear second-order differential equation with an integer- or non-integer-order mixed term. In the paper, an analytical procedure for solving this new type of strong nonlinear equation is developed. The approximate solution is assumed as the perturbed version of the exact solution in the form of a sine Ateb function. As a result, it is obtained that the amplitude, period, and frequency of vibration depend not only on the coefficient and order of nonlinearity, but also on the initial velocity. The procedure is tested on two examples: oscillator perturbed with small linear damping and small linear displacement functions. The analytically obtained results are compared with the exact numerical ones and show good agreement. It is concluded that the mathematical model and also the procedure developed in the paper would be convenient for prediction of motion for this type of oscillator without necessary experimental testing.

Suggested Citation

  • Livija Cveticanin, 2023. "Exact Closed-Form Solution for the Oscillator with a New Type of Mixed Nonlinear Restitution Force," Mathematics, MDPI, vol. 11(3), pages 1-11, January.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:3:p:596-:d:1044821
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    References listed on IDEAS

    as
    1. Livija Cveticanin & Nicolae Herisanu & Ivona Ninkov & Mladen Jovanovic, 2022. "New Closed-Form Solution for Quadratic Damped and Forced Nonlinear Oscillator with Position-Dependent Mass: Application in Grafted Skin Modeling," Mathematics, MDPI, vol. 10(15), pages 1-15, July.
    2. Richárd Horváth & Livija Cveticanin & Ivona Ninkov, 2022. "Prediction of Surface Roughness in Turning Applying the Model of Nonlinear Oscillator with Complex Deflection," Mathematics, MDPI, vol. 10(17), pages 1-15, September.
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    Cited by:

    1. Remus-Daniel Ene & Nicolina Pop & Rodica Badarau, 2023. "Heat and Mass Transfer Analysis for the Viscous Fluid Flow: Dual Approximate Solutions," Mathematics, MDPI, vol. 11(7), pages 1-22, March.

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