Periodic Solution of Nonlinear Conservative Systems

Big‐Alabo, Akuro; Ossia, Chinwuba Victor

doi:10.5772/intechopen.90282

Cited by 5 publications

(2 citation statements)

References 63 publications

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“…3,6,7,11 However, we observe that the model has a velocity squared term, which is a well-known feature of nonnatural oscillators. 4,12,13 This term does not introduce any dissipation into the system but rather acts as an elastic force that adds to the conservative restoring force. 4 As a result, the term has been called a "velocity-dependent elastic modulus" in Lev and Zagorodny.…”

Section: Coupled Theory For Phase Transitionmentioning

confidence: 99%

Exact analytical solution of a mechanical oscillator for phase transition involving spatially inhomogeneous distribution of the order parameter

Big‐Alabo

Ossia

Ekpruke

2021

Math Methods in App Sciences

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In this paper, we derive the exact analytical solution for the periodic oscillations of a nonlinear mechanical oscillator that is capable of describing phase transition phenomenon with spatially inhomogeneous distribution of the order parameter. The exact analytical solution was derived in terms of the elliptic integral of the third kind and covers the cases where the physical variable influencing the order parameter is negative or positive. The conditions for obtaining periodic oscillations for the two cases were discussed, and results were simulated for small-and large-amplitude nonlinear oscillations. The results also cover periodic responses with bistable oscillations, which indicate the existence of bifurcation in phase transition. Furthermore, the error of the numerical solution and other published approximate analytical solutions were analyzed and discussed. The present study can be viewed as a benchmark solution for the mechanical oscillator for phase transition and can be used to verify the accuracy of existing and future approximate solutions.

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Section: Coupled Theory For Phase Transitionmentioning

confidence: 99%

Exact analytical solution of a mechanical oscillator for phase transition involving spatially inhomogeneous distribution of the order parameter

Big‐Alabo

Ossia

Ekpruke

2021

Math Methods in App Sciences

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“…For most cases, approximate analytical or numerical solutions are normally employed. Many approximate analytical methods have been developed [11][12][13][14][15][16] and are capable of providing physical insight into the response of a nonlinear vibrating system. Springs or elastic members are an integral part of machines and in some cases a combination of these are required to design machine components.…”

Section: Introductionmentioning

confidence: 99%

Frequency response of a mass grounded by linear and nonlinear springs in series: An exact analysis

Big-Alabo

2023

Journal of Low Frequency Noise, Vibration and Active Control

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An exact analytical solution for the frequency response of a system consisting of a mass grounded by linear and nonlinear springs in series was derived. The system is applicable in the study of beams carrying intermediate rigid mass. The exact solution was derived by naturally transforming the integral of the governing nonlinear ODE into a form that can be expressed in terms of elliptic integrals. Hence, the exact frequency–amplitude solution was derived in terms of the complete elliptic integral of the first and third kinds. Periodic solutions were found to exist for all real values of [Formula: see text] and for [Formula: see text]. On the other hand, linear or weakly nonlinear frequency–amplitude responses were found to occur during small-amplitude vibrations ([Formula: see text]), very large-amplitude vibrations with strong hardening nonlinearity ([Formula: see text] and [Formula: see text]), and for all amplitudes when [Formula: see text]. Simulations showed that the system’s periodic response is significantly influenced by the nonlinearity parameter ([Formula: see text]), linearity parameter ([Formula: see text]), and amplitude of vibration ([Formula: see text]). Besides, the existence of bifurcation points at [Formula: see text] and different values of [Formula: see text] was confirmed. Lastly, an approximate frequency solution obtained using the He’s frequency–amplitude formulation was found to produce errors less than 1.0% for a wide range of input parameters. In conclusion, the present study provides a benchmark solution for verification of other approximate solutions, and the He’s frequency–amplitude formulation can be used to obtain fast and accurate solutions for complex nonlinear systems.

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Free nonlinear vibration analysis of a functionally graded microbeam resting on a three-layer elastic foundation using the continuous piecewise linearization method

Alfred,

Ossia,

Big-Alabo

2023

Arch Appl Mech

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Periodic Solution of Nonlinear Conservative Systems

Cited by 5 publications

References 63 publications

Exact analytical solution of a mechanical oscillator for phase transition involving spatially inhomogeneous distribution of the order parameter

Exact analytical solution of a mechanical oscillator for phase transition involving spatially inhomogeneous distribution of the order parameter

Frequency response of a mass grounded by linear and nonlinear springs in series: An exact analysis

Free nonlinear vibration analysis of a functionally graded microbeam resting on a three-layer elastic foundation using the continuous piecewise linearization method

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