Periodic Oscillation and Bifurcation Analysis of Pendulum with Spinning Support Using a Modified Continuous Piecewise Linearization Method

Big‐Alabo, Akuro; Ossia, Chinwuba Victor

doi:10.1007/s40819-019-0697-9

Cited by 11 publications

(12 citation statements)

References 26 publications

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“…However, the velocity profile for α = 1.0 shows the presence of bistable oscillations. 15 When α was increased to a value of 10.0, the bistable oscillations were absent. Hence, an increase in the inertia nonlinearity caused the behavior of the mechanical oscillator to change from a bistable oscillation with double-well potential to an oscillation with single-well potential (see phase plots in Figure 9).…”

Section: Resultsmentioning

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: Resultsmentioning

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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“…Then, substituting equations (11a)-(11c) into equation (13) and simplifying gives the approximate period of the pendulum as an explicit function of amplitude, as shown in equation (…”

Section: Approximate Analytical Expression For the Period Of The Simp...mentioning

confidence: 99%

“…In a previous paper [3], the author derived periodic solutions for the simple pendulum that are accurate for the entire range of possible amplitudes, using an iterative analytic algorithm which discretizes and linearizes the restoring force. The algorithm is known as the continuous piecewise linearization method (CPLM) and has proved to be effective in providing accurate periodic solutions for complex nonlinear oscillators [12,13]. It was shown that a maximum relative error of less than 0.20% can be obtained by using the CPLM to estimate the period of the simple pendulum for j   179 .…”

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confidence: 99%

Approximate period for large-amplitude oscillations of a simple pendulum based on quintication of the restoring force

Big‐Alabo

2019

Eur. J. Phys.

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A new approximate formula for estimating the period of the simple pendulum has been presented in this paper. The formula was derived based on ‘quintication’ of the restoring force of the pendulum, which replaces the original pendulum equation with an equivalent cubic–quintic oscillator. The coefficients of the equivalent oscillator are amplitude-dependent and were derived based on quasi-static equilibrium principles. Then, an approximate solution for the equivalent cubic–quintic oscillator was applied to derive the approximate formula used to estimate the pendulum period for the entire range of possible amplitudes of angular displacement, i.e. The formula is simple and very accurate with a maximum relative error of 0.421% for amplitudes up to 179.9°.

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“…Nonlinear vibration is a subject of immense scientific and engineering importance. Many physical phenomena such as chaos, 1 bifurcations, 2,3 jumps, 4 sub-harmonic resonance, 5 softening and hardening backbone response, 5,6 parametric response, 7 stability analysis, 3,5 and fractal oscillations 8–10 can only be understood through nonlinear vibration analysis. The nonlinearity in a vibrating system may arise from static and/or dynamic inertia effects, damping, size-dependent mechanics, fractal mechanics, elastic or spring stiffness, and external excitation.…”

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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Periodic Oscillation and Bifurcation Analysis of Pendulum with Spinning Support Using a Modified Continuous Piecewise Linearization Method

Cited by 11 publications

References 26 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

Approximate period for large-amplitude oscillations of a simple pendulum based on quintication of the restoring force

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

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