Analysis of Strongly Nonlinear Systems by Using HBM-AFT Method and Its Comparison with the Five-Order Runge–Kutta Method: Application to Duffing Oscillator and Disc Brake Model

Ghorbel, Ahmed; Zghal, Becem; Abdennadher, Moez; Walha, Lassâad; Haddar, Mohamed

doi:10.1007/s40819-020-0803-z

Cited by 4 publications

(2 citation statements)

References 35 publications

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“…Although it is unlikely that there is an explicit analytical solution for such a nonlinear-coupled vibrational system, an approximate analytical approach can be used to obtain the steady-state responses and reveal the roles of nonlinearity and thermomechanical coupling. With this hypothesis, the answers ( ) t  and ( ) Ttcan be developed into Fourier series [34] as:…”

Section: Steady State Solutionmentioning

confidence: 99%

Coupled Thermomechanical Dynamics of Phase Transitions in Shape Memory Alloys and Related Nonlinear Phenomena

Siryabe¹

2023

Mater. Plus

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The aim of this paper is to study the coupled non-linear dynamics of the behaviour of Shape Memory Alloys (SMA) under harmonic excitation. The thermomechanical model developed by Falk and based on Landau theory is used to describe the coupled thermomechanical behaviour of the SMA mass-damper-bar system. Fully coupled mechanical and thermal field equations are established to reveal the strong coupling phenomena arising from the phase transition. Then, the harmonic balance method is used to obtain the steady-state frequency responses. The effect of thermomechanical coupling on the frequency and time response curves is taken into account by modifying the heat transfer coefficient. The results show that deformation and temperature provide a stable sinusoidal response. The analysis of the results leads to different ways of controlling the nature and extent of the coupled and non-linear response of SMA-based oscillators. Comparison of the analytical results with experimental data from the literature validates the coupled thermomechanical nonlinear model. These results can be used effectively to control the external vibrations of various systems.

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Section: Steady State Solutionmentioning

confidence: 99%

Coupled Thermomechanical Dynamics of Phase Transitions in Shape Memory Alloys and Related Nonlinear Phenomena

Siryabe¹

2023

Mater. Plus

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“…railway systems [4], and non-smooth vibration systems [5,6]. Predominantly, issues in non-smooth vibrations often involve oscillatory systems characterized by nondifferentiable points [7][8][9], notably exemplified by oscillators incorporating fractionalpower restorative forces [10] and damping [11]. To address these intricate problems, scholars have developed an array of methodologies, including the Homotopy Perturbation Method [12][13][14][15], the modified Lindstedt-Poincaré method [16], the Piecewise Linear Approach [17], the Krylov-Bogoliubov method [18,19], and the Harmonic Balance Method [20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

Advanced Numerical Algorithm for Non-smoothness Differential Equations: Integrating Fractional Interpolation with Predictive- Corrective Techniques

Ye,

Chen,

Liu

et al. 2024

Preprint

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In this study, we investigate numerical methods for non-smooth differential equations (NSDEs), which are pivotal in simulating abrupt phenomena in natural and engineering systems. We introduce the fractional interpolation method (FIM), a novel technique that utilizes fractional power functions to approximate solutions at points where derivatives are infinite. This method’s principal innovation is its adept handling of NSDEs' inherent discontinuities, offering a stable and convergent solution framework. Our findings confirm that FIM is both theoretically sound and practically reliable. Through rigorous numerical experiments, we have demonstrated its superior performance compared to conventional high-order numerical methods and MATLAB’s built-in functions. To further affirm FIM’s practicality, we applied it to two distinct non-smooth system types: systems with dry friction and binary wing systems with clearances. These applications substantiate the effectiveness of FIM and highlight its potential to tackle real-world challenges. Furthermore, this research equips scientists and engineers with a robust new tool for addressing NSDEs, setting the stage for further exploration and practical uses, especially in scenarios requiring accurate simulation of abrupt system behaviors. We anticipate the broader application of FIM in analyzing and designing non-smooth systems and are enthusiastic about its role in enhancing our understanding and prediction of complex dynamics across various natural and technical systems.

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The origin point of the unstable solution area of a forced softening Duffing oscillator

Wawrzyński

2022

Sci Rep

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Each Duffing equation has an unstable solution area with a boundary, which is also a line of bifurcation. Generally, in a system that can be modeled by the Duffing equation, bifurcations can occur at frequencies lower than the origin point frequency of the unstable solution area for a softening system and at higher frequencies for a hardening system. The main goal of this research is to determine the analytical formulas for the origin point of the unstable solution area of a system described by a forced Duffing oscillator with softening stiffness, taking damping into account. To achieve this goal, two systems of softening Duffing oscillators that differ strongly in their nonlinearity factor value have been selected and tested. For each system, for three combinations of linear and nonlinear stiffness coefficients with the same nonlinearity factor, bistability areas and unstable solution areas were determined for a series of damping coefficient values. For each case, curves determined for different damping values were grouped to obtain the origin point curve of the unstable solution, ultimately developing the target formulas.

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Analysis of Strongly Nonlinear Systems by Using HBM-AFT Method and Its Comparison with the Five-Order Runge–Kutta Method: Application to Duffing Oscillator and Disc Brake Model

Cited by 4 publications

References 35 publications

Coupled Thermomechanical Dynamics of Phase Transitions in Shape Memory Alloys and Related Nonlinear Phenomena

Coupled Thermomechanical Dynamics of Phase Transitions in Shape Memory Alloys and Related Nonlinear Phenomena

Advanced Numerical Algorithm for Non-smoothness Differential Equations: Integrating Fractional Interpolation with Predictive- Corrective Techniques

The origin point of the unstable solution area of a forced softening Duffing oscillator

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