On Harmonic Balance Method-based Lagrangian contact formulations for vibro-impact problems

The dynamic characteristics of the fractional Duffing system with nonlinear time delay term are studied according to the average method and the harmonic balance method. First, the steady‐state solutions of the fractional Duffing system with a nonlinear time delay term are obtained by applying the average method and the harmonic balance method. The analytical expressions of the amplitude–frequency characteristic curves under the average method and the harmonic balance method are also established. Second, the stability criterion of the system is obtained according to the indirect method of studying the motion stability problem, and the concept of the stability condition parameter is proposed in this process. Third, the transient solution of the fractional Duffing system with a nonlinear time delay term is obtained utilizing the average method, and the approximate analytical solution of the fractional Duffing system with nonlinear time delay term is obtained. Finally, the amplitude–frequency characteristic curves of the different fractional differential terms, coefficients of different fractional differential terms, magnitude of different time delay quantities, and feedback strength of different delay are analyzed by numerical simulation. Furthermore, the amplitude–frequency characteristic curves obtained by the average method and the harmonic balance method under different order of fractional differential term are compared and analyzed by numerical simulation. The time sequence and phase diagrams of the system under different order of fractional differential term and different time delay quantities are analyzed by numerical simulation. In addition, the correctness of the analytical analysis is verified by comparing the analytical results with the numerical simulation results under the average method and the harmonic balance method, respectively, by numerical simulation.

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“…According to the harmonic balance principle [27–34], we can obtain

- m Λ ω^{2} + k Λ + \frac{3}{4} β {normalΛ}^{3} \cos ωτ + K Λ ω^{α} \cos \frac{italicαπ}{2} = f \cos γ,

c Λ ω - \frac{3}{4} β {normalΛ}^{3} \sin ωτ + K Λ ω^{α} \sin \frac{italicαπ}{2} = f \sin γ .

…”

Section: Steady‐state Solutionmentioning

confidence: 99%

Dynamics analysis for a class of fractional Duffing systems with nonlinear time delay terms

Yang

Jiao

2023

Math Methods in App Sciences

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“…They applied two simplified Newton steps to reduce the amount of linearization in the restring force function. Recently, Vadcard et al [54] conducted a study to compare two HB method-based numerical Lagrangian strategies, namely linear complementarity problem (LCP-HB) and dynamic Lagrangian frequency time (DLFT-HB), for analyzing the periodic response of a rod model subject to unilateral contact constrains. They found that the DLFT-HB is asymptotically equivalent to the LCP-HB.…”

Section: Introductionmentioning

confidence: 99%

Improved harmonic balance method for analyzing asymmetric restoring force functions in nonlinear vibration of mechanical systems

Mohammadian,

Ismail

2024

Phys. Scr.

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The non-linear differential equation governing some mechanical systems, such as the non-linear vibrations of FG beams resting on a nonlinear foundation, behaves similarly to the oscillations of a non-linear oscillator with an asymmetric restoring force function that includes both odd and non-odd non-linear terms. The objective of this research is to obtain higher-order approximate analytical solutions for such problems by introducing an enhanced harmonic balance method. By building upon the original problem, two new symmetric systems with odd non-linear terms are introduced, and their higher-order approximate analytical solutions are derived using a novel approach. In proposed method, the restoring force function is represented by its Fourier series expansion. Unlike previous papers, linearizing the equation or taking the first derivative of the Fourier series in the subsequent iteration is unnecessary. However, to enhance the solution’s accuracy, the remaining error from each iteration is utilized in the next one. Finally, by combining the results from the two introduced systems, the analytical period and corresponding periodic solution of the original problem can be obtained. This method is applied to the governing differential equation of FG beams resting on non-linear foundations, a physical non-natural oscillator, and conservative Toda oscillator. The key advantages of this approach are its simplicity and its ability to provide highly precise solutions for both small and large amplitudes in a single iteration.

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Large-scale harmonic balance simulations with Krylov subspace and preconditioner recycling

Kuether,

Steyer

2024

Nonlinear Dyn

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On Harmonic Balance Method-based Lagrangian contact formulations for vibro-impact problems

Cited by 11 publications

References 44 publications

Dynamics analysis for a class of fractional Duffing systems with nonlinear time delay terms

Dynamics analysis for a class of fractional Duffing systems with nonlinear time delay terms

Improved harmonic balance method for analyzing asymmetric restoring force functions in nonlinear vibration of mechanical systems

Large-scale harmonic balance simulations with Krylov subspace and preconditioner recycling

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