A modified harmonic balance method for solving forced vibration problems with strong nonlinearity

Ullah, M Wali; Rahman, Mohammad Shaifur; Uddin, Marjan

doi:10.1177/1461348420923433

Cited by 10 publications

(8 citation statements)

References 34 publications

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“…The trial solution of Eq. ( 1) is assumed as (Ullah et al, 2021) 𝑦 = α cos(𝜔𝑡) + 𝛽 𝑠𝑖𝑛(𝜔𝑡) + 𝛾 cos(3𝜔𝑡) + 𝛿 sin(3𝜔𝑡) + ⋯…”

Section: The Methodsmentioning

confidence: 99%

“…According to the truncated Fourier series, the guess solution of Eq. ( 1) is given by (Ullah et al, 2021) 𝑦 = α cos(𝜔𝑡) + 𝛽 𝑠𝑖𝑛(𝜔𝑡) + 𝛾 cos(3𝜔𝑡) + 𝛿 sin (3𝜔𝑡)…”

Section: Examplementioning

confidence: 99%

“…In such cases, the original nonlinear oscillatory systems must be solved straightway. Several researchers have developed analytical techniques to handle nonlinear oscillatory systems using various methods (Belendez et al, 2012;Kovacic and Mickens, 1986;Krylov and Bogoliubov, 1947;Mondal et al, 2019;Nayfeh, 1981;Liu, 2005;Lim and Lai, 2006;Guo and Leung, 2010;Guo and Ma, 2014;He, 1998He, , 1999He, , 2006Uddin al et., 2011Uddin al et., , 2012Uddin al et., , 2015Mishara et al, 2016;Khan, 2019;Yeasmin et al, 2020;Uddin and Sattar, 2010;Ullah et al, 2021). Perturbation techniques (Kovacic and Mickens, 1986;Krylov and Bogoliubov, 1947;Nayfeh, 1981;Uddin and Sattar, 2010), which are extensively applied tools.…”

Section: Introductionmentioning

confidence: 99%

“…Also, the derivations and calculations of these procedures are very tedious work. Recently, Ullah et al (2021) have developed a modified harmonic balance method for handling damped forced Duffing oscillators in the presence of cubic and quadratic nonlinearities.…”

Section: Introductionmentioning

confidence: 99%

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An analytical technique for handling forced Van der Pol vibration equation

Ullah¹,

Uddin

Rahman

2022

J Bangladesh Acad Sci

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Based on the modified harmonic balance method, an analytical method has been developed for handling the forced Van der Pol vibration equation. Usually, a system of nonlinear algebraic equations arises within the unfamiliar coefficients in several harmonics terms and the frequency of the forcing term. A numerical technique has been applied to handle those nonlinear algebraic equations in the classical harmonic balance method. In our study, a system of linear algebraic equations is calculated with the aid of a single nonlinear one. The solutions attained by the suggested scheme have been likened to the results acquired by the well-known Runge-Kutta method, and these results display very nice harmony with the result obtained by the mentioned method. Also, it is noticed that the proposed technique is straightforward and gives the desired results in the whole solution domain. J. Bangladesh Acad. Sci. 45(2); 231-240: December 2021

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“…The trial solution of Eq. ( 1) is assumed as (Ullah et al, 2021) 𝑦 = α cos(𝜔𝑡) + 𝛽 𝑠𝑖𝑛(𝜔𝑡) + 𝛾 cos(3𝜔𝑡) + 𝛿 sin(3𝜔𝑡) + ⋯…”

Section: The Methodsmentioning

confidence: 99%

Section: Examplementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An analytical technique for handling forced Van der Pol vibration equation

Ullah¹,

Uddin

Rahman

2022

J Bangladesh Acad Sci

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show abstract

“…Sinusoidal Input Describing Functions (HOSIDFs) 52 and many modified versions of HBM and IHB [53][54][55][56][57][58][59] in the 1990s, 2000s and 2010s.…”

mentioning

confidence: 99%

Determination of the characteristic curves of a nonlinear first order system from Fourier analysis

Gonzalez

2023

Sci Rep

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Based on Fourier analysis, we develop an expression for modeling and simulating nonlinear first order systems. This expression is associated to a nonlinear first order differential equation $$y=f(x)+g(x)x'$$ y = f ( x ) + g ( x ) x ′ , where $$x=x(t)$$ x = x ( t ) is the dynamical variable, $$y=y(t)$$ y = y ( t ) is the driving force, and the f and g functions are the characteristic curves which are associated to dissipative and memory elements, respectively. The model is obtained from the sinusoidal response, specifically by calculating the Fourier analysis of y(t) for $$x(t)=A_1\sin (\omega t)+A_0$$ x ( t ) = A 1 sin ( ω t ) + A 0 , where the model parameters are the Fourier coefficients of the response, and the values of $$A_0$$ A 0 , $$A_1$$ A 1 and $$A_1'=A_1\omega$$ A 1 ′ = A 1 ω . The same expression is used for two kinds of time-domain simulations: to calculate other driving force $$\hat{y}$$ y ^ based on a dynamical variable $$\hat{x}$$ x ^ ; and, to calculate the dynamical variable $$\hat{x}$$ x ^ based on a driving force $$\hat{y}$$ y ^ . In both cases, the dynamical variable must remain in the range $$\hat{x}\in [A_0-A_1,A_0+A_1]$$ x ^ ∈ [ A 0 - A 1 , A 0 + A 1 ] . By analyzing this expression, we found an equivalence between the Fourier coefficients and the polynomial regressions of the characteristic curves of f and g. This result allows us to obtain the system modeling and simulation based on the amplitude and phase Fourier spectrum obtained from the Fast Fourier Transform (FFT) of the sampled $$y_n$$ y n version of y(t). It is shown that this technique has a low computational complexity, and it is expected to be suitable for real-time applications for system modeling, simulation and control, in particular when the explicit expressions of the characteristic curves are unknown. Fourier analysis is a fundamental tool in electronics, mathematics and physics, but to the best of the author’s knowledge, no work has found this clear evidence of the connection between the Fourier analysis and a first order differential equation. The aim of this work is to initiate a systematic study on this topic.

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A harmonic balance-based method to predict nonlinear forced response and temperature rise of dry friction systems including frictional heat transfer

Gao,

Fan,

et al. 2023

Nonlinear Dyn

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A modified harmonic balance method for solving forced vibration problems with strong nonlinearity

Cited by 10 publications

References 34 publications

An analytical technique for handling forced Van der Pol vibration equation

An analytical technique for handling forced Van der Pol vibration equation

Determination of the characteristic curves of a nonlinear first order system from Fourier analysis

A harmonic balance-based method to predict nonlinear forced response and temperature rise of dry friction systems including frictional heat transfer

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