Free vibration analysis of nonlinear axially loaded beams using a modified harmonic balance method

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

doi:10.1016/j.padiff.2022.100414

Cited by 4 publications

(4 citation statements)

References 10 publications

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“…The solution of equation ( 1) is considered to be as per the truncated Fourier series is (Ullah et al 2021a(Ullah et al , 2021b(Ullah et al , 2021c(Ullah et al , 2022)…”

Section: Methodsmentioning

confidence: 99%

“…Scientists, physicists, applied mathematicians and engineers are forced to find the approximate solutions of the nonlinear physical problems rather than the numerical solutions. A lot of researchers have developed analytical procedures to handle nonlinear oscillatory problems (Alam 2002, 2003, Alam et al 2006, Belendezet al 2012, Guo and Ma 2014, Guo and Leung 2010, Uddin and Sattar 2010, He 1998, Krylov and Bogoliubov 1947, Kovacic and Mickens 2012, Khan 2019, Liu et al 2007, Mondal et al 2019, Mishara et al 2016, Nayfeh 1981, Uddin et al 2011, 2015, Ullah et al 2021a, 2021b, 2021c, 2022, Yeasmin 2020. Among of them, perturbation techniques (Alam 2002, 2003, Alam et al 2006, Uddin and Sattar 2010, Krylov and Bogoliubov 1947, Kovacic and Mickens 2012, Nayfeh 1981) are well-established and most popular methods.…”

Section: Introductionmentioning

confidence: 99%

“…in order to describe the action of the heart, Zduniak et al (2014) created a modified Van der Pol equation with delay. A modified harmonic balance method has recently been developed by Ullah et al (2021aUllah et al ( , 2021bUllah et al ( , 2021cUllah et al ( and 2022 to manage damped forced Duffing oscillators and forced Van der Pol equation. For handling the modified forced Van der Pol equation, a modified harmonic balance approach has been proposed in this paper.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Analytical Solution of Modified Forced Van Der Pol Vibration Equation Using Modified Harmonic Balance Method

Ullah¹,

Uddin

Rahman

2022

Khulna Univ. Stud.

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For obtaining analytical solutions to the modified forced Van der Pol equation, the modified harmonic balance method has been developed. In classical harmonic balance method, the numerical procedure is used for solving a set of nonlinear algebraic equations. But it requires laborious computational attempt and accurate primary guesses values which makes it very hard-working to calculate. According to our method, a set of nonlinear algebraic equations has been converted to a set of linear algebraic equations by using a nonlinear one instead of a set. As a result, it reduces the massive computational work. This method provides not only better results than the existing harmonic balance method but also provides very close solutions to the corresponding numerical results. It is noticeable that there is substantial similarity between the approximate and the numerical results attained by the fourth order Runge-Kutta method. Moreover, the method is facile and straightforward. This technique may play great role to handle strongly nonlinear damped systems with external forces.

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“…The solution of equation ( 1) is considered to be as per the truncated Fourier series is (Ullah et al 2021a(Ullah et al , 2021b(Ullah et al , 2021c(Ullah et al , 2022)…”

Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Analytical Solution of Modified Forced Van Der Pol Vibration Equation Using Modified Harmonic Balance Method

Ullah¹,

Uddin

Rahman

2022

Khulna Univ. Stud.

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show abstract

“…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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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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The Analysis of Higher Order Nonlinear Vibrations of an Elastic Beam with the Extended Galerkin Method

Lian

Meng

Jing

et al. 2023

J. Vib. Eng. Technol.

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Free vibration analysis of nonlinear axially loaded beams using a modified harmonic balance method

Cited by 4 publications

References 10 publications

Analytical Solution of Modified Forced Van Der Pol Vibration Equation Using Modified Harmonic Balance Method

Analytical Solution of Modified Forced Van Der Pol Vibration Equation Using Modified Harmonic Balance Method

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

The Analysis of Higher Order Nonlinear Vibrations of an Elastic Beam with the Extended Galerkin Method

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