He’s frequency–amplitude formulation for nonlinear oscillator with damping

Ren, Zhong-Fu; Wu, Jin‐Bin

doi:10.1177/1461348419851931

Cited by 10 publications

(8 citation statements)

References 36 publications

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“…There are two shortcomings in Equations ( 31)- (33). In each stage to find x j , j ≥ 1 by solving the second-order differential equation, it involves the second-order differentials of x 0 , .…”

Section: A Linearized Lindstedt-poincaré Methodsmentioning

confidence: 99%

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A Simplified Lindstedt-Poincaré Method for Saving Computational Cost to Determine Higher Order Nonlinear Free Vibrations

Liu

Chen

2021

Mathematics

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In order to improve the Lindstedt-Poincaré method to raise the accuracy and the performance for the application to strongly nonlinear oscillators, a new analytic method by engaging in advance a linearization technique in the nonlinear differential equation is developed, which is realized in terms of a weight factor to decompose the nonlinear term into two sides. We expand the constant preceding the displacement in powers of the introduced parameter so that the coefficients can be determined to avoid the appearance of secular solutions. The present linearized Lindstedt-Poincaré method is easily implemented to provide accurate higher order analytic solutions of nonlinear oscillators, such as Duffing and van Der Pol nonlinear oscillators. The accuracy of analytic solutions is evaluated by comparing to the numerical results obtained from the fourth-order Runge-Kotta method. The major novelty is that we can simplify the Lindstedt-Poincaré method to solve strongly a nonlinear oscillator with a large vibration amplitude.

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Section: A Linearized Lindstedt-poincaré Methodsmentioning

confidence: 99%

“…For complicated oscillators, He's frequency-amplitude formulations [30][31][32][33][34] were frequently employed to estimate the frequency of free vibration. For example, for the Duffing oscillator, it reads as:…”

Section: A Linearized Lindstedt-poincaré Methodsmentioning

confidence: 99%

A Simplified Lindstedt-Poincaré Method for Saving Computational Cost to Determine Higher Order Nonlinear Free Vibrations

Liu

Chen

2021

Mathematics

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“…where  is the frequency, A is the amplitude. He's frequency formulation has been caught much attention due to its simplest solution process and relatively high accuracy [27][28][29][30][31][32][33][34][35][36][37].…”

Section: Periodic Solutionmentioning

confidence: 99%

Periodic solution of fractal Phi-4 equation

Liu

2021

THERM SCI

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This paper focuses on a fractal Phi-4 equation with time-space fractal derivatives, though its solitary solutions have been deeply studied, its periodic solution was rarely revealed due to its strong nonlinearity. Now the condition is completely changed, He?s frequency formulation provides with a universal tool to having a deep insight into the periodic property of the fractal Phi-4 equation. The two-scale transform is used to convert approximately the fractal Phi-4 equation to a differential model, and a criterion is suggested for the existence of a periodic solution of the equation, the effect of fractal orders on the periodic property is also elucidated.

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“…Equation ( 1) can be effectively solved by some analytical methods, for examples, the variational iteration method [13][14][15][16], the variational approach [17][18][19], the homotopy perturbation method [20-28], He's frequency formulation [2][3][4][5][6][7][8][9][10][11][12], and Taylor series method Though the analytical methods can reveal the basic property of the vibration, it is impossible to present the packing system from vibration.…”

Section: Packing Dynamical Systemmentioning

confidence: 99%

“…The vibration can be described by the well-known hyperbolic tangent oscillator [1], and it is easy to predict accurately the frequency-amplitude relationship [2][3][4][5][6][7][8][9][10][11][12], which is the main factor affecting the safe loading. Though we have many technologies to absorb the kinetic energy of the loading goods, a safe loading is almost impossible unless the loading kinetic energy is zero when loaded.…”

Section: Introductionmentioning

confidence: 99%

A thermodynamic model for a packing dynamical system

Song

2020

THERM SCI

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So far shock and vibration are the inherent properties of all packing systems, and various mitigation measures have been considered using packing materials, which result in a large packing size. Now the thing changes, this paper gives an effective thermodynamic technology that can completely avoid shock and vibration. An aerial delivery system is used as an example to elucidate the novel packing system, which requires zero loading velocity when loaded. A thermodynamic model is established to reveal the main factors affecting the packing system.

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He’s frequency–amplitude formulation for nonlinear oscillator with damping

Abstract: He's frequency-amplitude formulation takes an active role in the analysis of nonlinear conservative oscillators. This paper shows that the formulation can be extended into damping cases, and the result is the same as that obtained by the homotopy perturbation method.

Cited by 10 publications

References 36 publications

A Simplified Lindstedt-Poincaré Method for Saving Computational Cost to Determine Higher Order Nonlinear Free Vibrations

A Simplified Lindstedt-Poincaré Method for Saving Computational Cost to Determine Higher Order Nonlinear Free Vibrations

Periodic solution of fractal Phi-4 equation

A thermodynamic model for a packing dynamical system

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