A fractal Boussinesq equation for nonlinear transverse vibration of a nanofiber-reinforced concrete pillar

Ji, Fei-Yu; He, Chun‐Hui; Zhang, Jingjing; He, Ji‐Huan

doi:10.1016/j.apm.2020.01.027

Cited by 93 publications

(59 citation statements)

References 47 publications

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“…The most general form of the solution of Equation by using the Exp‐function method is given in He and Wu, Wu and He, and Ji et al

U false(η false) = \frac{true \sum_{i = - 1}^{y} a_{i} \exp false(i η false)}{true \sum_{j = - r}^{s} b_{j} \exp false(j η false)},

where r , s , l and y are unknown positive integers, a i and b j are unknown constants. We assume that the solution of Equation has the form:

U false(η false) = \frac{a_{l} \exp false(l η false) + \dots + a_{- y} \exp false(- y η false)}{b_{r} \exp false(r η false) + \dots + b_{- s} \exp false(- s η false)} .

…”

Section: Methodsmentioning

confidence: 99%

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New exact solutions of a nonlinear integrable equation

Yıldız

Dağhan

2020

Math Methods in App Sciences

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Analytic solutions of the partial differential equations are needed to explain many phenomena seen in thermodynamics, aerodynamics, plasma physics, and other fields. In this paper, variational principle is analyzed of the integrable nonlinear Korteweg–de Vries (KdV) typed equation. In addition, exact solutions of this equation are obtained by using various methods such as direct integration, homogeneous balance method, Exp‐function method, and Kudryashov method.

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“…The most general form of the solution of Equation by using the Exp‐function method is given in He and Wu, Wu and He, and Ji et al

U false(η false) = \frac{true \sum_{i = - 1}^{y} a_{i} \exp false(i η false)}{true \sum_{j = - r}^{s} b_{j} \exp false(j η false)},

where r , s , l and y are unknown positive integers, a i and b j are unknown constants. We assume that the solution of Equation has the form:

U false(η false) = \frac{a_{l} \exp false(l η false) + \dots + a_{- y} \exp false(- y η false)}{b_{r} \exp false(r η false) + \dots + b_{- s} \exp false(- s η false)} .

…”

Section: Methodsmentioning

confidence: 99%

“…The elliptical integral derived here is analyzed. Furthermore, solutions of Equation is obtained by using homogeneous balance method (HBM), Exp‐function method, and Kudryashov method …”

Section: Introductionmentioning

confidence: 99%

New exact solutions of a nonlinear integrable equation

Yıldız

Dağhan

2020

Math Methods in App Sciences

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“…We convert equation 9to be an equation without any harmonic functions; the frequency x 2 0 is disappearing through the replacing process. Some useful operations are required to put equation (9) in an alternative solvable form. Operating on both sides of equation 9with D 2 t ; then adding an auxiliary part x 2 0 y to both sides, we have the following one…”

Section: The Fractional Derivativementioning

confidence: 99%

“…[6][7][8] The fractional term in equation (1) can describe the noise and uncertain properties of many practical problems and can model a nonlinear beam vibration, a bridge vibration, and other vibration systems arising in architectural engineering. Ji et al 9 found many attractive properties for nonlinear transverse vibration of a nanofiber reinforced concrete pillar, which cannot be revealed by any traditional differential models. He 10 demonstrated that a nonlinear wave equation might have some periodic properties and a nonlinear vibration equation might have solitary properties. Fractional calculus performs a popularization of standard differentiation and integration in arbitrary order.…”

Section: Introductionmentioning

confidence: 99%

A periodic solution of the fractional sine-Gordon equation arising in architectural engineering

Shen

El‐Dib

2020

Journal of Low Frequency Noise, Vibration and Active Control

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Many nonlinear vibrations arising in the engineering of architecture include noise and uncertain properties, and this paper suggests a fractional model to elucidate the properties. The fractional sine-Gordon equation with the Riemann-Liouville fractional derivative is used as an example to solve its periodic solution by the homotopy perturbation method. The frequency-amplitude relationship is obtained, and the effect of the fractional derivative order on the vibration property is discussed. Additionally, the harmonic resonance is also discussed. This preliminary research can be further extended to real applications.

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“…where k and w are constants. According to some researchers [32][33][34] such as Ji and He, the solution of equation 10is as follows…”

Section: Dynamic Modelmentioning

confidence: 99%

Research on piezoelectric vibration energy harvester with variable section circular beam

Ding

Wei

et al. 2020

Journal of Low Frequency Noise, Vibration and Active Control

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In order to reduce the natural frequency of the piezoelectric vibration energy harvester, improve performance of the piezoelectric vibration energy harvester, and meet the requirements of energy acquisition in the low-frequency vibration environment, a variable-section circular piezoelectric vibration energy harvester is presented. The dynamic model and electromechanical coupling model of variable-section circular piezoelectric vibration energy harvester are established. The main factors affecting the output performance of piezoelectric vibration energy harvester are analyzed. The structure parameters of piezoelectric vibration energy harvester are optimized by orthogonal experiment. An experimental platform is built to test output voltage and output power of piezoelectric vibration energy harvester. The experimental results show that when the number of energy harvester is 4 and the external load is 180KΩ, the parallel output power can reach 0.213mW, which can meet the requirements of micro-power device power supply.

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A fractal Boussinesq equation for nonlinear transverse vibration of a nanofiber-reinforced concrete pillar

Cited by 93 publications

References 47 publications

New exact solutions of a nonlinear integrable equation

New exact solutions of a nonlinear integrable equation

A periodic solution of the fractional sine-Gordon equation arising in architectural engineering

Research on piezoelectric vibration energy harvester with variable section circular beam

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