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Analytical scrutiny of nonlinear equation of hypocycloid motion by AGM

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Abstract

The curve produced by fixed point on the circumference of a small circle of specified radius rolling around the inside of a large circle with bigger radius than the other one is named hypocycloid. In the present paper, a semi-hypocycloid motion of a cylinder has been simulated by extracting the governing nonlinear differential equation of this especial motion, and then, the obtained equation has been solved completely by a simple and innovative approach which we have named it Akbari–Ganji’s method (AGM). On the basis of comparisons which have been made between the gained solutions by AGM, numerical method (Runge–Kutta 4th) and VIM, it is possible to indicate that AGM can be successfully applied for various differential equations. In this paper, a nonlinear vibrational equation has been solved by AGM, and afterward, the application of AGM will be shown in our own specified problem. It is noteworthy that this method has some valuable advantages, for instance in this approach, it is not necessary to utilize dimensionless parameters in order to simplify equation. So there is no need to convert the variables to new ones that heightens the complexity of the problem. Moreover by utilizing AGM, the shortage of boundary condition(s) for solving differential equation will be terminated by using derivatives of main differential equation(s). The results reveal that this method is very effective, simple, reliable and can be applied for many other nonlinear problems.

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Abbreviations

u(t):

Vibrational displacement \(\left( {\text{m}} \right)\)

\(\dot{u}(t)\) :

Vibrational velocity \(\left( {\frac{\text{m}}{\text{s}}} \right)\)

\({\ddot{u}}(t)\) :

Vibrational acceleration \(\left( {\frac{\text{m}}{{{\text{s}}^{2} }}} \right)\)

\(\theta \left( t \right)\) :

Angular position as a function of time \(\left( {\text{rad}} \right)\)

\(A\) :

Initial amplitude of vibration \(\left( {\text{m}} \right)\)

\(\omega\) :

Angular frequency \(\left( {\frac{\text{rad}}{\text{s}}} \right)\)

\(\ddot{\psi }(t)\) :

Angular acceleration \(\left( {\frac{\text{rad}}{{{\text{s}}^{2} }}} \right)\)

\(g\) :

Gravity acceleration \(\left( {\frac{\text{m}}{{{\text{s}}^{2} }}} \right)\)

\(m\) :

Mass of the cylinder \(\left( {\text{kg}} \right)\)

\(w\) :

Weight of the cylinder \(\left( {\text{N}} \right)\)

\(I_{\text{c}}\) :

Mass moment of inertia about instantaneous center \(\left( {{\text{kg}}\,{\text{m}}^{2} } \right)\)

\(\sum {M_{\text{c}} }\) :

Resultant moment about instantaneous center of rotation \(\left( {{\text{N}}\,{\text{m}}} \right)\)

\(R\) :

Radius of circular path \(\left( {\text{m}} \right)\)

\(r\) :

Radius of the cylinder \(\left( {\text{m}} \right)\)

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Acknowledgements

The authors are grateful to Alireza Ahmadi, who helped to improve and amend the paper with his useful and practical suggestions.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Babol Noshirvani University of Technology, P.O. Box 484, Babol, Iran

    Navid Balazadeh Meresht & D. D. Ganji

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  1. Navid Balazadeh Meresht
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  2. D. D. Ganji
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Correspondence to Navid Balazadeh Meresht.

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Meresht, N.B., Ganji, D.D. Analytical scrutiny of nonlinear equation of hypocycloid motion by AGM. Neural Comput & Applic 29, 1575–1582 (2018). https://doi.org/10.1007/s00521-016-2654-4

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