Analyzing the nonlinear vibrational wave differential equation for the simplified model of Tower Cranes by Algebraic Method

Abstract

In the current paper, a simplified model of Tower Cranes has been presented in order to investigate and analyze the nonlinear differential equation governing on the presented system in three different cases by Algebraic Method (AGM). Comparisons have been made between AGM and Numerical Solution, and these results have been indicated that this approach is very efficient and easy so it can be applied for other nonlinear equations. It is citable that there are some valuable advantages in this way of solving differential equations and also the answer of various sets of complicated differential equations can be achieved in this manner which in the other methods, so far, they have not had acceptable solutions. The simplification of the solution procedure in Algebraic Method and its application for solving a wide variety of differential equations not only in Vibrations but also in different fields of study such as fluid mechanics, chemical engineering, etc. make AGM be a powerful and useful role model for researchers in order to solve complicated nonlinear differential equations.

Key words： Algebraic Method (AGM); tower crane; oscillating system; angular frequency

Acknowledgments

The authors are grateful to the Ancient Persian mathematician, astronomer and geographer Muḥammadibn Musa Kharazmiwho was the one that his Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear and quadratic equations. Furthermore, he was considered as the original inventor of algebra and is the one whose book Europeans derive the term ALGEBRA from and also the expression ALGORITHM has been taken from his name (the Latin form of his name). Consequently, we dedicate this way of solving linear and nonlinear differential equations to this scientist.

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