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

doi:10.1007/s11465-014-0289-7

Front. Mech. Eng.

2014, Vol. 9

Issue (1) : 58-70 https://doi.org/10.1007/s11465-014-0289-7

RESEARCH ARTICLE

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

M.R. AKBARI¹,D.D. GANJI^2,^*(

),A.R. AHMADI³,Sayyid H. Hashemi KACHAPI²

1. Department of Civil Engineering, The University of Pardisan Mazandaran and Department of Chemical Engineering, University of Tehran, P.O. Box 66456-43516, Iran
2. Department of Mechanical Engineering, Babol University of Technology, P.O. Box 484, Babol, Iran
3. Department of Mechanical Engineering, Sari Branch, Islamic Azad University, Sari, P.O. Box 48161-19318, Iran

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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.

Keywords Algebraic Method (AGM) tower crane oscillating system angular frequency

Corresponding Author(s): D.D. GANJI

Issue Date: 16 May 2014

Cite this article:

M.R. AKBARI,D.D. GANJI,A.R. AHMADI, et al. Analyzing the nonlinear vibrational wave differential equation for the simplified model of Tower Cranes by Algebraic Method[J]. Front. Mech. Eng., 2014, 9(1): 58-70.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-014-0289-7
https://academic.hep.com.cn/fme/EN/Y2014/V9/I1/58

Tab.1 Nomenclature

Fig.1 The schematic diagram of the physical model

Fig.2 The displacement diagram of the mentioned system

Fig.3 The achieved solution, θ(t), by AGM

Fig.4 The resulted phase plane by AGM

Tab.2 The obtained results for θ(t) and its derivative according to the Numerical solution of Eq. (27)

Fig.5 Comparing the obtained solution by AGM and Numerical Method

Fig.6 A comparison between the resulted phase planes by AGM and Numerical Method

Fig.7 A comparison among the obtained phase planes by AGM on the basis of the different amplitudes of vibration

Fig.8 The obtained solution by AGM

Fig.9 The resulted phase planes by AGM

Tab.3 The obtained results for θ(t) and its derivative according to the Numerical Solution of Eq. (42)

Fig.10 Comparing the obtained solution by AGM and Numerical Method

Fig.11 A comparison between the resulted phase planes by AGM and Numerical Method

Fig.12 A comparison among the obtained phase planes by AGM based on the four different damping coefficients

Fig.13 The chart of the obtained solution by AGM

Fig.14 The resulted phase plane by AGM

Tab.4 The obtained results for θ(t) and its derivative according to the Numerical Solution of Eq. (56)

Fig.15 A comparison between the obtained solution by AGM and Numerical Method

Fig.16 Comparing the resulted phase planes by AGM and Numerical Method

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