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Local accuracy and error bounds of the improved Runge-Kutta numerical methods

Qureshi S., Memon Z., Shaikh A. A.

Journal of Applied Mathematics and Computational Mechanics

2018

Vol. 17, nr 4

73--84

In this paper, explicit Improved Runge-Kutta (IRK) methods with two, three and four stages have been analyzed in detail to derive the error estimates inherent in them whereas their convergence, order of local accuracy, stability and arithmetic complexity have been proved in the relevant literature. Using single and multivariate Taylor series expansion for a mathematical function of one and two variables respectively, slopes involved in the IRK methods have been expanded in order to obtain the general expression for the leading or principal term in the local truncation error of the methods. In addition to this, principal error functions of the methods have also been derived using the idea of Lotkin bounds which consequently gave rise to the error estimates for the IRK methods. Later, these error estimates were compared with error estimates of the two, three, and four-stage standard explicit Runge-Kutta (RK) methods to show the better performance of the IRK methods in terms of the error bounds on the constant step-size h used for solving the initial value problems in ordinary differential equations. Finally, a couple of initial value problems have been tested to determine the maximum absolute global errors, absolute errors at the final nodal point of the integration interval and the CPU times (seconds) for all the methods under consideration to get a better idea of how the methods behave in a particular situation especially when it comes to analyzing the error terms.

Lifting capacity enhancement of a crawler crane by improving stability

Shaikh A. A., Kumar D D.

Journal of Theoretical and Applied Mechanics

2016

Vol. 54 nr 1

219--227

The lifting capacity of a crawler crane is limited by its stability and structural strength. This paper analyzes the stability factor by calculating tipping loads at various load radii for a particular boom length. It shows that the tipping load decreases with an increase in the load radius. A new structural frame is proposed to extend out the superstructure counterweight of the crane. With such a proposed arrangement, it is shown that the lifting capacity of the crane, limited by stability, increases. Static structural analysis of the proposed structural frame is performed using ANSYS workbench software.