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Research Papers

Characteristic Values in the Scaling of Differential Equations in Engineering

[+] Author and Article Information
Patricio F. Mendez

Department of Chemical and Materials Engineering, University of Alberta, 9107 116th St., Edmonton, AB, T6G 2V4, Canadapmendez@ualberta.ca

J. Appl. Mech 77(6), 061017 (Sep 01, 2010) (12 pages) doi:10.1115/1.4001357 History: Received November 17, 2008; Revised December 02, 2009; Published September 01, 2010; Online September 01, 2010

This work introduces, for the first time, a formal approach to the estimation of characteristic values of differential and other related expressions in the scaling of engineering problems. The methodology introduced aims at overcoming the inability of the traditional approach to match the exact solution of asymptotic cases. This limitation of the traditional approach often leaves in doubt whether the scaling laws obtained actually represent the desired phenomena. The formal approach presented yields estimates with smaller error than traditional approaches; these improved estimates converge to the exact solution in simple asymptotic cases and do not diverge from the exact solution in cases in which the error of traditional approaches is unbounded. The significance of this contribution is that it extends the range of applicability of scaling estimates to problems for which traditional approaches were deemed unreliable, for example, cases in which the curvature of functions is large, or complex cases in which the accumulation of estimation errors exceeds reasonable limits. This research is part of a larger effort towards a computational implementation of scaling, and it is especially valuable for approximating multicoupled, multiphysics problems in continuum mechanics (e.g., coupled heat transfer, fluid flow, and electromagnetics) that are often difficult to analyze numerically or empirically.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

Mass, spring, and damper dynamical system

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Figure 2

Evolution of a mass, spring, and damper dynamical system with m=1 kg, c=0.1 N s/m, k=10 N/m, and u0=0.1 m

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Figure 3

Error in the estimation of the characteristic time of an underdamped mechanical oscillator. The curve etcˆ represents the error using the improved estimation approach assuming a sinusoidal evolution, the curve “etcˆ (parabola)” represents the estimation error if the improved estimation had assumed a parabolic variation, and the curve “etcˆ (traditional)” represents the error when the traditional approach is used.

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Figure 4

Evolution of an overdamped mass, spring, and damper dynamical system with m=1 kg, c=300 N s/m, k=700 N/m, and u0=0.1 m

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Figure 5

Error in the estimation of the characteristic time of an overdamped mechanical oscillator. The curve etcˆ represents the error using the improved estimation approach assuming an exponential decay and the curve etcˆ (traditional) represents the error when the traditional approach is used.

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Figure 6

Simple pendulum with large oscillations

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Figure 7

Evolution of a simple pendulum with large oscillations

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Figure 8

Error in the estimation of the characteristic time of simple pendulum with large oscillations. The curve etcˆ represents the error using the improved estimation approach assuming an exponential evolution and the curve etcˆ (traditional) represents the error when the traditional approach is used.

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