Though industrial processes are perceived to be dauntingly complex from a mathematical modelling perspective, simple mathematical models often provide remarkable insight. One of the reasons behind this apparent contradiction is that the mathematical models often involve small and/or large non-dimensional parameters and therefore are amenable to simplification through the use of perturbation analysis. In many cases, the leading order perturbation approximation provides the bulk of the information about the structure and behaviour of the solution and is often sufficient to obtain profound insight into the phenomenon under examination. This article illustrates such utility of the perturbation analysis by using examples in which leading order perturbation approximations provide substantial insight into the mode transition phenomena in the vibrational behaviour of curved beams and helices. The methodology described in this article can be used in a wide range of applications to reveal the simplest possible structure of the mathematical model that answers the question under examination. References R. S. Anderssen. Acceptance speech on the award of the George Szekeres Medal. October, 2004. C. M. Bender, and S. A. Orszag. Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill, 1978. F. R. de Hoog. Why are simple models often appropriate in industrial mathematics? In R.S. Anderssen et al., (editors), {18th World IMACS Congress and MODSIM09 International Congress on Modelling and Simulation}, pages 23--36, 2009, http://www.mssanz.org.au/modsim09/Plenary/deHoog.pdf T. Tarnopolskaya, F. R. de Hoog, N. H. Fletcher, and S. Thwaites. Asymptotic analysis of the free in-plane vibrations of beams with arbitrarily varying curvature and cross-section. Journal of Sound and Vibration, 196: 659--680, 1996, doi:10.1006/jsvi.1996.0507 T. Tarnopolskaya, F. R. de Hoog, A. Tarnopolsky, and N. H. Fletcher. Vibration of beam and helices with arbitrarily large uniform curvature. Journal of Sound and Vibration, 228(2): 305--332, 1999, doi:10.1006/jsvi.1999.2413 T. Tarnopolskaya, F. R. de Hoog, and N. H. Fletcher. Low-frequency mode transition in the free in-plane vibration of curved beams. Journal of Sound and Vibration, 228(1): 69--90, 1999, doi:10.1006/jsvi.1999.2400 N. H. Fletcher, T. Tarnopolskaya, and F. R. de Hoog. Wave propagation on helices and hyperhelices: a fractal regression. Proc. R. Soc. Lond. A, 457(1): 33-43, 2001, doi:10.1098/rspa.2000.0654 N. H. Fletcher. Hyperhelices: a classical analog for strings and hidden dimensions. Am. J. Phys., 72(5): 701--703, 2004, doi:10.1119/1.1652038 K. A. Seffen and E.Toews. Hyperhelical actuators: coils and coiled-coils. AIAA Structures, Structural Dynamics and Materials Conference, 2004, California. J. Ueda, T. W. Secord, and H. H. Asada. Large effective-strain piezoelectric actuators using nested cellular architecture with exponential strain amplification mechanisms. IEEE/ASME Transactions on Mechatronics, 15(5): 770--782, 2010, doi:10.1109/TMECH.2009.2034973
@article{5145,
title = {Practical insight through perturbation analysis},
journal = {ANZIAM Journal},
volume = {52},
year = {2012},
doi = {10.21914/anziamj.v53i0.5145},
language = {EN},
url = {http://dml.mathdoc.fr/item/5145}
}
Tarnopolskaya, Tanya; de Hoog, Frank. Practical insight through perturbation analysis. ANZIAM Journal, Tome 52 (2012) . doi : 10.21914/anziamj.v53i0.5145. http://gdmltest.u-ga.fr/item/5145/