Matrices and Shape Factors for Linear Empirical Similitude Method

Empirical Similitude Method (ESM) is a technique developed to enhance the applicability of similarity methods to incorporate more realistic experimental data and scaling effects. Much of the development has been motivated by the premise of forging a relationship between experiential information and physical systems that have inherent non – linear variables and factors affecting their response. We augment this procedure by incorporating the Toeplitz and Hankel matrices, and solve the modified linear ESM problem using the conjugate gradient method, highlighting the potential benefits in the process. A simple deflection example is illustrated for lucidity and a comparison is made between all the available methods for contrast. A final extension into the use of shape factors is made using a numerical example. INTRODUCTION Scaling and similarity methods stem from the idea of similitude introduced by [Rayliegh, 1915] where a complex system can be represented by an equivalent simpler model that is more readily analyzable. Such an approximation allows for easier experimentation of the model, the resulting test data of which can then be accurately and reliably scaled and transformed to the actual system (product). Buckingham [Bridgman, 1931] provided the initial framework to carry out such a scaling and detailed the development of the transformation process through the use of dimensionless π groups. Considering the fact that the applicability of this process is determined by the ability to determine the constants and the exponents of the scaling factors [Szirtes, 2003], the technique is limited by the degree of non – linearity and independence of the affecting geometric and material variables. While some headway can still be expected using more advanced methods, the associated experimental and computational effort also compounds relatively. In an effort to ease this effort, the ESM process [Cho, 1999] was developed that simplified the conversion by disassociating geometry and material properties and provided a means for independent individual scaling.