An Algorithm for Systematic Generation of Subscales With Application to a Two-Disk Friction System

Abstract

Abstract In this paper we propound an algorithm for development of scale models of complex physical systems. The algorithm systematically treats the three sub-tasks of scaling, which includes creation of dimensionless parameters, application of scaling law and similitude requirements, and determination of scale factors. Optimum set of dimensionless parameters are obtained in which the pertinent variables with only integer powers appear. This is achieved by seeking integer solutions to appropriate set of linear algebraic equations in the powers, using a simple iterative technique. Application of the scaling law and similitude requirements along with a logarithmic transformation leads to a set of linear algebraic equations in the scale factors. In many applications, over-constrained systems are resulted for which true scaling can not be achieved. Two techniques are introduced in the proposed algorithm which circumvent the difficulty in treating over-constrained systems. The first method is based on a relaxation technique, whereas the second method introduces the concept of auxiliary variables to obtain the least squares solutions to the new set of linear equations. The effectiveness of the algorithm is demonstrated through its application in finding scale models of a two disk friction system.