A New Measure of Shape Difference

Abstract

The Modal Assurance Criterion (MAC) is currently the most popular method for measuring whether or not two mode shapes are strongly correlated. In fact, MAC can be applied to any two sets of data that can be defined as a shape, e.g. mode shapes, Operating Deflection Shapes (ODS’s), or two time or frequency domain waveforms. When applied to two Frequency Response Functions (FRFs) MAC has been renamed FRAC (Allemang RJ, The modal assurance criterion (MAC): twenty years of use and abuse. In: Proceedings of the international modal analysis conference, 2002).

MAC values range between 0 and 1. If MAC = 1, the two shapes are identical. A “rule of thumb” is that two shapes are similar or strongly correlated if MAC > 0.9, and they are different or weakly correlated if MAC < 0.9.

MAC is a measure of the co - linearity of two shapes. That is, it measures whether or not two shapes lie together on the same straight line. MAC has two limitations however;

• 1. MAC does not measure the difference in values of two shapes.

• 2. MAC requires at least two shape components. The MAC value for two shapes with one component, i.e. two scalars, is always 1.

In this paper, a new measure, called the Shape Difference Indicator (SDI), is introduced which overcomes the two limitations of MAC. This new measure is more useful for machinery and structural health monitoring applications where, for instance, changes in vibration levels or temperatures are typically used to detect a fault.

An example is given showing how SDI indicates that shape pairs are different even when their MAC values indicate that they are the same, i.e. they are co-linear. A second example shows how SDI can be used not only to detect a fault, but also to correctly identify the fault by comparing its shape values with those in a database of known fault conditions.