Toward to Vibration Analysis by Knot Theory

Abstract

Vibration analysis is one of the most important aspects in the design of structures and mechanical systems, among others, subject to dynamic loads. As well as for the analysis of failures caused by vibratory aspects. A good performance of an industrial system is often associated with the availability of mathematical models of the dynamic behaviour of the system. In some situations, the complexity of the processes makes it difficult to have models that help us to analyse these processes. This paper proposes the use of knot theory, which is a topological tool, for vibration analysis. This topological tool, in this case, associates a topological invariant when there is a drastic change in vibrations. The present work is based on the fact that it is well known that the equations representing harmonic motion generate Lissajous figures. In knot theory, there are several classifications of knots, one of these classifications is known as Lissajous knots. The use of this tool is shown in the supposition that we have a system represented by three equations of the form f(t) = Acos(Bt + C) , where with the indicated parameters it generates a knot (being its nominal value). Making a change in the phase, which represents a fault, generates a different knot than the nominal knot. One of the advantages of this proposed method is that it is not necessary to have the model, and one of the disadvantages by nature of this method is that three signals are required to use this topological tool.