Delay differential equations via the matrix lambert w function and bifurcation analysis: application to machine tool chatter - Sun Yi; Patrick W. Nelson; A. Galip Ulsoy (Mathematical Biosciences and Engineering)
In a turning process modeled using delay differential equations(DDEs), we investigate the stability of the regenerative machinetool chatter problem. An approach using the matrix Lambert Wfunction for the analytical solution to systems of delaydifferential equations is applied to this problem and compared withthe result obtained using a bifurcation analysis. The Lambert Wfunction, known to be useful for solving scalar first-order DDEs,has recently been extended to a matrix Lambert W function approachto solve systems of DDEs. The essential advantages of the matrixLambert W approach are not only the similarity to the concept of thestate transition matrix in linear ordinary differential equations,enabling its use for general classes of linear delay differentialequations, but also the observation that we need only the principalbranch among an infinite number of roots to determine the stabilityof a system of DDEs. The bifurcation method combined with Sturmsequences provides an algorithm for determining the stability ofDDEs without restrictive geometric analysis. With this approach, onecan obtain the critical values of delay, which determine thestability of a system and hence the preferred operating spindlespeed without chatter. We apply both the matrix Lambert W functionand the bifurcation analysis approach to the problem of chatterstability in turning, and compare the results obtained to existingmethods. The two new approaches show excellent accuracy and certainother advantages, when compared to traditional graphical,computational and approximate methods.