On parametrically phase-coupled random harmonic processes

Abstract

In this paper, we will present the concept of a parametrically phase-coupled random harmonic process. It will be argued that such processes will be present in the vibration signature of a simple mechanical system, and as such would be of interest in the analysis of vibrations for condition-based maintenance. It is shown that conventional higher-order spectral (HOS) techniques reveal only limited features of parametrically phase-coupled processes and would prove of limited utility in the analysis of such signals. A modified second order cumulant spectrum is then introduced, which has the advantage of revealing all parametric phase relationships in the signal of interest. The results of testing using simulated vibration signals are presented. INTRODUCTION.A time series, { x(t)}, is said to be a random almostharmonic process if it can be expressed as a sum of sinusoids whose amplitudes are deterministic, but whose initial phases are random, i.e., where the {on} are distinct frequencies [ 11. In classical time series analysis, it is typically assumed that the {$n) are a set of i.i.d. random variables uniformly distributed over (0, 2n). It is straightforward to show that this assumption regarding { $n} implies that { x(t)} is widesense stationary. However, this assumption requires that no underlying phase relationship exist between the spectral components of {x(t)}. Now assume that the { a n } and the {$n) of (1-1) satisfy the relationship and where the set of {hn} are distinct, deterministic parameters, wo is some deterministic fundamental frequency, and $o is some random underlying initial phase uniformly distributed over (0, 2n). We will say that a random, almost harmonic, process satisfying (1-2) and (1-3) exhibits parametric phase coupling.As an example of a simple system which produces signals exhibiting parametric phase coupling, consider the cam and follower indicated schematically in Figure 1.1. It is assumed that the cam rotates about a fixed center, 0, at a constant rate of wo and with some random initial phase, q0. Let R denote angular displacement about 0 relative to some fixed radial line on the cam, and let p(Q) denote the surface of the cam at R. If we assume that the surface of the cam is continuous, then p(R) admits the Fourier series representation p(Q) = E rk cos(ks2) + qk sin(ks2) .(1-4) kThe radial displacement velocity of the follower, v(t), is given by dP dP a n v(t) =-=--, dt dR dt (1)(2)(3)(4)(5) or (1-6) -kr sin kw t + k 4 k ( 0 0 Note that the mechanical vibrations produced by this simple cam and follower arrangement will exhibit parametric phase coupling at harmonics of the shaft rate. In a more complicated rotating mechanical system, we may observe vibrations which exhibit parametric phase coupling at both harmonic and non-harmonic multiples of the applied shaft rate [2]. Moreover, these vibrations are often indicators of mechanical wear, and as such are of great interest in the area of condition-based maintenance