The identification of cutting states, associated with the orthogonal cutting of stiff cylinders, was realized in reference [1] through an analysis of the singular values of an unsymmetric Toeplitz matrix, R, of third order cumulants, r(i, j), of acceleration measurements. The ratio of the two dominant pairs of singular values of R, the R-ratio, was shown to differentiate between light cutting, medium cutting, pre-chatter and chatter states. The R matrix is the coefficient matrix in an autoregressive approximation of the bispectrum references [6,7].
On the basis of an analysis of five sequences of experiments with variable depth of cut and two sequences with variable turning frequency, a total of 42 cutting experiments, it was found that the R-ratio was approximately one for all cases of light cutting and two or more for chatter. For intermediate states the ratio increased as the chatter state was approached. The R-ratio was evaluated for q = 100; see equation ( 6) below.
Relationships between phase coupled trigonometric functions and the singular values of the corresponding R matrix were established in reference [1] through a numerical study of three specific phase coupled functions. However, no general assertions regarding relationships between properties of the R-ratio and those of the associated time series were made. Such a relationship is established in what follows for time sequences consisting of sums of phase coupled cosine functions.
The third order cumulant, r(i, j), of a time series consisting of the sum of phase coupled cosine functions may be expressed as a finite sum of cosine functions; see reference [6]. If the sums of cosine functions are periodic for some integral value of the argument of r(i, i), then the Toeplitz matrix R(r(i, j)), of dimension q + 1, is circulant for a sequence of values of q. In this case, a simple closed form representation for singular values of R is known (see references [2,4,8]), which yields an expression for the R-ratio in terms of the coefficient of the phase coupled cosine functions.
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Let r 3 (t 1 , t 2 ) be the third order cumulant of the real third order stationary random process X(k), k = 0, 2 1, 2 2, . . . . If the mean of X(k) vanishes, then r 3 (t 1 , t 2 ) = m 3 (t 1 , t 2 ), where m 3 (t 1 , t 2 ) = E(X(k) X(k + t 1 ) X(k + t 2 )). E is the expected