Using the Eigenvalues of Multivariate Spectral Matrices to Achieve Cutting Direction and Sensor Orientation Independence

2005 ◽  
Vol 128 (1) ◽  
pp. 350-354 ◽  
Author(s):  
John T. Roth
Keyword(s):  
Cutting Tools ◽  
Autoregressive Models ◽  
Triaxial Accelerometer ◽  
Industrial Environment ◽  
Monitoring Techniques ◽  
Monitoring Technique ◽  
Sensor Orientation ◽  
Multivariate Autoregressive Models ◽  
Cutting Direction ◽  
Multivariate Autoregressive

There is a strong need for monitoring techniques capable of tracking the health of cutting tools under varying conditions. Unfortunately, most monitoring techniques are dependent on the cutting direction and/or the sensor orientation, limiting their effectiveness in the typical industrial environment. With this in mind, this research develops a monitoring technique that is independent of both of these factors. This is accomplished by using multivariate autoregressive models that are fit to the output from a triaxial accelerometer. The work shows that the eigenvalues of multivariate spectral matrices, calculated at the machining frequencies, are not only sensitive to the condition of the tool but are also independent of the direction of cutting and the orientation of the sensor. This independence is verified experimentally through tests conducted under a variety of cutting directions and sensor orientations.

Using the Eigenvalues of Multivariate Spectral Matrices to Monitor Equipment

2001 ◽  
Author(s):  
John T. Roth
Keyword(s):  
Cutting Tools ◽  
Autoregressive Models ◽  
Industrial Environment ◽  
Monitoring Techniques ◽  
Monitoring Technique ◽  
Sensor Orientation ◽  
Multivariate Autoregressive Models ◽  
Cutting Direction ◽  
Multivariate Autoregressive

Abstract There is a strong need in industry for monitoring techniques that are capable of tracking the health of cutting tools under varying conditions. Unfortunately, most monitoring techniques that are currently available are dependent on the cutting direction and/or the sensor orientation, limiting their effectiveness in the typical industrial environment. With this in mind, this research focuses on developing a monitoring technique that is independent of both of these factors. This is accomplished by using multivariate autoregressive models that are fit to the output from a tri-axial accelerometer. The work shows that the eigenvalues of multivariate spectral matrices, calculated at the machining frequencies, are sensitive to the condition of the tool. Furthermore, it is theoretically demonstrated that these eigenvalues are independent of the direction of cutting and the orientation of the sensor. This independence is verified experimentally through tests conducted under a variety of cutting directions and sensor orientations.

Development of a Cutting Direction and Sensor Orientation Independent Monitoring Technique for End-Milling

1999 ◽  
Author(s):  
John T. Roth ◽  
Sudhakar M. Pandit
Keyword(s):  
End Milling ◽  
Multivariate Models ◽  
Actual Event ◽  
Monitoring Technique ◽  
Sensor Orientation ◽  
Acceleration Data ◽  
Life Tests ◽  
Multivariate Autoregressive Models ◽  
Cutting Direction ◽  
Warning Time

Abstract In the authors’ previous work, univariate models were fit to acceleration data to predict impending tool failure. Numerous end-milling life tests, conducted under a wide variety of cutting conditions, demonstrated that the method could consistently warn of impending failure between 6 inches (15 cm) and 8 inches (20 cm) prior to the actual event. This paper presents an improved method that increases the warning time and allows the technique to function independent of the cutting direction or sensor orientation. Using multivariate autoregressive models fit to tri-axial accelerometer signals, monitoring indices are developed, verified and the results are compared with those from the univariate models. The multivariate models detected impending failure 30 inches (76 cm) prior to its occurrence, 23.5 inches (60 cm) earlier than with the univariate models. Furthermore, the multivariate models are able to monitor the condition of the tool, regardless of the cutting direction or sensor orientation.

Development of a Cutting Direction and Sensor Orientation Independent Monitoring Technique for End-Milling

1999 ◽  
Vol 122 (4) ◽  
pp. 671-677 ◽  
Author(s):  
John T. Roth ◽  
Sudhakar M. Pandit
Keyword(s):  
End Milling ◽  
Multivariate Models ◽  
Actual Event ◽  
Monitoring Technique ◽  
Sensor Orientation ◽  
Acceleration Data ◽  
Life Tests ◽  
Multivariate Autoregressive Models ◽  
Cutting Direction ◽  
Warning Time

In the authors’ previous work, univariate models were fit to acceleration data to predict impending tool failure. Numerous end-milling life tests, conducted under a wide variety of cutting conditions, demonstrated that the method could consistently warn of impending failure between 6 inches (15 cm) and 8 inches (20 cm) prior to the actual event. This paper presents an improved method that increases the warning time and allows the technique to function independent of the cutting direction or sensor orientation. Using multivariate autoregressive models fit to tri-axial accelerometer signals, monitoring indices are developed, verified and the results are compared with those from the univariate models. The multivariate models detected impending failure 30 inches (76 cm) prior to its occurrence, 23.5 inches (60 cm) earlier than with the univariate models. Furthermore, the multivariate models are able to monitor the condition of the tool, regardless of the cutting direction or sensor orientation. [S1087-1357(00)01003-0]

On Asymptotic Prediction Problems for Multivariate Autoregressive Models in the Unstable Nonexplosive Case

1987 ◽  
Vol 36 (1-2) ◽  
pp. 29-38 ◽  
Author(s):  
A. K. Basu ◽  
S. Sen Roy
Keyword(s):  
Autoregressive Process ◽  
Autoregressive Models ◽  
Multivariate Autoregressive Models ◽  
Prediction Problems ◽  
Dependent Error ◽  
Meansquare Error ◽  
Multivariate Autoregressive

This paper considers the prediction problems of a k-dimensional, pth order autoregressive process with unstable but non-explosive roots and dependent error variables. The estimated predictor has been shown to be asymptotically equivalent to the optimal predictor. An expression for the meansquare error of the estimated predictor has also been derived .

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