Condition Monitoring of Unsteadily Operating

EquipmentJordan McBain, B.Eng.Mgt, EIT

Health Monitoring of steady speed/load machinery a well established practice

However, few techniques are available for monitoring unsteadily operating equipment

Techniques required for advanced equipment such as electromechanical shovel, variable duty hoists, etc.

Problem

Condition Monitoring Pattern Recognition Vibration Analysis Condition Monitoring of Unsteadily Operating

Equipment◦ Suggested Approach: Statistical Parameterization◦ Novelty Detection

Experimental Methodology◦ Classification Results

Conclusions Future Work

Outline

Machinery Maintenance Policy driven by:◦ Availability of resources (spare parts, pers., capital)◦ Importance of equipment◦ Availability of technology and expertise

Modern Maintenance Policy evolved through:◦ Run-to-Failure◦ Periodic Maintenance◦ Predictive Maintenance

Maintenance is delayed until some monitored parameter of the equipment becomes erratic

Proactive Balances resources

Condition Monitoring

Benefits:◦ Environment◦ Safety◦ Production◦ Staff Shortages/Costs◦ Scheduling◦ Spare Parts (JIT)◦ Insurance◦ Life Extension

Condition Monitoring

Faults in rotating machinery have very representative features in the frequency domain

Consider bearing:◦ Frequency Response a

function of Fault, Slippage, Noise

Vibration Analysis

Diagrams from: Randall, B. State of the Art in Machinery Monitoring, JSV

One branch of artificial-intelligence domain Usually involves representing a state or

object to be indentified with a vector of commensurate numerical values◦ E.g. In classifying fruit: weight, spectroscopic

values, etc. Representative vector called a “pattern” or

“classification object” Classification achieved by computing

decision surfaces around classes of objects

Pattern Recognition

Sensing

Segmentation

Feature

Extraction

Classification

Post-Processing

Pattern Recognition

Vibration Measurement

DividingVibration Signal

Choosing RepresentativeNumeric Values

CalculatingClassificationBoundaries

-Decision Support-Prognostics-Etc.

Condition Monitoring of Unsteadily Operating

EquipmentStatistical Parameterization

Explore a technique developed for monitoring health of structures first established in◦ K Worden, H Sohn, CR Farrar. Novelty detection in a

changing environment: Regression and inter polation approaches, J.Sound Vibrat. 258 (2002).

Essential idea: clustering of patterns will vary with modal parameters (speed/load/temp)

Technique improves the segmentation step; rendering classification almost trivial

Statistical Parameterization

Variable speed machinery◦ Elements of a machine’s vibratory response are

assumed to have a strong relation to the speed of the given machinery

Distribution for speeds:◦ Means vary with speed◦ Variances vary with resonance response

x

y

* C10

*C20

*C30

Segment vibration signal Group segments according to the machine’s

speed Calculate Gaussian parameters for small

segments of speed (sample statistics assumed to be population statistics)◦ Curse of Dimensionality

Interpolate or Regress each component of statistical parameters

Decision boundary function of speed (and other modal parameters)

Method

Sub problem of pattern recognition◦ Rather than train a classifier on all classes, we train on

the “normal” class and then signal an error when behaviour deviates from it

◦ Employed where knowledge of all classes (of faults) not practical to attain

Decision boundary encircles normal patterns A wide variety of techniques available Examine two:

◦ Boundaries containing a certain quantile of data (i.e. a discordance test)

◦ Boundaries derived by Support Vectors

Novelty Detection

For uni-variate data – a simple task:◦ Classify as normal if test pattern falls within nth

quantile of training data◦ Think confidence level

Novelty Detection: Discordance

(| | ) 0.95P x R | |1.96

x

1.96 1.96x

For multi-variate data:◦ Build multi-variate model from multiple uni-

variate ones – assuming independence

Novelty Detection: Discordance

( ) ( )* ( )P A B P A P B

Assuming independence

Novelty Detection: Discordance

2

21

1

1( )

2

1 1

1 1( ) ( ) ( )2 2

1

1( ) ( )

2

1 1

(2 ) | |(2 )

i i

i

di i t

ii

xd d

ii i i

xx x

d dd

ii

p x p x e

e e

Example:◦ Distr. #1 µ= 5 and σ=4◦ Distr. #2 µ= 10 and σ=8◦ Joint Distr. therefore has µ= [5,10] and

◦ The level curves of the distribution are determined by the Mahalanobis squared distance given by

Novelty Detection: Discordance

4 0

0 8

2 1( ) ( )tr x x

This is the equation of an ellipsoid

In practice, covariance matrix non-diagonal (ie. Cross terms present)◦ Consequence: ellipses not

aligned with central axis◦ PCA required to

determine orientation Decision boundary, for

d-dimensional problem, containing n-th quantile given by:

1 1 1 12 1

2 2 2 2

1

11 2

2

11 2

2

11 2

2

2 22 1 2

( ) ( )

54 05 10

100 8

10 545 10

1010

8

55 10

104 8

( 5) ( 10)

4 8

tx xr

x x

xx x

x

xx x

x

xx x

x

x xr

2 ( , )k chi inv q d

Is independence a reasonable assumption in the context of variable load/speed machinery?◦ Many spectral components of vib. Machinery

strongly related Consider Bearings Consider Gear Meshing Etc.

◦ Gaussian fit depends on independence of probabilities of individual parameters

◦ May prove poor in this context

Novelty Detection: Discordance

This ellipsoidal boundary is very rigid and will not work well if the data is not perfectly Gaussian

Rather than computing the quantile for a test patterns given speed◦ Center each speed bin’s data about the origin and alter

its distribution from ellipsoidal to spherical with the whitening transform

◦ Consequence: All modal data is centered at the origin with faulted data orbiting the healthy data

◦ Now draw a decision boundary around the healthy data: use Support Vectors

N.B. There is still some dependence on the assumption of a Gaussian fit

Novelty Detection: Support Vectors

x

y

* C10

*C20

*C30

x

y

Healthy Data for all Speeds

Faulted Data

Support Vector Technique: Tax’s Support Vector Data Description (for Novelty Detection)◦ Attempts to fit a sphere of minimal radius around

normal data◦ But a in a higher dimensional space (using the

“kernel trick”) Generates a very flexible decision boundary in the

input space

Support Vectors

Experimental Methodology

Dr. Timusk’s PhD data Spectraquest gear dynamics simulator

◦ Variable frequency drive ◦ Gearbox (two stage parallel reduction)

Subject to variable loads (particle brake) Data acquisition system: NI PXI

◦ Ceramic Shear ICP Accelerometers (0.5 to 6500 Hz)◦ Sampling 4kHz/channel

Faults: ◦ motor with bearing faults, broken rotor bars, rotor unbalance◦ gear faults: missing tooth, chipped pinion, outer race bearing

Appartus

Segment vibration data into segments of ‘steady’ speed and load◦ Segments defined by n-shaft rotations

Accounts for varying speed Ensures coherent signal

Windowed (Gaussian Window – 70% overlap)

Sensing

Segmentation

Feature

Extraction

Classification

Post-Processing

Steady speed/load not guaranteed◦ But can generate segments with reasonable steadiness

and variance can be computed Group vibration segments into bins of a selected

size◦ Size effects how many classification objects in each bin

curse of dimensionality balanced against need for very fine modal resolution

Segmentation

A number of parameters could be employed to represent a vibration segment◦ Crest factor, average power, kurtosis, impulse factor, etc.◦ Autoregressive Models (AR)

AR models◦ Think Root Locus Method from Control Systems: You

determine the placement of poles to shape the frequency response of the CS AR models control placement of poles to shape model’s

frequency response to be representative of a signal’s frequency response in the least squares sense

◦ User selects the number of poles The more poles, the more representative the signal is Balanced against the curse of dimensionality

Sensing

Segmentation

Feature

Extraction

Classification

Post-Processing

Segmentation step makes data almost perfectly separable

Sensing

Segmentation

Feature

Extraction

Classification

Post-Processing

Fit each component of each statistical parameter (mean and covariance matrix) to model

Components of mean vector could be fit with polynomial

Components of covariance matrix not traceable

Classification: Regression

Covariance matrix components vary wildly Additional concern:

◦ Covariance matrix derived from regression may not be positive semi-definite

Method available to deal with issue (added complexity)

Classification results are poor

0tx x

Instead, we must store each bin’s statistical parameters◦ Any bins which are ill-conditioned or under

sampled could then simply be interpolated over◦ Positive semi definitenessguaranteed◦ Good classification results

Classification: Interpolation

High acceptance rate of healthy data generates poor rejection rate of faulted data (ellipsoidal boundaries)

Interpolating over missing/ill-conditioned bins◦ One missing bin: interpolated statistics almost the

same as those of measured values◦ Three bins missing:

SVDD has one parameter – sigma◦ Integer value [1,inf)◦ Low values – Tight

bound Choice of sigma has

very little effect No frustrating trade

off between classification error on normal and faulted data

Superior classification

Classification: Whitening with Support Vectors

Too good to be true?◦ Tax explored variable load/speed machinery

without our segmentation steps Training SVDD over all speeds, he achieved an

average error of 8% Our average error of 2% is very plausible!

Segmentation step removes overlap between faulted data of one speed bin and healthy data of others

The errors shown on the right are based on data from one accelerometer

Faults are not all located near this accelerometer

Segmentation has made classifications sensitive enough so that accelerometers can measure spatially disparate faults

Plausible: Underwater warfare analogy

Key Observation

For a fixed amount of data, increasing the dimensionality of the space increase classification error

Statistical Parameterization is doubly cursed

Curse of Dimensionality

Statistical parameterization◦ Approach extends well to variable speed

machinery Gaussian/independence assumption not theoretically

correct but the data cluster well anyway◦ Prefer interpolation over regression

Memory requirements not a concern (but might try piece-wise linear regression in the

future)◦ Interpolation possible over missing/ill-conditioned

bins

Conclusion

◦ Whitened data with Support Vectors Statistics for each bin still required Produces a less rigid decision boundary Better classification results Still somewhat dependent on assumption of

Gaussianaity ◦ Segmentation essentially renders classification

stage trivial◦ Segmentation makes it possible for sensors to

detect faults on physically distant machinery components

◦ Suffers doubly from the curse of dimensionality

Verification of methodology on real world machinery (diamond drill head with dyno)

Develop classifier variants for multi-modal processes which are less susceptible to the curse of dimensionality

Develop ONLINE prognostics techniques◦ When will failure occur?◦ What is the probability a machine will fail at time x?

Develop economic means of measuring torsional load for this application

Develop complete software architecture (software engineering principles) and prototype

Future Work