Journal of Industrial Mathematics

Volume 2014, Article ID 543056, 10 pages

http://dx.doi.org/10.1155/2014/543056

## Cross Correlation for Condition Monitoring of Variable Load and Speed Gearboxes

Bharti School of Engineering, Laurentian University, Sudbury, ON, Canada P3E 2C6

Received 31 July 2014; Accepted 22 November 2014; Published 22 December 2014

Academic Editor: Domenico Vitulano

Copyright © 2014 Jordan McBain and Markus Timusk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The ability to identify incipient faults at an early stage in the operation of machinery has been demonstrated to provide substantial value to industry. These benefits for automated, in situ, and online monitoring of machinery, structures, and systems subject to varying operating conditions are difficult to achieve at present when they are run in operationally constrained environments that demand uninterrupted operation in this mode. This work focuses on developing a simple algorithm for this problem class; novelty detection is deployed on feature vectors generated from the cross correlation of vibration signals from sensors mounted on disparate locations in a power train. The behavior of these signals in a gearbox subject to varying load and speed is expected to remain in a commensurate state until a change in some physical aspect of the mechanical components, presumed to be indicative of gearbox failure. Cross correlation will be demonstrated to generate excellent classification results for a gearbox subject to independently changing load and speed. It eliminates the need to analyze the highly complex dynamics of this system; it generalizes well across untaught ranges of load and speed; it eliminates the need to identify and measure all predominant time-varying parameters; it is simple and computationally inexpensive.

#### 1. Introduction

The dynamics of the vibrations generated by a gearbox subject to changing load and speed are complex and nonlinear. Faults in bearings, gears, or other aspects of prime movers can easily be masked by the effects of these state changes alone when one fails to consider their effects on decision rules. The detection of faults in this class of machineries is a growing concern in the literature. In this work, we adapt a technique from sensor failure analysis to reduce this present problem’s complexity. A common approach in detecting failure in sensors employs decision rules based on the cross correlation of their signals; in broaching this technique to variable-state machinery, the authors note that vibrations at disparate locations in a power train should be correlated to one another (e.g., the spectra of vibrations from the output shaft of a gearbox are related to those of the input shaft by the gear ratio of the gearbox). Signals from disparate locations of a power train may contain similar vibration from components along the train; for instance, the load on the gearbox’s bearings is modulated by the meshing of the gear’s teeth and its vibrations or acoustics will be apparent at both the input and the output of the gearbox (and possibly at more distant locations in the train; see [1]). The cross correlation signal between these vibration signals should remain commensurate until components of the train change—a state presumed indicative of faults.

Under this hypothesis, the authors propose deploying standard novelty detection on feature vectors generated from the cross correlation signal generated between disparate vibration sensors. Past efforts by the authors focused on adapting either novelty-detection techniques or feature vectors in order to address this problem. These algorithms required the investigators to measure all predominant state parameters and to include them in the algorithm [1, 2]. While the proposed techniques were shown to work well, they suffered from various limitations. Some classification schemes work only for one changing system input parameter [1]. Others require measurement of a gearbox’s load which can be either a costly or cumbersome requirement when an inline load cell needs to be installed on a system not fitted with it. Finally, the computational complexity of others requires large processing facilities not typically available on distributed embedded systems employed in condition monitoring. The cross correlation technique should eliminate or mitigate all of these drawbacks. This approach should provide an excellent means of failure detection in systems whose dynamics are too complex for traditional approaches and consequently may extend well beyond the monitoring of variable load and speed gearboxes.

To validate these conclusions, the necessary theoretical background is first explored including a review of cross correlation and how it is presently employed in this field as well as an overview of other existing approaches for solving this class of problems. The underlying methodology is subsequently described, from a description of the employed mechanical test bench to the details of each of the steps in the classification problem. Finally, the results are demonstrated to establish the flexibility of this simple approach.

#### 2. Background

The mathematics of cross correlation is first reviewed followed by an overview of related existing techniques.

##### 2.1. Cross Correlation Analysis

Cross correlation analysis provides a signal representing the measure of the similarity between two signals as a function of time lag , defined as where denotes the cross correlation function and denotes complex conjugation; similarly, it can be expressed in discrete form It is used extensively in pattern-recognition for speech, fingerprint and face recognition, automatic target recognition, and so forth. In these applications, typically one cross correlates a reference pattern with a test pattern when the two patterns are expected to lack shift invariance. The cross correlation signal between two patterns will have a peak at the shifted value τ if they have some similarity.

##### 2.2. Cross Correlation of Systems Subject to Common Excitation

In this work, signals from disparate aspects of machinery, under common excitation, are cross correlated in order to simplify discerning the system’s health when the excitation is nonstationary. If two linear systems, with impulse response functions and , are commonly forced with some function , having equivalent frequency domain representation of , the particular solutions for the systems’ response will be the product of the forcing function and the system’s impulse response for all in ; that is, and . From elementary Laplace and Fourier transform theory, it is known that the frequency domain representation of the convolution of two signals is the product of their frequency domain representations. Cross correlation is equivalent to the convolution operation except without the folding operation; as such the frequency domain representation of the cross correlation of two signals is the product of the two signal’s frequency domain representations. Since the linear systems are forced with the same function, their output signals’ bandwidths overlap and the frequency domain representation of the cross correlation operation returns a product of the two system’s impulse response functions and the forcing function. The impulse response function for each system is determined by the system’s parameters (e.g., for a spring, the impulse response function is a function of the spring’s stiffness, the damping constant, etc.). The cross correlation of the two systems’ output therefore is a relation given by the system’s parameters. If any parameters of a system change, the cross correlation of the two systems’ output will change; it is on this basis that this work is advanced.

The vibration from a gearbox is inherently nonlinear and some of the assumptions of the foregoing therefore break down. Complex pattern-recognition techniques like novelty detection are engaged to handle these aspects.

Sampled systems are discrete in nature which was not presumed in the above analysis. The discrete systems under scrutiny herein are made discrete by sampling the continuous phenomena. The argumentation of the above is very similar in discrete form and a direct analogy can be made between the transforms of discretized form and the continuous form.

##### 2.3. Relevant Cross Correlation Techniques from the Literature

Cross correlation is used heavily in signal processing for denoising purposes. Several examples of denoising in the domain of fault detection can be found in the literature; in [3], the authors used cross correlation from two proximate vibration sources for signal-to-noise ratio improvement while [4] used cross and autocorrelation for denoising. The authors in [5] exploited the auto- and cross correlation of different variables for signal processing in developing a fault-detection technique.

Cross correlation is used in a similar vein as the present approach in the detection of failed sensors as was the case in [6] whose authors used cross correlation between two flow sensors along with neural networks to verify sensor accuracy. The work in [7] acknowledges the dynamic nature of a motor run by an adjustable speed drive and the resultant effects on monitored signals are one of the common factors that yield erroneous fault tracking and unstable fault detection; the authors employed matched filtering (i.e., cross correlation between expected fault signals and actual motor current signals) the result of which is fed through a statistical hypothesis-testing fault-detection regime. Statistical-process monitoring with spectral clustering was used to classify samples according to differences in correlation among measured variables in [8]. In [9] cross correlation of the fault-response echo in electrical-power transmission systems from test-input excitation was used to detect potentially faulted cables. Jiang et al. [10] used the correlation dimension (a type of fractal dimension) in gearbox fault diagnosis.

More directly related techniques can be found in a number of other works. For instance, Parlar employed a similar methodology to that of this thesis in the monitoring of vibrating screens in [11]. In [12] Napolitano et al. exploited cross correlation of an airplane’s pitch and yaw state variables along with neural networks for fault identification in airplane systems. Rajamani et al. found the cross correlation between healthy and faulted transformer winding signals that was used to generate statistical feature vectors for classification [13]. In [14], Wu and Sun used the cross correlation of energy performance of a variable-air-volume (VAV) unit in an HVAC system [15] and the outside temperature as the criteria to evaluate the VAV health.

Cross correlation is used heavily in this field but the methodology proposed herein on this particular class of problems does not appear to exist in the literature.

##### 2.4. Established Techniques

In the literature, there are a number of other algorithms focused on means other than correlation based fault detection for this complex class of machineries. Nonlinear principal-component analysis (NLPCA) in [16], advanced signal processing in [17, 18], adaptive filters in [19, 20], and adaptations to pattern-recognition techniques in [21–24] are all well established—each having differing strengths and weaknesses.

To provide a baseline for comparison for the approach advanced within, a comparison between a number of related techniques developed by the present authors will be undertaken. In [1] the authors explored expansions to the work by Worden et al. in [25]; Worden et al. suggested that vibration data from structures be grouped into discrete ranges of the time-changing parameters whose statistics (mean and covariance) are regressed or interpolated to develop a health rule as a function of the time-varying parameters. The work in [1] applied this approach to data from real gearbox vibrations along with an augmentation to Worden’s approach that focused on first whitening the statistical distribution so that any variant of novelty detection could be employed. Both techniques were subject to the assumption of normally distributed data and the double curse of dimensionality, a phenomenon occurring when there is a need not only to gather sufficient data to describe a complex high-dimensional problem space but also to do so for continuous changes in that problem space (e.g., in the form of changing speed or load). These initial investigations were conducted with only one time-changing parameter; in this work, two time-changing parameters are used (i.e., speed and load). While a large amount of data has been collected (nearly 20,000 feature vectors generated with ambitious segmentation), they are insufficient to accurately characterize the behavior of the gearbox with these approaches due to the double curse of dimensionality.

In an upcoming work, the present authors suggested the almost trivial approach of adding a gearbox’s average speed over a feature vector’s segment to that feature vector. The results generated with the same experimental data were found to be excellent; unfortunately, the fault-detection methodology does not extend beyond one time-varying parameter. The confusion eliminated in adding one time-varying parameter to the feature vector is again reintroduced when another time-varying parameter is added.

In a different upcoming work, the authors suggest using the parameters of a discrete state-space model as elements of the feature vector in the novelty-detection problem [26]. In a simple view, this state-space model can be regarded as the transfer function of a gearbox modeled as a torsional spring; the state-space model’s parameters are ultimately functions of the physical nature of the gear (i.e., stiffness, damping, geometric configuration, etc.). These parameters ought to be insensitive to changes in load and speed and should be highly indicative of incipient fault states. The model is generated by assuming that the gearbox’s input speed and load are the inputs to a MIMO system; the vibration signal at any point on the machine is used as the output signal and the MIMO model formed with ARMAX techniques [27]. While the vibration problem being modeled with this linear state-space approach is in reality nonlinear, the use of novelty detection to develop a boundary around a set of these linear models is shown to provide adequate adaptation to the underlying nonlinear problem. The approach was shown to eliminate the double curse of dimensionality and assumption of normally distributed data. As evidence of the model’s sound nature, the results demonstrated excellent generalization to speeds and loads not experienced during training. The only limitations to the approach are the need to collect speed and load signals (a potentially costly consideration) and the computationally intensive nature of the algorithms for generating these models.

#### 3. Experimental Configuration

This work focuses on the use of the parameters generated by cross correlating signals from sensors on disparate components of a machine. The pattern-recognition problem as advanced by [28] focuses on first collecting and conditioning signals (in this case, on a simulation test bench), segmenting them, and transforming them into -dimensional feature vectors that are ultimately fed into pattern-recognition solutions. The steps for this problem instance are described below.

##### 3.1. Apparatus

The fault-detection algorithm proposed herein was evaluated based on data collected from a gearbox under realistic load and speed as shown in Figure 1. The test bench is described in further detail in [29].