Structural Damage Detection Using AR-ARX Models
Conner Shane1, and Ratneshwar Jha2
Abstract
Damage can be detected in a system using natural frequencies and mode shapes as the damage sensitive
feature or by applying a time series analysis method on the response of the system to an excitation. This
study attempts to confirm results presented using mode shape analysis in a FEM simulated system by
using a procedure that uses time series analysis using autoregressive models. This method is shown to be
able to detect damage, but in this case it is unable to predict the location of that damage.
Introduction
Vibration based Structural Health Monitoring (SHM) is a field that attempts to detect damage in its initial
stage by analyzing the acceleration responses of several points in a system to a known or unknown
excitation. Damage is defined as changes to the material or geometric properties of a system that affect
that system’s performance. This damage can not be detected by visual inspection of the response,
meaning that a response plot of a damaged structure would look the same as that of a damaged structure.
In Jha et. al. [1] damage is detected by analyzing the changes in natural frequencies and mode shapes.
This was accomplished by applying the Hilbert-Huang Transform to acceleration data generated by a
Finite Element Model of a thin aluminum plate. This study attempts to confirm the results of Jha et. al.
[1] using time series analysis methods, specifically a procedure proposed by Sohn and Farrar [2].
Data
The data used in this study was generated by a FEM simulation of a thin aluminum plate (Figure 1). The
four nodes marked by a black dot represent vibration sensors and the responses from these four nodes
were recorded. The plate was excited by a 200 Hz sine wave applied at the left end of the plate. Damage
was simulated by a 50% reduction in stiffness applied to the elements marked in Fig. 1. Based on the
geometry of the plate only the signals from sensors 1 and 2 were used because the signals from sensors 3
and 4 are the same as the signals from sensors 1 and 2.
1
Class of 2005, Department of Mechanical and Aeronautical Engineering, Clarkson University
(Honors Program - Poster Presentation)
2
Assistant Professor, Department of Mechanical and Aeronautical Engineering, Clarkson University.
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Figure 1: Plate geometry with simulated damages and sensors marked.
Analysis Procedure
The procedure proposed by Sohn and Farrar [2] and used in this study uses time series analysis,
specifically auto-regressive (AR) and auto-regressive with exogenous input (ARX) models, to detect the
simulated damage. The basis for this procedure is the assumption that the prediction model used to
identify the undamaged case will not be able to sufficiently predict the damaged signals; the difference
between the models for the undamaged and damaged cases will be maximized at the location of the
damage.
The first step in the procedure is standardizing the time signals. This is accomplished as follows:
where x̂ is the standardized signal, x is the original signal,  x is the mean of the signal, and  x
is the standard deviation of the signal. ( x̂ is written as x below).
After of the signal has been standardized it is then input into a Partial Auto-Correlation (PACF) analysis
[3-4]. This shows how many past values of x correlate to the present value. Using the standardized signal
an AR model is constructed with an order p that corresponds to the results of the PACF. In this case that
order was set to 30. This model can be represented as:
Once the model has been constructed the residual error of the model,
is calculated by subtracting
the predicted data from the measured data. This residual error is then used as the input of an ARX model
represented as follows:
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The appropriate model orders (a and b) were chosen using the Schwartz-Bayesian Criterion (SBC) which
is calculated as follows:
SBC  N log( )  d log( N )
where N is the number of samples in the signal,  is the loss function of the model, and d is the number
of parameters in the model (a+b). The range of model structures tested was limited so that (a + b ≤ p).
The model structure that minimized SBC was chosen, for this case a model order of 14 and 15 was used
for sensor 1 and a model order of 18 and 7 was used for sensor 2 (Figs. 2 - 3). Once the ARX model has
been constructed the residual error
was calculated and was then used as the damage sensitive
feature. This procedure was repeated for all of the signals. The residuals from the undamaged case were
termed
while the residuals from the damaged cases were referred to as
. Damage is indicated
by the difference in the coefficients  i and  j between the undamaged models and the damaged models,
however, in order to show were the damage is located the following ratio was calculated:
 ( y )
 ( x )
Based on the assumptions of this procedure put forward by Sohn and Farrar [2] this ratio would be
maximized at the location of the damage (Figures 4 - 5).
Results and Discussions
Damage detection was accomplished using the above procedure. The large difference in coefficient
values between the undamaged and damaged cases for each sensor suggests that there is damage present
in the system. However, the ratios of standard deviations of the residuals were not sufficient to predict
the damage location in this case. Based on the assumptions put forward by Sohn and Farrar [2] and the
geometry of the system for the first damage case, the standard deviation ratio calculated as the last step in
the procedure should be greater for Sensor 1 than it is for Sensor 2 because it is closer to the damaged
elements. However, Figure 14 shows that for the first damage case Sensor 2 is actually more sensitive to
the damage than Sensor 1. For the second damage case, because the damaged elements are in between
Sensors 1 and 2 both sensors should show a similar sensitivity to the damage, Sensor 1 being a little
closer should show a slightly greater sensitivity. However Figure 15 shows that while both sensors show
a similar level of sensitivity, Sensor 2 is more sensitive.
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Measured and Predicted Output for ARX Model of Sensor 1
Undamaged Case
1
0.8
0.6
0.6
0.4
Normalized Acceleration
Normalized Acceleration
0.4
0.2
0
-0.2
0.2
0
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
-1
Measured and Predicted Output of ARX Model for Sensor 2
Undamaged Case
1
0.8
3
3.01
3.02
3.03
3.04
3.05
Time (sec)
3.06
3.07
3.08
3.09
-1
3.1
3
3.01
3.02
3.03
3.04
3.05
Time (sec)
3.06
3.07
3.08
3.09
3.1
Figures 2 & 3: Plots of Measured Data (dashed lines) and ARX Model predicted data (solid lines) for the Undamaged
Case (Left - Sensor 1, Right - Sensor 2).
1.4
Plot of Ratios of Standard Deviations of Residuals from ARX Model
Damage Case 1
1.4
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
Sensor 1
0
Sensor 2
Plot of Ratios of Standard Deviations of Residuals from ARX Model
Damage Case 2
Sensor 1
Sensor 2
Figures 4 & 5: Plots of the ratio  ( y ) for Damage Case 1 (left) and Case 2 (right).
 ( x )
Reference:
[1] Jha, R., Yan, F., Ahmadi, G. “Energy-Frequency-Time Analysis of Structural Vibrations Using
Hilbert-Huang Transform” 12th AIAA/ASME/AHS Adaptive Structures Confrence, 2004
[2] Sohn, H., Farrar, C. R., “Damage diagnosis using time series analysis of vibration signals” Smart
Materials and Structures v. 10 pg. 446-451, 2001
[3] Allen, D.W., Inmann, D.J., Farrar, C. R., “Optimization of Time Domain Models Applied to
Structural Health Monitoring.”
[4] Box, George E.P., Jenkins, Gwilym M., Reinsel Gregory C., Time Series Analysis Forecasting and
Control, 3rd Edition, Prentice Hall International, Inc., 1994
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