Structural Health Monitoring Using Linear and Non-Linear Time Domain Methods
Conner Shane, Class of 2006
Mentor: Ratneshwar Jha, Associate Professor
Department of Mechanical and Aeronautical Engineering
Clarkson University
Abstract:
Structural Health Monitoring attempts to detect and localize damage present in a
given structure. Current methods require that the location of the damage in that structure
to be known and that that location is accessible to be inspected. Using the vibration
signals from a structure, damage can be detected by applying a time series analysis
method on the response of the system to an excitation. Linear and non-linear time series
analysis methods are studied for their ability to predict damage as well as the location and
extent of that damage. The linear method makes use of auto-regressive and autoregressive with exogenous inputs (AR-ARX). The damage sensitive feature in this case
is the standard deviation of the residual error between the model and time series. The
non-linear method uses the assumption that an increasing level of damage increases the
level of non-linearity in the system. The maximum Lyapunov exponent is used to
characterize the level of non-linearity in the system. Preliminary results show that the
linear method has the ability to predict damage but is not yet refined enough to predict
the location of that damage.
1. INTRODUCTION
Structural Health Monitoring (SHM) is a field that attempts to detect damage in a
given structure. Damage is defined as changes to the material or geometric properties of
the structure that result in a loss of performance. Typical types of damage that these
methods detect range from the development of fatigue cracks to the degradation of
structural connections and bearing wear. Current SHM methods include acoustic and
ultrasonic methods, magnetic field methods, radiograph, eddy-current methods and
thermal field methods. While these methods are useful in detecting damage they require
that the location of the damage be know a priori as well as be accessible in order to test
of damage [1]. This study will explore a method known as Vibration-based Structural
Health Monitoring. This method has been shown to be able to not only predict damage
present in the structure but also the vicinity of the damage. This type of system uses the
response of the system to its natural operating environment, which means that a structure
does not need to be taken out of use to test. Uses for a SHM system that can diagnose
and localize damage in a system are far-ranging. Long-term SHM systems would
periodically output updated information on the ability of the structure to perform its
intended function [1]. These systems could be made to be part of the structure during
fabrication and could potentially be entirely automated, only notifying the user once
damage becomes present. The benefits of such systems in the aerospace, civil, and
mechanical engineering fields are clear. These systems will allow structural analysts to
design lighter, more efficient components which will lead to better performance in the
aerospace industry. Such systems will be constantly updating the lifespan of the
components which would mean that the period between costly overhauls could be
potentially extended.
Vibration-based SHM is a method 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. The basic premise of vibration-based SHM is that damage will alter
the stiffness, mass, or energy dissipation properties of the system which will result in
changes in the dynamic response of the system. Most vibration-based techniques have
focused on changes in the modal properties of the structure (natural frequencies, mode
shapes). A drawback from these approaches is that the natural excitations from ambient
sources tend to only excite the lower frequency modes that are generally insensitive to
local damage. Another drawback is that most structures experience a wide-ranging
operational environment. Varying operational and environmental conditions can produce
changes in the modal properties of the structure that can be mistakenly identified as being
caused by damage [7]. These studies also assumed that the structure can be modeled as a
linear system before and after damage is initiated. Damage is widely considered to
introduce an increasing level of non-linearity into a system which would not be captured
by a method that assumes a system can be modeled as linear. This study will focus on
two methods that do not rely on the modal properties of the system and therefore seem
more versatile in terms of SHM.
The first method is a linear method that involves time series analysis of the
acceleration of several points in the system [3]. The acceleration signals are fitted to
auto-regressive (AR) and auto-regressive with exogenous inputs (ARX) models. The
premise behind this method is that the AR and ARX model’s ability to re-create the
acceleration signal will be significantly reduced with the presence of damage in the
system. The difference between the actual and modeled signal, called the residual error,
will be maximized at the location of damage. The second method is a non-linear method
that follows the premise that damage initiates a level of non-linearity in a system. The
more a structure becomes damaged the more non-linear that structure becomes which
means that a parameter that characterizes the level of non-linearity in a system could be
used as a damage-sensitive feature. The parameter in this case is the maximum
Lyapunov exponent. The maximum Lyapunov exponent is a property of a time series
that indicates the level of chaos or non-linearity in the system. It is thought that for a
linear system (non-damaged) that the Lyapunov exponent will be negative. For a nonlinear system (damaged) the exponent will be positive and will be amplified by
increasing non-linearity [6]. Both of these methods need to be studied because of the
wide range of structures on which they can be applied. Simple structures may exhibit
highly linear behavior, for these systems the use of the linear methods is appropriate and
these methods are capable of predicting damage in those structures. However, many
more structures are much more complicated and do not behave in a linear fashion,
particularly structures made with composite materials. The use of linear time series
analysis methods on these structures will not produce favorable results as the linear
methods are unlikely to be able to capture the complete behavior of the system.
2. METHODOLOGY
Linear Time Series Analysis:
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 [3]. The first step in the procedure is
standardizing the time signals. This is accomplished as follows:
xˆ 
x  x
x
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 [4-5]. 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. This model can be represented as:
p
x(t )    xj x(t  j )  e x (t )
j 1
Once the model has been constructed the residual error of the model, ex(t), 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 (see Figures 2-9):
a
b
i 1
j 1
x(t )   i x(t  i)    j e x (t  j )   x (t )
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 [5].
Once the ARX model has been constructed the residual error ex(t) 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 ex(t) while the residuals
from the damaged cases were referred to as ey(t). Damage is indicated by the difference
in the coefficients i and i 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 this ratio would be maximized at the location
of the damage.
Non-linear Time Series Analysis:
This method focuses on systems that are inherently non-linear or chaotic. The
unpredictability of these systems is caused by what is known as sensitive dependence on
initial conditions. This means that slight deviations in the initial conditions applied to the
system are blown up after a few time steps. One concept that can be used to explain this
behavior is that nearby trajectories separate exponentially over time. The properly
averaged exponent of this increase is termed the Lyapunov exponent, 
this exponent
is defined as follows [8]:
 n   0 e n
Where 0 is the distance between two points Sn1 and Sn2 in state space, n is the distance
between the two trajectories emerging from Sn1 and Sn2 some time n ahead. This
relationship is only valid if n is much less than zero and if n is much greater than zero.
If the Lyapunov exponent is negative it indicates that there is a stable fixed point in the
system. If the exponent is equal to zero the system has a stable limit cycle. A positive
number ranging from 0 to infinity indicates that there is chaos present in the system and if
the exponent is infinite it indicates that the system consists of random noise. Applying
these conditions to the structures of interest in this study a negative exponent would
indicate that there is no damage present in the system and that it is behaving in a linear
fashion. A positive exponent indicates that a level of damage has been introduced to the
system causing it to behave non-linearly. Increasing values of the exponent would
indicate increasing damage [6].
3. PRELIMINARY RESULTS
Linear Time Series Analysis:
The data used for the linear methodology portion of 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.
Figure 1: Plate geometry with simulated damages and sensors marked.
Damage detection was accomplished using the linear time series analysis
procedure detailed in the above methodology. 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 [3] 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 20 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 21 shows that while both sensors show a similar level of
sensitivity, Sensor 2 is more sensitive.
Plot of Residuals from AR Model for Sensor 1
Undamaged Case
0.1
0.1
0.05
0.05
0
-0.05
0
-0.05
-0.1
-0.1
-0.15
-0.15
-0.2
0
1
2
3
4
5
Time (sec)
6
Plot of Residuals from AR Model for Sensor 1
Undamaged Case
0.15
Normalized Acceleration
Normalized Acceleration
0.15
7
8
9
-0.2
10
0
0.1
0.2
0.3
Time (sec)
0.4
0.5
0.6
Figures 2 & 3: Plots of the residual error from the AR model for Sensor 1 in the
undamaged case. Figure 3 (Right) shows the first second of Figure 2 (Left) more
clearly.
Plot of Residual of AR Model for Sensor 2
Undamaged Case
0.15
0.1
0.1
0.05
Normalized Acceleration
Normalized Acceleration
0.05
0
-0.05
0
-0.05
-0.1
-0.1
-0.15
-0.15
-0.2
Plot of Residual of AR Model for Sensor 2
Undamaged Case
0.15
0
1
2
3
4
5
Time (sec)
6
7
8
9
-0.2
10
0
0.1
0.2
0.3
Time (sec)
0.4
0.5
0.6
Figures 4 & 5: Plots of the residual error from the AR model for Sensor 2 in the
undamaged case. Figure 5 (Right) shows the first second of Figure 4 (Left) more
clearly.
Plot of Residual from AR Model for Sensor 1
Damage Case 1
0.1
0.1
0.05
0.05
0
-0.05
0
-0.05
-0.1
-0.1
-0.15
-0.15
-0.2
0
1
2
3
4
5
Time (sec)
6
7
Plot of Residual from AR Model for Sensor 1
Damage Case 1
0.15
Normalized Acceleration
Normalized Acceleration
0.15
8
9
10
-0.2
0
0.1
0.2
0.3
Time (sec)
0.4
0.5
0.6
Figures 6 & 7: Plots of the residual error from the AR model for Sensor 1 in Damage
Case 1. Figure 7 (Right) shows the first second of Figure 6 (Left) more clearly.
Plot of Residual from AR model for Sensor 2
Damage Case 2
0.15
0.1
0.1
0.05
Normalized Acceleration
Normalized Acceleration
0.05
0
-0.05
0
-0.05
-0.1
-0.1
-0.15
-0.15
-0.2
Plot of Residual from AR model for Sensor 2
Damage Case 2
0.15
-0.2
0
1
2
3
4
5
Time (sec)
6
7
8
9
10
0
0.1
0.2
0.3
Time (sec)
0.4
0.5
0.6
Figures 8 & 9: Plots of the residual error from the AR model for Sensor 2 in Damage
Case 1. Figure 9 (Right) shows the first second of Figure 8 (Left) more clearly.
Measured and Predicted Output for ARX Model of Sensor 1
Undamaged Case
0.8
0.8
0.6
0.6
0.4
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
3
3.01
3.02
3.03
3.04
3.05
Time (sec)
3.06
3.07
Measured and Predicted Output of ARX Model for Sensor 2
Undamaged Case
1
Normalized Acceleration
Normalized Acceleration
1
3.08
3.09
3.1
-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 14 & 15: Plots of Measured Data (dashed lines) and ARX Model predicted data (solid lines)
for the Undamaged Case (Left - Sensor 1, Right - Sensor 2).
Measured and Predicted Output from ARX Model for Sensor 1
Damage Case 1
1
Measured and Predicted Output from ARX Model for Sensor 2
Damage Case 2
1
0.8
0.8
0.6
0.6
0.4
Normalized Acceleration
Normalized Accleration
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
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 16 & 17: Plots of Measured Data (dashed lines) and ARX Model predicted data (solid lines)
for Damage Case 1, (Left - Sensor 1, Right - Sensor 2).
Measured and Predicted Output for Sensor 1
Damage Case 1
0.8
0.8
0.6
0.6
0.4
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
3
3.01
3.02
3.03
3.04
3.05
Time (sec)
3.06
3.07
Measured and Predicted Output from ARX Model for sensor 2
Damage Case 2
1
Normalized Acceleration
Normalized Acceleration
1
3.08
3.09
3.1
-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 18 & 19: Plots of Measured Data (dashed lines) and ARX Model predicted data (solid lines)
for Damage Case 2, (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
Sensor 2
0
Plot of Ratios of Standard Deviations of Residuals from ARX Model
Damage Case 2
Sensor 1
Sensor 2
Figures 20 & 21: Plots of the ratio
 ( y )
for Damage Cases 1 (Left) and Damage Case 2
 ( x )
(Right).
Non-linear Time Series Analysis:
In order to develop a more in-depth procedure to use the maximum Lyapunov
exponent as well as show that the Lyapunov exponent is sensitive to increasing levels of
damage an analysis will be performed on a simple single DOF spring-mass-damper
system. A non-linear softening spring is used to simulate non-linear damage in this
system. The governing equation for this system is as follows:
mx  cx  k ( x) * x  Fapplied
Where m is the mass of the system, c is the damping, and k(x) is defined as:
k ( x)  kx  k1 x 3
where k is the linear spring coefficient and k1 is the non-linear softening component. To
simulate damage in this system k1 was altered. A k1 value of 0 would indicate a linearundamaged system. Increasing levels of k1 would lead to increasing levels of damage
and non-linear behavior.
To generate the vibration signals needed to compute the Lyapunov exponents
MATLAB Simulink was used to model the spring-mass-damper system. The output of
this model was a time series of the position of the mass, x(t). These time series are then
input into the TISEAN software package. Future work will focus on learning how to use
the TISEAN package to calculate the Lyapunov exponents. Once these exponents can be
calculated using a reliable method then it is hoped that they will be able to predict
damage in the simulated spring-mass-damper system as well as in a test structure.
4. TIMELINE
Spring 2005 –Learn how to use the TISEAN package, specifically the lyap_k function, to
calculate the maximum Lyapunov exponent of the spring-mass-damper
data. Use results from the spring-mass-damper system to develop a
method to test more complicated systems. Complete literature search.
Summer 2005 –Test linear and non-linear methods on a more complicated beam
structure. Finish methodology section.
July 2005 – Submit abstract to 47th AIAA/ASME/ASCE/AHS/ASC Structures,
Structural Dynamics & Materials Conference.
Fall 2005/Spring 2006 – Complete all testing and data analysis, and finish writing thesis.
5. REFERENCES
[1] Farrar, Charles R., Sohn, Hoon, Fugate, Michael L., Czarnecki, Jerry J.. “Integrated
Structural Health Monitoring” SPIE’s 8th Annual International Symposium on Smart
Structures and Materials, Newport Beach, CA, March 4-8 2001.
[2] Jha, R., Yan, F., Ahmadi, G. “Energy-Frequency-Time Analysis of Structural
Vibrations Using Hilbert-Huang Transform” 12th AIAA/ASME/AHS Adaptive Structures
Conference, 2004
[3] Sohn, H., Farrar, C. R., “Damage diagnosis using time series analysis of vibration
signals” Smart Materials and Structures v. 10 pg. 446-451, 2001
[4] Allen, D.W., Inmann, D.J., Farrar, C. R., “Optimization of Time Domain Models
Applied to Structural Health Monitoring.”
[5] Box, George E.P., Jenkins, Gwilym M., Reinsel Gregory C., Time Series Analysis
Forecasting and Control, 3rd Edition, Prentice Hall International, Inc., 1994
[6] Trendafilova, I. Van Brussel, H. “Condition Monitoring of Robot Joints Using
Statistical and Nonlinear Dynamics Tools” Meccanica v. 38 n. 2 pg. 283-295, 2003.
[7] Sohn, Hoon, Worden, Keith, Farrar, Charles R. “Statistical Damage Classification
Under Changing Enviornmental and Operational Conditions” Journal of Intelligent
Material Systems and Structures v. 13, September 2004.
[8] Krantz, H. Schreiber, T.. Nonlinear Time Series Analysis, Cambridge University
Press, Cambridge, 1997.