Proceedings of the IMAC-XXVII February 9-12, 2009 Orlando, Florida USA ©2009 Society for Experimental Mechanics Inc.
Using Full-Field Vibration Measurement Techniques for Damage Detection
Mark N. Helfrick1, Pawan Pingle, Christopher Niezrecki, and Peter Avitabile, Structural Dynamics and Acoustic Systems Laboratory, Department of Mechanical Engineering, University of Massachusetts Lowell, One University Ave, Lowell, MA 01854 ABSTRACT The use of structural dynamic testing for structural health monitoring has shown that changes in the operating or mode shape can be used in damage detection algorithms for detecting damage of the structure under consideration. However, in order for damage to be detected, located, and quantified, a dense set of measurement points is required on the structure of interest. Digital Image Correlation (DIC) and Scanning Laser Vibrometer (SLV) techniques provide full-field measurements with the necessary measurement resolution. Within this work, both of these methods are employed to detect defects for a simple well-behaved structure. The tests performed were able to detect a crack across the width of a cast acrylic cantilever beam whose depth was equal to 40% of the beam thickness. INTRODUCTION Several methods for damage detection and Structural Health Monitoring (SHM) which use changes in modal parameters have been proposed over the past two decades. Past techniques to identify damage use changes in frequency, MAC and COMAC values [1] and the relative difference in experimental mode shapes [2]. Comparisons between a damaged and an undamaged condition using these methods have consistently detected only very severe damage. More recently, SHM research using vibration techniques has been focusing on mode shape derivatives as indications of damage. Three of these approaches are: • • •
Curvature Difference Method (CDM) which looks at differences in curvature (second derivative of experimental mode shape) and compares it to an undamaged state [3, 4] Gapped Smoothing Method (GSM) which looks at differences in experimental mode shapes or curvature and compares it to a polynomial approximation of that shape [5] Higher Order Derivative Discontinuity (HODD) approach which looks at peaks in the fourth derivative of the mode shape [6]
Analytical studies [7] have consistently shown that modal curvature of a damaged structure can point to the existence and location of the damage. Even though the curvature methods of damage detection appear promising, the transition from analytical to empirical is marred by sparse data sets due to traditional modal testing techniques and the noise in measurements. Curve-fitting and interpolation with a sparse set of data points will allow for the calculation of derivatives of mode shapes with sufficient resolution; however, most interpolation schemes will tend to smooth over any damaged areas. Two modern full-field measurement techniques which provide the necessary measurement resolution for damage detection are identified as possible tools for SHM in this paper. Three-dimensional (3D) Digital Image Correlation (DIC) has been proven to be able to measure the visible surface displacement of a vibrating structure with a measurement resolution never before possible [8] and hence DIC is proposed to be an appropriate experimental approach to SHM through vibration measurement. A Scanning Laser Vibrometer 1
[email protected]
(SLV) is capable of measuring out-of-plane velocity at any visible point on the surface of a structure. A SLV is also proposed to provide the measurement resolution necessary for damage detection. While DIC has never been used before with curvature damage detection methods, the SLV method has been investigated with cuts on a free-free steel beam [9] and for detecting delamination on a composite plate [10]. The ability of DIC and SLV methods to detect damage severity and damage location in a cantilever beam will be assessed in this paper using the GSM approach which has broader applications since it does not require a measurement of the test structure prior to damage. Experimental noise is expected from both measurement techniques and will be minimized through averaging and smoothing algorithms. DAMAGE DETECTION USING MODE SHAPE CURVATURE Curvature methods of damage detection are based on the basic characteristics of beam-like structures. The bending stiffness EI can be obtained by dividing the internal bending moment M by the corresponding curvature which is the second derivative of the displaced shape φ as shown by the flexure equation:
M
EI = dφ
(1)
2
dx
2
A local anomaly in the stiffness will be apparent as a local anomaly in the curvature. The curvature φ''i at the
i th measurement point can be calculated from the measured displacement φ i using the central difference approximation formula:
φ''i = (φ i +1 − 2φ i + φ i −1 ) h 2
(2)
where h is the uniform distance between the measurement points. Local peaks and valleys exist in the curvature at locations with sudden differences in stiffness. In an ideal situation, a measurement would have been made of the test structure while that structure was in an undamaged state. A direct comparison could be made between the test curvature and the curvature of the healthy structure. This method is referred to as the Curvature Difference Method (CDM). Unfortunately, a measurement of a healthy structure most likely has not been made (nor can the original state ever be assumed to be healthy). In the Gapped Smoothing Method (GSM) [5], the undamaged shape of a structure of constant stiffness can be approximated by a smooth polynomial curve. An algorithm can be created which compares the curvature to a best-fit polynomial of the curvature. Points containing the greatest difference between the calculated curvature and the best-fit polynomial curve are likely candidates for the location of damage. GSM fits a third order polynomial pi to the data points surrounding one curvature point. For example, the i th test point would be evaluated by fitting a third order curve to data points φ''i− 2 , φ''i−1 , φ''i+1 , and φ''i+ 2 . The point φ''i would be left out, or “gapped” over. The damage index value for a discrete point Di using GSM is evaluated as:
(
Di = p i − φ'i'
)
2
(3)
Depending on the size and severity of the damage, as well as the spatial density of the data set, variations on this approach may need to be implemented in order to achieve better results. For example, if three adjacent data points are located on the damage site, the gapped polynomial curve would reflect the effects of damage on the curvature and not a healthy structure. For this example, gapping over three points at a time would produce better results. An alternative approach which is easily adopted by any dense data set is to curve-fit the entire curvature calculation with a best-fit higher-order polynomial for direct comparison. This variation of GSM is what will be applied to the experimental data sets discussed later in this paper. Furthermore, the damage index value can be altered to accommodate several displaced shape measurements of the test structure. If a structure’s shape is measured under a variety of loads or excitations (such as at multiple natural frequencies) then all of these n measurements can be included in the damage index value through:
(
Di = p1i − φ'1'i
) + (p 2
2i
− φ'2' i
) K (p 2
ni
− φ'ni'
)
2
(4)
ANALYTICAL MODEL The ability for curvature to be used as an indicator of damage can be demonstrated with the preceeding equations with a finite element model of a cantilever beam (Figure 1). A Matlab [11] script is developed to find the curvature of a cast acrylic cantilever beam with 50 elements excited near its base by a sinusoidal forcing function. Damage is modeled as a thickness reduction of elements at the damage location thus changing the mass and stiffness properties of the damaged elements. The model contained the following properties:
length = 22 in
width = 4 in
thickness = 0.236 in
ρ = 0.043
lb in
3
E = 460000
lb in 2
x thickness reduction
electromechanical shaker
Figure 1: Graphical depiction of the finite element model with 50 elements showing boundary conditions and locations of excitation and damage.
For the cantilever beam studied, the curvature of the response was calculated for an excitation at the beam’s second natural frequency corresponding closely to its second bending mode. The GSM technique is rd demonstrated in Figure 2 using a 3 order polynomial gapped over 3 points at a time (the test location and one point on either side). The resulting damage index values indicate the location of damage for both a 10% and a 20% reduction in thickness. A similar result is seen with a direct comparison between the curvature and a higher-order best-fit polynomial curve as shown in Figure 3. This latter method is less computationally intensive while still being able to smooth over the peaks in the curvature caused by damage and will work with any data set containing a number of points significantly greater than the order of the best-fit polynomial. For these reasons, a higher-order best-fit polynomial fit to each data set will be used in all future investigations. The analytical results demonstrated thus far demonstrate the significance of curvature in damage detection schemes. However, it would be remiss to stop the analytical study here since pure displaced shapes do not take into account the noise which plagues experimental measurements. The noise floor in a DIC measurement in the out-of-plane direction is equal to approximately 1/50,000 of the width of the field of view when using a stereo pair of cameras each containing a pixel array of 1200 x 1600 pixels. This level of noise was introduced into the analytical model. The effect was such that damage could no longer be detected without averaging and smoothing the data. By including averaging and a moving average filter on the displacement data, the minimum amount of damage possible to detect with this structural configuration was a 40% reduction in thickness (see Figure 4). Filtering does remove some of the effects of noise but it also tends to smear the data. A level of filtering is desired which removes effects of noise yet does not remove the local effects of damage on the measurements.
a)
b)
-3
1.5
x 10
-3
x 10
2
curvature
curvature 1.5
poly-fit curve
1
1 curvature
0.5 curvature
poly-fit curve
0
0.5 0
-0.5 -0.5
-1
-1.5
-1
0
5 -3
20
0
1
5
10 15 distance along beam (in)
0
5
10 15 distance along beam (in)
20
25
5
10 15 distance along beam (in)
20
25
-3
x 10
0.5
0
-1.5
25
damage index
damage index
1
10 15 distance along beam (in)
20
x 10
0.5
0
25
0
Figure 2: Curvature of a finite element model of a damaged cantilever beam excited only at its second natural frequency with a damage index calculation using the GSM method for (a) 10% reduction in thickness; (b) 20% reduction in thickness.
a)
x 10
b)
-3
1.5
x 10
-3
2 curv ature
curv ature
poly -f it curv e
1
poly -f it curv e
1.5
1
curvature
curvature
0.5
0
0.5
0
-0.5 -0.5 -1
-1
-1.5 0 x 10
5
10 15 distance along beam (in)
20
-1.5
25
0
-3
x 10
10 15 distance along beam (in)
20
25
5
10 15 distance along beam (in)
20
25
1 damage index
damage index
1
5 -3
0.5
0
0.5
0 0
5
10 15 distance along beam (in)
20
25
0
Figure 3: Curvature of a finite element model of a damaged cantilever beam excited only at its second natural frequency with a damage index calculation using a higher-order best-fit polynomial curve for (a) 10% reduction in thickness; (b) 20% reduction in thickness.
b)
x 10
1
0
0
1
0
curvature
curvature
poly-fit curve
poly-fit curve -1
5 10 15 20 distance along beam (in) -3
0.5
0
0
5 10 15 20 distance along beam (in)
0
-3
2
1
0 curvature poly-fit curve -1
5 10 15 20 distance along beam (in)
0
0
5 10 15 20 distance along beam (in)
0
5 10 15 20 distance along beam (in) -3
x 10
0.5
x 10
1
-3
x 10
damage index
damage index
1
c)
x 10
1 damage index
-1
-3
2
curvature
-3
2
curvature
curvature
a)
x 10
0.5
0
0
5 10 15 20 distance along beam (in)
Figure 4: Curvature of a finite element model of a damaged cantilever beam excited only at its second natural frequency with added noise with a damage index calculation using a best fit higher-order polynomial curve for (a) no damage; (b) 20% reduction in thickness; (c) 40% reduction in thickness.
Although not described in detail here, it was determined that besides experimental noise and damage severity, a structure’s geometries, material properties, damage location, boundary conditions, and properties of the excitation signal all had a significant effect on how well the damage could be detected. For this reason, it is strongly suggested by the authors that any attempt to use curvature methods of damage detection is prefaced by an analytical study in order to understand the experimental conditions which would make for a successful test. EXPERIMENTAL DEMONSTRATION USING DIC A cast acrylic cantilever beam was designed with the same dimensions and properties as those used in the analytical model as shown in Figure 5. The beam was rigidly clamped to a solid test bench to simulate a built-in condition. Digital Image Correlation requires a speckle pattern in order for it to track surface motion. A speckle pattern was applied to the front surface of the beam with paint. The pair of cameras were set atop a tripod and placed in front of the setup (Figure 6). Aramis [12] correlation software was used for acquisition and processing. The cameras were set to trigger at the peak of the excited shape of the beam. Damage was accomplished through a cut across the width of the structure using a blade with a thickness of 0.125 in. Three damage conditions were tested: no damage, 17% reduction in thickness and 40% reduction in thickness as outlined in Table 1. For each damage condition, the structure was excited by means of an electro-mechanical shaker at each of its first three natural frequencies corresponding to the first three bending modes of the structure. There were twenty image pairs taken for each damage condition and excitation frequency combination for averaging purposes.
4”
damage location
22” 24”
13.125” shaker attachment point 1.5” clamped area
Figure 5: Geometries and dimensions describing the experimental setup of the cast acrylic beam.
a)
b)
c)
Figure 6: Photographs of the setup including (a) the view from the right camera; (b) the setup of the beam, shaker and amplifier; (c) cameras position atop a tripod in front of beam.
Table 1: Quantification of the Inflicted Damage Beam thickness at damage location (in) UNDAMAGED 0.236 DAMAGED 1 0.196 DAMAGED 2 0.1405 Test
Percent reduction 16.95% 40.47%
While data was collected across the entire visible surface of the beam, only data on a section line along the centerline of the beam was exported for analysis. This data sample contained 400 points. The data was brought into Matlab [11] for averaging, smoothing, differentiating, and comparing to a polynomial curve. An example of the measured shape along the centerline of the structure is shown in Figure 7a. Averaging and smoothing cleans up the measurements as seen in Figure 7b.
b) 0.04
0.04
0.03
0.03
0.02
displacement (in)
displacement (in)
a)
5.47 Hz
0.01 0 -0.01
37.11 Hz
109.13 Hz
5 10 15 distance along beam (in)
0.02
5.47 Hz
0.01 37.11 Hz
109.13 Hz
0
20
-0.01
5 10 15 distance along beam (in)
20
Figure 7: Measurements of the excited shape of the undamaged structure (a) before and (b) after averaging/smoothing.
Curvature is calculated for each damage condition and for each excitation frequency. The curvatures are compared to a higher-order best-fit polynomial curve (top nine plots in Figure 8). The damage index for this experiment only gave indications of the existence of damage when the beam thickness was reduced 40% at the damage location (bottom three plots in Figure 8). At 40% reduction in thickness, the presence of damage and its location can be correctly determined to be at 13.125 in from the fixed end. At 17% thickness reduction, the damage index plot contains mostly the same degree fluctuations as the undamaged condition. A few lesser peaks do appear at non-damaged locations in the damage index plots making the damage index plots difficult to interpret. Note also that the damage index plot incorrectly places large values at the edges of the measured area. This edge noise is attributed to the smoothing algorithm which performs poorly at the edges as expected. The curvatures for the case of 40% reduction in thickness (right-most plots in Figure 8) illustrate the need for multiple excitation frequencies since only the excitations at second and third natural frequencies were able to definitively detect damage. The effect of damage on the curvatures is dependent on the loading condition. Multiple loading conditions increase the likelihood of damage detection.
Curvatures and Poly-fit Curves from DIC Measurements no damage x 10
x 10 5.47 Hz
5.47 Hz
x 10
x 10 37.115Hz
-5
17% reduction in thickness
10
15
20 37.11 Hz
x 10 109.13 Hz
108.66 Hz
-5
x 10
40% reduction in thickness
5.47 Hz
x 10 35.555Hz
10
15
20
10
15
20
10
15
20
x 10 5 Hz 106.25
5
Damage Index Values for each Damagex 10 Condition no damage
5 10 15 20 distance along beam (in)
17% reduction in thickness
5 10 15 20 distance along beam (in)
40% reduction in thickness
5 10 15 20 distance along beam (in)
Figure 8: Curvature calculations (solid line) and polynomial best-fit curves (dottedline) from DIC measurements along with the damage index for the three tested damage conditions as shown by the three columns of plots with a vertical dashed line indicating the location of inflicted damage.
EXPERIMENTAL DEMONSTRATION USING A SCANNING LASER VIBROMETER The test using a Polytec Scanning Laser Vibrometer (SLV) [13] was done simultaneously with the DIC experiment, thus the same setup as used in the DIC testing was used with the SLV measurements (Figure 9). A reflective tape was placed along the centerline of the beam for optimal laser signal. A broadband excitation technique using a burst chirp was used to excite a range of frequencies including the frequencies associated with the first four bending modes. Surface velocity was measured sequentially at 400 points along the centerline of the beam. Ten to fifteen averages were taken at each point causing the test to take a total of 3 to 4 hours. The real and imaginary parts of displacement frequency response functions (FRFs) were exported into Matlab [11] for processing. Shape data was extracted from the real and imaginary parts of the FRFs at each of the first four natural frequencies corresponding to the first four bending modes. The results from the first natural frequency (5.47 Hz) appeared extremely noisy and were not used in the damage index calculation. This noise is likely due to the large displacements at this lower natural frequency causing the laser measurement point to move around. A damage index was created using the real and imaginary shapes at the second through fourth natural frequencies. The same algorithms used in the DIC tests were used with the SLV results.
Figure 9: Scanning laser vibrometer scanning head is set atop a tripod directly behind the DIC cameras. A reflective tape is placed on the cantilever beam to increase the laser signal.
Like the DIC results, only the 40% reduction in thickness resulted in a profound peak in the damage index calculation at the location of the damage. The scenario with 17% reduction in thickness did not show any increase in the damage index value at the damage location. In the 40% reduction in thickness test, curvature variations at the location of damage occurred in both the real and imaginary parts of the FRF. These results are illustrated in Figure 10.
Curvatures and Poly-fit Curves from SLV Measurements 17% reduction in thickness
no damage 37.11 Hz, real
40% reduction in thickness
37.11 Hz, real
37.11 Hz, real
37.11 Hz, imaginary
37.11 Hz, imaginary
109.13 Hz, real
109.13 Hz, real
109.13 Hz, real
109.13 Hz, imaginary
0109.135 Hz, imaginary 10 15
037.11 5Hz, imaginary 10 15
20
20
0 5 10 214.45 Hz, real
214.45 Hz, real
214.45 Hz, real
214.45 Hz, imaginary
0214.455 Hz, imaginary 10 15
109.13 Hz, imaginary
20
15
20
214.45 Hz, imaginary
Damage Index Values for each Damage Condition no damage
5 10 15 20 5 10 15 20 distance along beam (in)
17% reduction in thickness
5 10 15 20 5 10 15 20 distance along beam (in)
40% reduction in thickness
5 5 1010 1515 2020 distance along beam (in)
Figure 10: Curvature calculations (solid line) and polynomial best-fit curves (dotted line) from SLV measurements along with the damage index for the three tested damage conditions as shown by the three columns of plots with a vertical dashed line indicating the location of inflicted damage.
OBSERVATIONS AND THOUGHTS FOR FUTURE WORK Changes in curvature from full-field measurement techniques can be used to detect damage if the damage is of a significant level. These tests were able to detect a crack across the width of a cast acrylic cantilever beam whose depth was equal to 40% of the beam thickness. This level of damage is severe. These tests did not account for the variety of structure materials, boundary conditions, loading conditions, and damage types which exist. Future tests must account for some of these variations and determine which structures (if any) this technique could be applied to and what level of damage could be detected. Future developments in full-field measurement technology will also improve the prospects of this technique. CONCLUSIONS From the analytical models, it was shown that local damage causes a local variation in the curvature of a dynamically excited structure. This variation can be detected in a damage index algorithm which compares the curvature to a best-fit polynomial curve. Measurement noise and density of data has a profound affect on how well the damage index is able to locate damage. DIC and SLV are two methods which provide the necessary data density and each were able to correctly identify the location of a crack along the width of a cast acrylic cantilever beam when the crack depth was equal to 40% of the structure’s thickness. These experiments performed validate the theory. More work is needed to translate these approaches into field tests where a variety of damage conditions and structures exist. REFERENCES 1
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3
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4
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5
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6
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Whalen, T.M., “The Behavior of Higher Order Mode Shape Derivatives in Damaged, Beam-like Structures”, Journal of Sound and Vibration, Vol. 309, January 2008, pp. 426 – 464.
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Helfrick, M.N., Niezrecki, C., Avitabile, P., Schmidt, T., “3D Digital Image Correlation Methods for Full-Field Vibration Measurement”, Proceedings from the 26th International Modal Analysis Conference, Orlando, Florida, February 2008.
9
Ratcliffe, C., “A Frequency and Curvature Based Experimental Method for Locating Damage in Structures”, Journal of Vibration and Acoustics, Vol. 122, July 2000, pp. 324-329.
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