CHARACTERISTICS OF OPERATIONAL DISPLACEMENT AND CURVATURE
SHAPES AND THEIR EFFECT ON CURVATURE BASED DAMAGE
DETECTION ALGORITHMS
Colin P. Ratcliffe
United States Naval Academy
Annapolis, MD 21402
Roger M. Crane
Naval Surface Warfare Center Carderock Division
West Bethesda, MD 20817
ABSTRACT
Structural damage often results in a localized stiffness change, which affects the dynamic characteristics of the structure. This
often causes little change to displacement-based mode shapes, and therefore damage detection methods based on the
analysis of mode shapes are not sensitive. An alternative is to inspect curvature mode shapes. Curvature mode shapes can be
obtained by spatially differentiating displacement mode shapes. A localized stiffness change causes a feature in an otherwise
smooth curvature mode shape, and there is a variety of published methods that look for this feature in order to locate damage.
An extension which has been shown to increase sensitivity is to use the entire broadband data sets and to use Operational
Curvature Shapes, OCS, for damage identification. In this case the data are used irrespective of whether the structure is at
resonance or not. Mode shapes are solely dependent on the structure, and are independent of excitation. Conversely,
operating shapes depend on both the structure and the location of excitation or measurement. Whereas displacement and
curvature mode shapes are both continuous and continuously differentiable, the work reported here shows that even for well
behaved, simple structures, operational curvature shapes are continuous, but they are not always continuously differentiable.
The consequence is that algorithms that look for “unusual” features in operational (off resonant) curvature shapes may
incorrectly identify these theoretically expected features as damage. The work presented in this paper analytically inspects the
theoretical behavior of a beam with particular regard to the development of frequency and excitation position dependant
displacement and curvature operational shapes. The theoretical effect of a non-continuously differentiable curvature shape on
the performance of a curvature-based damage detection algorithm is presented. Experimental results from a full sized
composite structure show that the features generated by the theoretical issues presented in this paper are often more
significant than the features generated by real damage. A solution to this problem is presented.
Introduction
Structural damage often results in a localized stiffness change, which affects the dynamic characteristics of the structure. It is
well documented that structural damage causes little change to displacement-based mode shapes, Yuen [1]. Therefore,
experimental damage detection methods solely based on the analysis of experimentally determined mode shapes, e.g. Liang
et al. [2] and Chang et al. [3], often are not sensitive enough to identify damage locations. This is especially the case when
damage is located near nodes of displacement. The sensitivity of these procedures is also adversely affected by the errors
typically introduced during modal analysis curve fitting. An alternative approach is to inspect curvature mode shapes.
Curvature mode shapes can be obtained by spatially differentiating experimentally obtained displacement mode shapes.
Curvature and strain are intimately related, and therefore theoretical issues with curvature and strain based methods are often
associated. A localized stiffness change causes a feature in an otherwise smooth curvature mode shape, and there is a variety
of published methods that look for this feature in order to locate damage, e.g., Pandey et al. [4] and Ratcliffe [5]. An extension
to the modal curvature methods is to omit the need for the modal analysis curve fitting, and to use the entire broadband data
sets. Thus, the ‘raw’ data for these methods consists of broadband, frequency-dependant Operational Displacement Shapes
(ODS) or Operational Curvature Shapes (OCS). Frequency dependant ODS are commonly used for visualization of the
vibration pattern of a structure under given operating conditions. They are not, though, commonly used for damage
identification. There is an increasing interest in using broadband OCS for damage identification. In this category some
methods use data obtained at a single frequency which may be at or away from resonance, and some methods use data from
an entire broadband spectrum, Ratcliffe [6]. In this latter case the data are used irrespective of whether the structure is at
resonance or not.
Mode shapes are solely dependent on the structure, and are independent of excitation. Conversely, both ODS and OCS
depend on both the structure and the location of excitation or measurement. Whereas displacement and curvature mode
shapes are both continuous and continuously differentiable, the work reported here shows that even for well behaved, simple
structures, operational curvature shapes are continuous, but they are not always continuously differentiable. The consequence
is that algorithms that look for “unusual” features in operational curvature shapes may incorrectly identify these theoretically
expected features as damage. The work presented in this paper inspects the theoretical behavior of a beam with particular
regard to the development of frequency and excitation position dependant ODS and OCS. The analytical solution can also be
used to demonstrate the effects of discontinuities such as thickness or material changes. The theoretical effect of a curvature
shape that is not continuously differentiable on the performance of a curvature-based damage detection algorithm is
presented. Experimental results from a full sized composite structure show that the features generated by the theoretical
issues presented in this paper are often more significant than the features generated by real damage. A real-world solution to
this problem is presented and discussed.
Operating Shapes by Modal Superposition
Many displacement ODS studies calculate a structure’s displacement deflection shape using modal superposition. This is
because the eigenstructure consisting of natural frequencies and mode shapes is relatively easy to determine and superpose
in numerical studies. Often this eigenstructure will be determined from a finite element model of the structure. Typically in
numerical studies the superposition will converge after relatively few terms, perhaps 50 to 100 depending upon the frequency
of interest and accuracy requirements. The OCS is typically obtained by spatially differentiating the ODS, commonly with a
difference function approach. While the modal superposition approach has many uses and benefits, it is unfortunate that the
total procedure associated with obtaining an OCS can lead to analytical problems. For example:
The determined eigenstructure is subject to the tolerance of the analysis. Even though modal superposition may have
converged sufficiently for ODS analysis, there is still insufficient accuracy for OCS analysis. For example, in an unpublished
preliminary study the authors investigated a simply supported beam problem that converged for ODS analysis with about 50
modes. However, superposition of 1000 modes was required before sufficient accuracy was obtained for OCS analysis.
Using a finite element approach limits the analysis to mesh nodes, thus the mesh has to be damage-case specific.
Consequently, a new finite element model must be generated and analyzed for every structural variation that is to be
investigated. Differences between models can obscure and confuse OCS analysis results.
The above problems can be avoided if the OCS is developed as a continuous analytical solution.
Operating Shapes from an Analytical Model
For the work presented here it is beneficial to derive the flexural motion of a beam from first principles. The beam under
investigation is established as a series of uniform, orthotropic Euler-Bernoulli beam segments. There is no restriction on the
sectional properties of each segment. Thus, localized damage may be modeled as a short segment with a reduced flexural
stiffness. Transducer loading effects could be investigated by having a transducer-sized beam segment with appropriate mass
and flexural stiffness. The model can have harmonic force excitation applied at any or all boundaries between beam segments.
There is also no restriction on the material and geometric properties on either side of the force point. Excitation can be applied
to a uniform beam by setting the properties of the adjacent segments to be the same. It should be emphasized that, unlike
finite element methods, the number of beam segments required for the analysis is determined by the number of discontinuities
that need to be modeled. For example, if the problem is to analyze a uniform beam with single point force excitation, the entire
process only requires two beam segments connected at the excitation point. Conversely, a localized stiffness change can be
modeled by introducing a very short beam segment with reduced properties. The analytical model determined by the following
procedure will generate the beam response as continuous functions, and can thus be used to determine the response at any
point on the beam, not just at segment boundaries.
There is a separate coordinate system for each of the n+1 beam segments, as shown in Fig. 1. Each segment, numbered from
0 to n, has to be uniform. However, as described above, adjacent segments may have the same or different properties. The
i-th beam segment has length Li and its lateral motion yi is measured at distance xi from the left end of the segment.
y0
y1
yn
x0
x1
xn
P.sin(ωt)
LT
Figure 1: Coordinate system
The general displacement (ODS) and curvature (OCS) functions for each beam segment can be shown in the form of equation
(1), where Ai, Bi, Ci and Di are arbitrary constants to be determined from the boundary conditions and continuity constraints.
y i ( xi , ki , t ) =  Ai cos ( k i xi ) + Bi sin ( k i x i ) + Ci cosh ( k i xi ) + Di sinh ( k i xi )  sin (ω t )

1/ 4
2

 ρ A ω 
wave number k i =  i i 
(1)

∂ y i ( xi , ki , t )
OCS :
= k i2  − Ai cos ( k i xi ) − Bi sin ( k i xi ) + Ci cosh ( k i xi ) + Di sinh ( k i xi )  sin (ω t ) 
 Ei Ii 
2
∂xi

Note that the wave numbers, ki, are continuous functions of frequency, unlike eigen-analysis, where the wave numbers are
discrete numbers associated with each eigenvector. The frequency dependence of the ODS and OCS functions is thus
established. The following restrictions apply:
ODS:
2
0≤i ≤n
n
LT = ∑ Li
∀i : 0 ≤ xi ≤ Li
i =0
While the analytical process can be applied to a beam with any boundary conditions, the specific example used for this study
is a simply supported beam with boundary conditions of zero displacement and zero bending moment at each end. Applying
these boundary conditions to the left end of the beam, where x0 = 0 , establishes equation (2).
A0 = 0
C0 = 0
(2)
We now turn to the connectivity between each of the beam segments, and establish equations of continuity between the
segments. The connection between the (i-1)-th and i-th beam segments, with 1 ≤ i ≤ n , is defined by xi −1 = Li −1 and xi = 0 .
Displacement continuity at this location gives:
y i −1 ( xi −1, k i −1, t ) = y i ( xi , ki , t )
(3)
Ai −1 cos ( k i −1Li −1 ) + Bi −1 sin ( k i −1Li −1 ) + Ci −1 cosh ( k i −1Li −1 ) + Di −1 sinh ( ki −1Li −1 ) = Ai + Ci
Slope continuity at this location gives:
∂y i −1 ( xi −1, k i −1, t )
∂xi −1
=
∂y i ( xi , k i , t )
∂xi
k i −1 {− Ai −1 sin ( k i −1Li −1 ) + Bi −1 cos ( k i −1Li −1 ) + Ci −1 sinh ( k i −1Li −1 ) + Di −1 cosh ( k i −1Li −1 )} = k i {Bi + Di }
Bending moment continuity at this location gives:
Ei −1Ii −1
∂ 2 y i −1 ( xi −1, k i −1, t )
∂x i −1
2
= E i Ii
(4)
∂ 2 y i ( xi , k i , t )
∂xi2
Ei −1Ii −1k i2−1 {− Ai −1 cos ( ki −1Li −1 ) − Bi −1 sin ( ki −1Li −1 ) + Ci −1 cosh ( ki −1Li −1 ) + Di −1 sinh ( ki −1Li −1 )}
(5)
= Ei Ii k i {− Ai + Ci }
2
Shear force is continuous, except at segment connections where there is an externally applied force. In which case:
∂ 3 y i −1 ( xi −1, k i −1, t )
∂ 3 y i ( xi , k i , t )
Ei −1Ii −1
= Ei Ii
+ Pi
3
∂xi −1
∂xi3
Ei −1Ii −1k i3−1 { Ai −1 sin ( k i −1Li −1 ) − Bi −1 cos ( k i −1Li −1 ) + Ci −1 sinh ( k i −1Li −1 ) + Di −1 cosh ( k i −1Li −1 )}
(6)
= Ei Ii k i {Bi + Di } + Pi
3
Finally, the displacement and bending moment boundary conditions for the right end of the beam, where xn = Ln, yield:
An cos ( k n Ln ) + Bn sin ( k n Ln ) + Cn cosh ( k n Ln ) + Dn sinh ( k nLn ) = 0
(7 a, b)
k n2 {− An cos ( k nLn ) − Bn sin ( k n Ln ) + Cn cosh ( k nLn ) + Dn sinh ( k n Ln )} = 0
There are ( n + 1) beam segments, with four unknown constants per segment. Thus the 4 + 4n equations (2-7) can be solved to
determine the unknown constants Ai , Bi ,C i , Di with 0 ≤ i ≤ n , generating ( n + 1) Equations (1) that, in combination, yield the
beam’s ODS and OCS.
EXAMPLES
The numerical examples presented here consider a beam with a nominal length of 1 meter, comprised of a number of different
length segments. Whereas each segment is, in itself, uniform along its length, the geometric and material properties of
different segments are arbitrary. Testing of a large number of different scenarios reveals consistent trends that are
demonstrated by the results presented in this paper. For all the examples presented here the excitation was applied at a
position 0.3-m from the left end of the beam. This ensured the excitation was not at a node point for any of the lower natural
frequencies. Similar results pertain for other excitation points. Details of the beam model are shown in Table 1.
Table 1: Details of the beam model
Property
Segments
Dimensions
Length 1-meter, uniform
thickness of 0.007-meter,
uniform width of 0.025-m
Material
Modulus 210 × 109 Pa
properties
Density 7843kg / m 3
Excitation
Damage
Notes
The basis model is a rectangular cross section prismatic bar, with simple
boundary conditions at each end. The width is arbitrary.
Steel is assumed, but not necessary.
Harmonic, 0.3-m from the left end
For damaged models, the thickness was reduced to 0.00665-m (5%
reduction) for the section 0.59 to 0.61-m from the left end
ODS
ODS
Undamaged models. The undamaged uniform beam model comprises two beam segments which, other than their lengths,
have identical properties. The segments are joined at a position 0.3-m from the left end, which is the point where harmonic
excitation is applied. Figures 2 and 3 show the displacement and curvature shapes when the excitation is very close to the 1st
and 4th natural frequencies, 103.2 rad/s and 1651.2 rad/s for this model respectively. Analysis exactly at a natural frequency is
not possible because the model does not include damping or energy dissipation terms. The figures show that the procedure
correctly determines mode shapes. Also, the curvature shapes at resonance are smooth and continuous. In the figures the
point of excitation, which is coincident with the boundary between the 2 segments, is marked with an arrow.
0
0.2
0.4
0.6
0
0.2
0.4
0.4
0.6
0.8
Position
0.6
0.8
OCS
OCS
0.2
0.8
0.8
Position
0
0.6
Figure 2: Displacement and curvature mode
shapes – Mode #1
0
0.2
0.4
Figure 3: Displacement and curvature mode
shapes – Mode #1
ODS
ODS
If the excitation frequency is away from resonance, the ODS remains continuous and continuously differentiable (smooth).
However, while the OCS is continuous, it is not smooth at the point of excitation. This aspect is demonstrated in Figures 4 and
5 where the excitation frequency is close to, but away from resonance. The figures show there is a discontinuity in slope of the
OCS at the point of excitation. The small change in excitation frequency has caused a significant change to the curvature
shape. While displacement and curvature are both continuous at the excitation point, the curvature is not continuously
differentiable at this point.
0
0.2
0.4
0.6
0
0.2
0.4
0.4
0.6
0.8
Position
0.6
0.8
Figure 4: Uniform beam excited at 110%
of the 1st natural frequency
OCS
OCS
0.2
0.8
0.8
Position
0
0.6
0
0.2
0.4
Figure 5: Uniform beam excited at 90%
of the 4th natural frequency
ODS
ODS
When excitation is significantly far from resonance, the ODS remains continuous and smooth. Visually, it can be difficult to
identify on the displacement shape where the excitation is applied. However, the OCS develops a very strong feature, showing
it is not continuously differentiable. Examples are shown for two different excitation frequencies in Figs. 6 and 7.
0
0.2
0.4
0.6
0.8
0
0.2
0.4
0
0.2
0.4
0.8
0.6
0.8
Position
0.6
OCS
OCS
Position
0.6
0.8
Figure 6: Excitation at 300 rad/s
0
0.2
0.4
Figure 7: Excitation at 1200 rad/s
The reason the curvature is not continuously differentiable at the point of excitation is because the derivative of the curvature
function is representative of shear force, and shear force is not continuous at the location of application of a point force. The
special case of resonance is different because classic vibration theory shows that an undamped structure can vibrate at
resonance without the need for any externally applied force. Thus, Pi is zero and equation (6) directs that shear force is
continuous. Therefore resonant curvature is continuously differentiable, whereas off-resonant curvature is not.
Effect of Damage and Damage Detection
This work has demonstrated that resonant curvature shapes and non-resonant curvature shapes have different mathematical
properties. Consequently, damage detection methods that operate on resonant curvature shapes (mode shapes) may not
transition directly to using nonresonant data. The concept of using curvature (or strain related) vibration data to locate damage
is that damage causes a localized reduction in stiffness. While this causes minimal change to the displacement mode shape, a
feature will appear in the curvature mode shape at the point of stiffness change, as demonstrated in Figure 8. The damage for
this figure was introduced into the model by having a beam segment near position 0.6-m that had a 5% reduction in thickness.
The length of this small ‘damaged’ segment was 1% of the overall beam length, and is shown in the figure with the ‘H’ marker.
The curvature feature in Figure 8 is typical of that caused by damage, and curvature-based damage detection algorithms
typically try to locate this feature. The concern is that when the off-resonant situation is considered, the algorithm may
incorrectly identify the expected curvature features formed at the excitation point. The algorithm chosen for demonstration in
this paper is SIDER, Ratcliffe [6] and Ratcliffe et al [7], which is a broadband damage detection method that operates on OCS.
Prior to completing the SIDER analysis, the continuous analytical functions determined above were sampled at 48 points along
the beam in order to simulate real world experiments where data can only be acquired at discrete points on a structure.
ODS
For the resonant case shown in Figure 8, the result using SIDER is shown in Figure 9. This figure has a strong identifier at the
location of damage, indicating successful damage detection and location. When SIDER is applied to the off-resonant example
in Figure 10, the result is shown in Figure 11. The algorithm has incorrectly identified the excitation point as a potential
damage site. It can be expected that curvature-based damage detection algorithms, when applied to nonresonant deflection
shapes, will identify the points of excitation as damage locations.
0
0.2
0.4
0.6
0.8
0.6
0.8
0
0.2
0.4
SIDER
OCS
Position
0
Figure 8: Mode 1 with a small amount of damage
introduced near beam position 0.6
0.2
0.4
0.6
0.8
Figure 9: SIDER output for the data in Figure 8
where the feature is coincident with damage
ODS
0
0.2
0.4
0.6
0.8
0.6
0.8
0
0.2
0.4
SIDER
OCS
Position
0
Figure 10: ODS and OCS for an excitation frequency of 300 rad/s
with damage near position 0.6
0.2
0.4
0.6
0.8
Figure 11: SIDER output for the data in Figure 10
Experimental Demonstration
An 8 x 8 feet composite panel, with a 4-inch thick foam core and two 9/32-inch glass/epoxy face sheets was vibration tested
and the results analyzed using the broadband SIDER method. During this testing the rule of reciprocity was applied; rather
than exciting at a fixed point and measuring the response across the structure, the exercise was inverted. The structure was
excited sequentially at a mesh of test points, and the response was measured at two locations. Thus, the theoretical issues
discussed above are now associated with the accelerometer locations. Unlike the SIDER results for the one-dimensional beam
shown earlier in this paper, SIDER results for a surface are shown on a contour plot. The results of the first analysis of the
panel are shown in Figure 12. There are two strong features that dominate this plot. These features are exactly coincident with
the two accelerometers that were used for this experiment. This result shows that the lack of smoothness of curvature at
reference points generated damage indicators greater than all other features for this panel.
One solution to this problem is to limit analysis to areas away from the ODS/OCS reference transducer or excitation locations.
The obvious disadvantage of having to ignore results near transducers is that procedures that analyze off-resonant structural
curvature will not be able to identify damage in these areas. However, this limitation can be overcome by installing several
transducers at different locations on the structure, thus forming an overlapping pattern of results.
The results for the composite panel analyzed using this overlapping pattern approach, and ignoring data near each transducer,
are shown in Figure 13. The strong features in Figure 12 caused by the theoretically expected lack of smoothness in the OCS
have been removed from the results, which now show the expected features near areas of structural concern.
Figure 12: SIDER finds the transducers on a square
composite panel
Figure 13: When the region around the transducer is
ignored, SIDER correctly locates areas
of structural concern
Conclusions
Several vibration based damage detection methods analyze structural curvature shapes in order to locate damage. This paper
has shown that if the analysis is conducted using resonant data, both the modal displacement and modal curvature functions
are continuous and smooth. In this case, features in curvature caused by damage can be located without ambiguity. When
analysis is conducted away from resonance, whether it be at a single frequency or across a range of frequencies, operational
displacement shapes (ODS) remain continuous and smooth. However, the curvature shapes (OCS) are continuous, but not
smooth. Since the OCS are not continuously differentiable, damage detection algorithms may incorrectly identify transducer
locations as potential damage sites.
The consequence is that nonresonant curvature based damage detection algorithms cannot be used to locate damage that is
near a transducer. The paper introduced a method whereby results from several transducers can be combined, thus providing
complete coverage of the structure under test.
Acknowledgments
This research was funded by Dr. Ignacio Perez, Office of Naval Research, Code 332.
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