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The 14th IFToMM World Congress, Taipei, Taiwan, October 25-30, 2015 DOI Number: 10.6567/IFToMM.14TH.WC.OS19.019
Buckling Profile Acquisition for Modal Analysis Based on Computer Vision
Yen-Hao Chang1 Chun-Lung Chang2 Jen-Yuan (James) Chang3
National Tsing Hua University Industrial Technology Research Institute National Tsing Hua University
Hsinchu, Taiwan Hsinchu, Taiwan Hsinchu, Taiwan
[email protected] [email protected] [email protected]
Abstract: As opposed to classical method of acquiring test data
point by point in experimental modal analysis, computer vision
method is adopted and studied in this work to offer full-field test
data at once. The proposed method is validated by capturing
geometric data of three buckled steel plates by camera, of which
data is then used to generate the plates’ vibration modes.
Comparing to the ideal geometry of the manufactured plates used
in finite element analysis, the proposed method is validated to be effective in the study of vibrations of buckled components.
Keywords: Buckling, Modal Analysis, Finite Element
Analysis, Computer Vision
1. Introduction
In common engineering practice, experimental modal
analysis is usually conducted by measuring structure
dynamic responses through accelerometers or laser
Doppler vibrometers. In order to get mode shapes of the
structure, researchers have to divide areas of the structure
into grids and acquire vibration through accelerometers at
these different locations to reconstruct the desired mode
shapes. The aforementioned procedure is apparently labor-
intensive and also time-consuming. Moreover, limited by
number of equipment available, it is impossible to get
vibration data at all grid points at once. On the other hand,
multiple times of excitation are needed to measure the
structure response at each location. A laser Doppler
vibrometer also has this kind of problem. Although its
scanning version can improve, it will never offer structure
vibration data at all points at the same time. Therefore, the
digital image correlation (DIC) method is developed to
offer an alternative in conducting the experimental modal
analysis. Different from the previous two methods, DIC
method can acquire the response of multiple locations at
the same time. By doing so, researchers do not have to
spend too much time on the experiment, and they can focus
on the analysis of the data. Not on the DIC method, the
present work is emphasized on the FEA portion by using
the model reconstructed from the shapes acquired from the
camera. Whether the reconstructed model is good enough
for the modal analysis is one of the key points in this paper.
1.1 The Elastica
In the present study, the method will be tested and
validated by using buckled elastic components, the
components having largely deformed elastic shape which
is commonly referred to elastica. The problem of elastica
is first discussed by Euler. Through the efforts of several
mathematicians and physicists, the exact shape of buckling
bars is understood. The calculation of the solution depends
on the path integral. To calculate the path integral, the
complete and incomplete elliptic integrals are needed to
fulfill the task.
1.2 The Mathematics of Elastica
The knowledge about the elastica in the paper is based
on the information provided in [1]. The diagram and
symbols of the elastica are shown in figure 1. In figure 1, 𝑙 represents the length of the bar with its Young’s modulus
being E, and moment of inertia being I. At the free end, the
angle between the bar and the vertical axis is 𝛼, and a force
P is exerted on the free end in the vertical direction. 𝑥𝛼 and
𝑦𝛼 are the vertical and horizontal coordinates of the free
end relative to the fixed end, respectively. In the derivation
of the equations of the buckling shape, s is defined to be
the distance along the axis of the bar from the free end, and
𝜃 is the angle between the bar and the vertical axis at the
position s. The moment of the free end is zero.
Fig. 1. The diagram of a buckling bar [1]
The length of the buckling bar and the coordinates of
the free end relative to the fixed end are computed by the
following equations:
𝑙 =1
𝑘∫
𝑑𝜙
√1−𝑝2 sin2 𝜙
𝜋
20
=1
𝑘𝐾(𝑝) (1)
yα =2p
k (2)
xα =2
k𝐸(𝑝) − 𝑙 (3)
In the above equation, K (p) is the complete elliptic integral
of the first kind with p being the variable, and E (p) is the
complete elliptic integral of the second kind with p being
the variable. In the above equations, p is equal to 𝑠𝑖𝑛 (𝛼
2).
In this paper, the plate’s buckling shape is assembled from
4 equal parts of the buckling shape in figure 1, because the
moment of an inflection point is also zero in the buckling
plate, and it is exactly the same situation as the free end.
Figure 2 demonstrates the result of buckling shape plotted
in Matlab program using the elastica model to simulate the
buckling shape of a fixed-fixed plate.
Fig. 2. The buckling shape with 10mm bulging
In figure 2, the units of both axes are in millimeter. The
coordinates of intermediate points along the curve can be
computed from the incomplete elliptic integrals of the first
and the second kind.
1.3 Dimension and Material Properties
The type of plate is named by the quantity of bulging of
buckling as indicated in table 1. The weight of the
accelerometer that is attached to the plates is 0.5 gram,
which is less than 0.5 percent of the plates used in the
experimental study. Therefore, the error due to the mass
loading effect of the accelerometer on the structure plate
can be ignored.
Each plate is approximately 120 mm in length, 3 mm in
thickness, and 100 mm in width. Due to manufacturing
tolerance, inevitable subtle difference from design in
dimension and weight can occur. The Young’s modulus of
the plates is 187 GPa, and the Poisson’s ratio is 0.3. In the
analysis, the density of the plates is set to be 7750 kg/m3,
which is acquired from the manufacturer’s data sheet.
Type Flat 10mm 20mm
Weight 283g 270g 254g
Table 1. The weight of each plate
Three steel plates with different buckling shapes are
juxtaposed with each other as shown in figure 3, and the
picture is taken from the top view. The difference of shapes
is easily perceived from the figure.
Fig. 3. The flat and curved plates
In figure 4, two metal blocks are stacked, and they are
the remaining parts of the curved plates after wire electric
charge machining.
Fig. 4. Remaining parts after wire electric discharge
machining
2. Methodology
In this section, the shape acquisition method will be
elaborated.
2.1 Feature
In order to calculate the shape of plates by computer
vision, addition of features on the plates is necessary for
image processing. With the known dimension of the plates,
the required feature can be easily created by computer-
aided design (CAD) software. The feature is attached to the
plates by convenient and low-cost water-soluble glue. The
feature and glue can be seen in figure 5.
Fig. 5. The dot-shaped feature and glue
Fig. 6. The feature attached to one of the curved plates
Figure 6 shows the effect of attaching feature to one of
the plates, where paper containing red dot-shaped feature
array almost perfectly covers the surface area of the plate.
The reason why red dot array is applied is because the
depth of view of the camera is not enough to cover all the
object when distance between the camera and the structure
specimen is short. With this kind of dot-shaped features,
we do not have to worry about losing the ability to identify
coordinates or the dimensions of the feature points.
Moreover, to facilitate modal analysis based on finite
element method, a regular pattern such as the dot-shaped
pattern is the best choice. On the other hand, shall ones
prefer applying other kind of irregular pattern, ones shall
use other than the present simple algorithm to reconstruct
the regular keypoints. And of course, by doing so, extra
computational time is required. Here, we just apply the
simple pattern to demonstrate the feasibility of the
proposed method.
2.2 Stereo Vision
After the features are attached, we can start taking
pictures of the specimen. The method to calculate
coordinates of the feature points is based on the stereo
vision similar to that of human eyes. When the object is
closer, the disparity goes down. Whereas when the object
is farther, the disparity will goes up. This method is the
most accurate method for calculating the distance between
the camera and the object. The equation [2] expressing the
relation between the disparity and the distance can be
written as:
𝑍 =𝑇
𝑑𝑓, 𝑑 = |𝑥𝑙 − 𝑥𝑟| (4),
where Z represents the distance, T is the translational
quantity between the two camera positions, f is the focal
length of the camera, and d is the disparity calculated by
the horizontal coordinates of the left and the right images.
2.2.1 Calibration
The lens of camera are not possible to be the perfectly
shaped to design specifications in the manufacturing.
Therefore, whenever a camera is applied to conduct
experiment, one must calibrate the images taken from the
camera so to minimize the error induced by the
manufacturing tolerances, which potentially lead to the
distortion of images. The most widely adopted method to
calibrate the images taken from a camera is through a
predefined calibration board with known dimensions of the
patterns. Figure 7 shows the calibration board which is used
in the present study.
Fig. 7. The calibration board
2.2.2 System setting
In our experiment, a linear guideway with scale as
shown in the left hand side of figure 7 is used to hold the
camera and display the position of the camera.
2.2.3 Coordinate calculation
Following the method described in literature [2], the
coordinates of the feature points are calculated based on the
captured images which are then compared between left and
right images for in the calibration process. The left and the
right images of the same 20 mm plate with features are
illustrated in figure 8, of which images, one can easily
observe distortion features due to different of lens in the
corresponding camera.
Fig. 8. The left and right images of 20 mm plate
2.3 Setting for experimental modal testing
To carry out the free-free vibration testing, two sheets
of Polyethylene (PE) foam are used to hold up the plates
and absorbed the vibration wave propagates to the
boundaries. The excitation point is located inside the red
circle as shown in figure 9, and the excitation is done by an
impact hammer. In this paper, the focus is placed on modes
which can be excited from the impact point. Furthermore,
because the plate is symmetry in both direction, we just
measure one quarter of the area of the plate. There are 13
columns and 6 rows of red dots, thus only 21 positions of
these dots needed to be measured. Figure 10 shows the
numbers of the measurement points, which cover one
quarter of the surface of the curved plates as shown in
figure 9.
Fig. 9. The setting of boundaries for the measurement
Fig. 10. The numbers of measurement points
3. Modal Analysis
The modal analysis contains results of three parts,
namely the finite element modal analysis using ideal and
computer vision measured geometry models of the curved
plates, and the experimental modal analysis using the
curved plates.
3.1 Meshing
The commercial software adopted in this study is the
most widely applied one, ANSYS. The element type of the
model is set to be Solid186 which element contains 20
nodes. For meshing setting, the ideal plate model is divided
into 3000 elements. In terms of the boundary conditions,
we set both of the boundaries free. The meshed models of
ideal flat, 10mm curved and 20 curved plates are illustrated
in figure 11 and figure 12, respectively.
Fig. 11. The meshed ideal flat plate and 10mm curved plate
Fig. 12. The meshed ideal 20mm curved plate
Fig. 13. The mesh of the real 20mm curved plate
Using the captured images as mentioned in the
previous session, the volume of the curved plate is then
reconstructed, which is further meshed with the same
element as the ideal case. As shown in figure 13, with the
proposed method, once the geometry of the structure
specimen, the 20mm curved plate as shown, to enhance
accuracy, more elements can then be put in finer mesh of
the model. In the present work, the preprocessing part of
classical finite element commercial software is now
replaced by computer vision for capturing real geometry of
the target structure specimen, Matlab algorithm for image
calibration and assigning element nodes, element type and
material properties. The ANSYS is pretty much the
calculator for executing finite element computation.
3.2 Frequency response
In this section, frequency responses of the three plates
were obtained through transfer function analyses when the
accelerometer is placed at point 14 while impact hammer
hits at other points. From the frequency responses for the
flat plate, the 10mm curved plate, the 20mm curved plate
as shown in feature 14-16, respectively. It is observed that
frequencies of low frequency modes (around and less than
2kHz) are quite close for the three plates, whereas quite
different results are found for higher frequency modes, of
which modes buckling shape of the plate may be the major
contributor for the frequency shifting.
3.3 Modes
Because there are too many modes of the plates, only
two of the modes with the largest peaks will be presented
in this paper for comparison. The first mode cannot be
excited from the impact point chosen due to the prescribed
free-free boundary conditions. Therefore, the second and
the third modes are chosen instead for discussion. Table 2
and 3 list the modes calculate by the ANSYS, and Table 4
lists the modes from the measurement data. The modes
which are going to be compared are marked in red text.
Apparently, the frequency variation is less than 10% which
is acceptable in the present study.
Fig. 14. The frequency response of flat plate
Fig. 15. The frequency response of 10mm plate
Fig. 16. The frequency response of 20mm plate
number\type flat (Hz) 10mm (Hz) 20mm (Hz)
1 786.08 775.43 741.9
2 1039.7 1032.4 904.21
3 1590.2 1794.8 1516.7
4 1909.6 2267.8 2460.3
5 2181.1 2408 2694.8
Table 2. The calculated modes using ideal geometry
number\type flat (Hz) 10mm (Hz) 20mm (Hz)
1 796.82 791.16 758.53
2 1050.8 1042.1 896.11
3 1626.2 1814.1 1512.6
4 1933.4 2338.1 2466.8
5 2217.5 2507.1 2712.6
Table 3. The calculated modes using captured geometry
number\type flat (Hz) 10mm (Hz) 20mm (Hz)
2 1096 1056 920
3 1680 1848 1560
4 2024
Table 4. The measured modes
Fig. 17. Second measured mode of 10mm plate at 1056 Hz
Fig. 18. Third measured mode of 10mm plate at 1848 Hz
Fig. 19. Second measured mode of 20mm plate at 920 Hz
Fig. 20. Third measured mode of 20mm plate at 1560 Hz
Fig. 21. Second measured mode of flat plate at 1096 Hz
Fig. 22. Third measured mode of flat plate at 1680 Hz
Fig. 23. Fourth measured mode of flat plate at 2024 Hz
3.4 Mode shapes
The measured and calculated mode shapes are presented
from figure 17 to 37, and the measured mode shapes are
composed of dense contour lines. The method to
reconstruct the mode shapes from the data of measurement
is by extracting the imaginary part of the frequency
response and combine the imaginary part of each sample
point to form the mode shape at different frequency [3].
From the shown mode shape plots, we can see that the
measured and calculated mode shapes are similar to each
other at the same mode.
Fig. 24. Second mode of ideal 10mm plate at 1032 Hz
Fig. 25. Third mode of ideal 10mm plate at 1794 Hz
Fig. 26. Second mode of ideal 20mm plate at 904 Hz
Fig. 27. Third mode of ideal 20mm plate at 1516 Hz
Fig. 28. Second mode of ideal flat plate at 1039 Hz
Fig. 29. Third mode of ideal plate at 1590 Hz
Fig. 30. Fourth mode of ideal flat plate at 1909 Hz
Fig. 31. Second mode of real 10mm plate at 1042 Hz
Fig. 32. Third mode of real 10mm plate at 1814 Hz
Fig. 33. Second mode of real 20mm plate at 896 Hz
Fig. 34. Third mode of real 20mm plate at 1512 Hz
Fig. 35. Second mode of real flat plate at 1050 Hz
Fig. 36. Third mode real flat plate at 1626 Hz
Fig. 37. Fourth mode of real flat plate at 1933 Hz
4. Conclusions
Effect of buckling profile on structure vibration is studied
experimentally and numerically by using computer vision
method in acquiring structure geometry models in this
paper. It is found that with the computer vision,
experimental modal analysis can be much accelerated as
the structure’s full-field data can be acquired at once and
be used to conduct modal analysis computation
immediately. From the computation using the captured
structure geometries, both frequency and mode shape of
each natural mode can be easily generated. From the
present study, it is observed that as the plate’s buckling
profile deviates from flat case, the frequencies of the
structure’s second and the third mode decrease under the
free-free boundary condition. Given that all modes can be
provided once the real geometry data is captured, the
method proposed in this paper demonstrates that the error
of the results of the real model is acceptable, and even
closer to the measured results, which apparently has
engineering implementation and offers a low-cost approach
as compared to expensive DIC method.
References [1] Stephen P. T. and James M. G. Theory of Elastic Stability,
Second ed., pp. 76-81, 1936.
[2] Yen-Hao C. and Jen-Yuan C. Model Acquisition for Modal Analysis of Flexible Media Based on Stereo Vision. ASME ISPS conference, 2014.
[3] The fundamental of modal testing. Application Note 243 – 3. Agilent Technologies.