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

james.chang411@gmail.com VincentChang@itri.org.tw jychang@pme.nthu.edu.tw

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.