David Browne

NT090461M

Applied Mechanics ME4211

Deformation Measurement of Circular Plate Using ESPI

1 Introduction

The objective of this experiment is to measure the deflection of a circular plate when a point load is

applied to it and compare the results to what would be expected by applying plate bending theory.

The method used to measure the deflection will be Electronic Speckle Pattern Interferometry. This

technique is widely used in industry for its qualitive yet quantitive quality (1). In other words it provides

relatively high quality results at relatively high speed. The technique has a wide range of applications

for industrial use. One use is observing the stress and strain in an object undergoing either

mechanical or thermal loading. This is useful not only for locating stress concentrations in an object

but also to calculate the materials Young’s Modulus and Poisons Ratio or the objects stiffness.

Another use is for locating cracks in an object. ESPI is highly sensitive to local variations in

deformations, meaning that when an object is placed under load, the fringe pattern reveals cracks (2)

(Appendix 4). ESPI is also useful for studying vibrations, such as the vibrating walls of a flute during

playing (4).

2 Background and Theory

2.1 Circular Plate Bending Theory

Fig 1: Diagram of circular plate

When a central load is applied is applied to a circular plate it can be assumed that the plate

undergoes “Axisymmetrical bending” whereby the deflection at a point on the plate will only be

dependent on its radial position and not on its angular position. This results in a simplified equation for

the deflection of the plate:

� =����

��(ln − 1) +

�

��� � + �� ln + �� � ... (2.1)

When a circular plate is “fully clamped” it can also be assumed to have zero deflection and rate of

change of deflection with respect to radial distance at its edge:

���� = 0 … (2.2)

��

�� ���= 0 … (2.3)

It is also known that at the centre of the plate always has a finite displacement:

���� < ∞ ∴ �� = 0 … (2.4)

From (2.3):

�� = −!

�� (2 ln # − 1) … (2.5)

Taking into account (2.5), for (2.1) to fulfil (2.2) and (2.3) it can be calculated that c3 must have the

following value:

�� =����

�$�

… (2.6)

Finally, inserting eq2.5 and eq2.6 into eq2.1, it can be seen that the deflection is calculated to be:

� =����

��ln

�

�+

!

�$�(#� − �)

… (2.7)

2.2 Electronic Speckle Pattern Interferometry

When the rough surface of an object is illumined light is scattered back in all directions. Being ‘rough’,

the surface has a height variation. This means that each wave that is reflected back off of the surface

has travelled a different distance and therefore possesses a phase difference (Fig 2). The scattered

light consists of the superposition of each individual wave, forming a series of bright and dark spots

randomly distributed in space. In regular white light this behavior is scarcely observable but when

expanded laser light is used a granular appearance known as a speckle pattern is observed (2).

Fig 2: Diagram demonstrating the phase difference of scattered light

As discussed in the introduction, it is often of interest to measure small displacements of a surface.

When a surface undergoes some displacement there is also a corresponding displacement in the

observed speckle pattern. Electrical Speckle Pattern Interferometry (ESPI) can be used to measure

the displacement of the speckle pattern so that the deflection of the surface can be calculated.

In ESPI the laser light source is split into an object beam and a reference beam which are used to

illuminate an object of interest and a reference object respectively. When the scattered light from the

surfaces of both objects interferes with one another they are superimposed to form a further speckle

pattern called an Interferogram (3).

Speckle Pattern Interferometry can be split into two catergories (1);

- Class 1: The Reference beam is reflected as single wavefront and the Object beam is

scattered back as a speckle pattern.

- Class 2: Both the object and reference beam are scattered back as speckle patterns.

A camera is used to obtain a reference and a live image of the interferogram. When the object of

interest is displaced the object beam is required to travel further and therefore each wave experiences

a different phase difference resulting in an interferogram that is different to the reference image. A

computer then digitally subtracts the live image of the disc with the reference image to create a new

image. Any change in pattern between the live and reference image is highlighted by ‘fringes’ on the

new image. The relative displacement between two fringes is equal to half of the wavelength of light

illuminating it.

3 Experimental Procedures and Results

3.1 Experimental Apparatus

Fig 3: Diagram showing apparatus

(A) ESPI Setup

For the experiment a Class 2 ESPI system is used (a) with a He-Ne laser. The wave length of the light

is 0.6328μm. The system is coupled to a loading device which is used to displace the specimen. The

device consists of a Micrometer head (c), a frame and a proving ring. When the micrometer head is

turned, a load is transmitted through the proving ring to the specimen. Two strain gauges are attached

to the proving ring (b1 &b2), both of which generate an electrical strain signal which to later be used

to quantify the corresponding load.

The specimen used is a thin aluminum alloy disc with the properties specified in table 1.

Table 1

Specimen Radius (A) 32.65 mm

Thickness (t) 1.23 mm

Youngs Modulus (E) 79.71 Gpa

Poissons Ratio (v) 0.334

Flexual Rigidity (D) 13.91287

(B1)

(B2)

(C)

3.2 Experimental Technique

3.21 Preliminary

To calculate the resolution of pixels to mm for the current apparatus an image of a blank screen with 2

vertical lines marked on is taken. The distance between the 2 lines corresponds to the Specimen

No.3’s diameter which is known to be 65.3mm. Using imaging software the distance between the two

lines can be measured by noting the number of pixels between the left hand side of the image and

each line. This was found to be 27 and 727, meaning there is a distance of 700pixels between the two

lines.

65.3

700= 0.093285714 ≈ 0.093 //. 012345�

3.22 Experiment

The micrometer head on the loading device is turned slowly and the corresponding strain is recorded.

The fringe pattern at this point is then also recorded by capturing an image using the CCD camera.

3.23 Post Processing

The captured fringe pattern is opened using imaging software on a computer. It is desired to record

the location in pixels of each fringe.

By constructing a right angle triangle (red) with all 3 corners lying on the circumference of a circle, it is

known that the hypotenuse is the diameter. By taking the mean of the X and Y pixel locations of the

adjacent and opposite corner, the centre point of the fringe pattern can be calculated. A horizontal line

(yellow) is then constructed through the centre point and the pixel X coordinate of each fringe

interception is recorded (Fig 4). Starting from the outside of the fringe pattern and heading towards

the centre each dark fringe is given a sequential ‘order’.

Fig 4: Diagram demonstrating processing technique

1 2 3 (C) 3 2 1

4 Error Analysis and Discussions

4.1 Result Interpretation

In the experiment 4 different fringe patterns at a load corresponding to a micro-strain of 40, 72, 89 and

110 were captured (Appendix 3). The displacement of the plate was the calculated by considering the

previously recorded X coordinates of each fringe. The displacement of the plate at the location of

each dark fringe is equal to;

� =67

�

… (4.1)

Where N is the order of the fringe and λ is the wavelength of the light.

Using the load cell specimen calibration chart it is known that the micro-strain is related to the load (N)

by the equation:

9 = 1.7489 : … (4.2)

The theoretical deflection at each point on the plate can then be calculated using equation (2.7). The

results of this calculation are shown in Appendix 1.

Finally, the theoretical and experimental deflections can be plotted alongside each other on a graph

against the radial distance from the centre of the circular plate. These graphs are shown in Appendix

2.

4.2 Error Sources

It can be seen in Fig5a to 5d that the experimental deflections observed in this investigation are very

close to the expected theoretical values. It is perhaps notable that in Fig 5c there is more noticeable

error between the two values at points towards to edge of the disc. This may be due to the actual

experiment not fully fulfilling the boundary conditions that are specified for the theoretical

displacements: Zero displacement and displacement gradient at the edge.

There are several explanations for any small deviations between the experimental and theoretical

deflections;

- The theoretical value for the deflection of the disc actually relies on the experimental value for

the load being exerted on it. This particular experimental value is itself subject to several

errors and limitations;

o The strain was displayed with a resolution of 1μ – approximately 1% of the total value.

o The displayed strain result would regularly fluctuate between several values, whilst

only one was recorded.

o The calibration chart to convert the displayed strain into a force was also generated

experimentally. A series of loads with known magnitudes were applied to the proving

ring and the corresponding strains were recorded. A line of best fit was the

constructed between each experimental point to generate equation (4.2). This

equation is therefore also subject to any errors which occurred during the calibration

experiment.

- The recording of the pixel coordinate for each fringe on the fringe pattern was subject to some

human error. An attempt was made to take each measurement at the centre of each fringe

but the centre of the fringe was only located by eye. Similarly, an attempt was made to

simplify the readings of the displacement in pixels between each fringe by constructing a

horizontal line through the centre of the fringe pattern so as to isolate the x co-ordinate.

However, it can be seen that this horizontal line did not always go through what would seem

to be the centre of the pattern either because the fringe pattern was not exactly circular or

because the right angle triangle was not constructed accurately. This means that the recorded

displacement is likely to be smaller than the actual value.

- ESPI is vulnerable to environmental noise and vibrations. This means that the recorded fringe

patterns are likely to of been influenced by any sound or movement in the room.

4.3 Future Recommendations

Without altering the equipment used in the experiment the best recommendation for improving the

results would be to refine the method of calculating the displacement between each fringe in a fringe

pattern. Ideally a specific computer software would be used that could read the range of

displacements in each angular direction around the fringe pattern and take an average.

5. Conclusion

The results of this experiment show such small deviations from what would be expected theoretically

that they can be seen to validate equation (2.7). The largest errors between theoretical and

experimental values are observed towards the edges of the plate. The evaluation of the results leads

to the conclusion that these particular errors are likely to be present in the theoretical results rather

than the experimental results. The reasoning behind this is that the theoretical results are built upon

the assumed boundary conditions, equation (2.2) and (2.3), which states there is zero deflection or

deflection gradient at the clamped edge. In reality these assumptions are not likely to be entirely

accurate, which explains why the errors are localized around the plate edge. To calculate a more

accurate set of theoretical results it may be more appropriate to use a numerical method such as finite

element.

References

1. Malacara, D. (2001) Handbook of optical Engineering, New York: Marcel Dekker Inc

2. Gasvik, J. (2002) Optical metrology, London: John Wiley & Sons Inc

3. Dantec Dynamics. (2009) Electronic Speckle Pattern Interferometry – ESPI,

http://www.dantecdynamics.com/Default.aspx?ID=1029, Date accessed 11/11/09.

4. Lokberg, OJ. and Ledang, OK. (1984) “Vibration of flutes studied by electronic speckle

pattern interferometry”, Applied Optics, Vol. 23, Issue 18, pp. 3052-3056

Appendix 1 S = 40 1.00 2.00 3.00 4.00 5.00 6.00 P

Position relative to left (P) 170.00 250.00 329.00 440.00 515.00 595.00 0.70

Position relative to centre (P) -210.00 -130.00 -51.00 60.00 135.00 215.00

Position relative to centre (mm) -19.59 -12.13 -4.76 5.60 12.59 20.06

Fringe order 1.00 2.00 3.00 3.00 2.00 1.00

Experimental deflection (;;;;m) 0.32 0.63 0.95 0.95 0.63 0.32

Theoretical deflection (;;;;m) 0.29 0.63 0.96 0.92 0.61 0.27

Points

S = 72 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 11.00 12.00 P

Position relative to left (P) 117.00 172.00 218.00 259.00 299.00 348.00 411.00 462.00 504.00 542.00 583.00 630.00 1.26

Position relative to centre (P) -259.00 -204.00 -158.00 -117.00 -77.00 -28.00 35.00 86.00 128.00 166.00 207.00 254.00

Position relative to centre (mm) -24.16 -19.03 -14.74 -10.91 -7.18 -2.61 3.26 8.02 11.94 15.49 19.31 23.69

Fringe order 1.00 2.00 3.00 4.00 5.00 6.00 6.00 5.00 4.00 3.00 2.00 1.00

Experimental deflection (;;;;m) 0.32 0.63 0.95 1.27 1.58 1.90 1.90 1.58 1.27 0.95 0.63 0.32

Theoretical deflection (;;;;m) 0.24 0.56 0.91 1.23 1.54 1.85 1.81 1.48 1.15 0.84 0.54 0.26

Points

S = 89 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 11.00 12.00 13.00 14.00 P

Position relative to left (P) 90.00 149.00 194.00 228.00 262.00 298.00 336.00 428.00 470.00 504.00 538.00 570.00 608.00 660.00 1.56

Position relative to centre (P) -285.00 -226.00 -181.00 -147.00 -113.00 -77.00 -39.00 53.00 95.00 129.00 163.00 195.00 233.00 285.00

Position relative to centre (mm) -26.59 -21.08 -16.88 -13.71 -10.54 -7.18 -3.64 4.94 8.86 12.03 15.21 18.19 21.74 26.59

Fringe order 1.00 2.00 3.00 4.00 5.00 6.00 7.00 7.00 6.00 5.00 4.00 3.00 2.00 1.00

Experimental deflection (;;;;m) 0.32 0.63 0.95 1.27 1.58 1.90 2.21 2.21 1.90 1.58 1.27 0.95 0.63 0.32

Theoretical deflection (;;;;m) 0.15 0.52 0.90 1.23 1.57 1.91 2.21 2.11 1.74 1.41 1.07 0.77 0.47 0.15

Points

S = 110 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 11.00 12.00 13.00 14.00 15.00 16.00 17.00 18.00 P

Position relative to left (P) 105.00 148.00 183.00 212.00 241.00 268.00 296.00 325.00 356.00 410.00 448.00 476.00 504.00 530.00 557.00 583.00 614.00 651.00 1.92

Position relative to centre (P) -273.50 -230.50 -195.50 -166.50 -137.50 -110.50 -82.50 -53.50 -22.50 31.50 69.50 97.50 125.50 151.50 178.50 204.50 235.50 272.50

Position relative to centre (mm) -25.51 -21.50 -18.24 -15.53 -12.83 -10.31 -7.70 -4.99 -2.10 2.94 6.48 9.10 11.71 14.13 16.65 19.08 21.97 25.42

Fringe order 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 9.00 8.00 7.00 6.00 5.00 4.00 3.00 2.00 1.00

Experimental deflection (;;;;m) 0.32 0.63 0.95 1.27 1.58 1.90 2.21 2.53 2.85 2.85 2.53 2.21 1.90 1.58 1.27 0.95 0.63 0.32

Theoretical deflection (;;;;m) 0.26 0.60 0.95 1.28 1.63 1.97 2.30 2.61 2.85 2.79 2.44 2.12 1.78 1.46 1.14 0.86 0.55 0.27

Appendix 2

Fig 5a – Fig5d: Graph plotting the Experimental and Theoretical deflection of a circular plate

versus the position relative to the centre

Fig 5a

Fig 5b

0

0.2

0.4

0.6

0.8

1

1.2

-40 -20 0 20 40

Defl

ecti

on

(µ

m)

Position relative to centre (mm)

Strain=40uExperimental

Theoretical

0

0.5

1

1.5

2

2.5

-40 -20 0 20 40

Defl

ecti

on

(µ

m)

Position relative to centre (mm)

Strain=72uExperimental

Theoretical

Fig 5c

Fig 5d

0

0.5

1

1.5

2

2.5

-40 -20 0 20 40

Defl

ecti

on

(µ

m)

Position relative to centre (mm)

Strain=89uExperimental

Theoretical

0

0.5

1

1.5

2

2.5

3

3.5

-40 -20 0 20 40

Defl

ecti

on

(µ

m)

Position relative to centre (mm)

Strain=110uExperimental

Theoretical

Appendix 3

Fig 6a: Strain = 40;;;;

Fig 6b: Strain = 72;;;;

Fig 6c: Strain = 89;;;;

Fig 6d: Strain = 110;;;;

Appendix 4

Fig 7: Diagram showing a) a vertical crack seen in ESPI a b) a vertical crack not detected in TV

holography (2)