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The Development and Applications of Amplitude Fluctuation ESPI Method 349

Based on Eqn. (9), the vibration amplitude of the specimen can be obtained by a

set of

, which can null Eqn. (9); then the amplitude is determined by

)cos1(2

θπ

λζ

(10)

3 Experiments and Discussions

To perform the experiments, a self-assembled ESPI system as shown in Fig. 1 was

used. The system is composed of two parts, the out-of-plane optical setup and the

image processing system. As for the ESPI out-of-plane optical setup, a 35 mW He-

Ne laser of wavelength 632.8 nm was used as the light source. Regarding the image

processing subsystem, the commercial software Intelliwave [16] was utilized for im-

age capturing and analysis. To excite the specimen with a sinusoidal force, a shaker

(LDS Co., U.K.) was used. The exciting force was detected by a force sensor (PCB

Co., U.S.A.) and the force history was monitored and recorded by a Signal Doctor

Spectrum Analyzer (ProWave Co., Taiwan, R.O.C). As for the test specimen, an

aluminum alloy plate of length 220 mm, width 130 mm and thickness 2.2 mm was

used. As shown in Fig. 4, 40 mm of the specimen along the longitudinal direction is

fastened by a vise to simulate clamped edge boundary condition.

Fig. 4. An aluminum plate with described geometrical dimensions was used as the speci-

men to evaluate TA ESPI vibration measurement [15]

As described in Eqn. (5), the TA ESPI displacement and fringes can be de-

termined by the function

[]

1)( −kAJ

. The TA ESPI fringe patterns show bright

fringes when

[]

1)( −kAJ

are local maximums and become dark ones as

[]

1)( −kAJ

350 W.-C. Wang and C.-H. Hwang

are local minimums. According to Eqn. (3), the dark fringes can be observed

along the nodal lines, the nodal points and fixed edges where the associated dis-

placements are zero and should be time invariant. With improvements of the im-

age processing system, ESPI method can be performed in frame rate and fringe

patterns can be recorded continuously. To re-examine the match between Eqn. (3)

and experimental results, in this paper, TA was used with the out-of-plane ESPI

optical setup. Since nodal lines of both 2

and 4

resonant modes are easily

determined, a video recording system was also used to record the 2

and the 4

resonant mode fringe patterns of the aluminum plate. As observed from the re-

corded images, the TA ESPI fringe patterns vary with time. In fact, this time

varying phenomenon cannot be easily observed without the frame rate image

processing and/or continuously image recording aperture is adopted for image

processing. For discussion purpose, as shown respectively in Fig. 5 and Fig. 6,

five images were extracted from each of the recorded images of the 2

and 4

modes without denotative time.

Fig. 5. The time variation of TA ESPI fringe patterns of the 2

mode. The red line is used

to highlight the area for gray level analysis.

Fig. 6. The time variation of TA ESPI fringe patterns of the 4

mode. The red line is used

to highlight the area for gray level analysis

In Fig. 5, two phenomena can be observed. Firstly, the visibility of the fringe

patterns is lower in images captured at time intervals T2, T3 and T4 with respect

to images captured at time intervals T1 and T5. Secondly, the nodal line of the

The Development and Applications of Amplitude Fluctuation ESPI Method 351

2nd mode is a dark band in image captured at time interval T1 and becomes a

bright one at time interval T5. In addition, fringe pattern at clamped edge of the

specimen also turns bright. Similar results can also be observed for 4

mode (Fig.

6), i.e. the visibility of fringe patterns first becomes lower (time intervals T2 and

T3) and then recovers as time increases (time intervals T4 and T5), and in particu-

lar, the nodal lines and clamped edge become bright.

To identify the variation of gray levels with time, the gray levels along the red

line across the fringe patterns of Figs. 5 and 6 were processed by fast Fourier

transform (FFT) and low pass filter was utilized to filter out speckle noises [13].

The associated speckle-free normalized gray levels are plotted in Figs. 7 and 8 for

the 2

and 4

modes, respectively. According to the obtained normalized gray

level distributions, without any doubt, images captured at time intervals T1 and T5

give better fringe visibility than the other extracted images. Besides, from the

geometrical symmetry of the specimen and the normalized gray level plots, the

nodal lines of 2

mode and 4

mode should be located at around 150

pixels

counting from the left edge of the specimen. From the normalized gray level

plots, the gray levels at 150

pixels of both images extracted at time interval T1

are lowest, and the normalized gray levels changes into the highest ones for both

images captured at time interval T5. In short, the gray levels of fringe patterns ex-

tracted at time intervals T1 and T5 are opposite to each other, i.e. the bright bands

in images of time interval T1 become dark bands for images of time interval T5.

Even though the gray levels of the two fringe patterns at time intervals T1 and T5

are opposite to each other, the extreme values of gray levels at time intervals T1

and T5 occur almost at the same location.

Fig. 7. The distribution of normalized gray levels of TA ESPI fringe patterns of the alumi-

num plate excited at 2

mode at different time intervals

352 W.-C. Wang and C.-H. Hwang

Fig. 8. The distribution of normalized gray levels of TA ESPI fringe patterns of the alumi-

num plate excited at 4

mode at different time intervals

As for the AF ESPI, the fringe patterns were also processed in frame rate and

recorded. Without extracting images at particular time, as shown in Fig. 9, the

images of five different time intervals are rather consistent; however, the visibility

of images extracted at time T3 and T4 are a little bit lower than those of the other

three time intervals. The normalized gray levels of fringes after FFT and low pass

filter operations are shown in Fig. 10. By comparing with Figs. 7 and 8, the nor-

malized gray levels of the AF ESPI are relatively stable with respect to those of

the TA ESPI.

Fig. 9. The time variation of TA ESPI fringe patterns of the 2

mode. The red line is used

to highlight the area for gray level analysis

The Development and Applications of Amplitude Fluctuation ESPI Method 353

Fig. 10. The distribution of normalized gray levels of AF ESPI fringe patterns of the alumi-

num plate excited at 2

mode at different time intervals

To explain why fringe patterns obtained by TA ESPI method change with time,

the effect of environmental noise on the optical setup should be first considered.

Assuming the environmental noise acts on the optical setup and introduces the ad-

ditional phase change

),,( tyx

; Eqn. (1) can then be modified as

[]

),,(),,(),(cos2),,( tyxtyxyxIIIItyxI

roro

δϕ

+Δ+++= (11)

The environmental noise spectrum covers wide frequency ranges; however, in this

paper, only low frequency environmental noise was discussed. Because of filter-

ing or damping low frequency environmental noises cannot be done easily. As

mentioned earlier, for an ESPI system, the CCD will accumulate the incoming

light during each shutter opening time interval; then the general output voltages

which are converted from the CCD camera detected light intensity, becomes

∫

+Δ+++=

δϕα

)]cos(2[ dtIIIIV

rorog

(12)

In the TA ESPI method, the reference image is captured from a specimen not sub-

ject to periodic driving force. The phase term,

introduced by the input vibration

can thus be dropped, Eqn. (12) then becomes

∫

+++=

δϕα

)]cos(2[ dtIIIIV

rorog

(13)

Assuming the CCD camera’s frame frequency (number of frames can be captured

per second) is much higher than the frequencies of environmental noise acting on

354 W.-C. Wang and C.-H. Hwang

the ESPI optical system, the phase ),,( tyx

is simplified as a linear function of

time [17], i.e.

tyxyxtyx ),(),(),,(

βδ

+Φ= (14)

where ),( yxΦ is the initial phase term and ),( yx

is the time-averaged phase

change. Then

V can be approximated as

()()

[

]

2/cos2

βτϕατ

+Φ+++=

rorog

IIIIV (15)

Similarly, the captured image of the specimen under vibration can be determined

[18], i.e.

()

)(

1*cos

*)sin(

*sin

*cos2

)(

kAJII

IIV

oro

rovibg

⎥

⎦

⎤

⎢

⎣

⎡

−

Φ++Φ+

++=

−

τβ

ϕα

ατ

(16)

It should be noted that that the images under vibration are captured at different

time intervals with respect to the reference image, i.e. the initial phase term and

the time-averaged phase change are different. In Eqn. (16), the superscript * de-

notes the environmental noise which is different from the reference one.

If the environmental noise does not change significantly, then both

βτ

and

are small, Eqns. (15) and (16) can be respectively simplified into

()

[

]

Φ+++=

ϕατ

cos2

rorog

IIIIV (17)

()

[

]

)(*cos2 kAJIIIIV

ororovibg

Φ+++=

−

ϕατ

(18)

Assuming the initial phases of reference and object images are related by

Φ+Φ=Φ

* (19)

Then the traditional TA ESPI can be remodeled and the output voltage of TA

ESPI method can be modified as [14]

()

[]

()

[]

)sin(sin)(1cos)(2

2/1

ςφδδατ

+Φ+Φ+−Φ=

−=

−

kAJkAJII

VVV

ooro

gvibgoutput

(20)

where

()

[]

1cos)(

sin)(

tan

−Φ

−

kAJ

The Development and Applications of Amplitude Fluctuation ESPI Method 355

The fringe patterns of TA ESPI method can be determined by the func-

tion ),,( Φ

Akf , i.e.

()

[]

()

[]

2/1

sin)(1cos)(),,( Φ+−Φ=Φ

δδδ

kAJkAJAkf

. (21)

The fringe patterns are not only determined by vibration amplitude and ESPI opti-

cal setup but also determined by the environmental noise introduced equivalent in-

itial phase difference,

. When

0=Φ

, Eqn. (21) becomes

[]

1)(),,( −=Φ kAJAkf

, this is the same as the traditional TA ESPI, the nodal

lines and the clamped edge in the fringe patterns should appear in dark.

When

=Φ

, Eqn. (21) becomes

1)(),,(

+=Φ kAJAkf

. Different from the

case of

0=Φ

, the fringe patterns along nodal lines and clamped edge become the

brightest ones. The normalized gray levels plot of Eqn. (21) is shown in Fig. 11.

There are dark fringes along the nodal lines and clamped edge as

increases

Fig. 11. Modified fringe formula with environmental noise introduced equivalent initial

phase difference,

, considered

356 W.-C. Wang and C.-H. Hwang

from 0 to

, and then the fringes become bright ones as

increases from

. Besides, the visibility of the fringe pattern becomes worst as

in-

creases from 0 up to

, and then the visibility increases as

increases

from

. The above results well describe the phenomenon shown in

Figs. 5 and 6, i.e. the visibility of fringe patterns first becomes lower and then re-

covers as time increases, and in particular, the nodal lines and clamped edge be-

come bright.

4 Engineering Applications of the AF ESPI Method

The AF ESPI method was first used to investigate the fracture behavior of the

edge-cracked composite plates [19, 20] and determine modified mode III stress in-

tensity factors (SIFs) of different resonant frequencies. Vibration behavior was

investigated for composite plates containing circular defect at different depths

[21], shadow masks [22] and printed circuit boards [23]. The repair efficiency of

composite patching of edge- and center-cracked aluminum and carbon fiber rein-

forced plastic (CFRP) plates [24, 25] was evaluated by the AF-ESPI method. In

addition, nondestructive inspection of composite plates containing defects [26-28]

was also performed by the AF ESPI method.

5 Conclusions

In this paper, by continuously recording TA ESPI and AF ESPI fringe patterns,

the traditional time-averaged ESPI was re-examined. The recorded fringe patterns

obtained by TA ESPI showed that they vary from time to time. For the extreme

case, the gray levels of two fringe patterns at different time intervals are opposite

to each other. No matter how the gray level changes, fortunately, the extreme val-

ues of the gray level still occur at the same location. On the contrary, AF ESPI

fringe patterns remain essentially stable with almost nil visibility difference as

time increases. A new mathematical model was proposed to describe the TA ESPI

method. With environmental noises considered, the new model has the

form

()

[]

()

[]

sin)(1cos)( Φ+−Φ

δδ

kAJkAJ

. Based on the proposed new

model, the time variation as well as the decreasing and recovering of visibility of

the fringe pattern can be well predicated by environmental noise introduced

equivalent initial phase difference,

. The results indicate that the importance

between the reference image and the object image as TA ESPI is adopted

for vibration measurement. At the end of this paper, a summary of the applica-

tions of AF ESPI method is provided.

The Development and Applications of Amplitude Fluctuation ESPI Method 357

Acknowledgement

This research was supported in part by the National Science Council of the Republic

of China (grant nos. NSC95-2221-E007-150 and NSC95-2221-E007-011-MY3) and

Instrumentation Engineering Division of Instrument Technology Research Center,

National Applied Research Laboratories, Taiwan, Republic of China.

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