Kounadis A.N., Gdoutos E.E. (Eds.) Recent Advances in Mechanics

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Application of DSPI in Determination of Elastic Modulus Using Plate Vibration 339

Table 2. Evaluation of elastic modulus based on data of vibrating square plate under differ-

ent boundary conditions.

5 Conclusion

DSPI and vibration analysis of plate can be effectively utilized to evaluate the

elastic modulus. Plate has unique mode shape of vibration and there is no ambigu-

ity in identifying these mode shapes when observed by DSPI. To establish the

method a large number of experiments were conducted on square aluminum plates

fixed at one edge and keeping other edges free. Experimental results reveal that if

preformed carefully, a single observation is sufficient to determine the elastic

modulus of the plate material. Applying the boundary condition one edge fixed

and other edges free is relatively easier. It was observed that for the boundary

condition, one edge fixed and other edges free, the frequencies corresponding to

first torsional mode obtained from DSPI are almost same when calculated using

classical theory or by the FEA. So the single observation of frequency at first

340 C. Shakher and R. Kumar

torsional mode is sufficient to evaluate the elastic modulus for all practical

purposes with measurement error less than 1%. Preliminary results with other

boundary conditions also reveal that this technique can effectively be used for the

purpose.

References

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Application of DSPI in Determination of Elastic Modulus Using Plate Vibration 341

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The Development and Applications of

Amplitude Fluctuation Electronic Speckle

Pattern Interferometry Method

Wei-Chung Wang

and Chi-Hung Hwang

Department of Power Mechanical Engineering, National Tsing Hua University,

Taiwan, Republic of China

wcwang@pme.nthu.edu.tw

Instrument Technology Research Center, National Applied Research Laboratories,

Taiwan, Republic of China

chhwang@itrc.org.tw

Abstract. In vibration measurement, traditional time averaged (TA) electronic

speckle pattern interferometry (ESPI) method was essentially used for obtaining

modal shapes rather for quantitative analysis. In 1996, the authors first reported

that the driving force acting on the specimen would be fluctuated due to environ-

mental disturbances and vibration fringe patterns obtained by TA ESPI method

can be significantly improved if both the reference and object images were cap-

tured under vibration load. This new TA ESPI method was named as amplitude

fluctuation (AF) ESPI method. In this paper, the development and successive

improvement of the AF ESPI method was first introduced. The effects of envi-

ronmental noise and vibration characteristics on the ESPI fringe pattern were then

investigated. Theoretical derivation on the effect of environmental noise was per-

formed and the time varying brightness of the traditional time averaged (TA) ESPI

fringe patterns was successfully explained. In addition, applications of the AF

ESPI method were briefly reviewed.

1 Introduction

In 1992, during the execution period of an investigation project of applications of

composite materials of Taiwan’s bicycle and motorcycle industries, the authors

were aware that the inlet springs of the inlet valve of a 4-stoke motorcycle engine

were made of carbon fiber reinforced plastic (CFRP) [0/90] plates as shown in

Fig. 1 [1]. In practice, the engine of a motorcycle completes 3000~6000 combus-

tion cycles per minute, i.e. the springs are forced to vibrate at about 50~100 Hz.

To achieve the best efficiency of the engine, the inlet valve should be completely

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

closed. If the combustion frequency is happened to be the same as the modal fre-

quency of inlet springs, the inlet valve may not be closed completely and the effi-

ciency of the engine will be decreased. Moreover, the free ends of the inlet

springs will impact on the head of the inlet valve and cracks may be produced in

the springs. Besides, the inlet valve is mounted on the top of the combustion

chamber and therefore the inlet springs are operated under high temperature,

surface defects may also be produced on the plate springs. While defects were in-

troduced, the dynamic properties of damaged plate springs are changed and the as-

sociated modal modes will also be different from defect-free ones. Structures may

be failed because of crack propagation and large displacement introduced by

crack-induced free surfaces.

Fig. 1. An inlet-valve cover consists of three CFRP plates was used on a 4-stoke motorcy-

cle engine [1]

Many experimental methods can be employed for vibration measurement, such

as holography, modal testing method, shadow moiré method, laser Doppler vi-

brometer, shearography, electronic speckle pattern interferometry (ESPI), etc.

Among all, the ESPI is an optical method which can provide real-time, whole-

field and non-contact measurement with no special surface treatment required.

The ESPI can be used for static and dynamic displacement measurement ranges

within 5~100μm [2], the measurement sensitivity is as good as holography and no

chemical development required. Besides, orthogonal displacement information

can be obtained individually by proper optical setup.

The ESPI is also known as the TV-holography and the “Digital Speckle Pattern

Interferometry” (DSPI) as video sensor and image processing hardware are util-

ized respectively. In 1971, Butters and Leendertz [3] demonstrated ESPI with

some engineering applications including the vibration of a disc. However, detail

analysis was not well described in their paper. In fact, vibration measurement is

The Development and Applications of Amplitude Fluctuation ESPI Method 345

one of the important applications of the ESPI method. The ESPI has been applied

on vibration measurement in corporation with time-averaged (TA) method [4],

stroboscopic method [4], the use of a pulsed laser [5] and TA based reference

wave modulation method [6-8]. Because of the lower cost and simpler measure-

ment system set up, the TA method is the easiest method to be implemented for

vibration measurement. Theoretically, the absolute displacement field can be ob-

tained by the TA ESPI method; however, the quality of the fringe patterns ob-

tained is not good enough for accurate quantitative analysis.

In the past, many research works have been done to improve the quality of TA

ESPI fringe pattern. Kerr and Tyrer [9] summarized that there are five different

methods which were proposed to solve the problem of poor quality of the ESPI

fringe patterns. The proposed methods are fringe tracking, spatial synchronous

detection, frequency heterodyning, digital transformation and phase-steeping me-

thod. In 1989, Vikhagen [10] adopted two digital image processing methods, the

max–min scanning method and the normalized max–min scanning method, to op-

timize fringe contrast and quality of vibration fringe patterns. It was found that

the contrast of a fringe pattern can be improved by the normalized max–min scan-

ning method by eliminating the speckle noise. Joenathan [11] adapted phase-step

method with TA ESPI method and showed that the fringe patterns can be im-

proved; in his paper, the associated S/N ratios were all developed for various

phase-step methods.

In 1996, the major author and his research group [1, 12] proposed a new TA

ESPI method, i.e. amplitude fluctuation (AF) ESPI method for the vibration meas-

urement. Distinct from the conventional TA ESPI methods, the reference image

of the AF ESPI method need not be obtained from the static condition but of the

same nominal driving force level as the object image. In addition, higher resolu-

tion and better contrast of ESPI fringe patterns can be obtained by the AF ESPI

method. The AF ESPI method is easy to implement, no extra hardware required,

higher fringe density and better fringe pattern contrast can be obtained. It can be

applied for quantitative displacement analysis and has been successfully applied in

many engineering applications, however, how environmental disturbances and vi-

bration characteristics affect the quality of the ESPI fringe patterns are still need to

be investigated [13-15]. Besides, thanks to the improvement of computation

speed, the obtained fringe patterns can be displayed in video frame rate (i.e. at

least 24 images per second). Since the resolution of the ESPI is the same as the

holography and the real-time fringe patterns can be taken by the ESPI, the ESPI is

thus an excellent tool to investigate the fundamental problems of the TA method.

In this paper, with proper mathematical modeling and experimental design, the

effect of environmental noise on the test specimen and the ESPI optical system

were formulated as a function of zero order Bessel function. The formulation pro-

vides rational explanation on the time varying phenomenon of the TA ESPI im-

ages and the superiority of the AF ESPI method on the vibration measurement.

The vibration characteristics were also experimentally and numerically studied.

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

2 Principle of Time-Averaged ESPI Method

2.1 The Traditional TA ESPI Method

A typical out-of-plane ESPI setup is shown in Fig. 1. When the specimen is under

periodic motion, the light intensity detected by a CCD camera at time

can be

expressed as [2]

[]

),,(),(cos2),,( tyxyxIIIItyxI

roro

Δ+++=

(1)

where

I is the light intensity of the object light beam;

I is the light intensity of

the reference light beam;

λωθπ

t)/)(Acoscos+(12=Δ ;

is the wavelength of

the light source;

A is the vibration amplitude;

is the vibration frequency;

the illumination angle of the object light beam; and

(, )xy

is the random phase

resulting from the surface roughness. Since a CCD camera itself accumulates the

incoming light during each shutter opening time interval,

, the output voltage,

which was converted from the light intensity detected by the CCD camera,

becomes

[]

∫

Δ+++=

ϕα

)cos(2 dtIIIIV

rorovib

(2)

where

is the slope of the CCD camera’s sensitivity curve. Assuming

is set to

be a complete integral vibration period to avoid possible mismatch between the

observed fringe patterns and the output voltage signal [1, 12], then

[

]

ϕατ

cos)(2 kAJIIIIV

ororovib

++= (3)

where

/)cos1(2 +=k

For traditional TA ESPI method, the image of the still object is taken as the ref-

erence image. Then from Eqn. (1), the reference image captured by the CCD can

be expressed as

[

]

ϕατ

cos2

rororeference

IIIIV ++=

(4)

In order to eliminate the background brightness, the image subtraction method is

utilized. That is, the fringe patterns can be described by the following equation

[]

ϕατ

cos1)(2 −=−= kAJIIVVV

ororeferenceviboutput

(5)

The Development and Applications of Amplitude Fluctuation ESPI Method 347

Fig. 2. Typical out-of-plane ESPI optical setup

2.2 The AF ESPI Method

The AF ESPI method for vibration measurement is also based on the traditional

signal subtraction method. The essential feature of the AF ESPI lies on that both

reference and object images are captured under the same nominal vibration force

level. In other words, to execute the AF ESPI method, two images both subject to

the vibration loading are captured. The first image is taken as the reference image

and the voltage converted from the total light intensity recorded by the CCD cam-

era is also expressed as Eqn. (3).

Since the environmental and electronic noises do exist all the time, it is ap-

propriate to assume that the input driving force changes during each cycle. This

assumption was experimentally proved [1, 12] and shown in Fig. 3 for a typical

time history record of the excited force (expressed by voltage) versus time. Note

that the desired input exciting force is a periodic function. However, owing to ex-

ternal noises, the input to the structure to be vibrated is not exactly periodic during

the whole vibration period. Therefore, the amplitude of second image can be as-

sumed to change from A into AA Δ+ . The output video signal of the second im-

age now becomes

() ()( )

⎥

⎦

⎤

⎢

⎣

⎡

⎟

⎠

⎞

⎜

⎝

⎛

+Δ+++=

ϕθ

ατ

coscos1

2 AAJIIIIV

ororo

disturbed

vib

(6)

By using the image processing system, the resulting voltage signal

()

vib

disturbed

viboutput

VVV −=

can be expressed as

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

()()()()

[

]

ϕατ

cos2 kAJAAkJIIV

oorooutput

−Δ+=

(7)

In fact,

AΔ is one of the undetermined parameters. As indicated in Eqn. (7), the

output voltage

output

is expressed by the zero-order Bessel function. It is impos-

sible to express concisely the relationship between the output voltage signal and

the undisturbed vibration amplitude

A . Hence, additional mathematical modifica-

tion must be used. Considering

AΔ is rather small, dividing each side of Eqn. (7)

AΔ , and evaluating the limits of both sides, Eqn. (7) is then simplified and ex-

pressed as a function of vibration amplitude only, i.e.

[]

ϕατ

cos)(2

kAJII

output

−=

(8)

Eqn. (8) can be interpreted as the video-signal sensitivity of the vibration ampli-

tude. To avoid all possible mistakes introduced when signal is transformed into

the gray level; the negative voltage value must be rectified into the positive value,

then the output signal becomes

[]

cos)(2

ϕατ

kAJIIV

roAFESPI

= (9)

Fig. 3. Typical excited force-time histogram recorded by a force sensor attached to the driv-

ing rod with input sinusoidal exciting force. The diagram shows that the driving force fluc-

tuates about the nominal value [1, 12].