In-plane vibration displacement measurement using fiber-optical speckle interferometry Ming Chang Department of Mechanical Engineering, Chung Yuan Christian University, Chung Li, Taiwan, Republic of China

An optical fiber sensor system based on dual beam speckle interferometry is presented. The system provides a noncontact method of very small vibration displacement real-time measurement. It takes advantage of simple optical components and fibers through a signal processing technique capable of measuring vibration displacement components. The theoretical analysis and experimental systems are described. Experimental results show that the in- plane vibration spectrum can be easily established.

Keywords: in-plane vibration; speckle interferometry

Introduction The research in measuring small mechanical vibra- tion displacements of devices and components such as quartz and piezo-electric vibrators, is most frequently undertaken using optical noncontact methods with their potential in nondestructive evaluation and remote, high-speed, automated measurements. The primary techniques of optical measurement are shadow moire interferometry, 1 holographic interferometry, 2 speckle interferome- try, 3 and laser Doppler effect. 4 The development of these methods can have good qualitative observa- tion of full field vibration in principle. There are some disadvantages in these methods, such as lack of high-frequency vibration measurement devices, insensitivity to in-plane vibration measurement, quantitative analysis of interferogram consuming too much time, resolution being limited, or the opti- cal device being too complicated to be used practi- cally. It is hoped that a technique with much improved performance, especially a wider variety of vibration measurements and higher frequency devices, can be developed.

The method of speckle effect was first intro- duced by Leendertz 5 and Archbold et al. 6 to mea- sure in-plane displacement. There was electronic speckle pattern interferometry, 7 which fetches the interference pattern by TV camera with analog video signal processing, to observe the mechanical vibration pattern at a later stage. Recently, the

Address reprint requests to Ming Chang, Department of Mechan- ical Engineering, Chung Yuan Christian University, Chung Li, Taiwan 32023, Republic of China. © 1994 Butterworth-Heinemann

charge-coupled device (CCD) array camera an~t microprocessor were invented, and digitai speckle pattern interferometry 8 was developed to fetch the speckle interference pattern with digital signal pro- cessing. These techniques allow good observation of full-field mechanical vibration. However, quanti- tative processing is not straightforward, and data ambiguity may occur because the mathematical form of the fringe pattern is expressed by a Bessei function.

In this article, a single-point, real-time vibratior~ measurement technique based on the idea of tradi- tional dual-beam speckle interferometry is pre sented. The method takes advantage of the opticai fibers and components to develop a compact, sta- ble, and easily operated fiber-optical interferome- ter. It is equipped with a flexible optical fiber sensor head capable of separating in-plane and out-of- plane vibration displacement components. Move- ment of the specimen causes a phase change pro- portional to the displacement. The phase change causes a proportional intensity change, which is measured. The constant of proport ional i ty is de n duced by modulating the phase at a different f re quency using a f iber-wrapped piezoelectric crystal. Maximum sensitivity is achieved with a static phase difference of ~-/2 between the two arms of the inter- ferometer. The technique avoids the shortcomings of traditional optical methods and can measure very small mechanical vibration displacement point by point under higher vibration frequency. Experimen- tal measurements are made in the in-plane vibra- tion amplitude of a piezoelectric transducer (PZT). The measured vibration frequency range is 500 Hz to 110 kHz at present. The first- and second-order-

36 JANUARY !994 VOL 16 NO ',

( × ,Y

BEAM2 /T~

.z ) ~

LEN @

X

Z j =y

VIBRATOR

W

Figure 1 Schematic diagram of dual beam speckle interferometry for measuring in-plane displace- ment

in-plane vibration natural frequencies of the PZT were found. The results show that the system is a practical means for a variety of vibration measure- ments.

Measuring principle The schematic diagram of in-plane displacement component measurement is shown in Figure 1. The specimen is il luminated by two coherent beams and is thus known as the dual beam method. As shown, the object is i l luminated by two collimated beams equally inclined at a certain angle ~ to the yz plane. It will have a random speckle distribution caused by random interference on the image plane. When the object is moved, the displacement of the points causes a relative phase change between the two speckle patterns. The relative phase change is only due to the in-plane displacement in the x direction given by

~b = (USin 6) 4 ~ (1)

where 6 is the phase change due to a movement, U is the displacement component in the x direction,

is the il luminating angle, and ,k is the wavelength of light.

When the object to be tested is a harmonic mo- tion vibrator, and in-plane displacement USin~st in the x direction is introduced (where U is the vibra- tion displacement amplitude and oJ s is the vibration frequency of the vibrator), the phase change caused by the vibration of the vibrator is shown as

& = (USin e) 4-~ S in~ t

= ~bs S in~ t (2)

where ~bs is equal to (USine)4~-/x. In the case of combined in-plane and out-of-plane vibration, as most vibration is, the out-of-plane vibration in the z direction and the in-plane vibration in the y direc- tion do not contribute to the phase changes.

The same principle can also be used to measure the out-of-plane displacement by slightly changing the optical arrangement. A wel l-known way to do this is to make the two il luminating beams inclined

Chang: Vibration displacement

on the same side with two different angles with respect to the viewing direction. When the illuminat- ing angles are close to tr/2, and realizing that out-of- plane displacement is much greater than in-plane displacement in the problem, the contribution of the relative phase change due to the in-plane dis- placement may be neglected. Therefore, the tech- nique measures the out-of-plane displacement only.

Measuring system and algorithm Figure 2 is the schematic of the fiber-optical system set-up for measuring the in-plane vibration dis- placement component. As shown, a hel ium-neon (He-Ne) laser beam is divided into two beams. The two beams are coupled into the single-mode fibers separately through the optical couplers. One of the beams is modulated by a phase modulator PZT. The phase change in the beam due to the phase modulator is ~bmSin~mt, where ~b,~ and OJn~ are the vibration amplitude and frequency of the modula- tor, respectively. The laser beams through single mode fiber are then focused at angles -+8 onto one point on the surface of the test object, a vibrator. The diffusely reflected light normal to the surface is collected and guided through a lens to a photodiode detector. The vibrator is driven by a function gener- ator with voltage VSin~st, which introduces the vibrator to an in-plane displacement USin~st. The relative phase change caused is shown in Equa- tion (2).

According to the principle of dual-beam inter- ferometry, the phase of the interference term is the summation of ~bsSin~t and 6mSin~mt. Therefore, the resulting intensity of the interference pattern is transferred to an electrical current signal through the photodiode detector and can be expressed as

/ = ~[1 + vCos(~b' + ~b~Sin~0st + ~bmSinoJmt)] (3)

where ~b' is the static phase difference of the two beams introduced by OPD, and s and v are the aver-

~EAMS~UTTER

VIBRMOR r L INTEGPAfO~ LPF

RE~(?RD[R

Schematic of the optical-fiber measuring Figure 2 system

PRECISION ENGINEERING 37

Chang: Vibrat ion d isp lacement

age intensity and the v is ib i l i ty of the interference pattern, respectively.

The term Cos((/)' + q~sSin~t + ~mSin~mt) can be expanded to the Bessel function as fol lows:

Cos(~' + ~S in~s t + ~mSin~mt)

: C o s & ' {Jo(#Ps)Jo(d)m)

+ 2J0(~m) ~ J2/(6~)Cos2i~t) i - 1

+ 2J0(~) ~ J2i(d)m)Cos2i~mt) i:1

+ 2 ~ J2jl4,m)J2~(d,,) i - l j = 1

x [ C o s ( 2 i o ) s t - 2 jco~t ) + C o s ( 2 i ~ t ÷ 2 j ~ , , , t ) j

- 2 ~ ~ J2j_l(~m)J2i l((/~s) i - l j : 1

x [Cos ( (2 i - 1 ) ~ t - (2j .... 1)~o~t)

-- C o s ( ( 2 i - 1 )~%t + (2j .... 1 ) ~ m t ) ] }

- Sin~'{2J0(6m) ~ J2, ~(~)Sin(2i 1)to.~t) i=1

+ 2J0(~ s) ~_~ J2i_l(~m)Sin(2j - 1)~mt) j = l

+ 2 ~. J2i_l(~m)J2ildPs) i = 1 j = 1

x [ S i n ( 2 i % t + (2 / -~. 1 ) co~t )

- S i n ( 2 i ~ , t - (2/" 1 ) ~ m t ) ]

+ 2 ~ ~ J2i(d)m)J2, l((~)s) i = 1 j = 1

x [Sin((2i - 1)~,t + 2j~n~t)

+ S i n ( ( 2 i - 1 )~ t - 2j~,,t)]J. (4)

The induced electric current is amplif ied and converted to the voltage signal. It is then connected to two high Q bandpass filters (BPF) of which reso- nant frequencies are working on ~s and o~,, sepa- rately. Thus, the two output voltages of the filters are

Vs : 2gse [Cos6 ' ~ { J2 i (~ )J< 2, 1 ~'> OZ((~mi L / ' : :1

+ J < 2,, 1 ~" c~(6m)J2,((h~)

- J2i ~(~)J2 <: + ~ > c~((h~,,)

- J2/~(q~m)J2. l((~S ) }

- Sin(/)' {J0(q~m)J~ (d~)

- \~ IJ; , , (~b~)J .... a ( (b . , :

and

~- J2/~:(d,,,;)J; -.i~b.)ji i '

X/,. :: 2g~'v t Cos~b' ~ '£ {J~, ,.d~)J.,.~ . (,~) ;

Sin d, :J , ( , , . : ~J~ (d)

i

J.:. ! (,!~.).. .

j . i d)m / J'.'i " t(."~ J :~,

where ~ is equal to (,~,.(~,,,, d is the gain factor ot the amplifier, and V m and V.. are the referenced and measured voltage signals, respectively.

For small vibrat ion displacement measure- ment, the Bessel function terms higher than an or. der of one can be neglected because ~hs and ~,,, are much smaller than unity, and

el) Jl(#) = } ,

where 4) fore, the output signals can be shown as

V• = g•v ~sSin~ ' SinoJ,t,

and

V m ::= gsv ~6mSind)'Sin(,,~ ~

is a parameter representing ~s or #~. There

i8!

(9i

38 ~ANUARY t994 VOL 16 NO :

BEAMSPLITFER , FRINGE

_ _

PZT LENS DIN

Figure 3 Determination of 6.7 using a Mach- Zehnder interferometer

Then 6s can be obtained from

Vs 6~ = 6~ vm' (10)

and the vibration displacement component to be measured is given by

U - X V~ 4~rSin~ 6m ~--~. (11)

Here, the reference phase change 6,, is modu- lated by a phase modulator, i.e., the quantity of 6m is controlled by controlling the phase modulator. V s and V m are determined from the outputs of the bandpass filters. From Equation (11), the displace- ment component U can be determined by measur- ing Vs/Vm from one point to another on the vibration surface with 6,, kept constant. A single measure- ment can be taken in as little as a few microseconds. The outputs can be generated in real time.

For optimizing the sensitivity of the detection, the technique of phase tracking homodyne with di- rect current (PTDC) 9 was used to lock the static phase difference 6 ' to be ~r/2. Here, the output sig- nal from the PIN photodiode is passed through a Iowpass filter (LPF) and connected to the reference arm to offer an appropriate compensation phase A(t), whereA(t ) is the phase obtained by a feedback bias voltage to the piezoelectric stretching device that stretches the reference arm. If an appropriate feedback voltage can be produced and applied to the PZT, the quadrature condition 6' - A(t) can be locked to approximately rr/2, so that the phase A(t) can be made to drive the error signal to zero.

Determination of ~m The reference phase change amplitude 6m can be changed in a discrete manner by the PZT by applying a discrete voltage to the device. The in- duced phase shift is

6m = hVin (12)

where 6~ is the modulated vibration amplitude that causes phase changes, Vi, is the applied voltage,

Chang: Vibration displacement

and h is the sensitivity factor of the device that is dependent on the coupled length of the fiber around the PZT.

Figure 3 shows the schematic diagram of a Mach-Zehender interferometer set-up for determin- ing the sensitivity factor of a device. Again, the tech- nique of phase tracking homodyne with DC was used. In this case, the output signal from the PIN photodiode is given by

/ m = s[1 + v C o s ( 6 ' - A ( t ) +6mSin~mt)] (13)

According to the expansion of Bessel function,

Cos(6' - A(t) + 6~SinoJmt)

= C°s86IJ°(6m) + 2 ~

- Sin66 2 ~, J2, l(6m)Sin(2k - 1)~mt) k = l

(14)

where 86 is equal to 6 ' - A(t), and the output signal from the bandpass filter is

V~z = 2gavJl(6m)

= 2gavJl(h Vin) (15)

The values of Vmz and Vin can be detected directly, thus h can be calculated from Equation (15), and the quantity of 6m is determined and can be controlled through controll ing the phase modulator.

Vibrations due to multiple frequencies The vibration of a mechanical part is usually gener- ated by multiple frequencies resulting from several vibrating sources. Thus, the measured signal from the PIN photodiode is the summation of various phase changes. The output current can be shown a s

/ = a 1 + z, Cos ' + 6skSin%kt + 6mSincomt k = l

(16)

When the PTDC technique is used, the static phase shift 6 ' can be adjusted to be approximately ~-/2, 6s, and 6m can be seen as much smaller than unity, and the related voltage of the signal can be simpli- fied to

V = g~v 6skSin~skt + 6mSin~mt (17)

Again, the outputs from the two high Q bandpass filters of which resonant frequencies are working on OJsk and (Ji m are

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Chang: Vibration displacement

DISPLACEMENT V,S VOLTAGE GRAPH 5 0 0 H z

DISPLACEMENT V.S VOLTAGE GRAPH 0 k H :~

S

~4

~ q

S ~ 2 .=,

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0 52 "

0 0 1

1 1 3

54

1 94

D R I V E N - V O L T A G E OF V I B R A T O R ( V p )

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~O

6S ho

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[

D R I V E N - V O l T A G E 0~ v l B R A T O R [ V p

DISPLACEMENT V.S VOLTAGE GRAPH 2 0 k H z

DISPLACEMENT V.S VOLTAGE GRAPH " I M t

9

C

=,5.7

B

~ 2 i

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0 0 1

I 7 6 .

b ( 9

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D R I V E N - V O L T A G E OF V I B R A T O R ( V p t

DISPLACEMENT V,S VOLTAGE GRAPH 6 5 k H z

C

Lu

4 , , u

c~ co

0;, •

OF D R I V E N - V O L ] AGE V I B R A T O R ( V p

DISPLACEMENT V.S VOLTAGE GRAPH I Q k H z

24

e 21

O

u. O 1 2

~ ,

~ 6 2 9 5

1 5 ~ - i I •

0 : 0 I

Figure 4

cE4,•

9 3 1

s 22 5 66

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D R I V E N - V O L T A G E OF V I B R A T O R ( V p ) D R I V E N - V O L T A G E OF V I S R A T O R ( V p )

The re la t ionsh ips b e t w e e n m e a s u r e d d i s p l a c e m e n t a m p l i t u d e and driven v o l t a g e s at vibration frequencies of (a) 500 Hz, (b) 10 kHz, (c) 20 kHz, (d) 41 kHz, (e) 65 kHz, and (f) 110 kHz

Vsk = gsv~s ,S in~kt (18) and

V m = gsv~mSinemt (19) and the related vibration ampli tude for a certain

40

frequency o~.~k is given by

A Vs~

J A N U A R Y 1994 VOL 16 NO '

DISPLACEMENT V,S FREQUENCY GRAPH 60O

~ , 500 '

4oo i

~ ~oo 'j

N 1oo

* 10 -Vo l t s

8 Vo l t s

• 6 -Vo l t s

4 Vo l t s

1 4 2 *Vo l t s

° 1 -Vo l t s

0 . . . . . . . . 0 20 40 60 80 100 120

DRIVEN-FREQUENCY OF VIBRATOR(kHz)

Figure 5 In-plane vibration spectrum of a PZT at some driven voltages

Experimental results

Experimental measurements were made for in- plane displacement generated by a piezoelectric transducer. As shown in Figure 2, two laser beams are focused at angles -+45 ° onto the PZT surface. One of the two beams is modulated by a low- frequency phase modulator. The modulator was made by wrapping the fiber 15 times around a PZT cylinder. The sensitivity factor of the modulator is determined first by using the Mach-Zender interfer- ometer mentioned earlier, and the conversion value of ~-/14.9 rad/V was obtained. Because the wave- length of the He-Ne laser is 632.8 nm, the displace- ment component U can then be derived from

632.8 f ~- ~ Vs u-- vm

vs = 1 5 . 1 - (21) vm

The working frequency range of the vibrator is 500 Hz to 110 kHz. Many sampling frequencies were made in these experiments. Some of the measured results are shown in Figure 4, which is a series of plots based on our measurements to show the lin- ear relationships between the driving voltage and the displacement amplitude of the vibrator at fre- quency values including 500 Hz, 10 kHz, 20 kHz, 41 kHz, 65 kHz, and 110 kHz. These plots can be made from a single data set or from the average of many data sets. Some fitted lines do not exactly pass through the coordinate origins, possibly due to ex- perimental errors such as system errors and numer- ical fitting errors. The relationship between the driven voltage and the measured displacement is linear and agrees with the theoretical analysis. The minimum detectable displacement was about 5 nm with an estimated accuracy of 0.5 nm. The sensitiv- ity may be reduced by properly varying the illumina- tion angle 0, the reference phase ~m, and the light wavelength X.

Chang: Vibrat/on displacement

Figure 5 illustrates the relationships between the measured in-plane displacement and the driven frequency of the vibrator under some certain driven voltages. Obviously, the spectrum shows that 10 kHz and 41 kHz are the first and second in-plane vibration natural frequencies of the vibrator, respec- tively. Note that the in-plane vibration spectrum is not easily detected when using other techniques.

The maximum measurable displacement is de- pendent on the speckle size (usually several mi- crometers) because the formation of the fringe pat- tern is based on the speckle correlation principle. The theoretical detectable highest frequency is de- pendent on the detector limitation and the response range of electronic parts.

Conclusions

A fiber-optical method for high-frequency small vi- bration displacement measurement is developed. The feedback system is controlled by the PTDC tech- nique. Experimental measurements were made for in-plane vibration amplitudes using a PZT. The tested frequencies ranged from 500 Hz to 110 kHz. The measured results were in agreement with the theoretical analysis. In contrast to other optical methods, the in-plane vibration spectrum and natu- ral frequencies are more easily obtained with the presented technique, and the measurements can be made via computer control to perform automated detections. Accordingly, this technique could be one of practical and fundamental means for a vari- ety of vibration measurements.

Acknowledgment

I thank the National Science Council of the Republic of China for financial support.

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time shadow moire vibration measurement: method featur- ing simple setup, high sensitivity, and exact calibration," Appl Opt 1986, 25, 3785-3788

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3 Tiziani, H. J. and Klenk, J. "Vibration analysis by speckle techniques in real time," Appl Opt 1981, 20, 1467-1470

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9 Giallorenzi, T. G., Bucaro, J. A., Dandridge, A., S/gel, G. H., Cole, J. H., Rashleigh, S. C. and Priest, R. G. "Optical fiber sensor technology," IEEE 1982, QE18, 626-665

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