axisymmetric vibrations of a cylindrical resonator measured by holographic interferometry

Axisymmetric Vibrations of a Cylindrical Resonator Measured by Holographic Interferometry

Oscillatory displacements of a special sonic resonator, measured by holographic interferometry, are compared with theoretical results

by P. A. Tuschak and R. A. Allaire

ABSTRACT--A circular, cylindrical, ultrasonic resonator excited at one of its resonant frequencies is studied by holographic interferometry. Displacement distributions as- sociated with the axisymmetric oscillations of the resonator are measured with the aid of time-average holograms, and are compared with a simple one-dimensional theory of rod vibrations, corrected for radial inertia. Analysis shows the overall error bounds on measured displacements to be "4"9 percent of the maximum displacement at the resonator tip. Although the accuracy of measurements could be increased by refinements in experimental techniques, the work. report- ed here represents substantial improvement in measuring the vibratory motion characteristics of ultrasonic devices over the point-by-point technique used heretofore.

List of Symbols A = in tegra t ion constant

a - -

B = E =

f n = I "--

Ist ~-"

1r - -

L - "

A l,i. l - - -

A It o ---

N, -----

Pn ---~

resonator radius, m

in tegrat ion constant Young's modulus, N / m ~ nth na tu ra l frequency, Hz l ight in tens i ty of v ib ra t ing point on recon- s t ructed image, W/ma l ight in tens i ty of s ta t ionary point on r e - constructed image, W / m 2 equivalent spr ing constant of piezoelectr ic ceramics, N / m half length of resonator , m

unit vector f rom a surface point t oward source of i l lumina t ion

unit vector f rom a surface point t o w a r d point of observat ion radia l component of sens i t iv i ty vector axial component of sens i t iv i ty vector nth e igenvalue of resonator model

P. A. Tuschalr and R. A. Allaire were Assistant Professor and Teach- ing Associate, respectively, Department of Engineering Mechanics, The Ohio State University, Columbus, OH 43210, when this paper was prepared. P. A. Tuschalr is now Development Engineer, Engi- neering Development Laboratory, E. I. du Pont de Neraours and C o , Wilmington, DE 19898 and R. A. Allaire is with Coming Glass Co , Coming, N.Y: Paper was presented at 1973 SESA Fall Meeting held in Indianapolis, IN on October 16- t9 . Original manuscript submitted: May 23, 1973. Bevised version re- ceived: /une 20, 1974.

A ' r - -

U ( z ) =

U =

W (z ) = u ( z , t ) = w ( z , t ) =

A

A =

# =

r - - -

p - - -

A

~o=

C o =

unit vec to r in rad ia l d i rec t ion t ime r ad ia l -d i sp l acemen t ampl i t ude at i a t e r ~ surface, m

long i tud ina i -d i spIacement ampl i tude, m rad ia l d i sp lacement at l a t e ra l surface, m longi tud ina l disl~lacement, m

uni t vec tor in axia l d i rect ion phase change due to d i sp lacement of su r - face point , rad. correct ion mul t ip l ier , kl ---- 1 wave leng th of coherent light, 632.8 n m Lam~'s constant, N / m 2 Lam~'s constant, N / m 2 nth c i rcular frequency, 1/s density, k g / m s

uni t vec tor in c i rcumferent ia l d i rect ion A A

angle enclosed b y n~ and z ," ,%

angle enclosed b y nt and r , at e = 0 A A

angle enclosed by no and z A A

angle encIosed by no and r, at o = 0 Poisson's ra t io

Introduction Holographic in t e r f e romet ry is an effective tool in in- ves t iga t ing cer ta in v ib ra t ion problems in engineer ing. Most of its appl icat ions to da te have occurred in con- nect ion wi th t h in -wa l l ed s t ruc tura l elements, such as plates, shells, membranes and thin beams. F r e q u e n t l y invest igators seek only the qual i ta t ive mode shapes by holographic means, and when quant i t a t ive da t a are desired, the d isp lacement direct ions are of ten known. In the present paper , holographic i n t e r f e rom- e t ry is appl ied in the s tudy of a solid, t h r e e - d i m e n - sional object execut ing ax isymmetr ic , ha rmonic v i - brat ions. There are two significant d i sp lacement components at each point on the surface of the object and

Experimental Mechanics I 81

41.28 mm

1 2 7 m m

- - P I E Z O E L E C T R I C

D I B K B

s.3-~ ~ ~ e L S C T R = O S mm ~, "-6.3 mm

~ l a 7 rnm

2 6 8 . 2 r n m ( a )

L

I I I O U R C E O F O B l i E R V I N a

I L L U M I N A T I O N P O I N T

Fig. 1--The cylindrical resonator. (a) Dimensions, (b) coordinate system. Pi and Po are contained in the ABCD plane

the direction of the total displacement vector is not known a priori.

The present study was under taken in relat ion to the application of h igh-f requency (10 kHz), high- power, sonic vibrat ions to a wide range of industr ia l processes. Recent interest in power sonics led to the construction of electromechanical transducers having power ratings in the range of 700 W-30 kW. These devices are shaped, resonant assemblies convert ing electrical energy to mechanical vibrat ions via the ac- t ion of piezoelectric ceramics.

While many successful developments have occurred in power sonics, major problems have been encoun- tered in the design and application of sonic t ransduc- ers. In general, t ransducer designs are based on the one-dimensional theory of longi tudinal vibrations of bars, and are either half-wavelength, s tepped-horn devices, or fu l l -wavelength assemblies of a horn and a composite resonator. Considerations of assembly, ceramic prestressing, mount ing fixtures, and provi- sions for electrical insulat ion result in final t ransducer configurations which radically depart from the simple, homogeneous systems of the design principle. In some cases, the lateral dimensions of these devices are comparable to the longi tudinal wavelengths. Un- der such conditions, la te ra l - iner t ia effects become im- portant, and the simple one-dimensional wave theory used for design purposes breaks down.

Since t ransducer behavior is different from that predicted by the best theoretical model current ly available, the vibrat ional characteristics of these d e -

vices must be determined experimental ly. A number of methods have been employed to obtain some vibra- tional information. Most of these have been poin t -by- point measurements which yielded only a crude es t i - m a t e of the basic mode shapes and general ly required the at tachment of some instruments, such as strain gages, accelerometers, etc., to the surface of the t rans- ducers. It became clear that a whole-field measure- ment technique was needed to obtain accurate information on the resonant modes of these transducers. Holographic in ter ferometry is one experimental method capable of yielding the desired data.

The subject of s tudy in the present work was a special cylindrical resonator, Fig. 1, which in most re- spects operates precisely the same way as a typical, h igh-power sonic transducer. The one notable difference is the geometry. For this study, the simplicity of a circular cyl inder offers obvious advantages in that an adequate mathematical model describing resonator motion can be obtained with relative ease. At the same time, the lateral dimensions of the resonator are sufficiently large that radial inert ia effects are significant, as they are in most current, power- sonic transducers. Consequently, successful experience gained in the present study could be used con- fidently in any fu ture investigations of more practical t ransducer configurations.

Theory Analysis of Resonator Motion

A circular, cylindrical bar of radius a (Fig. 1), and length 2L is harmonical ly excited in the axial direc- t ion by piezoelectric ceramics. The ceramics are placed such that the bar is divided into two sym- metric halves. The resul t ing assembly oscillates sym- metrical ly with respect to the center electrode and, therefore, can be analyzed as a bar of length L connected to a rigid wall by the prestressed ceramics. The latter will be modeled as elastic springs 1 whose spring constant can be determined from material properties by the s t rength of materials approach (Fig. 2). Pre l iminary t ime-average holograms showed that, in addition to the longi tudinal oscillations of the resonator, substantial radial displacements also occur. In order to account for these, a modified Mindl in-Herr - mann theory 2 was used to model resonator motion. In the modified theory, the effects of radial inert ia were included, while radial shear was neglected. In this way, the analysis accounts for the principal modes of motion, yet undue complications are avoided.

With the above assumptions, the equations of mo- t ion are

-- 8K:~(~. + ~)u - - 4a~lXW' = pa2u, (1)

2Kla~u' + a2(~ + 2~)w" = pa~w (2)

where w is the longi tudinal displacement (m), u the radial displacement at the lateral surface (m), ~: a correction constant taken as 1, ~, ~ Lam~'s constants (N/m2), p, the mater ia l density (kg/mS), and a the bar radius (m). A prime denotes differentiation with respect to axial position, and a dot denotes differentiation with respect to time. A s s u m i n g

u = U(z) sin~t , w = W ( z ) sin~t

82 I March 1975

n = 'Z z.k Fig. 2--The resonator is modeled as a cylindrical rod in series with a spring of constant k(N/m}

and el iminat ing u from eqs (1) and (2), the equation for de termining the normal modes is given as

W" + p,2 W = 0, (3) w h e r e

p2a4w.4 - - 8pa2c~n2(k --]-/~) p,2 = (4)

pa4~,~(k + 2~) - - 8aS~(3X + 2~)

and =n ( l / s ) is the n th na tura l f requency of the system. The solution of eq (3) is subject to the following boundary conditions:

at z----0 W'----0, (5)

at z = L :~[2a~ .U+a20 .+2~)W'] = - - k W , (6)

where k ( N / m ) is the equivalent spring constant of the ceramics, and

4akW' U = (7)

pa2~n 2 - - 8 (~. + /z)

Equations (5) - (7) express the conditions of force balance at the ends of the bar.

The general solution of eq (3) is

W---- A sin Pn z + B cos Pn z. (8)

Subst i tut ing eq (8) into the boundary conditions yields

A = 0, (9)

and the frequency equation

kL tan pnL -- (10)

~a2E(1 -- 5) (pnL)

where E is Young's modulus, and 5 (O~n) is a small correction term due to radial inertia. Solving eqs ( 1 0 ) and (4) numer ica l ly with k = 8.8 • 109 N / m and n = 3 the frequency obtained is

=s -- 158,000 1/s (ii)

w3 f8 ---- = 25,100 kHz

2~

This agrees well with the th i rd-mode frequency ob- served exper imenta l ly by nonholographic means. Once o~s is determined, the third normal mode is given by

W ---- cos 0.79z (12)

U = --0.25 sin 0.79 z. (13)

I R ~

M R ' *

Fig. 3--Experimental arrangement for recording holograms, Laser (A), beam-splitter (B), spatial filter (C), mirror for illuminating beam (MI), mirror for reference beam (MC), hologram (H), resonator (R)

Equations (12) and (13) will be compared with nor - malized exper imental displacement distr ibutions.

Holographic In terJerometry

Holography is a technique of image forming and reconstruct ion in which both the phase and the ampli tude of a wavefront scattered from an object i l lu- minated by coherent l ight are recorded. A coherent beam of light is split into two beams which are ex- panded with the aid of spatial filters. One beam is used to i l luminate the object, the other beam is re- tained for use as a reference. A photographic plate is placed in such a way that light scattered by the object and the reference beam interfere at the plane of the plate exposing the emulsion to a spat ial ly va ry - ing light intensity. The developed plate, the hologram, contains dark and bright fringes in accordance with the interference of the reference and object beams. In the second step of the holographic process, either a v i r tual or a real image of the object may be reconstructed when either the same reference beam or its conjugate, respectively, i l luminate the hologram. The dark and light interference fringes now act as a diffraction grat ing which recreates the wave- form that was scattered from the object in the image- forming process). 4

If several sl ightly different positions of an object are recorded in the same hologram, upon reconstruction, the wavefronts corresponding to each position will interfere with each Other. If the hologram is recorded while the object in question executes harmonic vibrations, there is a continuous change of positions for each surface point between the max ima of v ibra- tion. Powell and Stetson showed 5 that the resul t ing hologram is equivalent to a hologram made with the object occupying all of its positions from peak to peak, s imultaneously. The reconstruct ion of such a " t ime-average" hologram wil l show the object cov- ered with a set of bright and dark fr inges and the in tens i ty at a point (x, y, z) on the image is given by

I ( x , y, z) = Jo2(~r Is t(x , Y, z) (14)

where J0 is the zero-order Bessel function, ~b (rad) the phase difference at point (x, y, z) caused by the


m a x i m u m displacement of the point, and Ist (x, y, z) is the in tensi ty of the point on an image of the sta- t ionary object. When I (x , y, z) is zero, a dark fringe appears on the image; when it is a relative maximum, a bright fringe appears . Dark fringes appear when- ever ~r is a root of the zero-order Bessel function; consequently, the fringe pat tern visible on the image is related to wavefront phase changes at points on the surface of the object. These phase changes are given by Ref. 6,

2 ~ A A ~r = U �9 ( ni + no) (15)

A

where A is the wavelength of the coherent light, U A

is the zero to peak displacement vector at a point, nt is a uni t vector from the moving point toward the

A

source of i l lumination, and no is a uni t vector from the point toward the observer. One can see that he

A A is max imum when the angle enclosed by n~ -t- no and U is a minimum. In order to obtain three components of the displacement vector at a given point, it is only necessary to know the phase change due to oscillations of the point, corresponding to three different points of observation. Equat ion (15) can then be

A A wri t ten three times, once for each value of nt and no, yielding a set of three simultaneous, l inear equations in the three components of the vector, U.

Experimental Procedure

Resonator

The dimensions of the resonator are given in Fig. 1 (a), and the coordinates locating points on its surface in Fig. 1 (b). The resonator was fabricated from cold-roUed steel with a Young's modulus, E = 20.3 • 104 MPa (2'9 • 10 '6 psi), and Poisson's ratio ~ = 0.29. The dr iving elements were two piezoelectric ceramic rings symmetr ical ly placed relative to a copper washer which acted as an electrode. The ceramics and the copper washer are electrically insulated from the s t ructural elements of the resonator everywhere except at the two outside flat surfaces of the ceramic rings, where these are in contact with the steel body

of the resonator. The driving voltage is applied between the copper washer and the main resonator body. In assembling the resOnator, a prestress is applied to the piezoelectric ceramics. The driving voltage causes the ceramic stress to oscillate about this prestress level, and contact between ceramic and resonator is never broken.

The resonator was mounted on knife edges whose locations coincided with two longi tudinal displacement nodes when the resonator was excited in its third mode (Fig. 4). Since radial motion also o c c u r s ,

the resonator undergoes some vertical r igid-body displacements dur ing operation. This motion is small, and is essentially perpendicular to the "sensitivity

A A vector", (ni -t- no) in eq (15), therefore it does not affect the results substant ia l ly in t ime-average holographic interferometry.

Only th i rd-mode response of the resonator was studied because the main purpose of this investiga- tion was not to collect data on the resonator, bu t to determine whether accurate quant i ta t ive measurements could be obtained by t ime-average holographic interferometry. The third mode possesses a desirable geometry by vir tue of the symmetry of the nodes, and it was chosen for convenience.

Electronics

A wide-range, 5-Hz-600-kHz, oscillator connected through an ultrasonic power amplifier excited the resonator at 30 V peak- to-peak input level. Paral lel connections to a f requency counter and dua l -beam oscilloscope helped monitor the input. A magnetic transducer, shown in Fig. 4, aided in tun ing the resonator at its third resonant frequency, 25.1 kHz.

Holographic Sys tem

The ar rangement of holographic components is shown in Fig. 3. A 15 mW, HeNe, cont inuous-wave laser (A) producing coherent l ight at 632.8-mm wavelength was used with a beam spl i t ter -a t tenuator (B), two spatial filters (C), two mirrors (MR and MI), and the hologram plate holder (E). The beam reflected from MR was used as the reference wave, that reflected from MI was the i l luminat ing wave. The experiments were conducted on a heavy, v ibra- tion-isolated, steel plate. Holograms were recorded on Agfa-Gevaert 10E75 emulsion wi th 4-s exposures.

Fig. 4--Experimental arrangement for mounting the resonator and ,monitoring the frequency

84 [ March 1975

The plates were developed, stopped, washed, fixed, dried and reconstructed with the o r ig ina l reference wave. For observation of the vir tual image, a tele- scope and various conventional cameras were used. Data were taken from photographs of the reconstructed wavefronts.

Data Reduction The motion of the cylindrical resonator was ob-

served to be axisymmetric. This means that each surface point had an axial, and a radia l -displacement component. The dis tr ibut ion of these displacement components along any generator of the resonator surface was identical to the distr ibution along any other generator. Therefore, the authors took data along a single generator, l ine AB, Fig. 1 (b).

In order to obtain the displacement components from eq (15), the phase difference, z$~, must be known along line AB as a function of z. This information was obtained from the holograms by locating each dark fringe by its z coordinate and assigning to each fringe location the corresponding value of the phase difference, ~ , from the relationship,

J02 ( ~ ) = 0.

Between dark fringes, the phase difference was determined by interpolation.

The two components of displacement U ( z ) and W (z) can be calculated if one has two equations such as eq (15). Dhir and Sikora s showed that, for a general, three-dimensional displacement field, results obtained from exactly three equations contained very large errors. On the other hand, an overdetermined set of l inear s imultaneous equations, solved by the method of least squares, el iminated most of the error. The accuracy of the results increased with the n u m - ber of equations used in calculating the displacements, however, even a set of four equations yielded sub- s tant ial ly bet ter accuracy than any combinat ion of three would have. In the present work, three holograms of the resonator were made in order to provide three substant ia l ly different observation points for the same i l luminat ion point. This way, the phase equations obtained from the various holograms would have sufficiently different parameters in order that i l l - conditioned systems of equations might be avoided. The average sensit ivity vectors enclosed approximately 20-deg, 40-deg, and 60-deg angles with the z axis. Thus, three systems of two simultaneous equations each were obtained for the calculation of the two displacement components. Appendix A gives details of the least-squares solution of these equations.

To determine fringe position on the object surface, a grid system was scribed d i rec t ly on the object at known positions. With the object mounted in its test position a photograph was taken of the object with the camera in the same location as used for the holographic data record. This photograph clearly showed the scribed lines and made it possible to plot the photographic position vs. the actual surface position. The fringes could now be accurately located anywhere on the object surface, the photographic distortions having been eliminated.

The i l luminat ion and observation in this project were of a spherical nature; thus the sensit ivity vector varied from point to point along l ine AB, Fig. 1 (b).

In calculating the displacement components, this var i - ation of the sensi t ivi ty vector was taken into account. The coordinates of the points of i l lumina t ion and observation were noted and along with coordinates on the resonator (along l ine AB) were entered into a

A A computer program. The uni t vectors n~ and no, and

A ^ the sensit ivi ty vector ( n~ -t- no), were then electroni- cally calculated for each surface point of interest.

Results Figures (5) and (6) show reconstructions of holo-

grams with average sensit ivity vectors incl ined at 20 deg and 40 deg to the z axis, respectively. A third hologram with a 60-deg sensit ivi ty vector, used in data reduction, is not shown. Figure 7 shows the reconstructed image of a hologram made with a 90-deg sensit ivi ty vector. It is clear from the lat ter that the radial displacement components are not negligible.

The measured displacement dis tr ibut ions are compared to theoretical results in Fig. 8. In order to br ing these two results, theory and experiment, to a com- mon scale, the exper imenta l displacement ampli tudes

Fig. 5--Photograph of a time-average hologram of the resonator excited at its 3rd natural frequency, f3 ---- 25.1 kHz. The "sensitivity vector" and the resonator axis en- close a 40-deg angle

Fig. 6--Time-average hologram of the resonator excited at f8 ---- 25.1 kHz. Sensitivity vector and resonator axis en- close a 20-deg angle. Note the increase in the number of fringes visible compared with Fig. 5

Experimental Mechanics ] 85

were divided by the measured longi tudinal ampli tude of the resonator tip, 8.4 X 10 -4 mm. In the theory, the tip ampli tude was taken as 1. The errors involved in the measurements were analyzed following a scheme presented by Matsumoto, e t al . 9 In this ana ly- sis, errors in de termining the fringe numbers and errors in the sett ing zoordinates were considered. Sett ing errors were found to be small, less than • 1.3 percent of tip displacement, and fringe errors varied along the resonator axis depending on the sharpness of the fringes. The m a x i m u m error due to fringe reading was found to be • 7.5 percent of tip displacement. Details of the error analysis are given in Ap- pendix B.

The theoretical predictions are everywhere wi thin the total e r ro r bounds of • 8.8 percent except at z = 100 ram, where the agreement between theory and exper iment is poor. This can be explained, at least in part, by the fact that the resonator is not homogeneous in this section, but has a threaded hole into which a connecting stud is inserted, Fig. l ( a ) ; the theory, on the other hand, is based on the assumption of homogeneity. The shape of the experimental curve for radial displacements agrees less well with the theoretical curve than the longi tudinal curve does, but the magni tude of predicted, radial displacements is everywhere wi thin the error bounds.

In order to draw conclusions on whether the results presented here represent good agreement between theory and experiment, or not, one must know what the purpose of the measurements was in the first place. Admittedly, more accurate holographic results have been obtained by other investigators, and refinements of the authors ' exper imental techniques in this project probably would have yielded better results. However, the main purpose in s tudying the sonic resonator of this paper was to gain experience in apply- ing holographic techniques in the design and development of other sonic, and ultrasonic devices. Inasmuch as no whole-f ield-displacement information was available for the type of device studied here before the current work, even errors of _.+ 9 percent are ae-

Fig. 7--Hologram of resonator excited identically with Figs. 5 and 6, but with the sensitivity vector almost normal to the resonator axis. The influence of radial motion is shown by the long horizontal portion Of the dark fringe

ceptable and reasonable. Since completing the resonator study, the authors carried out similar studies of actual, practical sonic transducers, developed at The Ohio State Univers i ty Sonic Power Laboratory, and the results of these later investigations have already been utilized in fur ther t ransducer developments.

References 1. Feng, C. C., "'Analysis of Sonic Transducers," PhD Diss.,

The Ohio State Univ. (1973). 2. Mindlin, R. D. and Herrmann, G., "'.4 One-Dimensional Theory

of Compressional Waves in an Elastic Rod," Proc. 1st U. S. Nat. Cong. AppL Mech., 187-19I (1951).

3. Collier, R. f , Burekhardt, C. B. and [,in, L. H., O~tlcal Holog- raphy, Academic Press, Inc., New York and London (1971).

4. Brown, G. M., Grant, R. M. and Stroke, G. W. , "'Theory of Holographic Interferometry,'" J. Acoust. Soc. Am., 45 (5), 1166- 1179 (1969).

5. Powell, R. L. and Stetson, K. A,, "'Interferometric Vibration Analysi~ by Wave#ont Reconstruction," f. Opt. Soy. Am., 55 (12), 1593-1598 (Dec. 1965).

6. Stetson, K. A., "'A Rigorous Treatment of the Fringes ot Holo-

Fig. 8--Experimental and theoretical longitudinal and radial-displacement distributions. Measured displacements were normalized for this comparison

F- Z W

W 0

0.~

--- 0 a a I,i N "-i_O5

0 Z _ I

I

,,,%\wcz

\

u(z)

~x

0 25.4

THE ORY o--_:! EXf'E RIMENT

~/ERROR BOUNDSq

50.8 76.2 101.6 AXIAL POSITION, z (mrn)

127

86 ] March 1975

gram Interferometry,'" Optik, 29 (4), 386-400 (1969). 7. Dhir, S. K. and Sikora, 1. P., "'An Improved Method for Ob-

taining the General-displacement Field from a Ho',ographie Inter- ferogram,'" EXpEI~I1VKENTAL MECHANICS, 1Q (7), 323"327 (1972).

Two equations are obtained in this way in terms of the two unknowns, W and U. Solving these equations we obtain for W and U:

W =

A U ~ ,

2~

A I' l i ] I i l I 1 2 - Nr,, 2 Nz,,,i {=1 / = 1 t = 1 i = 1

N z t N n - - Nz~ 2 N,.iz i = 1 i = 1 / = 1

8

[ ~ Nz~Nn ~=i

(A-4)

( A - 5 )

8. Matsumoto, T,, Iwata, K. and Nagata, R., "'Measuring Ac- curacy of Three-Dimensional Displacements in Holographic Infer- ferometry,'" Appl. Optics, 12 (5), 961-967 (1973).

APPENDIX A

Least-squares Solution of the Phase Equations In order to obtain an overdetermined set of l inear

simultaneous equations for the calculation of two orthogonal displacement components, three holograms

A were made, each with different average values of n~

^ and no. Each hologram yielded a different phase relationship, eq (15); thus, there were three equations to solve for two unknowns, U ( z ) and W ( z ) . The three equat ions were solved by the me thod of least squares. In eq (15), let

A A A A n~ + no - - N z z + N r r

A A where z and r are unit vectors, and Nz and Nr are

A A components of the vector ( n~ + no) in the axial and radial directions, respectively. The phase equations then become

2 ~ Ar = ( W N z ~ + U N t O i = 1, 2, or 3. ( A - l )

A

The two sides of eqs A-1 are not exact ly equal, but differ by an amount d~:

A d ~ = W N z i W U N n ~ A~ i - - l , 2, or3.

(A-2)

The d{ are the results of exper imenta l error. The principle of least squares requires the quanti ty,

d,,= wN.i+ , , , , , i = 1 i = 1

(A-3)

to be minimized with respect to W and U. To accom- plish this, first differentiate both sides of eq (A-3) with respect to W , and set the de r iva t ive equal to zero, then repeat the same thing wi th respect to U.

The displacements calculated f rom eqs (A-4) and (A-5) were plotted in Fig. 8, af ter dividing through by the tip displacement, 8.4 • 10 -4 mm, in order to compare the results to nondimensional ized theoret ical predictions.

APPENDIX B

Error Analysis Recent ly Matsumoto, e t al. 9 invest igated the errors

of th ree-d imens ional displacements in holographic interferometry. The fol lowing analysis is based on their work.

Two sources of error will be considered. One is the error in reading fract ional fringes. In the current work these proved to be the most significant. Errors due to sett ing the i l luminat ion and observat ion d i rec- tions were of lesser consequence.

Fringe-reading Errors The optical phase on the surface of the resonator

was determined in the fol lowing way. The in teger - valued fringes were found by observat ion and were located re la t ive to a grid superimposed on the resonator surface. The phase difference A~, at the in teger - valued fr inge locations was de termined and was plot - ted vs. axial position (Fig. B - l ) . The phase difference for fract ional fr inge numbers was de termined by l inear interpolation. The average error i n locating an in teger -va lued fr inge was est imated to be • 1.3 ram. The errors due to l inear in terpolat ion were small along the resonator axis except in the v ic in i ty of z = i 0 0 mm. There the phase curves, Fig. B- l , are s t rongly nonl inear and l inear interpolat ion is not v e r y accurate. It should be ment ioned that the measured displacements in the region around z -- I00 mm agree poorly wi th theoret ical predictions.

A Let the i l luminat ing and observing directions, nl

A and no, form the angles ~J, ~i j and ~o j, ~o j wi th the z and x axes, respectively. Then,

A A A noJ ---- COS ~o j r --~ cos ~o j Z

(B-l) A A A n~J = cos 42 r + cos hJ z


~4

12

~ s

~ o ~_ UJ

~ LL

~ . 8

IO

12

14

16

18

0 25.4 50.8 762. I01~ 127 AXIAL POSITION, z (ram)

Fig. B-l'Phase-difference distribution for three different holograms of the resonator

A A where r and z are un i t vectors in the x and z directions. Equat ion (15) can now be wr i t ten in the following form

A h~# : A �9 U (B-2)

where

U(z) }

u = W(z) ' {h~, }

and

A = f COS ~il -h COS ~ol cos ~'tl -h coS ~'ol t cos ~2 + cos ~o2 cos ~2 + cos ~o~

If the reading error in the phase difference is 8(h,r

then the following equat ion may be wri t ten:

A -- [4r -5 8 (h~)] = A �9 (U + 8U) (B-3) 2~ . . . . .

In view of eq (B-2), we have A

~v = A - 1 . ~ ( ~ ) 2-'L ( s -4 )

where

~ = ~ w ' b ( h ~ ) = O(h~2)

and A -1 is the inverse of A. Assuming that 18U I =

[SW] __- 8u and ~(hr ---- 8(hr = 8(hr and using eq (11) of Ref. 9

A ~u_~ 5 ( h ~ ) I IA -11t2 - - (B-5)

2~

where IIA-~[12 is a norm of the mat r ix ~A -1, defined

as follows:

f ]B1:1~ = (max imum eigenvalue of BrS)1/%

When the observing and i l lumina t ing s y s t e m s a r e determined and the reading error 8 (he) is estimated, the resul t ing error in the displacement components, 8u, can be calculated. Three sets of data were col-

A A lected with three sets of nt -- s and no -- s. The three sets of data can be combined into three groups of two sets each, which subst i tuted into eq (15) yield three systems of l inear s imultaneous equations in the two variables U(z) and W(z). Of the three sets of data, the ones with average sensit ivi ty vectors at 40 deg and 60 deg yielded the least accurate results compared with the least-squares solution of Appendix A. Con- sequently, the 40-deg and 60-deg data were used for de te rmin ing the typical reading error. The sett ing angles for this case were:

h I : 120 ~ ~o 1 = 140 ~ ~'t I = 210 ~ ~'o I = 230 ~

er o, eo 2_160 o, ~r o, fo ~ = 2 5 0 . o

With these angles, I IA-1]]2 _-- 2.08. With f r inge-place-

ment errors of _ 1.3 mm the average error in phase difference was 0.3 rad. Subst i tu t ing these values into eq (B-5), we find the error in displacement to be

5 u --~ 6.3 • 10 - s m m ,

or _+ 7.5 percent of the tip displacement.

Setting Errors of the Illuminating and Observing Directions

In a typical setup, such as the one with a 20-deg sensit ivity vector, the point of i l luminat ion and point of observation are each approximately 610 mm from a point on the resonator. It was estimated that the coordinates of points on the holographic bench can be determined with an error of -+- 2.6 mm. Accord- ingly, the typical error in sett ing angles is he -- 0.0041 rad. Equat ion (2) of Ref. 9 gives the error in displacement due to a setting error of he:

where

~U

I[U112 h, ]L_A-fll2 [EhAo]I2

------- ( B - 6 ) W

[[~11~-- (r~-5 w2)z/= and

I ]s in ~oll -5 [sin ~11 ]sin ~1] -t-Isin roZl 1 hAo

Isin ~o2l-5 ]sin ~r Isin Cr 2] -5 ]sin Co2lJ Subst i tu t ing numerical values into eq (B-6) we have

llhAoll2 = 2.73

and 8 u

~ 0.0134 ll~UIl~

or the error is _ 1.3 percent. Combining this result with the estimated reading error, it appears that the overall expected error in displacements is within _ 9 percent of tip displacement, Fig. 8.

88 t March 1975

axisymmetric vibrations of a cylindrical resonator measured by holographic interferometry

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