Ultrasonics 37 (1999) 231–238

Extracting modal parameters of ultrasonic bar hornsfrom ESPI FRF data

G. Graham a, J.N. Petzing b,*, M. Lucas ca Department of Mechanical Engineering, Loughborough University, Loughborough, Leicestershire, LE11 3TU, UK

b Department of Manufacturing Engineering, Loughborough University, Loughborough, Leicestershire, LE11 3TU, UKc Department of Mechanical Engineering, University of Glasgow, Glasgow, G12 8QQ, UK

Received 4 July 1997; received in revised form 4 August 1998

Abstract

This paper demonstrates the problems associated with determining the modal parameters of curved surface structures whichexhibit high modal density and a predominant longitudinal response. The inadequacies of inferring modal parameters solely fromconventional normal-to-surface measurements, typically using a laser Doppler vibrometer (LDV ), are highlighted. A measurementsolution is proposed which relies on extracting a grid of surface in-plane vibration amplitude and phase data, which is resolvedfrom two laser speckle fringe patterns using electronic speckle pattern interferometry (ESPI ).

In order to obtain frequency spectrum data by in-plane ESPI, successive surface responses are captured through a swept-sinetest and fringe processing algorithms are used to calculate a vibration amplitude and phase map at chosen grid positions on thestructure’s surface, at selected frequency steps. Further, by measuring an electrical signal input to the driving transducer, frequencyresponse function data is obtainable such that modes of vibration estimated by modal analysis can be correlated with finiteelement model data. The technique is demonstrated by modal analysis of ultrasonic bar horns such that the longitudinal modesare successfully identified. © 1999 Elsevier Science B.V. All rights reserved.

Keywords: Analysis; ESPI; Interferometry; Machining; Optics; Ultrasound

1. Introduction problem with ultrasonic systems due to often encoun-tered high modal density around the operating frequency

The development and application of high power ultra- and the problem is exacerbated when multiple resonantsonic techniques in manufacturing processes such as elements are required between the transducer and thecutting, welding or machining, generally requires the workpiece.use of specifically designed ultrasonic components to The ultrasonic components examined during thiscorrectly transmit the energy from the transducer to the study are tuned longitudinally resonant bar horn trans-tool/material interface. Whilst the design detail of the mission units used in multi-component ultrasonic sys-ultrasonic tools are task dependent, the common criteria tems, nominally designed to operate at 20 kHz and tois that all components are required to resonate with transmit ultrasonic energy across large distancesspecific mode shapes, and will have commonly tuned (metres), compared with more traditional systems. It isresonant lengths (generally half wavelengths). essential that the modal characteristics of these compo-Maintaining the tuned operational mode with the nents are determined (mode shapes, frequencies, damp-required working amplitude is critical to the successful ing values) in order to identify problematic couplingperformance of the ultrasonic system. Any form of modes (flexural, torsional ). Traditional modal analysiscorruption of the desired operational features due to using accelerometer-based experimentation is prohibitedmodal coupling will lead to the less efficient operation

in this instance by the large surface accelerations whichof the high power ultrasonic system. This is a common

tend to dislodge the transducers, and small radii ofcurvatures causing mounting difficulties for the trans-ducers. Previous studies of high power ultrasonic compo-* Corresponding author. Fax: +44 1509 223934;

e-mail: j.petzing@lboro.ac.uk nents have therefore relied upon laser Doppler

0041-624X/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved.PII: S0041-624X ( 98 ) 00051-1

232 G. Graham et al. / Ultrasonics 37 (1999) 231–238

vibrometer (LDV ) analysis to monitor the normal-to- in-plane and out-of-plane response from the highlycurved surface in a format which is transferable tosurface (out-of-plane) component of vibration [1].

Measurements taken from a grid of positions along the commercial modal analysis software packages.sides and ends of the bar horns, have been used toinitially identify longitudinal and flexural modes. 2.2. FRF measurementsHowever, verification of modal behaviour is difficult inthis case because the components are designed to exhibit Finite element analysis (FEA) provides a powerful

computational tool in the design of high power ultra-a predominantly longitudinal response, hence the mea-surable normal-to-surface component of motion can be sonic horns. However, predicted component behaviour

can have a poor correlation with actual componenteasily misidentified as a flexural mode response.Accurate identification of modal characteristics is behaviour and therefore, to improve model confidence,

it is necessary to generate experimental results whichdependent upon having comprehensive analysis dataand this necessitates the measurement of out-of-plane will allow a comparison of the estimated modal parame-

ters. Hence, modal analysis data is used to provide(OOP) and in-plane (IP) surface motion components.LDV instruments capable of in-plane response measure- validation and improvement of the FEA models of

dynamic components being studied. Most commonly,ments are now commercially available but their accuracyin modal analysis relies on taking measurements from a modal parameters are estimated from measured fre-

quency response functions (FRFs) by adopting anpredefined grid of data points and alignment errors canbecome significant for surfaces with a small radius of appropriate curve fitting method using commercial

modal analysis software. Modal frequencies and damp-curvature. Electronic speckle pattern interferometry(ESPI) has previously been used on a qualitative basis ing values can be compared directly with FEA predic-

tions and usually mode shapes are compared using anto measure wholefield vibration displacement of ultra-sonic components and structures [2]. This work has appropriate correlation such as modal assurance crite-

rion (MAC ).now been extended to provide quantitative analyses ofOOP and IP displacement components, which can be This study was therefore concerned with manipulating

data from ESPI measurements in a form which wastransformed into a format suitable for further dataprocessing using commercial modal analysis software, comparable with conventional modal test measurements.

The requirement is, therefore, that ESPI measurementsgenerating frequency response functions (FRF ) anddamping values. In this instance, the analysis is made are formatted as FRF data, so that modal parameter

estimations can be performed using modal analysismore challenging because of the curvature of the barhorns, which must be taken into account when calibrat- software and ESPI can be used along with, or as an

alternative to, LDV and accelerometer measurements.ing the wholefield measurement data. ESPI can thereforenow provide an alternative or complementary techniqueto LDV FRF measurements. 2.3. In-plane and out-of-plane measurements using ESPI

Electronic speckle pattern interferometry (ESPI), orless commonly TV-holography, is the name given to2. Requirements of ESPI for modal analysisseveral laser speckle-based techniques which are usedfor measuring discrete displacement components. Two2.1. Ultrasonic componentsdifferent optical designs are available, one to produceout-of-plane (OOP) displacement data and the other toUltrasonic machining systems of all types have

common design foundations. Components are usually produce horizontal and vertical in-plane (HIP and VIP,respectively) displacement data, all of which rely onhalf-wavelength units or multiples thereof, generally

based around the axial longitudinal velocity of sound CCD-TV cameras at the image plane for the recordingof data [4,5]. A schematic of the out-of-plane ESPI[3] within the respective component material (usually

an aluminium or titanium alloy). The component exam- system used for the modal analysis is shown in Fig. 2,showing the illuminating object wavefront, the camerained in this study (Fig. 1) was a cylindrical bar horn

(22 mm diameter×122 mm long) manufactured from axis and the reference wavefront. The speckle patternobserved on the bar horn is recorded at the image planean HE15 aluminium alloy, this being one of 16 used in

the ultrasonic system. Each bar horn is designed to have of the CCD-TV camera and is then post-processed inorder to generate the correlation fringes describing out-in-plane longitudinal displacement antinodes at the ends

of the unit, with a node at the centre of the unit. The of-plane displacement (Z-axis).A different optical design is required for the measure-length of the bar horn depends on the speed of sound

within the aluminium alloy. The X, Y, Z axis set used ment of the in-plane displacement components and isshown in Fig. 3. In this system, two illuminating wave-to describe motion is defined in Fig. 1. For successful

modal analysis, ESPI must be capable of measuring the fronts are used at equal but opposite angles to the

233G. Graham et al. / Ultrasonics 37 (1999) 231–238

Fig. 1. Bar horn schematic.

Fig. 2. Out-of-plane ESPI system schematic.

surface normal camera axis. As in the previous optical sity of the light (I ) for any point at the image plane, isproportional to the square of the total light field, i.e.design, the speckle pattern observed on the bar horn is

recorded at the image plane of the CCD-TV camera I3UTUT* , which, via mathematical manipulation, canbe reduced to a term synonymous with speckle met-and is then post-processed in order to generate the

correlation fringes describing in-plane displacement. The rology:in-plane interferometer may be used with the objectbeams horizontally orientated to record horizontal IA=I

1+I

2+2EI

1I2

cos y. (1)in-plane displacement components (X-axis) and with theobject beams vertically orientated to record vertical In this instance, the descriptor IA is used as a means of

identifying the intensity distribution of the object in itsinplane displacement components (Y-axis).For both forms of interferometer, the total light initial state.

If the object receives a dynamic displacement (but areceived at the image plane from the two illuminatingwavefronts is U1+U2=UT. More importantly, the inten- pulsed laser is used to strobe the surface with the correct

234 G. Graham et al. / Ultrasonics 37 (1999) 231–238

Fig. 3. In-plane ESPI system schematic.

synchronisation), an optical phase change D(x, y) is displacement (w), such that:introduced into the object wavefront due to the displace-ment. Using the same arguments as for the initial object w=

nl

2(4)

state, the new intensity distribution at any given pointcan be described as:

where n is the fringe order number and l is the laserwavelength.IB=I

1+I

2+2EI

1I2

cos(y+D ). (2)Similarly, for the in-plane interferometer, the optical

phase change (and fringe order) can be related to theThe interferometer data is displayed on a TV monitorhorizontal and vertical in-plane displacements (u and v,as correlation fringes, which may be formed via arespectively), such that:subtraction process using software or hardware-based

frame stores. For subtraction fringes, the change ofu=

nl

2 sin hv=

nl

2 sin h. (5)intensity between the initial object state and the dis-

placed state (dI=IA−IB) is produced by storing theinitial undisplaced intensity distribution as a referenceimage and subtracting all subsequent displaced intensitydistributions from this reference. This process is 2.4. Fringe analysis and data manipulationdescribed mathematically as:

Quantitative wholefield data is generated by applyingphase-stepping techniques to the subtraction ESPI corre-dI=−4EI

1I2

sin(y+D ) sinD

2. (3)

lation fringes [6 ], although this methodology assumesthat the dynamic condition of the target object isresonantly steady state. Initially, modulo 2p grey scaleThis brief summary of laser speckle physics describes

the formation of the correlation fringes. Knowing the maps (0–255) of ‘wrapped’ optical phase are produced,which are then ‘unwrapped’ using suitable mathematicalinterferometer sensitivity function, the optical phase

change (and fringe order) for the out-of-plane interfer- algorithms based on cosine transforms. The opticalphase data requires calibration with respect to theometer can now be directly related to out-of-plane

235G. Graham et al. / Ultrasonics 37 (1999) 231–238

Fig. 4. Schematic of the experimental equipment.

interferometer used (Eqs. (4) and (5)) and the geometry The wholefield analysis of the bar horn used an out-of-plane (OOP) ESPI interferometer and an in-planeof the object being examined, resulting in wholefield

calibrated displacement and vibration phase data sets (IP) ESPI interferometer with a pulsed Nd: YAG laseras the illumination source (l=532 nm at 25 Hz). Aand plots. Specifically, the out-of-plane data requires

sensitivity adjustment with respect to the constant curva- schematic of the experimental apparatus is shown inFig. 4. The pulsed laser was used in order to avoid theture of the bar horn, because the interferometer will

only measure true out-of-plane response along the line generation of time-averaged interferograms typical of acontinuous wave (CW ) laser-based analysis, althoughnormal to the camera axis. As the object curves away

from this line, the interferometer becomes less sensitive synchronization of the pulsed laser with the TV cameraand the bar horn resonant frequency was necessary, into the object motion.

The wholefield data sets produced are 512×512 in order to obtain consistent stationary fringe patterns.The phase stepping operations were controlled from asize, but the modal analysis software used (Star Modal )

could only import and handle a maximum of 200 mesh PC-based ITI 15040 image processing system, whichcalculated the optical phase distribution and producedpoints. Consequently, the interferometric data was

passed through subsampling routines developed in the calibrated 512×512 data files. The software usedfor the modal analysis was Star Modal, which wasMatlab which also correctly configured the data formats

for the modal analysis package. This methodology was limited to a total of 200 input data points with the barhorn model only requiring 66 data points. Therefore,applied to the amplitude and vibration phase data

derived from the wholefield optical analysis. the wholefield optical results required sub-sampling toproduce a 6×11 mesh of data, via conversion routinescompleted in Matlab.

The bar horns were excited through a range of fre-3. Experimental techniquequencies around resonance (20 480–20 492 Hz) in 121 Hz steps, using a standard input voltage of 40 V. ThisA stack of two bar horns was mounted on a 2 kW

Branson ultrasonic transducer, with an FFR Autotuner frequency was higher than the designed value of 20 kHzbecause the system was operating without a block hornused as the input power and frequency control source.

Electrical energy and frequency input control was pro- component which would lower the system first orderlongitudinal frequency response to this value. Out-of-vided by an Hewlett Packard HP3330A frequency syn-

thesiser via an FFR Autotuner, acting as a power plane phase-stepped subtraction correlation fringe pat-terns were obtained for each frequency. This processamplifier. The frequency synthesiser also provided an

input reference signal for monitoring and for pulsed was then repeated for the bar horn using the in-planeESPI interferometers, generating a total of 48 data sets,laser synchronisation.

236 G. Graham et al. / Ultrasonics 37 (1999) 231–238

Fig. 5. Out-of-plane bar horn resonant mode fringe pattern.

24 displacement amplitude and 24 vibration phase sets, the component is a long thin bar which occupies approxi-mately 20% of the field of view. Portions of this compo-which were passed into the Star Modal software.nent can be magnified to fill the whole field of view andstudied in detail, but to observe the complete componentbehaviour, it is necessary to sacrifice 80% of the spatial4. Experimental resultsresolution of the instrumentation. Consequently, theinterferometric data is recorded over a reduced area4.1. Wholefield optical analysiswhich tends to reduce the dynamic range of theinstrumentation.Only two displacement components are presented for

this high power ultrasonic component; the out-of-plane Processing the correlation speckle information via theimage processing equipment leads to calibrated displace-(Z-axis) and the horizontal in-plane (X-axis). The vertical

in-plane displacement component (Y-axis) was found, ment and vibration phase data with respect to interfer-ometer sensitivity and bar horn geometry. It should beas expected, to have a negligible contribution to the first

order longitudinal mode shape. Fig. 5 provides an exam- noted that the vibration phase data is an implicit func-tion of the correlation fringes. Uncertainties containedple of the out-of-plane ESPI analysis, which shows a

subtraction correlation fringe pattern superimposed on within these data sets are primarily a function of fringecontrast and the inherent speckle noise associated withthe bar horn, each fringe delineating constant out-of-

plane displacement. Similar data was produced from the the wholefield optical technique. Low pass Fourier-based filtering was used before optical phase calculationin-plane ESPI interferometer and an example of this

data is shown in Fig. 6. An important aspect of this in order to improve the fringe signal-to-noise ratio(SNR), and reduce the influence of the speckle noise onapplication is to consider the ratio of the component

geometry to the CCD-TV array geometry. In this case, the final result. However, some noise is still a feature of

Fig. 6. Horizontal in-plane bar horn resonant mode fringe pattern.

237G. Graham et al. / Ultrasonics 37 (1999) 231–238

the wholefield displacement data sets in terms of local Although a single set of ESPI measurements of thecomponent at resonance can be manipulated to extractvariations in the surface displacement form, but these

are small with respect to the amplitude of the motion the mode shape successfully, an accurate determinationof the modal frequency and a measure of the modalrecorded.

The primary benefit of the ESPI analysis is the damping cannot be evaluated from a single frequencydata set. For LDV measurements, on the other hand, awholefield generation of modal data during the course

of one experiment. This can provide a maximum of single response measurement allows estimation of modalfrequency and damping but a set of measurements is262 000 discrete contiguous data points (if using a

512×512 processing array system) and allows analysis required to produce mode shapes. This is because oneESPI measurement contains wholefield data at a singleup to the boundaries of the ultrasonic bar horn, although

the bar horn geometry reduces the data density to a frequency, whereas one LDV measurement contains aresponse for the entire frequency range of interest frommesh of 100×512 in this case. The simplest form of

analysis of the bar horn (OOP and IP) required two a single point on the target surface.ESPI measurements are therefore required over awholefield data maps, with the analysis performed

sequentially and data obtained inside several seconds. frequency range around resonance and these must beprocessed in the form of a frequency response functionFor the results presented here, the analysis was extended

to 12 measurement sets in order to obtain FRF data. in order to extract all of the modal parameters. FRFsfor the OOP and IP wholefield bar horn analyses areshown in Fig. 8. A comparison of modal parameters4.2. Modal analysis from the wholefield optical dataestimated from ESPI FRF data and LDV FRF data ispresented in Table 1, where it can be seen that modalThe first function of the modal analysis was the

generation of dynamic mesh models of the bar horn frequencies and mode shapes are successfully predicted,although data quality is still dependant upon the instru-longitudinal resonance. Fig. 7 depicts the out-of-plane,

in-plane movement and combined displacements of the mentation used. In particular, the mode shape of thetuned bar horn mode was very difficult to interpretbar horn with respect to the static undeformed original

model mesh, which gives an indication as to the complex adequately from the out-of-plane vibrometer measure-ments and has consequently been labelled as ‘undeter-movement of the unit. Whilst the ESPI fringe patterns

provide excellent qualitative understanding of simple mined longitudinal (UL)’, whereas the ESPImeasurements provided an accurate and full picture ofmodal behaviour, the software representation of the

data furthers the understanding of the dynamic response, the participation of flexural motion in the longitudinalmode. However, compared with LDV data, the smallespecially when mixed mode response occurs, such as

flexural or torsional components interfering with the bandwidth of the wholefield data set (13 Hz) did notproduce a high enough frequency resolution to providedominant longitudinal response of the bar horn.

Fig. 7. Bar horn modal mesh plots.

238 G. Graham et al. / Ultrasonics 37 (1999) 231–238

5. Conclusions

Wholefield modal analysis has been performed on anultrasonic bar horn operating at 20 480 Hz, by integ-rating wholefield electronic speckle pattern interferome-try data with a commercial modal analysis softwarepackage. The generation of out-of-plane and in-planesurface displacement data has allowed the characterisa-tion of ultrasonic bar horn modal behaviour for itstuned mode (first order longitudinal ), from the compo-nent surface. The wholefield data has been calibratedwith respect to interferometer sensitivity and componentgeometry to produce displacement and vibration phasemaps. A high resolution three-dimensional surface pic-ture of modal response at resonance (even from highlycurved surfaces) is obtained by this technique.

The wholefield modal data has been processed inmodal analysis software, generating dynamic meshmodels of the bar horn describing the vector displace-ment motion and the total component motion.

Fig. 8. Frequency response functions.Frequency response functions were derived from theESPI data, allowing the estimation of modal parameters.Table 1Mode shapes and modal frequencies were accuratelyComparison of modal parameter estimationsidentified but damping calculations were limited by the

Mode shape Modal frequency (Hz) Modal damping (%) sampling resolution of the ESPI data. It is expected thatcurrent developments in instrumentation automation toESPI 1L+2F 20485 2.1increase data density will result in further improvementsLDV UL 20480 1.8in the FRF measurement technique to provide accurate

1L: 1st order longitudinal mode; 2F: 2nd order flexural mode; UL: estimation of all modal parameters.undetermined longitudinal mode.

accurate estimates of damping coefficients, which arecalculated from the half-power points of the FRF pro- Acknowledgementsviding an estimation of 2.1% damping, as comparedwith 1.8% obtained from automated polynomial curve- The authors wish to acknowledge the support offitting of the LDV data. EPSRC and Nestle Plc for this research.

The larger damping value measured using the ESPIsystem is a function of the coarse graduations of theESPI FRF and the limited frequency bandwidth.Increasing the accuracy of modal parameter extraction Referencesfrom ESPI FRFs is a function of extending the datadensity and resolution by increasing the frequency band [1] M. Lucas, G. Graham, A.C. Smith, Enhanced vibration control ofand reducing the width of the frequency steps. At an ultrasonic cutting process, Ultrasonics 34 (2–4) (1996) 205–211.

[2] M.C. Shellabear, J.R. Tyrer, Application of ESPI to three-dimen-present, high resolution is possible, and merely requiressional vibration measurements, Optics and Lasers in Engineeringmeasurements at a very large number of discrete frequen-15 (1991) 43–56.cies. It is intended that further developments of the

[3] L.D. Rosenberg, V.F. Kazantsev, L.O. Marakov, D.F. Yakhimov-instrumentation will potentially automate this process, ich, Ultrasonic Cutting, Consultants Bureau Enterprises, Newallowing improved FRF data sets to provide more York, 1964.

[4] R. Jones, C. Wykes, Holographic and Speckle Interferometry, 2ndaccurate damping estimations. The virtue of the whole-ed., Cambridge University Press, Cambridge, UK, 1989.field analysis is that it provides an implicit understanding

[5] D.C. Williams, Optical Methods in Engineering Metrology, Chap-of mode shapes in three dimensions, an understandingman and Hall, London, 1993.

which is difficult to obtain from single point analysis [6 ] D.W. Robinson, G.T. Reid, Interferogram Analysis; Digital Fringewithout considerable repetitive experimentation, with a Pattern Measurement Techniques, Institute of Physics Publishing,

Bristol, UK, 1993.further ability to derive FRFs from the data.