XIX IMEKO World Congress Fundamenal and Applied Merology Sepember 6 11, 9, Lisbon, Porugal THERMOELASTIC SIGNAL PROCESSING USING AN FFT LOCK-IN BASED ALGORITHM ON EXTENDED SAMPLED DATA L. D Acquiso 1, A. Normanno, G. Piarresi 1, A. M. Siddiolo 2 1 Diparimeno di Meccanica, Universià degli Sudi di Palermo, Palermo, Ialy, dacquiso@dima.unipa.i 2 siddiolo@gmail.com Absrac A fas infrared scanner is used o acquire he hermoelasic effec induced emperaure changes along a line on he surface of cyclically loaded ensile samples. The raser scanning movemen of he single deecor allows he sampling of emperaure versus ime. This daa are hen pos-processed by means of a lock-in algorihm coupled wih 1D and 2D FFT analyses in order o filer ou he hermoelasic signal from he noisy measured signal. A daa exension algorihm is proposed which uses he informaion from differen acquired frames o exend he daa sampling window. The whole signal processing seup is evaluaed on experimenal daa wih successful resuls, proposing a poenial ool for low cos Thermoelasic Sress Analysis. Keywords: Thermoelasic Sress Analysis, Lock-In, FFT analysis. 1. INTRODUCTION Among NDE echniques nowadays used o invesigae loaded elemens, hermoelasic sress analysis has become a well-esablished echnique o measure he full-field sress disribuion of componens under cyclic load [1]. Thermoelasic Sress Analysis echniques (TSA) perform his invesigaion by measuring emperaure changes on he surface of bodies when simple hermoelasic laws apply [2]. The Thermoelasic Effec describes he emperaure changes in loaded maer as induced by he volume changes under linear elasic behaviour. When such ransformaion is adiabaic and he maerial is homogenous and isoropic, a simple linear relaionship can be derived where he emperaure change, measured on he free surface of a body, is proporional o he firs sress invarian as follows: o ( x + σ y ) wih K α ρc p T = T K σ = (1) where T o is he absolue emperaure, K he hermoelasic consan funcion of he coefficien of hermal linear expansion α he densiy ρ and he specific hea C p. In order o deec very small emperaure changes (abou.1 K in seel sressed a MPa), hermoelasic sress measuremens require suiable equipmen; his is usually obained by means of highly sensiive infrared deecors and a lock-in amplifier able o deec ha par of he emperaure signal ha is coheren wih he load, i.e. he hermoelasic signal. Cyclically varying loads are also rouinely applied in order o achieve adiabaic condiions able o freeze he peak o peak ampliude of he hermoelasic effec induced emperaure change, which is hen modulaed according wih loading cycle. General pracise of hese echniques is usually carried on by means of on purpose developed commercial hardware, wih mos modern IR hermocameras using focal plane array deecors able o provide full-field maps of he hermoelasic signal in few seconds. While such ools are so far available a elevaed coss and wih a limied choice of suppliers and configuraions, he presen work employs a mehodology using low cos IR scanners o measure he hermoelasic signal [3,4]. A daa exension algorihm able o exend he sampling window is in paricular presened and applied ogeher wih 1D and 2D FFT based lock-in procedures o opimise he signal filering operaions for he evaluaion of he hermoelasic signal. 2. EXPERIMENTAL SETUP In his work a low resoluion ermocamera (NETD=.12 K) Varioscan 322 by Jenopik GmbH was used. This sysem employs a single hermoelecrically cooled MCT IR deecor assised wih wo roaing mirrors collecing he signal horizonal-wise and verical-wise on a full field frame array of 36 24 pixels. Tensile Specimen scanned line force ime daa frame sampling scheme (24 rows colleced in ~.9 sec) col 1 col 2 col n col 36 daa frame image row 1 row 2 row 24 Fig. 1. Scheme showing he line scanning sampling mechanism and a daa frame image a 6 Hz sine loading.
When daa acquisiion is performed as line scan, a line along he surface under invesigaion is repeaedly raser scanned ino 36 poins acquired in 1/27 s. A single hermogram is hen obained as a daa frame array by piling up 24 rows consecuively acquired from he same line. The emperaure from a pixel on he scanned line is acquired periodically whenever he opical pah of he sensor focuses on i. The informaion from one pixel locaion is hen sored in a column of he daa frame array. The columnwise informaion is hen a measure of emperaure versus ime on he same poin of he sample surface. This enables a procedure by which he emperaure-ime variaion is sampled over 36 aligned poins for a oal ime of.9 s (i.e. he frame rae). If he body is loaded a sufficienly slow frequencies, he hermoelasic effec induced emperaure change can be sampled over a finie number of periods during a single frame acquisiion. A scheme of his sampling mechanism for a uniform sress field generaed in a ensile sample is given in Fig. 1. I is possible o disinguish greyscale horizonal fringes deermined by he hermoelasic effec periodically induced emperaure changes due o cyclic loading frequency f L on he specimen under loading. A simple relaionship can be esablished beween he number of horizonal hermal fringes shown in one frame, n S, and he camera frame rae f R : n S (. 888 s) = f f = f (2) L R where.888 s is he exac ime spen by he camera employed in his work o acquire a frame. The ime shif on he hermoelasic signal beween differen columns is negligible given he high speed of he sensor line swap. 3. DATA PROCESSING The daa processing approach adoped in his work for an effecive recovery of he measuremen informaion conveyed by he noisy emperaure signal relies on a preliminary processing of grabbed hermograms for daa enhancemen and on signal processing using 1D or 2D FFT lock-in based algorihms. L The daa enhancemen procedure applied in his paper follows a ime domain approach, aiming a obaining, from several consecuive acquired se of hermograms, a sample of daa conaining a greaer number of periods of he fringe paern described. On purpose developed commercial hermocameras implemening lock-in signal analysis echniques use lock-in amplifiers o deec and measure very small AC signals deeply buried in noise. Lock-in amplificaion is radiionally accomplished wih expensive, monolihic hardware ha requires a frequency reference. Typically, an experimen is excied a a fixed frequency (from an oscillaor or funcion generaor), and he lock-in deecs he response from he experimen a he reference frequency. In he general case, he inpu consiss of signal plus noise. Noise is represened as varying signals a all frequencies. The ideal lock-in only responds o noise a he reference frequency. Noise a oher frequencies is removed by he low pass filer following he mixer (see Fig. 2). Acually, engineers and scieniss are going beyond wha radiional black-box DSP implemenaions can do by defining he hardware funcionaliy in sofware o creae virual insrumenaion []. In his work, a peculiar lock-in sofware procedure was se up using low-cos compuerbased echnologies for accurae measuremen of he hermoelasic emperaure change under cyclic loading, by means of a sandard hermocamera. In his work a self-referenced lock-in daa processing echnique was developed, in which he reference signal was direcly obained from specral filering of he global emperaure signal (S in Fig. 2). Once he main carrier frequency is exraced from he raw daa, his is hen used o build a noiseless reference carrier signal modulaed wih he obained reference frequency. The exracion of a reference signal as described above will enable o perform he analysis wihou using any exernal loading signal (his possibiliy is also shown in he scheme of Fig. 2 wih he doed line link). As shown in Fig. 2 he core of he lock-in echnique consiss in mixing he measured signal, named ST, wih he reference signal which is a carrier of he loading frequency ω R, i.e. he useful frequency. S T =Thermoelasic Signal S T =A sin(ω S+φ) S = measured noisy signal S=S T+noise S T Principal carrier frequency from FFT band filer ω R = ω S F Mixer Phase Shif φ S T*F FFT Low Pass Filer X/2 X=Acosφ Y=Asinφ inpu daa reference signal ω = ω R G Mixer FFT Low Pass Filer Y/2 A, φ S T S T*G Fig. 2. Schemaic represenaion of he lock-in signal processing.
Once he loading frequency ω R is obained, wo pure cyclic reference signals named F and G, wih F in quadraure wih G, are buil and muliplied o S (for simpliciy diagrams shown in Fig. 2 perform his operaion on he clean hermoelasic harmonic componen S T). This operaion resuls in wo signals which conain he informaion of ineres as DC signals, and hence a low-pass filering operaion performed wih FFT analysis is able o filer ou such componens named X and Y. These represen he hermoelasic signal componens in phase and in quadraure wih he reference signal; by combining hem he hermoelasic signal ampliude A is readily obained. The above procedure is implemened in his work wih an algorihm using he Fas Fourier Transform funcions available in Malab. The approach here proposed implemens a 1D FFT analysis on each column of he daa frame array, and resuls are also compared wih a previously implemened 2D FFT approach proposed in [4]. This 1D lock-in reamen allows he hermoelasic signal o be derived on each scanned poin independenly, enabling is immediae applicaion o he case of a 2D sress field analysis. In fac alhough he 2D FFT based reamen from [4] is believed o be more effecive in noise rejecion, i is more elaborae for analyzing 2D sress fields. A key issue influencing significanly he signal qualiy of he lock-in processed hermograms is he frequency resoluion of he FFT Low Pass Filers employed as shown in Fig. 2. The peculiar feaures of he frame grabbing of he hermograms, due o he uneven ime shif beween odd and even rows of he acquired hermogram (see Fig. 1), do no saisfy he basic condiion ha he sampled daa are equally spaced in ime, required o apply FFT analysis. As a consequence of his available daa are halved, by considering only he even rows or he odd rows from a hermogram acquired as line scan. Wihin each hermogram he oal number of rows is hen reduced from 24 o 1. The corresponding ime inerval becomes equal o one oscillaion period of he mirror, i.e. =2 (1/27) s; hen f S =1/ =13 Hz is sampling frequency. The maximum loading frequency applicable for he FFT analysis o work, i.e. he Nyquis frequency, will be half he sampling frequency, f Ny=f S/2=67. Hz. Usually in TSA he hermoelasic signal is significanly corruped by noise and hen a reduced number of poins from a single frame (as in he seup here adoped), canno be adequae o perform effecively he filering operaions based on FFT which are shown in Fig.2. A simple averaging over ime of several subsequenly acquired frames has no been considered because i would reduce he noise by narrowing he bandwidh, i.e. gaining improved noise rejecion bu worsening ime response. I is oherwise plain ha if he oal sampling period were exended, a higher n s value and a higher frequency resoluion would be obained. This would also gain smaller influences of border effecs and a more effecive noise o signal reducion. A sample daa exension would represen an improvemen in daa signal processing collecion, enhancing he qualiy of boh 1D and 2D-FFT filered resuls. The daa enhancemen echnique proposed in his work, for he measuremen of he hermoelasic signal on cyclically loaded elemens, is based on a daa exension procedure performed in he ime domain. A cusom synchronizaion algorihm o sick ogeher several acquired hermoelasic frames is herefore proposed o obain sample daa over a ime window longer han a single frame. There is however a small ime gap beween wo subsequen frames during which he hermocamera is no acquiring he signal. So if hese are joined by simply appending one afer he oher, a disconinuiy is hen insered in he emperaure versus ime daa of he new array a he poin of aachmen of he wo original frames, inroducing furher noise which can affec he analysis. The algorihm proposed in his work overcomes his problem by means of an ieraive procedure. Two subsequenly acquired frames are enaively spliced and hen analysed by FFT, by ieraively removing one more line from he beginning of he second frame before joining he wo frames. All performed rials of joining he wo subsequen frames are evaluaed by comparing heir ampliude specra; he enaive joining operaion which shows he highes peak in ampliude specrum a he loading frequency ω R is chosen as he bes joining of he wo frames. This is illusraed in Fig.3 where four enaive joining operaions are shown, ogeher wih heir ampliude specra. The hird case is he opimal because i shows he highes ampliude peak a exacly he loading frequency of he sample (3,38 Hz). (c) (d) 2 Y = 8.33 2 4 6 8 2 Y = 17.9 2 4 6 8 2 Y = 22.3 2 4 6 8 2 Y = 16 2 4 6 8 Fig. 3. Differen splice of he same pair of hermoelasic frames (R= 3,38 Hz) - a) simple queuing of he wo frames, rejeced splice; b) and d) rejeced splices; c) opimal splice.
4. RESULTS Polycarbonae ensile samples undergoing sinusoidal load have been invesigaed by recording he emperaure signal a differen racion-racion load ampliudes. An average load of 2 N was applied, wih a peak-o peak load ampliude ranging from 7 o 22 N. Two principal aspecs of he proposed hermoelasic signal processing have been aken ino accoun o evaluae experimenal resuls: a) he synergic enhancemen effec produced by he daa exension procedure and he FFT lockin based algorihm, b) he differen filering effeciveness of he 1D FFT and 2D FFT algorihms. Fourier Transform is usually applied in column-wise order on he array of pixels conaining he fringe paern irradiance: his is referred o as 1D-Fourier Transform. of combining he 1D-FFT lock-in filering and he daa exension procedure. The above illusraed improvemen in he qualiy of he hermoelasic signal, processed as proposed in his paper, is also confirmed by he improvemen in he linear correlaion beween he peak-o-peak load applied and he measured ampliude of he hermoelasic signal as shown in Fig. a. [Uncalibraed Gray Scale unis] 18 16 14 12 8 6 4 24 frames analysis Filered signal Linear regression 2 4 6 7 8 9 11 Tensile sress ampliude [MPa] [uncalibraed unis] Thermoelasic signal Phase shif φ [deg] 2 Single frame analysis 2 8 6 4 Single frame analysis [uncalibraed unis] Thermoelasic signal Phase shif φ [deg] 2 32 frames analysis 2 8 6 4 32 frames analysis Correlaion Coefficien 1.9.8.7.6..4.3 Single frame analysis.2 4 6 8 1 14 Sampled poins along he scaned line Correlaion coefficien 1.9.8.7.6..4.3 24 frames analysis.2 4 6 8 1 14 Sampled poins along he scaned line Fig.. Thermoelasic signal measured versus sress ampliude; Correlaion coefficien of he linear regression for all sampled poins: single frame; (c) 24 frames. (c) 2 2 Fig. 4. Processed emperaure daa from a ensile sample loaded a abou 3.4 Hz. Thermoelasic signal ampliude and angular phase shif (from reference signal) for: a) a single frame, and b) 32 joined frames. A hermogram recorded on he sample specimen, column-wise processed by means of he proposed 1D-FFT lock-in algorihm is shown on op of Fig. 4a. Due o he limied exension of he daa se, efficiency of he 1D-FFT based lock-in filering algorihm is no a is bes and an appreciable noisy paern is sill presen. This is observed in he plos a he boom of Fig. 4a ha show he ampliude A and phase φ of he hermoelasic signal, along all columns in he frame. Daa processing of exended emperaure sample daa from he same ensile specimen was performed on several joined daa frames. Plos in Fig. 4b show he resuls for 32 joined daa frames, evidencing an improvemen in he qualiy of he hermoelasic signal, if compared o resuls shown in Fig. 4a. This confirms he posiive synergic effec Since a linear relaionship beween he hermoelasic signal and he loading ampliude is prediced by he hermoelasic effec law (see eq. 1), he correlaion coefficien of he linear regression can be used as a parameer o quanify he abiliy of he processed experimenal daa o follow he ideal linear behaviour, and hence he qualiy of resuls. Resuls in Fig. b) are referred o a single grabbed frame and in Fig. c) o an exended sampling window of 24 joined frames. I is seen ha performing he analysis on a higher number of joined frames dramaically improves he linear rend beween he hermoelasic signal and sress ampliude. I mus be noed ha for he presen case of a uniform sress field each scanned poin should deliver he same hermoelasic signal. Experimenal aciviy has moved furher o invesigae he benefis of implemening a 2D FFT based lock-in analysis of he acquired frames [4], coupled wih he proposed daa exension procedure. The same experimenal daa frames recorded on polycarbonae ensile samples undergoing sinusoidal load a differen racion-racion load ampliudes wih peak-o-peak load ranging from 7 o 22 N, have been iniially processed o obain exended daa se joining up o 24 single daa frames. Then each daa se conaining differen number
of single joined daa frames (1-4 - 8-16 - 24) has been processed by 2D FFT lock-in analysis. This will enable o compare he effeciveness of 1D FFT based lock-in analysis o ha of he 2D-FFT, and o check o wha exen boh he 1D and he 2D-FFT can benefi of he preliminary ime domain daa exension procedure. Given ha he naure of he noise affecing he hermoelasic signal (e.g. NETD of he hermocamera) is mainly sochasic, he analysis of he mean and sandard deviaion values of he hermoelasic signal can be useful o highligh he differen performances of boh he 1D and he 2D-FFT lock-in based filering procedure applied o he invesigaed signal. highes number of concaenaed frames. I is also worh o observe ha a he lower loading ampliude (7 N), he mean value of he hermoelasic signal processed by he 1D- FFT, is almos halved moving from a single frame o 24 frames, herefore showing he lower qualiy of he 1D-FFT processed daa. Analogous consideraions can be formulaed on he ground of resuls shown in Fig. 7, for he sandard deviaion daa calculaed on he same hermograms. Sandard deviaion values calculaed on he hermograms filered by he 2D FFT lock-in based algorihm are always lower han hose obained by he 1D-FFT. Sandard dev. - 7 N load 8, 7, 6,, 4, Mean values - 7 N load 3, 2 19, 18, 17, 16,, 14, 13, Mean values - 22 N load Mean 1D Mean 2D Mean 1D Mean 2D 12, 2 Fig.6. Mean values of hermoelasic signal filered wih he 1D-FFT and 2D FFT algorihms applied on exended frames versus he number of concaenaed frames and for peak o peak load ampliude of a) 7 N; b) 22 N. 2, 2, 1, 1,, DevSd 1D DevSd 2D, 2 3, 2, 2, 1, 1,, Sandard dev. - 22 N load DevSd 1D DevSd 2D, 2 Fig.7. Sandard deviaion of hermoelasic signal (load 7 N) Sandard deviaion of hermoelasic signal (load 22 N) In Fig. 6 he rend of he mean value obained when he sample is subjeced o minimum and maximum value of he peak-o-peak load, respecively 7 and 22 N, are shown for differen processed signal obained a differen number of joined frames. The comparison show ha resuls obained by 2D-FFT have a significanly smaller variaion as he number of concaenaed frames increases, and can be subsanially considered sable when processing signals from eigh or more concaenaed frames; his performance is obained boh a he lower (fig. 6a) and a he higher (fig. 6b) peak-opeak loading values. On he conrary, he signals processed by 1D-FFT show mean values significanly dependen on he number of concaenaed frames and presening a high range of variaion, wihou seling a a sable value even for he Fig.8. Influence of he 2D-FFT band pass filer on he emperaure maps a load ampliudes of 7 N, 22 N.
Figure 8 shows he resul of he applicaion of a 2D-FFT band pass filer around he load carrier frequency on he emperaure maps acquired a peak-o-peak load ampliudes of 7 N and 22 N. I is ineresing o noice he high qualiy already gained by he filered maps in fig. 8, afer he iniial 2D-FFT band pass filer and before he mixing wih he F and G signals and subsequen final low-pass filering operaion (see scheme in fig. 2). The 1D and 2D FFT based filering approaches are also compared in erms of heir abiliy o recover he linear rend of he hermoelasic signal wih he load ampliude. This comparison is shown in Fig. 9 where hermoelasic signal is ploed versus he peak-o-peak load ampliude. I is noed ha he hermoelasic signal uncalibraed uni is he grey level ampliude as measured in he original hermogram (an eigh bi greyscale image). Given he loading condiion of he invesigaed ensile samples, he grey level ampliude of he hermogram has in heory a linear relaionship wih he ensile load applied (as already poined ou in fig. ). Resuls obained afer processing by means of boh he 1D and 2D FFT based filering procedures are shown in Fig. 9 for a single frame daa se and in Fig. 9 for a 24 frames daa se. The daa processed by he 1D FFT show a lack of qualiy a he lower level of he applied load (7 N) when he daa se consiss of a single frame, whose se of resuls differs appreciably from ha obained from he daa se conaining 24 frames. Daa processed by he 2D FFT show insead an almos negligible difference beween hose obained by processing a single daa frame and hose obained by processing a 24 frames daa se. In fac he comparison of heir linear inerpolaions provides resuls very similar in boh cases (1 and 24 frames) and wih a higher coefficien of correlaion compared o hose of 1D- FFT. more complex wo-dimensional sress fields, since he signal from each acquired poin is reaed separaely. However, in order for he 1D FFT approach o provide qualiy resuls i is essenial o exend sampled daa over he ime domain. The presen work has demonsraed he poenial of he presened mehodology o process hermoelasic signals using low cos IR scanners. Furher work is being devoed o he applicaion of he mehodology o he analysis of wo dimensional sress disribuions. 18 16 14 12 8 6 y =.81x -.428 R 2 =.9919 y =.6x + 1.9992 R 2 =.98 4 7 17 2 19 17 13 11 9 7 Uncalibraed Gray Level 1D-Algorihm 2D-Algorihm Lineare (2D-Algorihm) Lineare (1D-Algorihm) Uncalibraed Gray Level 1D-Algorihm 2D-Algorihm Lineare (2D-Algorihm) Lineare (1D-Algorihm) y =.79x -.734 R 2 =.989 y =.71x - 1.7823 R 2 =.96 Force [N]. CONCLUSIONS Force [N] In he presen paper a lock-in algorihm is presened o exrac he hermoelasic signal from emperaure daa acquired versus ime by means of a linear infrared scanner. The lock-in procedure implemens 1D or 2D FFT based filering algorihms and a procedure o obain exended sampled daa over ime. The whole signal processing sraegy has been esed on experimenal emperaure maps recorded on a ensile specimen undergoing ension-ension sinusoidal load. The ime domain daa exension procedure has been proposed o enhance qualiy of he daa se, acquired using a low cos IR scanner wih a 1 Hz frame rae and a 36 24 pixels single frame exension. Resuls obained have evidenced he effeciveness of he proposed daa exension procedure, which can be observed a a differen exen when coupled o 1D or 2D FFT daa filering implemened in he lock-in scheme. The 2D FFT has resuled in a more effecive and robus filering approach, fi o process he case of a uniform sress field. The 1D FFT approach, hough less effecive in filering, is a presen more sraighforward o apply in he analysis of 3 7 9 1 13 17 19 2 23 Fig.9. Measured Thermoelasic Signal versus sress ampliude: single frame; 24 frames. REFERENCES [1] J. M. Dulieu-Baron and P. Sanley, Developmen and applicaions of hermoelasic sress analysis, J Srain Anal Eng, 33-2, pp. 93-4 (1998). [2] G. Piarresi and E. A. Paerson, A review of he general heory of hermoelasic effec, J Srain Anal Eng, 38-, pp. 4-417 (3). [3] A. L. Audenino, V. Crupi and E. M. Zanei, Thermoelasic and elasoplasic effecs measured by means of a sandard hermocamera, Exp Tech, 28, pp. 23-28 (4). [4] G. Piarresi, L. D Acquiso and A. M. Siddiolo, Thermoelasic Sress Analysis by means of an IR scanner and a 2D-FFT based Lock-In echnique, J. Srain Analysis 43 pp. 493 (8). [] T. Sakagami, T. Nishimura, and S. Kubo, Developmen of self-reference lock-in hermography and is applicaion o crack monioring, Proc. SPIE 782, 379 ().