Application of Signal Analysis to Internal Friction Measurements

A fully automatized internal friction (IF) measuring system was constructed. It measures IF by proceeding a waveform analysis of the free decay vibration. The choosen method eliminates disturbances due to parasitic motions of specimen and shows a weak dispersion even at low strain amplitudes. It allows to measure short free decay keeping the specimen vibrating in quasi stationnary conditions. This system was used to measure IF of specimens vibrating in torsional mode in the Hz range and also of specimens vibrating in flexural or cantilever mode in the kHz range. It is driven by a desktop computer which controls all experimental parameters. Results are displayed in real time on a graphic screen and stored on a magnetic medium. Experimental results are presented. INTRODUCTION Many mechanical properties of metals like plastic deformation or damping capacity are due to interactions between dislocations and point defects. A good method to study these interactions is the measurement of internal friction(1F). These measurements are particularly difficult due to the fact that very small changes of IF should be detected on a wide range of strain amplitude. Therefore a good accuracy of IF measurements is desired, specially in the interesting region of very small strain amplitudes(+10-7). Another problem characterizing the study of these interactions is the mobility of the point defects near the dislocations [1,2]. During an IF measurement, the dislocations oscillate around their equilibrium position, which induces a reorganization of the spatial distribution of the mobile point defects surroundingthedislocations. Using free decay to measure IF will disturb the spatial distributionof these defects. Therefore it is necessary to keep the strain amplitude as constant as possible during an IF measurement, so that the measured specimen stays in stationary conditions. Above mentioned contradictory requirements, i.e. good accuracy, small strain amplitudes and constant amplitude during measurement show that the traditional method using a wave height analysis during a free decay is not suitable for our studies. A fully automatized IF measuring system using a waveform analysis was constructed. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983952 C9-358 JOURNAL DE PHYSIQUE This technique desc r ibed by Yoshida and a 1 [31 c a l c u l a t e s I F by Four ie r t ransform of the o s c i l l a t i o n s of t h e specimen v i b r a t i n g i n f r e e decay. The waveform a n a l y s i s g i v e s a very good accuracy on IF measurements even a t low s t r a i n amplitudes where poor s i gnal t o n o i s e r a t i o i s achieved. The choosen method e l imina tes p e r t u r b a t i o n s due t o p a r a s i t i c motions of t h e specimen. Furthermore it al lows t o decrease t h e v a r i a t i o n of s t r a i n amplitude dur ing t h e f r e e decay. This measuring system i s d r iven by a desktop computer which d i sp lays t h e experiment a l r e s u l t s i n r e a l t ime. I t was used t o measure t h e IF of specimens v i b r a t i n g i n t o r s i o n a l mode i n t h e Hz range and a l s o of specimens v i b r a t i n g i n f l e x u r a l o r c a n t i l e v e r mode i n t h e kHz range. WAVEFORM ANALYSIS OF FREE DECAY SIGNAL The method of I F measurement by a waveform a n a l y s i s i s based on t h e f a c t t h a t t h e frequency spectrum of damped o s c i l l a t i o n s shows an enlarged peak. The s tudy of t h e peak shape permits t o c a l c u l a t e t h e I F . During a f r e e decay, t h e specimen motion can be assumed t o be : where f , ( t ) i s t h e main component of t h e damped o s c i l l a t i o n and can be w r i t t e n : f 1 ( t ) = A exp ( t / t l ) cos (mot + Arg A) = 4 [A exp ( j w l t ) + A* exp ( jwl t ) ) where t l i s t h e t ime cons tan t of damping and wl is t h e complex angular frequency def ined a s : I F is then simply : f 2 ( t ) r e p r e s e n t s con t r ibu t ions due t o p a r a s i t i c motions such a s e.g. a p recess iona l one i n t h e case of t o r s i o n a l v i b r a t i o n s . These motions a r e supposed t o c o n s i s t of sev e r a l s imple harmonic motions wi th angular f r equenc ies f a r away from wo. The t h i r d term f , of eq. (1) r e p r e s e n t s a l l components of t h e specimen motion o t h e r than f ( t ) and f a ( t ) . It i s considered as a smal l random no i se . The d i s c r e t e Four ie r t ransform (DFT) of f ( t ) r evea l s a main peak a t t h e angular f r e quency wo and s e v e r a l o t h e r peaks due t o p a r a s i t i c motions f 2 ( t ) . Yoshida and a 1 [3] have shown t h a t i n t h e v i c i n i t y of t h e main frequency peak t h e DFT bf f ( t ) i n a f i n i t e time i n t e r n a l t g i s approximated by : assuming t h a t 2n >> wot3/2n >> 1 where 2n i s t h e number of t ime domain samples. Th i s assumption means t h a t t h e number N of samples pe r per iod of v i b r a t i o n s a t i s f i e s t h e cond i t ion : 2n >> N >> 1. The term B + C's r e p r e s e n t s a l i n e a r approximation f o r t h e c o n t r i b u t i o n s of p a r a s i t i c motions f 2 ( t ) and f 3 ( t ) t o the main peak of t h e frequency spectrum. Using four consec u t i v e va lues of F ( s ) (3'(sl? t o ~(s, ,)) around s%wot3 /2n w e c a n e l i m i n a t e t h e const a n t s B and C . I F i s then given by r ref .31 : F ( S ~ ) ~ . F ( S > ) + F ( ~ ~ ) where R = F(S,)-~.F(S,)+F(S,) This calculation of IF uses frequency domain data which are close to the frequency wo of the main peak. It is then easy to see that this method suppresses all disturbances whose frequency is fairly distant from the main one. This is very frequent in practice. In some cases, however, the parasitic motions can give additional peaks very close to the main one. It happens particularly when parasitic motions induce an amplitude modulation of the free decay signal. In these cases, the above described method calculates too large values of IF and then must be used with caution. Thegreatadvantage of the waveform analysis for calculating IF is that the effective signal to noise ratio is highly improved by effect of statistical treatment of noise. This method calculates IF from several data per period of specimen vibration. In the classical method using wave height analysis only one point per period is considered. This fact explains why the waveform analysis allows to measure shorter free decay to obtain similar precision to the one in the traditional method. The duration of free decay i.e. variation of strain amplitude and the number of samples per period cannot be choosen arbitrarily. They are limited by the assumption in equation (2). This condition, however, is not very restrictive. Generally the sampling of free decay during 10 periods of specimen oscillation satisfies this assumption. A free decay of 10 period means for example that strain amplitude decreases 15% in case of an IF equal to 5.10-~. REALISATION OF MEASURING SYSTEM The main part of the system is a rapid DC Voltmeter which samples the free decay signal with sufficient rate. The data are then transmitted from the voltmeter to a BASIC programmable desktop computer where a Fast Fourier Transform (FFT) is proceeded. Consequently the frequency domain results are used to calculate the IF. All the experimental results like IF, frequency, temperature and others are then displayed on the graphic screen of the computer and stored on a magnetic medium. The computercontrols entirely all experimental parameters. The use of standard interconnection bus (IEEE 488-HPIB) greatly simplifies main part of the development. Our measuring system was connected to an inverted torsional pendulum. Fig. 1 shows the block diagram of the system. The choosen FFT algorithm using 256 time domain data allows to calculate IF each 40 s. the IF results can be estimated as lo-'+. For this application, additional connections were added to the measuring system which allow to measure the electrical resistivityofthe specimen. This enhancement in conTORSIONAL nection with a measurement of the torsional component of the shape change of the specimen is particularly useful for studying phase transitions. Typical experimental results without post-processing are shown in Fig. 2. Fig. 2a gives results for a precipitation peakinan Ag A1 13% alloy and Fig. 2b for a Fig. 1 : Block diagram of electronic part for martensitic transformation in a torsional pendulum Ti Ni alloy. The weak dispersion of HP 85 COMPUTER PERIPHERIALS HP 3L97A DATA ACO U N ~ T Hp