An empirical method for correcting dispersion in pressure bar measurements of impact stress冲击应力
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Home Search Collections Journals About Contact us My IOPscienceAn empirical method for correcting dispersion in pressure bar measurements of impact stressThis article has been downloaded from IOPscience. Please scroll down to see the full text article.1996 Meas. Sci. Technol. 7 1227(/0957-0233/7/9/006)View the table of contents for this issue, or go to the journal homepage for moreDownload details:IP Address: 166.111.120.103The article was downloaded on 28/06/2011 at 01:22Please note that terms and conditions apply.Meas.Sci.Technol.7(1996)1227–1232.Printed in the UKAn empirical method for correcting dispersion in pressure barmeasurements of impact stressD A Gorham and X J WuFaculty of Technology,The Open University,Walton Hall,Milton Keynes MK76AA,UKReceived 2May 1996,accepted for publication 21June 1996Abstract.The dispersion of stress waves in pressure bars distorts a propagating stress pulse,and hence limits the accuracy with which dynamic stress can bemeasured.This dispersion is caused by a frequency dependence of phase velocity,and previous methods of dispersion correction have adjusted the phase shift of dispersed frequency components using a theoretical bar characteristic.However,the effectiveness of these schemes with more than a small amount of dispersive distortion is very limited.A new method of dispersion correction described in this paper is based on a bar phase characteristic which is derived from measuredstress pulses arising from the impact of very small spheres.Examples are given of reconstructing impact stress from a range of particle sizes,showing that the method is very effective even with large amounts of dispersive distortion.1.IntroductionA common method to monitor stress waves generated by impact is the instrumented pressure bar.This consists of a long,cylindrical rod (figure 1),usually with strain gauges attached to record the propagating stress waves.Two gauges are used,arranged so that flexural stress wave components (which are antisymmetric)are cancelled out,and only the compression pulse propagating from the impact site is recorded.In this form the pressure bar is a geometrically simple and experimentally convenient coupling between the impact event and the primary transducer.The propagation of longitudinal stress waves is,however,dispersive for wavelengths which are of the same order of magnitude as the bar diameter.This causes distortion of short stress pulses and acts as a limitation to the accuracy with which impact stress pulses containing high-frequency components can be measured.Since the dispersive distortion is created by the spread of phase velocities over the signal spectrum,it should in principle be possible to reverse these effects by applying appropriate phase shifts to each frequency component.Hsieh and Kolsky (1958),Gorham (1983),Follansbee and Frantz (1983),Gong et al (1990),Safford (1992),Lee and Crawford (1993)and Lifshitz and Leber (1994)have carried out this procedure,using phase corrections derived from the theoretical analysis of wave propagation in bars.Thistechnique is successful for moderately dispersed signals,but becomes inaccurate for large amounts of dispersion.Figure 1.Schematic of a pressure bar instrumented withstrain gauges.Those working in acoustic emission have taken very sophisticated approaches using numerical models to predict path dispersion,mainly in plates (for example,see the works of Ceranoglu and Pao (1981)and Buttle and Scruby (1990)).This approach,using magnitude as well as phase to correct for dispersion,is more reliable than the analytical methods,but relies on the accuracy of the numerical models.The present work was aimed at looking more closely at measured signals in pressure bars to establish better methods for the correction of force measurements from the impact of small particles.An empirically based method has been developed that is in practice much more accurate for particle impact signals than the techniques using wave propagation theory or numerical models.Although developed on a large (30mm diameter)pressure bar,it is applicable to any size,and so has particular applications for the measurement of impact forces from very small particles on miniature bars.0957-0233/96/091227+06$19.50c1996IOP Publishing Ltd 1227D A Gorham and X J Wu2.Instrumentation and data acquisitionThe work was carried out with a vertically mounted pressure bar.Steel spheres in the range of0.5mm to 10mm diameter were dropped from a height of less than 0.5m onto the centre of the end face of the bar.Collision velocities were around2m s−1,a low enough velocity for impacts to be largely elastic.To achieve a large amount of dispersive distortion with these particle sizes,a pressure bar of30mm diameter was used.The material chosen for the bar was Ti6Al4V alloy,which has a high yield strength and is non-magnetostrictive.The1.5m long bar was instrumented with unbacked, silicon strain gauges located approximately300mm from the impact end.The gauges had a physical length of1mm, a gauge length of0.5mm and a width of0.15mm.The rise time of their response,about0.1µs,is fast enough not to create a bandwidth limitation for the signals measured in this work.Two gauges were mounted on opposite sides of the bar,and connected in series so that unsymmetrical bending waves would be cancelled out.The gauges were energized from a constant dc voltage source in series with a resistance,and the output signal fed through a preamplifier with a bandwidth of1MHz.Digitization and storage were carried out with an8-bit resolution200MS s−1signal recorder.Data were transferred by disk to a PC system for processing with Microsoft QuickBasic and Matlab analysis software.3.The ideal pulseThe removal of dispersive effects will be carried out using phase adjustment in the frequency domain,so thefirst stage is to examine the frequency spectrum of an ideal,non-dispersed pulse.Classical Hertzian theory predicts that the force history arising from an elastic collision between a sphere and aflat is approximately of cosine shape,as sketched infigure2.The magnitude and phase spectra obtained from a fast Fourier transform(FFT)operation on this form are shown infigure3,where the frequency axis is marked in units of component number.Only the first200points are shown here,each point representing a frequency interval of1.53kHz,and so giving a maximum abscissa value of305kHz.Notice that the magnitude spectrum in thefigure displays a series of well defined minima,while the phase consists of a series ofπsteps which correspond to these magnitude minima.This simple variation of phase is easy to understand: groups of frequency components between the steps add up in phase,to create a symmetrical sum with either a negative going maximum at time zero(φ=π)or a positive going maximum(φ=0or2π).The exact shape of the symmetrical pulse,which is the total of all these contributing sums,is then determined by the positions at which theπsteps occur.Note that theseπsteps correspond to the minima in the amplitude spectrum.Impact velocities in the present work are above the threshold for plasticflow to take place,introducing the possibility that the force history is unsymmetrical.To illustrate the consequences of this,the phase spectrumfrom Figure2.An ideal cosine force pulse,predicted from elastic(Hertzian)theory.Figure3.Phase and magnitude spectra for the cosine pulse offigure2.The abscissa is the frequency component number.Units of magnitude arearbitrary.Figure4.A phase spectrum from a slightly unsymmetrical cosine pulse:the fall time is4%less than the rise time.a slightly asymmetric version offigure2,with a fall time 4%less than the rise time,is shown infigure4.This shows a structure of steps similar to the symmetrical case of figure3,but with an increasing variation of phase between the steps.These simple forms of phase spectra found with model pulses and their link to the amplitude spectrum provide the key to empirical deconvolution of experimental force pulses.1228Dispersion correction of pressure barsignalsFigure5.Stress pulses recorded from the30mm pressure bar,arising from the impact of steel spheres at approximately2m s−1.(a)10mm sphere;(b)4mm sphere;(c)1mm sphere.Units of force are arbitrary.4.Dispersed pulsesA series of signals recorded with the30mm pressure bar,from impacts of steel spheres of1mm,4mm and 10mm in diameter,are shown infigure5.The largest size (10mm)shows comparatively small effects of dispersion, with only a small oscillatory overshoot after the main pulse.However,the smallest size(1mm)is dominated by its effects,with large oscillations carrying on for a long time after a barely discernible main pulse.Thus in many experiments,dispersion adds considerable error to determining the main characteristics of the force record, particularly the amplitude and duration of the main pulse.Since these dispersive oscillations are caused mainly by the progressive phase lag of higher frequency components, resulting from their lower velocity of propagation, correction for dispersion involves careful adjustment of the phase of each component to reverse these effects.However, two other numerical procedures are also needed before the distinctive characteristics of the phase data can be seen.First,the pulses are time shifted so that t=0 corresponds to the peak of the main pulse.If this time shift is not carried out,there is a large,linear change of phase with frequency that obscures the effect of dispersion on the phase spectra.Second,the phase spectrum needs to be subjected to the numerical procedure known as unwrapping.Because the phase shift is caused by the increasing propagation delay of higher frequency components,it should then be a monotonically increasing function of frequency.However convention dictates that the arctan function used to calculate phase,which is periodic in2π,always returns values that are within the interval−πtoπor0to2π.This artificial and unnecessary constraint obscures the pattern behind the phase variation.Hence the unwrapping process adds an appropriate number of2πincrements to each calculated phase value,to yield a monotonic change of phase with frequency.Figure6shows the FFT spectra of magnitude and unwrapped phase for time shifted versions of each of the signals shown infigure5.The magnitude spectrum for the10mm sphere(figure6(a))clearly shows a structure of successive minima similar to that of the ideal pulse offigure3.This is less clearly defined in the smaller sizes,but in all cases in our experiments at least one or two main minima could be identified from the envelope of the magnitude spectrum.Note that the1mm phase data infigure6(c)are plotted with a different horizontal and vertical scale.The unwrapped phase spectra all display a very similar shape of curve,extending to approximately100radians over thefirst250components.In fact,close comparison of these curves shows that they do all follow almost exactly the same path apart from steps of2πor ofπ.The major point here is that this large,underlying variation of phase is the effect of the dispersion process.The2πsteps represent inaccuracies in the unwrapping process:they do not affect the subsequent computation,but can be removed individually if necessary.The smaller steps of aboutπand other small scale variations of phase define the pulse shape, as in the ideal case offigures3and4.Thus the process of dispersion correction involves cancelling out the large,smooth underlying variation of phase that is common to all these curves,leaving the detailed structure that enables the pulse to be reconstructed. The next section describes how this phase variation due to propagation can be separated from the phase structure of the pulse.5.An experimentally derived phase response Figure3showed thatπsteps in the phase spectrum of an ideal pulse correspond to minima in the amplitude spectrum.Further simulations have shown that the frequency at which thefirst step occurs is inversely proportional to the pulse width.The positions of the first minima in the amplitude spectrum for the10and 4mm spheres are approximately at points38and93 respectively andπsteps at these values can be seen in the phase plots offigure6.These results predict that the first minima for the1mm and0.5mm spheres(which are not so clearly defined)should be at about points380 and760respectively.(Note that only512components can be obtained from the1024point FFT used in this work.)Therefore,assuming that the original pulses were symmetrical about their peak value,the non-dispersed phase spectrum up to these values would have been a constant value ofπ,as infigure3.Hence,the observed phase curve for these small projectile sizes would represent quite closely1229D A Gorham and X JWuFigure 6.Magnitude and unwrapped phase spectra for each of the pulses illustrated in figure 5.(a )10mm sphere;(b )4mm sphere;(c )1mm sphere.Units of magnitude are arbitrary.the phase shifts introduced by the dispersive propagation along the bar.If the non-dispersed pulses were not symmetrical,then,following the example of figure 4,the measured curves will also contain phase variations between steps that depend on the exact form of the pulse.However this is thought not to be a very significant effect.The phase curves for all sizes of sphere coincide so closely over a wide range of phase that,apart from 2πand πsteps,any variations that are specific to one size must be very small.For example,the phase curve in figure 6(a )shows no irregularity up to the first πstep at component number 38.In particular,the phase curves for many impacts with the 1mm and 0.5mm spheres are all very close and reproducible.The signal from the 1mm size is stronger and hence less noisy than that from the 0.5mm,and so at least the first 380components may be regarded as a good approximation to the phase shift added by dispersive propagation.However,close examination of the phase data from these small particles reveals irregularities in the first few phase values,which oscillate by a few tenths of a radian about the expected smooth curve.These oscillations in the phase of low-frequency components are thought to be the result of aliasing from noise and from high-frequency signal components,and they seriously corrupt the correction of dispersed pulses.The irregularities have been removed by manual editing.The modified phase data are then an accurate enough representation to high enough frequencies to achieve good dispersion correction of experimental signals from larger particles,and some results to demonstrate this are described in the next section.6.Phase adjustment from the empirical impulse responseIf the set of values from the empirical phase data described above is P r (i),then a corrected set of values P c (i)can be obtained from any other measured set P m (i)byP c (i)=P m (i)−P r (i)+π.(1)Figure 7shows measured signals from 10mm,7mm,4mm and 3mm particle impacts on the 30mm pressure bar,together with reconstructed pulses after this method of phase correction has been carried out.Note that the time origin,which was in the centre of each pulse for the numerical analysis,has been shifted back for these waveforms to be printed.The effects of dispersion have been very successfully eliminated from all four pulses.Oscillations have been almost eliminated from the largest three,and the amplitude and duration of these pulses are now accurately measurable.There are still some significant oscillations to either side of the 3mm pulse,but their effect on the measured values of amplitude and duration will still be small.This particle size,one tenth of the bar diameter,probably represents the limit of successful correction using the present data.The residual phase curve after subtraction of the empirical phase data from the phase spectrum of the 10mm particle is shown in figure 8.This figure reveals the phase to be very irregular on a small scale,but πsteps which form the shape of the pulse and 2πsteps which represent inconsistencies in the unwrapping process are1230Dispersion correction of pressure barsignalsFigure7.Results of dispersion correction with the empirical phase data.(a)10mm sphere;(b)7mm sphere;(c)4mm sphere;(d)3mm sphere.In each case the original signal is above,and the recovered form below.Units of force are arbitrary.clearly visible.Similar forms of curve are obtained from the other sizes of sphere.Smoothing the peaks and other irregularities in the residual phase curves has a comparatively small effect upon the reconstructed pulse.No systematic method of improving the accuracy of the dispersion correction by individual adjustments to the residual phase curve has been found.Most of the impact events illustrated so far have taken place close to the centre of the bar end.However a number of experiments were carried out to see what effect there was on the accuracy of dispersion correction due to variations in impact position.In general there was very little effect, and good corrections can be made for impacts anywhere on the end of the bar.Off-axis impacts do,however,set up bending waves,and it is important that each pair of gauges is accurately mounted so that these unsymmetrical modes are cancelled out.The present method of extracting phase data from small particle impacts has also been successfully used to correct for dispersion in split Hopkinson pressure barexperiments.Figure8.Residual phase spectrum for the dispersion corrected10mm sphere signal offigure7.This work will be published shortly.The procedures outlined in this paper have involved1231D A Gorham and X J Wucorrections only to the phase of the signal.It can be seen from the magnitude spectra that there are features common to all particle sizes that must be arising from wave propagation effects or from the instrumentation.For example,there is often a significant dip in the amplitude response around components140–145which coincides with a small irregularity in the phase response.However, the extent of this varies between records.Adjusting the magnitude spectrum to compensate for changes like this does not have a large effect on the reconstructed signal, and this topic has not been pursued.7.ConclusionThe correction results infigure7illustrate a very successful method to recover accurate information on the peak amplitude and duration of highly dispersed stress pulses arising from the impact of particles.The present phase data allow useful corrections to be made for particles sizes down to approximately one tenth of the pressure bar diameter,i.e. a3mm particle on a30mm pressure bar.The method is directly applicable to other sizes,and so would be reliable for a100µm particle on a1mm pressure bar.Other forms of signal,for example from material testing experiments, can also be corrected with the phase data derived from small particle impacts.The key to this method of measuring bar characteristics is to use low-velocity impacts of very small spherical particles,making sure that the impact conditions are essentially elastic.The phase spectrum of these symmetrical signals then has a simple structure of approximately constant phase betweenπsteps,enabling the phase change due to dispersive propagation to be identified.The other important steps which have enabled this empirical approach to be developed are as follows.First, the dispersed signals are time shifted so that time zero corresponds to the peak of the main pulse.This simplifies the phase structure of dispersed and ideal signals considerably.Second,the phase is unwrapped so that it is a continuous function of component number.This allows comparison between signals,and simplifies identification of the large-scale phase changes that result from dispersive propagation.The method depends entirely upon detailed observation of measured signals,and does not depend upon analytical or numerical models of wave propagation or impact.This is an important benefit.The phase corrections needed for significant components may be up to100radians,and these corrections must be achieved with an accuracy of better than 0.1radians.Analytical or numerical models are not capable of this level of accuracy,since in general they are based on approximations of the realistic situation.Thus,empirically based approaches,which include methods to adjust the corrections to suit the precise conditions of the individual signal,are likely to remain the only accurate means to remove dispersive distortion from measured signals. AcknowledgmentsThe work was supported by grants from the EPSRC Process Engineering and Materials Programmes,with additional equipment funding from the Open University.R Murray is thanked for useful contributions to the apparatus and analysis.ReferencesButtle D J and Scruby C B1990Wear13763–90Ceranoglu A N and Pao Y-H1981Trans ASME:J Appl.Mech.48125–47Follansbee P S and Frantz C E1983ASME J.Eng.Mater.Technol.10561–6Gong J C,Malvern L E and Jenkins D A1990Trans.ASME J.Eng.Mater.Technol.112309–14Gorham D A1983J.Phys.E:Sci.Instrum.16477–9Hsieh D Y and Kolsky H1958Proc.Phys.Soc.71608–12Lee C K B and Crawford R C1993Meas.Sci.Technol.4931–7 Lifshitz J M and Leber H1994Int.J.Impact Eng.15(6)723–33 Safford N A1992Proc.Int.Symp.on Intense Dynamic Loading and its Effects(Chengdu,China)pp378–831232。
