errors in dynamic force measurement

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UDC: 531.2, 53.088.2 Errors in dynamic force measurement by M. Dlxon, DTI, Div. Mechanical & Optical Metrokyy, NPL, Teddngton, Middlesex, TW 11 OLW Loadcells for materials testing machines are calibrated statically by comparison with a transfer standard, traceable to the National Physical Laboratory. There is evidence to suggest that, under certain conditions, this static calibration may be insulyicient for dynamic testing. The possibility of an error being generated by the inertia of the mass between the loadcell and the specimen is discussed and the results of experiments to measure this error are presented. For the case chosen it is shown that the inertia error may be predicted by calculation. Errors may also arise from the method of dynamic force measurement. The conventional DVM and the analogue peak hold voltmeter are widely used but are subject to errors in a practical machine situation due to their method of operation. Bandwidth limitations are also illustrated for a number of instruments with differenr sampling rates. A better, although more expensive and complicated solution is to digitise the signal and to use some form of spectral analysis such as the Fast Fourier Transform or the Cross Correlation Integral. The theory for the latter is presented and its merits discussed. Key words: Dynamic force measurement, inertia error, spectral analysis Introduction The modern servo-hydraulic materials testing machine has a loadcell and instrumentation which, in order to comply with British Standard 1610, are calibrated annually by comparison with a transfer standard. This transfer standard, itself traceable to the National Physical Laboratory (NPL), consists of a precision loadcell or proving ring with its own instrumentation and is calibrated in one of the force deadweight machines at NPL by the static application of a range of forces. The procedure is well established and documented and provides a reliable check of the static load being applied to the specimens in the testing machine. The servo-hydraulic machine however is designed to operate dynamically as well as statically and there is considerable evidence to suggest that the static calibration of the force measurement system may be insufficient under these conditions.*Jp3 The comments in this paper refer directly to the application of sinusoidal forces over the frequency range of 0.1 to 100 Hz, as investigated by the author at the NPL. However the information is relevant to other waveforms (triangle, haversine etc), to higher frequencies, and to the increasingly popular random or spectrum loading tests. Static calibration may be insufficient under dynamic conditions for two reasons. Firstly, inertia forces due to any mass between the loadcell element and the specimen will give rise to an erroneous reading from the loadcell. Secondly, the conditioning electronics and readout system for the loadcell may not be capable of measuring the dynamic force correctly. This paper looks at typical servo- hydraulic testing machine procedures, quantifies the errors, and considers how the situation may be improved. The inertia error A typical machine load smng might consist of a specimen gripped in a pair of hydraulic wedge grips, connected in turn by adaptors to the loadcell and actuator. In a high cycle fatigue test it is desired to find the number of cycles to failure for a selected sinusoidal load and frequency. The test is usually run in load control at constant amplitude and the force range applied (the peak to trough load) is monitored via the peak monitor on the machine console. The the case above, the first main source of error will arise from the mass between the loadcell element and the specimen. This mass is quite considerable; an hydraulic grip of 100 kN static capacity may weigh approximately 70 Kg. To this must be added the mass of the connecting adaptors and the mass of that part of the loadcell element between the strain gauges and the adaptor. For the purposes of this discussion, a total mass of 100 Kg is assumed. When running dynamically, this mass is moving, relative to the machine crosshead, by an amount determined by the stiffness of the loadcell element. Typically, for an applied peak to peak force of 15 kN, this deflection could be .01 mm. The resulting inertia force from the motion of this mass may readily be calculated: Let and where o = circular frequency X = the amplitude of oscillation, M = the effective insertia mass x = X sinot be the applied motion then X = -X~sinwt (1) which has a maximum value of -wZX As F=Ma, thenF=-MozX. (2) For the numbers cited above, at a frequency of 100 Hz, the maximum force, corresponding to the point of maximum acceleration of the mass is 355 N, or 2.4 % of the applied force. This force will be registered by the machine loadcell but not experienced by the specimen. This effect has been verified experimentally by attaching a 'Strain'. Nownihcr 1988 I39

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Page 1: Errors in dynamic force measurement

UDC: 531.2, 53.088.2

Errors in dynamic force measurement

by M. Dlxon, DTI, Div. Mechanical & Optical Metrokyy, NPL, Teddngton, Middlesex, TW 11 OLW

Loadcells for materials testing machines are calibrated statically by comparison with a transfer standard, traceable to the National Physical Laboratory. There is evidence to suggest that, under certain conditions, this static calibration may be insulyicient for dynamic testing. The possibility of an error being generated by the inertia of the mass between the loadcell and the specimen is discussed and the results of experiments to measure this error are presented. For the case chosen it is shown that the inertia error may be predicted by calculation. Errors may also arise from the method of dynamic f o r c e measurement. The conventional DVM and the analogue peak hold voltmeter are widely used but are subject to errors in a practical machine situation due to their method of operation. Bandwidth limitations are also illustrated for a number of instruments with differenr sampling rates. A better, although more expensive and complicated solution is to digitise the signal and to use some form of spectral analysis such as the Fast Fourier Transform or the Cross Correlation Integral. The theory for the latter is presented and its merits discussed.

Key words: Dynamic force measurement, inertia error, spectral analysis

Introduction

The modern servo-hydraulic materials testing machine has a loadcell and instrumentation which, in order to comply with British Standard 1610, are calibrated annually by comparison with a transfer standard. This transfer standard, itself traceable to the National Physical Laboratory (NPL), consists of a precision loadcell or proving ring with its own instrumentation and is calibrated in one of the force deadweight machines at NPL by the static application of a range of forces. The procedure is well established and documented and provides a reliable check of the static load being applied to the specimens in the testing machine.

The servo-hydraulic machine however is designed to operate dynamically as well as statically and there is considerable evidence to suggest that the static calibration of the force measurement system may be insufficient under these conditions.*Jp3 The comments in this paper refer directly to the application of sinusoidal forces over the frequency range of 0.1 to 100 Hz, as investigated by the author at the NPL. However the information is relevant to other waveforms (triangle, haversine etc), to higher frequencies, and to the increasingly popular random or spectrum loading tests.

Static calibration may be insufficient under dynamic conditions for two reasons. Firstly, inertia forces due to any mass between the loadcell element and the specimen will give rise to an erroneous reading from the loadcell. Secondly, the conditioning electronics and readout system for the loadcell may not be capable of measuring the dynamic force correctly. This paper looks at typical servo- hydraulic testing machine procedures, quantifies the errors, and considers how the situation may be improved.

The inertia error

A typical machine load smng might consist of a specimen gripped in a pair of hydraulic wedge grips, connected in turn by adaptors to the loadcell and actuator. In a high cycle fatigue test it is desired to find the number of cycles to failure for a selected sinusoidal load and frequency. The test is usually run in load control at constant amplitude and the force range applied (the peak to trough load) is monitored via the peak monitor on the machine console.

The the case above, the first main source of error will arise from the mass between the loadcell element and the specimen. This mass is quite considerable; an hydraulic grip of 100 kN static capacity may weigh approximately 70 Kg. To this must be added the mass of the connecting adaptors and the mass of that part of the loadcell element between the strain gauges and the adaptor. For the purposes of this discussion, a total mass of 100 Kg is assumed. When running dynamically, this mass is moving, relative to the machine crosshead, by an amount determined by the stiffness of the loadcell element. Typically, for an applied peak to peak force of 15 kN, this deflection could be .01 mm.

The resulting inertia force from the motion of this mass may readily be calculated:

Let

and where o = circular frequency

X = the amplitude of oscillation, M = the effective insertia mass x = X sinot be the applied motion

then X = - X ~ s i n w t (1)

which has a maximum value of -wZX

As F=Ma, thenF=-MozX. (2)

For the numbers cited above, at a frequency of 100 Hz, the maximum force, corresponding to the point of maximum acceleration of the mass is 355 N, or 2.4 % of the applied force. This force will be registered by the machine loadcell but not experienced by the specimen.

This effect has been verified experimentally by attaching a

'Strain'. Nownihcr 1988 I39

Page 2: Errors in dynamic force measurement

strain gauge bridge to a specimen and calibrating it statically in the same manner as a loadcell. The specimen had threaded ends to allow direct attachment to the loadcell and actuator and a selection of different masses attached to each end to simulate the presence of grips (Figure 1).

I4ACtI 1 NO L A D C E L L I * STEEL BLOCKS

1 nI _] 1 1 BRIDGE

S P E C I M E N W I T H S T R A I N G A U G E

L O C K R I NG

A C T U A T O R

:;QUARE P L A r h b bOR A T l A C H M O N T OF MASSES

P L A T E N

F q . I Arranganmt of spbcimcn and grip mamx in 'Ming machine.

The machine was then operated dynamically, over a range of frequencies and amplitudes and the outputs from the strain gauge bridge on the specimen and that of the machine loadcell recorded. If it is assumed that the specimen records the true dynamic force, then the difference between its output and that from the machine loadcell is the inertia error.

If the deflection of the gnp mass for a given applied force is known then, using equation 2, the error may also be calculated. A dial test indicator between the lower flange of the loadcell and the machine crosshead was therefore used to measure the deflection for a series of statically applied loads.

In Figure 2 the measured and calculated errors for two sets of test conditions are plotted. These demonstrate that, in this configuration at least, the error is both significant and calculable. The third test condition, that of no added mass, was also run, but has not been plotted as there was no measurable error throughout the frequency range (as predicted by the theory).

1 EkEU ( H z )

Fig 2 2/

Measured and calculated mefia errors for two upper gnp rnasscs

This error is obviously undesirable as in the example above, the force actually applied to the specimen is smaller

than that indicated by the machine and will therefore load to a optimistic determination of load for a given fatigue life. The analysis above neglects the possibility that the testing machine itself is moving up and down in reaction to the actuator (and lower grip mass). Such movement will depend on the relative masses of the actuator piston and the machine, and on the method of attachment to the floor. In the experiment above, it was found that the presence of the second grip mass, between the actuator and the specimen, made no difference to the results. As the added mass was of a similar magnitude to that of the actuator piston, this indicates that the motion of the machine itself is negligible.

This method, of strain gauging a specimen similar to the one under test and using it to determine the transfer function with frequency, forms the basis of a draft British Standard for dynamic force measurement, which is currently under review 4 . An alternative method, used by some manufacturers of testing machines is to mount an accelerometer on the mass that has significant inertia and to subtract a proportion of the signal generated from the loadcell signal, thereby compensating for the error. Calibration is usually achieved by cycling the machine without a specimen but with the second grip mass and varying the amount of signal subtracted until the reading from the loadcell remains as close to zero as possible, over the frequency range of interest.

Instrumentation errors

The second source of error may arise from the method of measurement of the dynamic force range. A lX signal displayed on a digital voltmeter (DVM) is measured by integration over a period of time (the integration period) and hence any noise or fluctuations due to the servo-load are averaged out over this time interval. The integration period will vary with the resolution of display selected.

When running dynamically, however, a different approach must be adopted. Either the integration times must be very short in order to detect the peak, and an internal memory employed to retain this value and compare it with the next (the max/min DVM). or a continuous monitoring system, known as the peak hold voltmeter can be used.

The peak hold voltmeter may be thought of as the completely analogue solution as there is no sampling interval. In its simplest form the incoming signal charges up a capacitor, the charge on the plates being related to the maximum voltage applied. Resetting the instrument simply discharges the capacitor.

Alternatively, the DVM with a max/min facility converts the continuous signal into discrete samples and is therefore subject to bandwidth limitations. For the typical test frequencies used in a materials testing machine this often severely limits the choice of instrument. In Figure 3 the "frequency response" of a number of DVMs (their ability to detect the peak to peak value of a voltage at hfferent frequencies) is plotted with the sampling speak used in each case. This method should always be used to check the performance of a DVM before use in a dynamic situation.

Page 3: Errors in dynamic force measurement

Both the max/min DVM and the peak hold meter are subject to the same drawback when used in a practical machine situation; any noise on the load signal or fluctuation i n either mean level or amplitude from the servo-loop will result in an increase in the peak to peak force recorded. This error arises from the fact that this sampling method contains no method of averaging the results; it is always the highest peak and the lowest trough that are recorded. For this reason the error will generally increase with the number of cycles scanned. The results will often be repeatable, even though they are caused largely by random rather than systematic effects and hence the user may often be unaware that the answer is incorrect. Obviously, the better the testing machine, the lower the noise level on the load channel and the less the fluctuations in load from machine control will be.

k l J l . < i ( I 1 7 1

Frequency response of a DVM opaawd at vanous s a m p h g speeds Fig 3

The one advantage of both these measurement systems (especially the peak hold voltmeter) is that they are relatively cheap and easy to use. The equipment, particularly the DVM, is often already available in the laboratory. A substantial increase in accuracy, at an equivalent increase in cost, could be obtained by recording a number of successive peaks and troughs and then averaging them. This is not possible with manually operated equipment unless the frequency of operation is so low that the equipment can be reset for each cycle. It is, however, possible to address a high quality DVM from a computer and so to measure and record individual peaks (although probably not successive ones) but a more obvious method is to record several cycles of data digitally either on a digital storage oscilloscope (DSO) or a transient recorder. A computer may then be used to detect and average the peaks and troughs after the event. Alternatively the digital cursors available on some DSOs could be used directly.

An alternative method of measurement

The hardware level has now been reached for what may be recorded as the optimum solution, making use of all the information in the waveform instead of just the peak value. But before describing this, it is necessary to consider the precise content of a typical load waveform.

If the input to the control loop is a noise free sinusoid of a single frequency then the output from the machine loadcell will consist of a large component at this fundamental frequency, probably some harmonics of the fundamental with much smaller amplitudes, and some noise. This noise will be made up of specific frequencies, such as the mains supply frequency and its harmonics and some white noise spread across the spectrum. The envelope containing the spectrum will be defined by the bandwidth of the instrumenta tion.

This spectrum could be measured, although with finite frequency resolution by a specmm analyser running a Fast Fourier Transform 0, the frequency resolution depends on the number of data points in the original sample set. The operator of the materials testing machine however knows, or can measure the precise frequency of the excitation waveform. Furthermore, it is a safe assumption (in metals testing) that this frequency will be by far the largest component of the output spectrum. (It should be noted that this may not be true for specialised fields such as elastomeric testing).

The time series for the output spectrum can therefore be generated by cross correlation against internally generated sine and cosine waveforms of the input waveform frequency and its harmonics, up to a frequency determined by the bandwidth of the instrumentation. White noise is rejected from all the measurements, while the contribution of the mains supply frequency and its harmonics (the first of which is often at least as important as the fundamental) can be evaluated by a separate correlation against those frequencies. The algorithm for this approach is presented in Appendix 1.

Note that correlation against a constant results in the integral of the signal over the period of the fundamental, or the mean level.

It is evident from the above that knowledge of the exact excitation frequency is necessary to allow cross correlation with a reference sinusoid. Under these controlled circumstances, this makes the technique more accurate than the more common FFT analysis with its finite frequency resolution.

The cross correlation integral uses all the information contained in the sample set for each cycle instead of just the peak values. Determinations of the fundamental amplitude are therefore much more repeatable and accurate than those from peak monitoring methods and totally unaffected by fluctuations in the applied mean level and amplitude.

It is important to note that the cross correlation technique is measuring a different quantity from the earlier peak monitoring methods. Neglecting noise, the peak to peak value (from the maxlmin DVM) will only be equal to twice the amplitude (from the correlation analysis) if there is zero harmonic content. Normally, for metallic specimens, the harmonic content is low enough for the difference to be negligible but when it is not, the best approach may be to take the average of several peaks.

141

Page 4: Errors in dynamic force measurement

Finally the bandwidth of the loadcell instrumentation should be considered, as any low pass or mains notch filters will limit the frequency response.

Conclusions

The two main sources of error from dynamic testing in a materials testing machine are the inertia error from mass between the specimen and the loadcell and the inherent limitations in the peak and trough measurement systems. Early experiments at NPL have indicated that, for the case chosen, i t is possible to predict the inertia error by calculation. Alternatively, as this error is dependent on the acceleration of the mass, the error may be subtracted out from a suitably placed accelerometer or its magnitude may be reduced by the use of a stiffer testing machine.

The accuracy of the force measurement system is dependent on its bandwidth, which can be measured, and on the possible fluctuations in mean level and amplitude of the machine, which are more difficult to quantify. The most satisfactory solution is to digitise the signal and to measure its true amplitude using some form of spectral analysis.

Acknowledgement

The author wishes to express his thanks to colleagues, at the NPL, at Instron Ltd and to Mr P Turner of Brunel [Jniversity for advice freely given during the preparation of this paper.

APPENDIX 1:

The cross correlation algorithm.

The general cross correlation function may be used:

( 3 )

The special case where T=O is taken and the waveform for analysis, V,(t), correlated against Vb(t), where Vb(t) takes the form of sin(ot) and cos(wt).

The two cross correlations become;

V sin = Va(t). sin (ot)dt I (4)

and

V,;n and V,,, are, in fact the Fourier coefficients of the first term of the time series, fundamental frequency ~/2rc.

The signal amplitude is

and the phase angle between the signal and the reference is

(7)

Correlation against higher harmonics of the signal is also possible, yielding the time series. Care must be taken not to attempt correlation above the Nyquist frequency. This is the frequency, determined by the sampling rate of the analogue to digital converter, above which aliasing of any harmonic information will take place. For this reason an anti-aliasing filter should always be used before the data is sampled. Correlation above the Nyguist frequency will result in the number of frequency domain points generated exceeding the number of the original time domain sample set.

References

(1) Weibull, W., 'Fatigue testing and the analysis of results', Pergammon Press, Oxford, (1961), 63-65.

(2) Robertson, N.I:D., 'Dynamic force measurement,' Transducer Technology, 9,2, (March 1986), 14-17.

(3) Robinson, D.C., 'Determination of dynamic loads in a high frequency direct stress fatigue machine', National Bureau of Standards, Technical Note TN 578, (June 1971)

(4) British Standards Institution; 'Method for dynamic force calibration and verification of uniaxial load fatigue testing systems',Draft for approval, Document Reference 87/40876, British Standards Institution, London, (1987).

'Stroiri', Novtjmher 1988