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Modelling accelerometers for transient signals using calibration measurements upon sinusoidal excitation
Measurement 40 (2007) 928–935 www.elsevier.com/locate/measurement
Modelling accelerometers for transient signals using calibration measurements upon sinusoidal excitation A. Link
a,*
, A. Ta¨ubner b, W. Wabinski b, T. Bruns b, C. Elster
a
a
b
Physikalisch-Technische Bundesanstalt Berlin, Abbestrasse 2-12, 10587 Berlin, Germany Physikalisch-Technische Bundesanstalt Braunschweig, Bundesallee 100, 38116 Braunschweig, Germany Received 23 June 2006; received in revised form 13 September 2006; accepted 23 October 2006 Available online 1 November 2006
Abstract A recently proposed accelerometer model is applied for determining the accelerometer’s output to transient accelerations. The model consists of a linear, second-order differential equation with unknown coefficients. It is proposed to estimate these model parameters from sinusoidal calibration measurements, and an estimation procedure based on linear least-squares is presented. In addition, the uncertainties associated with the estimated parameters are determined utilizing a Monte Carlo simulation technique. The performance of the proposed modelling approach was tested by its application to calibration measurements of two back-to-back accelerometers (ENDEVCO type 2270 and Bru¨el & Kjær type 8305). For each of the two accelerometers, the model was first estimated from sinusoidal calibration measurements and then used to predict the accelerometer’s behaviour for two shock calibration measurements. Measured and predicted shock sensitivities were found consistent with differences below 1% in most cases which confirms the benefit of the proposed modelling approach. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Calibration; Sinusoidal excitation; Shock excitation; Modelling; Uncertainty; IIR filter
1. Introduction Piezoelectric accelerometers can be considered as linear electromechanical transducers designed for defined amplitude and frequency ranges. For sinusoidal excitations, the input/output behaviour of accelerometers is characterized in terms of the complex sensitivity. This sensitivity is for primary calibrations determined by the established sine* Corresponding author. Tel.: +49 3034817489; fax: +49 3034817490. E-mail address: [email protected] (A. Link).
approximation method [1] resulting in estimates of the magnitude and phase response of an accelerometer at discrete frequencies over the frequency range of interest. In addition, uncertainties associated with these estimates are determined by the calibration procedure. According to an agreed international standard [2], an accelerometer is characterized for shock excitations by its shock sensitivity. The shock sensitivity is defined as the ratio of the peak values of the output voltage and the applied input acceleration. Such an approach is appropriate for comparing different accelerometers with respect to well-defined excita-
0263-2241/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.measurement.2006.10.011
A. Link et al. / Measurement 40 (2007) 928–935
tions. However, it does not allow to predict an accelerometer’s output to shock excitations with spectral content being significantly different from that of the shock excitation used for the calibration. Hence, the shock sensitivity does not characterize an accelerometer in general but only with respect to a specific shock excitation. In order to overcome this limitation, characterization of accelerometers in terms of a linear, second-order differential equation has been proposed recently [3,4]. The parameters of this model are the unknown coefficients of the differential equation which have to be estimated from calibration measurements. In the recent works [3,4] an identification procedure has been presented for this estimation based on shock calibration measurements. In this paper, an identification procedure is proposed on the basis of sinusoidal calibration measurements. The proposed proceeding has the advantage that model estimation is based on the measured frequency response without the need to use a discretized solution to the differential equation. More important, since the established sineapproximation method is well understood and reveals reliable uncertainties associated with the measurements, this proceeding also allows the application of a v2-test to check the validity of the applied model. Furthermore, uncertainties of the estimated model (and quantities derived from it) can be determined on the basis of the available uncertainties for the measured frequency response. The proposed identification procedure is based on (weighted) linear least-squares and allows easy and stable parameter estimation. For uncertainty calculation, a Monte Carlo simulation technique is described and applied. The prediction of the accelerometer to transient shock excitations is done by means of a recursive digital filter which is derived from the differential equation (after their coefficients have been estimated). Application of this filter then allows to integrate the differential equation, i.e. to predict the accelerometer’s output to an applied excitation, fast and easily. The proposed modelling procedure is illustrated by its application to sinusoidal calibration measurements ranging from for 40 Hz to 20 kHz for two accelerometers. The derived models are used to predict the behaviour of the two accelerometers for two different shock excitations, and the predictions are checked by calibration measurements for these shock excitations. Predicted and measured shock sensitivities are compared and discussed.
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2. Accelerometer model The mass spring system of a piezoelectric accelerometer [5] is described by the linear differential equation €xðtÞ þ 2dx0 x_ ðtÞ þ x20 xðtÞ ¼ qaðtÞ;
ð1Þ
where d, x0, q denote the physical model parameters damping, resonant frequency x0 = 2pf0, and transformation constant of the input acceleration a(t). The complex sensitivity G(x) is the complex frequency response of the linear time invariant system (1) q GðxÞ ¼ 2 2 ðx0 x Þ þ 2jdxx0 ¼ SðxÞ exp½juðxÞ;
ð2Þ
where S(x) and u(x) denote magnitude and phase delay of the system. For the determination of accelerations, accelerometers are applied in combination with a connected charge amplifier whose input is the accelerometer’s output. The measurement signal then is the amplified accelerometer’s output. The frequency response of the charge amplifier – determined by additional calibration measurements – is typically known with high accuracy. 2.1. Estimation of model parameters by sinusoidal calibration measurements For the estimation of the model parameters in (1) it is proposed to use calibration measurements for sinusoidal excitations [1] which directly determine the complex sensitivity G(x) in (2), taking into account the known frequency response of the connected charge amplifier. From (2) the linear relation G1 ðxÞ ¼ S 1 ðxÞ exp½juðxÞ ¼ l1 þ l2 2jx l3 x2 ¼ f T ðxÞl
ð3Þ
is derived for the combined parameter vector lT ¼ ðl1 ; l2 ; l3 Þ ¼ ðx20 =q; dx0 =q; 1=qÞ with fT(x) = (1, 2jx, x 2). The relation (3) is used to first estimate the parameters li by linear least-squares, and then to calculate the physical parameters according to the relation pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi q ¼ 1=l3 ; x0 ¼ l1 =l3 ; d ¼ l2 = l1 l3 : ð4Þ The sinusoidal calibration measurements are well established (i.e. validated by CIPM key comparisons) and they provide reliable uncertainties associated with the frequency response [6]. Let Sm = S(xm),
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A. Link et al. / Measurement 40 (2007) 928–935
um = u(xm) denote the resulting amplitudes and phases of the frequency response with associated uncertainties u(Sm), u(um) at frequencies xm, m = 1, 2, . . . , L. From these measurements the vector 1 yT ¼ ðRe S 1 1 expðju1 Þ; . . . ; Re S L expðjuL Þ; 1 Im S 1 1 expðju1 Þ; . . . ; Im S L expðjuL ÞÞ
ð5Þ of length 2L is constructed. In addition, the covariance matrix Vy of y is determined utilizing a Monte Carlo simulation technique, see Appendix A. Let H denote the 2L times 3 matrix 1 0 Re f T ðx1 Þ C B. C B .. C B C B B Re f T ðxL Þ C C: B ð6Þ H ¼B C T B Im f ðx1 Þ C C B C B .. A @. Im f T ðxL Þ The parameters li are estimated by weighted linear least-squares according to n o ^ ¼ arg min ðy HlÞT V 1 l ðy HlÞ y l
1
T 1 ¼ ðH T V 1 y HÞ H V y y:
ð7Þ
^ then the model parameters of (1) are calcuFrom l lated according to (4). The uncertainties of the ^i , i.e. the covariance matrix parameter estimates l ^, are given by associated with l 1
V l^ ¼ ðH T V 1 y HÞ :
ð8Þ
Again, the determination of uncertainties associated ^ 0; q ^ calculated from l ^ is with the estimates ^ d; x done by Monte Carlo simulations, see Appendix A. In this way, the well-established sinusoidal calibration measurements including their uncertainties are used to estimate the physical model (1) including its associated uncertainty. In order to check the consistency of model and data it is proposed to check the criterion n o T v2m;p=2 6 min ðy HlÞ V 1 ðy HlÞ 6 v2m;1p=2 ; y l
ð9Þ 2
where v2m;p denotes the pth quantile of a v -distribution with m = 2L 3 degrees of freedom and with p chosen small, e.g. p = 0.05.
2.2. Prediction of accelerometer output for transient excitations To predict the accelerometer’s output to an excitation a(t), e.g. a shock, the differential equation (1) has to be integrated. For this, the differential equation can be transformed into the difference equation xðkÞ þ c1 xðk 1Þ þ c2 xðk 2Þ ¼ b½aðkÞ þ 2aðk 1Þ þ aðk 2Þ; k ¼ 0; 1; 2; . . . ; ð10Þ see Appendix B; in (10) x(k) = x(kTs) and a(k) = a(kTs) denote the accelerometer’s response and the assumed excitation evaluated at time instants kTs. The approximation error depends on Ts which has to be chosen sufficiently small. The model parameters in (10) are determined from the physical parameters according to b ¼ 0:25qT 2s =K;
ð11Þ
c1 ¼ ð0:5x20 T 2s 2Þ=K; c2 ¼ ð1 dx0 T s þ
0:25x20 T 2s Þ=K
ð12Þ ð13Þ
with K ¼ 1 þ dx0 T s þ 0:25x20 T 2s :
ð14Þ
The difference equation (10) can be identified as a recursive filter algorithm supplying for a given input sequences a(k) the accelerometer output x(k) for k = 0, 1, 2, . . . By again using Monte Carlo simulation techniques, the uncertainties associated with estimates derived from the predicted accelerometer’s output such as shock sensitivities can be determined. 2.3. Influence of charge amplifier The output of the accelerometer is fed into a charge amplifier whose output then yields the recorded data. Hence, the influence of the charge amplifier has to be accounted for. The charge amplifier behaviour is characterized by its frequency response which is usually determined with high accuracy. For sinusoidal excitations, the influence of the charge amplifier simply is to multiply the complex sensitivity of the accelerometer (2) by its frequency response. Such proceeding is, however, no longer possible for transient signals. In this case, the following procedure is proposed: The influence of the charge amplifier frequency response is modelled by a digital filter which is designed such that
A. Link et al. / Measurement 40 (2007) 928–935
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Magnitude
1 0.95 0.9 0.85 0.8 0
10
20
30
40
50
10
20
30
40
50
Phase in degree
0 –10 –20 –30 –40 0
Frequency in kHz
Fig. 1. Frequency response (solid line) of used charge amplifier together with the chosen Butterworth filter model (dotted line) for discretetime processing.
its frequency response approximately equals that of the charge amplifier within the relevant frequency range. Subsequent application of this digital filter to the predicted digital accelerometer output then yields the corresponding predicted measurement signal. Fig. 1 shows the frequency response of the charge amplifier which was used together with that of the designed digital Butterworth filter of firstorder. For the results presented in this paper the influence of this modelling stage on derived uncertainties could be neglected since (i) the frequency response of the charge amplifier was known with high accuracy and (ii) the difference between measured and modelled frequency response (Fig. 1) was not relevant with respect to the results presented in Section 4. The latter was checked by correspondingly varying the frequency response of the digital filter.
20 kHz using the sine-approximation method [1]. In addition, measurements were performed for two types of shocks: Low intensity shocks (LI) with peak values of 5 km/s2 and high intensity shocks (HI) with peak values between 10 km/s2 and 20 km/s2. For each of the two types of shocks six repeated measurements were performed. Output acceleration signals were sampled with a rate of 107 samples/s. The input accelerations were derived from interferometric measurements [3] at the same sampling rate as the output acceleration. Figs. 2
5 4.5 4
3. Measurement data Primary calibration measurements of two backto-back accelerometers (ENDEVCO type 2270 and Bru¨el & Kjær type 8305) were carried out using the calibration standards of the PhysikalischTechnische Bundesanstalt [8]. Sinusoidal excitations with an acceleration amplitude of approximately 100 m/s2 were used to determine the complex sensitivity at 62 different frequencies between 40 Hz and
a(t) in km/s2
3.5 3 2.5 2 1.5 1 0.5 0
1
2
3 4 Time in ms
5
6
Fig. 2. Low intensity input shock acceleration.
7
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15
a(t) in km/s2
10
5
0
–5
–10 1
2
3 Time in ms
4
5
6
Fig. 3. High intensity input shock acceleration.
and 3 show typical input accelerations for the two types of shocks. 4. Results For each of the two accelerometers, model parameters were estimated from the sinusoidal calibration experiments as described in Section 2.1. The uncertainties associated with the sinusoidal calibration measurements are listed in Table 1. They reflect the calibration and measurement capabilities of PTB using the sine-approximation method [6]. Fig. 4a and b shows the magnitude and phase of the complex frequency response determined from the sinusoidal calibration measurements and from the estimated model. The consistency criterion (9) was (for p = 0.05) passed for the ENDEVCO type 2270 ðv2min 144Þ and it was failed for the Bru¨el & Kjær type 8305 accelerometer ðv2min 67Þ which is smaller than ðv2121;0:05 ¼ 96:6Þ. The reason for this failure is probably due to a slight overestimation of the uncertainties of the sinusoidal calibration measurements for this accelerometer and not due to a model failure. Table 1 Standard uncertainties of sinusoidal calibration measurements for the accelerometers ENDEVCO type 2270 and Bru¨el & Kjær type 8305 Frequency range (kHz)
urel(Sm) Æ 103
u(um) (deg.)
65 5 10 10 15 15 20
0.5 1.5 2.5 5
0.25 0.5 0.5 0.5
Hence, the results predicted by the model for this accelerometer were considered as well. Note that a slight overestimation of the uncertainties of the sinusoidal calibration measurements for this accelerometer will in turn lead to slightly overestimated uncertainties associated with the estimated model and results derived from it. The resulting model parameters including derived uncertainties for the two accelerometers are listed in Table 2. For both accelerometers the obtained models were used to predict the measured output upon shock excitations, where the influence of the connected charge amplifier was taken into account as described in Section 2.3. As input excitations the individual acceleration measured by Laser interferometry during the LI and HI shock measurements were taken. This allowed for a quantitative comparison between model prediction and measurement result for each individual shock. Since for shockshaped signals the shock sensitivity is used to describe the behaviour of an accelerometer, comparison between prediction and measurement was based on these sensitivities. The corresponding results are shown in Fig. 5. Observed differences between predicted and measured shock sensitivities are in most cases smaller than 1% and in accordance with the uncertainties; for the predicted shock sensitivities uncertainties were calculated as described in Appendix A. While the observed differences between predicted and measured shock sensitivities are well accounted for by the associated uncertainties, the signs of these differences are all the same for the high intensity shocks. This indicates the presence of systematic differences between measured and predicted shock peak values for high intensity shocks. This finding motivates future investigations to resolve these systematic effects and to further reduce associated uncertainties. Note that the uncertainties associated with the shock sensitivities predicted by the model are smaller than those associated with the measurements. Note further that for each accelerometer the shock sensitivities themselves differ significantly for the two different shocks. The results demonstrate that the proposed accelerometer model estimated from sinusoidal calibration measurements allows for the highly accurate prediction of the accelerometer’s output to shock excitations. The shock sensitivity typically used to describe the behaviour of an accelerometer to a shock excitation strongly depends on the form of
A. Link et al. / Measurement 40 (2007) 928–935
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S in pC/ms2
0.18
0.16
0.14
0.12
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8 10 12 Frequency in kHz
14
16
18
20
0
2
4
6
8
12
14
16
18
20
0
2
4
6
8 10 12 Frequency in kHz
14
16
18
20
φ in degree
2
0
–2
–4
0.27
S in pC/ms2
0.26 0.25 0.24 0.23 0.22
10
φ in degree
2
0
–2
–4
Fig. 4. Magnitude and phase of measured frequency response together with expanded uncertainties (k = 2) for the Bru¨el & Kjær type 8305 (a) and ENDEVCO type 2270 (b) accelerometers. The solid lines show the obtained model response.
Table 2 Model parameter estimates together with associated standard uncertainties for the accelerometers ENDEVCO type 2270 and Bru¨el & Kjær type 8305 d
S 0 ¼ q=x20 (pC/(m/s2))
f0 (kHz)
Type
0.0055 (0.001) 0.0610 (0.002)
0.1240 (1.3e005) 0.2254 (2.5e005)
35.97 (0.04) 52.80 (0.22)
B8305 BD47
the shock. Hence, the shock sensitivity does not uniquely characterize an accelerometer. The proposed model, on the other hand, enables the predic-
tion to any kind of shock excitation and thus characterizes the accelerometer’s behaviour. 5. Summary and conclusion The input output behaviour of an accelerometer was modelled by a linear, second-order differential equation. For the determination of the unknown coefficients of the differential equation a linear least-squares approach was proposed based on established sinusoidal calibration measurements.
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A. Link et al. / Measurement 40 (2007) 928–935
0.135
Low intensity shocks
High intensity shocks
Peak ratio in pC/(m/s2)
measured predicted
0.13
0.125
0.12
0.235
1
2
3
4
5
Low intensity shocks
6 1 Shock number
2
3
4
5
6
High intensity shocks
Peak ratio in pC/(m/s2)
measured predicted
0.23
Appendix A. Calculation of uncertainties by Monte Carlo simulation The data vector y (5) entering into the linear least-squares estimation (7) is constructed by a non-linear transformation from the amplitude and phase measurements Sm = S(xm), um = u(xm), m = 1, . . . , L of the frequency response. For the amplitude and phase measurements, uncertainties u(Sm), u(um), m = 1, . . . , L are available according to [1]. In order to properly weight the least-squares adjustment (7) of y, the covariance matrix Vy of y has to be determined. This was done by Monte Carlo simulation [7] as follows: 1. For k = 1:MC: Calculate S km ¼ S m þ uðS m Þekm , ukm ¼ um þ uðum Þlkm , m = 1, . . . , L where the ekm, lkm denote independent random numbers drawn from a standard normal distribution. For each k, calculate the vector yk from the S km , ukm , m = 1, . . . , L according to (5). 2. The covariance matrix Vy is determined as the covariance matrix of the vectors yk, k = 1, . . . , MC.
0.225
0.22
1
2
3
4
5
6 1 2 Shock number
3
4
5
6
Fig. 5. Measured and predicted shock sensitivities together with expanded uncertainties (k = 2) for the Bru¨el & Kjær type 8305 (a) and ENDEVCO type 2270 (b).
Utilizing a recursive digital filter derived from the considered differential equation, the accelerometer’s behaviour upon shock-shaped (transient) excitations was predicted. For the calculation of associated uncertainties a Monte Carlo simulation technique was employed. The proposed modelling approach was applied to sinusoidal calibration measurements and the derived model was validated by means of additional calibration measurements using shock excitations. As a result, predicted and measured shock sensitivities were consistent with most differences being below 1%. These results thus confirm the proposed modelling approach and encourage its use for determining accelerometer output signals also for input accelerations differing from those used during the calibration measurements.
For the number of Monte Carlo trials we used MC = 106 which was found large enough such that the results remained essentially unchanged when repeating the whole analysis. A Gaussian distribution was used due to lack of distributional knowledge. Since no information was available on a possible correlation between the measurements Sm = S(xm), um = u(xm), m = 1, . . . , L, no correlation has been considered. Note that knowledge of correlation can immediately be taken into account. In a similar way uncertainties associated with the estimated model were derived based on the covari1 ance matrix V l^ ¼ ðH T V 1 (8) associated with y HÞ ^. For instance, uncertainties (and the covariances) l associated with the resulting physical parameter ^ 0; q ^ were calculated as follows: estimates ^d; x ^k ¼ l ^ þ ek where ek 1. For k = 1:MC: Calculate l denotes a three-dimensional random vector drawn from a multivariate normal distribution with zero mean and covariance matrix 1 V l^ ¼ ðH T V 1 For each k, calculate y HÞ . ^dk ; x ^ k0 ; q ^k from l ^k according to (4). ^ 0; q ^ is 2. The covariance matrix associated with ^d; x then determined as the covariance matrix of the ^ k0 ; q ^k ÞT ; k ¼ 1; . . . ; MC, with the vectors ð^dk ; x diagonal elements of the covariance matrix being
A. Link et al. / Measurement 40 (2007) 928–935
the squared standard uncertainties. (Note that the ^ k0 ; q ^k ÞT ; k ¼ 1; . . . ; MC mean of the vectors ð^ dk ; x ^ 0; q ^.) essentially coincided with ^ d; x Similarly, the uncertainties associated with estimates derived from the model such as shock sensitivities were calculated. Appendix B. Approximation of a linear continuoustime system by a discrete-time system Application of the Laplace transformation to the differential equation (1) provides the continuoustime system function G(s). From G(s), a corresponding discrete-time system function D(z) can be obtained by substituting s according to 2 1 z1 s¼ ðB:1Þ T 1 þ z1 that is 2 1 z1 DðzÞ ¼ G : T 1 þ z1
ðB:2Þ
The system function D(z) has the algebraic form PM bk zk ðB:3Þ DðzÞ ¼ Pk¼0 N k k¼0 ck z satisfying a difference equation of the form N X k¼0
ck yðn kÞ ¼
M X k¼0
bk xðn kÞ:
ðB:4Þ
935
Hence, it is straightforward to obtain the corresponding difference equation (B.4) from the system function D(z). For further details see [9]. References [1] ISO, International Standard 16063-11 Methods for the calibration of vibration and shock transducers – part 11: Primary vibration calibration by laser interferometry, International Organization for Standardization, Geneva, 1999. [2] ISO, International Standard 16063-13 Methods for the calibration of vibration and shock transducers – part 13: Primary shock calibration by laser interferometry, International Organization for Standardization, Geneva, 2001. [3] A. Link, H.J. von Martens, Accelerometer identification using shock excitation, Measurement 35 (2004) 191–199. [4] A. Link, A. Ta¨ubner, W. Wabinski, T. Bruns, C. Elster, Calibration of accelerometers: determination of amplitude and phase response upon shock excitation, Measurement Science and Technology 17 (2006) 1888–1894. [5] Bru¨el & Kjær, Instructions and Applications, Naerum, Denmark, 1975. [6] Calibration and measurement capabilities, Acoustics, Ultrasound and Vibration, PTB, Germany. Available from:. [7] BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, OIML, Guide to the Expression of Uncertainty in Measurement. Supplement 1 – Propagation of Distributions Using a Monte Carlo Method, Draft, 2005. [8] H.J. von Martens, A. Link, H.J. Schlaak, A. Ta¨ubner, W. Wabinski, U. Go¨bel, Recent advances in vibration and shock measurements and calibrations using laser interferometry, in: Sixth International Conference on Vibration Measurements by Laser Techniques: Advances and Applications, Proc. SPIE 5503, 2004, pp. 1–19. [9] A.V. Oppenheim, R.W. Schafer, Discrete-Time Signal Processing, Prentice Hall, Englewood Cliffs, NJ, 1989.
Modelling accelerometers for transient signals using calibration measurements upon sinusoidal excitation A. Link
a,*
, A. Ta¨ubner b, W. Wabinski b, T. Bruns b, C. Elster
a
a
b
Physikalisch-Technische Bundesanstalt Berlin, Abbestrasse 2-12, 10587 Berlin, Germany Physikalisch-Technische Bundesanstalt Braunschweig, Bundesallee 100, 38116 Braunschweig, Germany Received 23 June 2006; received in revised form 13 September 2006; accepted 23 October 2006 Available online 1 November 2006
Abstract A recently proposed accelerometer model is applied for determining the accelerometer’s output to transient accelerations. The model consists of a linear, second-order differential equation with unknown coefficients. It is proposed to estimate these model parameters from sinusoidal calibration measurements, and an estimation procedure based on linear least-squares is presented. In addition, the uncertainties associated with the estimated parameters are determined utilizing a Monte Carlo simulation technique. The performance of the proposed modelling approach was tested by its application to calibration measurements of two back-to-back accelerometers (ENDEVCO type 2270 and Bru¨el & Kjær type 8305). For each of the two accelerometers, the model was first estimated from sinusoidal calibration measurements and then used to predict the accelerometer’s behaviour for two shock calibration measurements. Measured and predicted shock sensitivities were found consistent with differences below 1% in most cases which confirms the benefit of the proposed modelling approach. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Calibration; Sinusoidal excitation; Shock excitation; Modelling; Uncertainty; IIR filter
1. Introduction Piezoelectric accelerometers can be considered as linear electromechanical transducers designed for defined amplitude and frequency ranges. For sinusoidal excitations, the input/output behaviour of accelerometers is characterized in terms of the complex sensitivity. This sensitivity is for primary calibrations determined by the established sine* Corresponding author. Tel.: +49 3034817489; fax: +49 3034817490. E-mail address: [email protected] (A. Link).
approximation method [1] resulting in estimates of the magnitude and phase response of an accelerometer at discrete frequencies over the frequency range of interest. In addition, uncertainties associated with these estimates are determined by the calibration procedure. According to an agreed international standard [2], an accelerometer is characterized for shock excitations by its shock sensitivity. The shock sensitivity is defined as the ratio of the peak values of the output voltage and the applied input acceleration. Such an approach is appropriate for comparing different accelerometers with respect to well-defined excita-
0263-2241/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.measurement.2006.10.011
A. Link et al. / Measurement 40 (2007) 928–935
tions. However, it does not allow to predict an accelerometer’s output to shock excitations with spectral content being significantly different from that of the shock excitation used for the calibration. Hence, the shock sensitivity does not characterize an accelerometer in general but only with respect to a specific shock excitation. In order to overcome this limitation, characterization of accelerometers in terms of a linear, second-order differential equation has been proposed recently [3,4]. The parameters of this model are the unknown coefficients of the differential equation which have to be estimated from calibration measurements. In the recent works [3,4] an identification procedure has been presented for this estimation based on shock calibration measurements. In this paper, an identification procedure is proposed on the basis of sinusoidal calibration measurements. The proposed proceeding has the advantage that model estimation is based on the measured frequency response without the need to use a discretized solution to the differential equation. More important, since the established sineapproximation method is well understood and reveals reliable uncertainties associated with the measurements, this proceeding also allows the application of a v2-test to check the validity of the applied model. Furthermore, uncertainties of the estimated model (and quantities derived from it) can be determined on the basis of the available uncertainties for the measured frequency response. The proposed identification procedure is based on (weighted) linear least-squares and allows easy and stable parameter estimation. For uncertainty calculation, a Monte Carlo simulation technique is described and applied. The prediction of the accelerometer to transient shock excitations is done by means of a recursive digital filter which is derived from the differential equation (after their coefficients have been estimated). Application of this filter then allows to integrate the differential equation, i.e. to predict the accelerometer’s output to an applied excitation, fast and easily. The proposed modelling procedure is illustrated by its application to sinusoidal calibration measurements ranging from for 40 Hz to 20 kHz for two accelerometers. The derived models are used to predict the behaviour of the two accelerometers for two different shock excitations, and the predictions are checked by calibration measurements for these shock excitations. Predicted and measured shock sensitivities are compared and discussed.
929
2. Accelerometer model The mass spring system of a piezoelectric accelerometer [5] is described by the linear differential equation €xðtÞ þ 2dx0 x_ ðtÞ þ x20 xðtÞ ¼ qaðtÞ;
ð1Þ
where d, x0, q denote the physical model parameters damping, resonant frequency x0 = 2pf0, and transformation constant of the input acceleration a(t). The complex sensitivity G(x) is the complex frequency response of the linear time invariant system (1) q GðxÞ ¼ 2 2 ðx0 x Þ þ 2jdxx0 ¼ SðxÞ exp½juðxÞ;
ð2Þ
where S(x) and u(x) denote magnitude and phase delay of the system. For the determination of accelerations, accelerometers are applied in combination with a connected charge amplifier whose input is the accelerometer’s output. The measurement signal then is the amplified accelerometer’s output. The frequency response of the charge amplifier – determined by additional calibration measurements – is typically known with high accuracy. 2.1. Estimation of model parameters by sinusoidal calibration measurements For the estimation of the model parameters in (1) it is proposed to use calibration measurements for sinusoidal excitations [1] which directly determine the complex sensitivity G(x) in (2), taking into account the known frequency response of the connected charge amplifier. From (2) the linear relation G1 ðxÞ ¼ S 1 ðxÞ exp½juðxÞ ¼ l1 þ l2 2jx l3 x2 ¼ f T ðxÞl
ð3Þ
is derived for the combined parameter vector lT ¼ ðl1 ; l2 ; l3 Þ ¼ ðx20 =q; dx0 =q; 1=qÞ with fT(x) = (1, 2jx, x 2). The relation (3) is used to first estimate the parameters li by linear least-squares, and then to calculate the physical parameters according to the relation pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi q ¼ 1=l3 ; x0 ¼ l1 =l3 ; d ¼ l2 = l1 l3 : ð4Þ The sinusoidal calibration measurements are well established (i.e. validated by CIPM key comparisons) and they provide reliable uncertainties associated with the frequency response [6]. Let Sm = S(xm),
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A. Link et al. / Measurement 40 (2007) 928–935
um = u(xm) denote the resulting amplitudes and phases of the frequency response with associated uncertainties u(Sm), u(um) at frequencies xm, m = 1, 2, . . . , L. From these measurements the vector 1 yT ¼ ðRe S 1 1 expðju1 Þ; . . . ; Re S L expðjuL Þ; 1 Im S 1 1 expðju1 Þ; . . . ; Im S L expðjuL ÞÞ
ð5Þ of length 2L is constructed. In addition, the covariance matrix Vy of y is determined utilizing a Monte Carlo simulation technique, see Appendix A. Let H denote the 2L times 3 matrix 1 0 Re f T ðx1 Þ C B. C B .. C B C B B Re f T ðxL Þ C C: B ð6Þ H ¼B C T B Im f ðx1 Þ C C B C B .. A @. Im f T ðxL Þ The parameters li are estimated by weighted linear least-squares according to n o ^ ¼ arg min ðy HlÞT V 1 l ðy HlÞ y l
1
T 1 ¼ ðH T V 1 y HÞ H V y y:
ð7Þ
^ then the model parameters of (1) are calcuFrom l lated according to (4). The uncertainties of the ^i , i.e. the covariance matrix parameter estimates l ^, are given by associated with l 1
V l^ ¼ ðH T V 1 y HÞ :
ð8Þ
Again, the determination of uncertainties associated ^ 0; q ^ calculated from l ^ is with the estimates ^ d; x done by Monte Carlo simulations, see Appendix A. In this way, the well-established sinusoidal calibration measurements including their uncertainties are used to estimate the physical model (1) including its associated uncertainty. In order to check the consistency of model and data it is proposed to check the criterion n o T v2m;p=2 6 min ðy HlÞ V 1 ðy HlÞ 6 v2m;1p=2 ; y l
ð9Þ 2
where v2m;p denotes the pth quantile of a v -distribution with m = 2L 3 degrees of freedom and with p chosen small, e.g. p = 0.05.
2.2. Prediction of accelerometer output for transient excitations To predict the accelerometer’s output to an excitation a(t), e.g. a shock, the differential equation (1) has to be integrated. For this, the differential equation can be transformed into the difference equation xðkÞ þ c1 xðk 1Þ þ c2 xðk 2Þ ¼ b½aðkÞ þ 2aðk 1Þ þ aðk 2Þ; k ¼ 0; 1; 2; . . . ; ð10Þ see Appendix B; in (10) x(k) = x(kTs) and a(k) = a(kTs) denote the accelerometer’s response and the assumed excitation evaluated at time instants kTs. The approximation error depends on Ts which has to be chosen sufficiently small. The model parameters in (10) are determined from the physical parameters according to b ¼ 0:25qT 2s =K;
ð11Þ
c1 ¼ ð0:5x20 T 2s 2Þ=K; c2 ¼ ð1 dx0 T s þ
0:25x20 T 2s Þ=K
ð12Þ ð13Þ
with K ¼ 1 þ dx0 T s þ 0:25x20 T 2s :
ð14Þ
The difference equation (10) can be identified as a recursive filter algorithm supplying for a given input sequences a(k) the accelerometer output x(k) for k = 0, 1, 2, . . . By again using Monte Carlo simulation techniques, the uncertainties associated with estimates derived from the predicted accelerometer’s output such as shock sensitivities can be determined. 2.3. Influence of charge amplifier The output of the accelerometer is fed into a charge amplifier whose output then yields the recorded data. Hence, the influence of the charge amplifier has to be accounted for. The charge amplifier behaviour is characterized by its frequency response which is usually determined with high accuracy. For sinusoidal excitations, the influence of the charge amplifier simply is to multiply the complex sensitivity of the accelerometer (2) by its frequency response. Such proceeding is, however, no longer possible for transient signals. In this case, the following procedure is proposed: The influence of the charge amplifier frequency response is modelled by a digital filter which is designed such that
A. Link et al. / Measurement 40 (2007) 928–935
931
Magnitude
1 0.95 0.9 0.85 0.8 0
10
20
30
40
50
10
20
30
40
50
Phase in degree
0 –10 –20 –30 –40 0
Frequency in kHz
Fig. 1. Frequency response (solid line) of used charge amplifier together with the chosen Butterworth filter model (dotted line) for discretetime processing.
its frequency response approximately equals that of the charge amplifier within the relevant frequency range. Subsequent application of this digital filter to the predicted digital accelerometer output then yields the corresponding predicted measurement signal. Fig. 1 shows the frequency response of the charge amplifier which was used together with that of the designed digital Butterworth filter of firstorder. For the results presented in this paper the influence of this modelling stage on derived uncertainties could be neglected since (i) the frequency response of the charge amplifier was known with high accuracy and (ii) the difference between measured and modelled frequency response (Fig. 1) was not relevant with respect to the results presented in Section 4. The latter was checked by correspondingly varying the frequency response of the digital filter.
20 kHz using the sine-approximation method [1]. In addition, measurements were performed for two types of shocks: Low intensity shocks (LI) with peak values of 5 km/s2 and high intensity shocks (HI) with peak values between 10 km/s2 and 20 km/s2. For each of the two types of shocks six repeated measurements were performed. Output acceleration signals were sampled with a rate of 107 samples/s. The input accelerations were derived from interferometric measurements [3] at the same sampling rate as the output acceleration. Figs. 2
5 4.5 4
3. Measurement data Primary calibration measurements of two backto-back accelerometers (ENDEVCO type 2270 and Bru¨el & Kjær type 8305) were carried out using the calibration standards of the PhysikalischTechnische Bundesanstalt [8]. Sinusoidal excitations with an acceleration amplitude of approximately 100 m/s2 were used to determine the complex sensitivity at 62 different frequencies between 40 Hz and
a(t) in km/s2
3.5 3 2.5 2 1.5 1 0.5 0
1
2
3 4 Time in ms
5
6
Fig. 2. Low intensity input shock acceleration.
7
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15
a(t) in km/s2
10
5
0
–5
–10 1
2
3 Time in ms
4
5
6
Fig. 3. High intensity input shock acceleration.
and 3 show typical input accelerations for the two types of shocks. 4. Results For each of the two accelerometers, model parameters were estimated from the sinusoidal calibration experiments as described in Section 2.1. The uncertainties associated with the sinusoidal calibration measurements are listed in Table 1. They reflect the calibration and measurement capabilities of PTB using the sine-approximation method [6]. Fig. 4a and b shows the magnitude and phase of the complex frequency response determined from the sinusoidal calibration measurements and from the estimated model. The consistency criterion (9) was (for p = 0.05) passed for the ENDEVCO type 2270 ðv2min 144Þ and it was failed for the Bru¨el & Kjær type 8305 accelerometer ðv2min 67Þ which is smaller than ðv2121;0:05 ¼ 96:6Þ. The reason for this failure is probably due to a slight overestimation of the uncertainties of the sinusoidal calibration measurements for this accelerometer and not due to a model failure. Table 1 Standard uncertainties of sinusoidal calibration measurements for the accelerometers ENDEVCO type 2270 and Bru¨el & Kjær type 8305 Frequency range (kHz)
urel(Sm) Æ 103
u(um) (deg.)
65 5 10 10 15 15 20
0.5 1.5 2.5 5
0.25 0.5 0.5 0.5
Hence, the results predicted by the model for this accelerometer were considered as well. Note that a slight overestimation of the uncertainties of the sinusoidal calibration measurements for this accelerometer will in turn lead to slightly overestimated uncertainties associated with the estimated model and results derived from it. The resulting model parameters including derived uncertainties for the two accelerometers are listed in Table 2. For both accelerometers the obtained models were used to predict the measured output upon shock excitations, where the influence of the connected charge amplifier was taken into account as described in Section 2.3. As input excitations the individual acceleration measured by Laser interferometry during the LI and HI shock measurements were taken. This allowed for a quantitative comparison between model prediction and measurement result for each individual shock. Since for shockshaped signals the shock sensitivity is used to describe the behaviour of an accelerometer, comparison between prediction and measurement was based on these sensitivities. The corresponding results are shown in Fig. 5. Observed differences between predicted and measured shock sensitivities are in most cases smaller than 1% and in accordance with the uncertainties; for the predicted shock sensitivities uncertainties were calculated as described in Appendix A. While the observed differences between predicted and measured shock sensitivities are well accounted for by the associated uncertainties, the signs of these differences are all the same for the high intensity shocks. This indicates the presence of systematic differences between measured and predicted shock peak values for high intensity shocks. This finding motivates future investigations to resolve these systematic effects and to further reduce associated uncertainties. Note that the uncertainties associated with the shock sensitivities predicted by the model are smaller than those associated with the measurements. Note further that for each accelerometer the shock sensitivities themselves differ significantly for the two different shocks. The results demonstrate that the proposed accelerometer model estimated from sinusoidal calibration measurements allows for the highly accurate prediction of the accelerometer’s output to shock excitations. The shock sensitivity typically used to describe the behaviour of an accelerometer to a shock excitation strongly depends on the form of
A. Link et al. / Measurement 40 (2007) 928–935
933
S in pC/ms2
0.18
0.16
0.14
0.12
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8 10 12 Frequency in kHz
14
16
18
20
0
2
4
6
8
12
14
16
18
20
0
2
4
6
8 10 12 Frequency in kHz
14
16
18
20
φ in degree
2
0
–2
–4
0.27
S in pC/ms2
0.26 0.25 0.24 0.23 0.22
10
φ in degree
2
0
–2
–4
Fig. 4. Magnitude and phase of measured frequency response together with expanded uncertainties (k = 2) for the Bru¨el & Kjær type 8305 (a) and ENDEVCO type 2270 (b) accelerometers. The solid lines show the obtained model response.
Table 2 Model parameter estimates together with associated standard uncertainties for the accelerometers ENDEVCO type 2270 and Bru¨el & Kjær type 8305 d
S 0 ¼ q=x20 (pC/(m/s2))
f0 (kHz)
Type
0.0055 (0.001) 0.0610 (0.002)
0.1240 (1.3e005) 0.2254 (2.5e005)
35.97 (0.04) 52.80 (0.22)
B8305 BD47
the shock. Hence, the shock sensitivity does not uniquely characterize an accelerometer. The proposed model, on the other hand, enables the predic-
tion to any kind of shock excitation and thus characterizes the accelerometer’s behaviour. 5. Summary and conclusion The input output behaviour of an accelerometer was modelled by a linear, second-order differential equation. For the determination of the unknown coefficients of the differential equation a linear least-squares approach was proposed based on established sinusoidal calibration measurements.
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0.135
Low intensity shocks
High intensity shocks
Peak ratio in pC/(m/s2)
measured predicted
0.13
0.125
0.12
0.235
1
2
3
4
5
Low intensity shocks
6 1 Shock number
2
3
4
5
6
High intensity shocks
Peak ratio in pC/(m/s2)
measured predicted
0.23
Appendix A. Calculation of uncertainties by Monte Carlo simulation The data vector y (5) entering into the linear least-squares estimation (7) is constructed by a non-linear transformation from the amplitude and phase measurements Sm = S(xm), um = u(xm), m = 1, . . . , L of the frequency response. For the amplitude and phase measurements, uncertainties u(Sm), u(um), m = 1, . . . , L are available according to [1]. In order to properly weight the least-squares adjustment (7) of y, the covariance matrix Vy of y has to be determined. This was done by Monte Carlo simulation [7] as follows: 1. For k = 1:MC: Calculate S km ¼ S m þ uðS m Þekm , ukm ¼ um þ uðum Þlkm , m = 1, . . . , L where the ekm, lkm denote independent random numbers drawn from a standard normal distribution. For each k, calculate the vector yk from the S km , ukm , m = 1, . . . , L according to (5). 2. The covariance matrix Vy is determined as the covariance matrix of the vectors yk, k = 1, . . . , MC.
0.225
0.22
1
2
3
4
5
6 1 2 Shock number
3
4
5
6
Fig. 5. Measured and predicted shock sensitivities together with expanded uncertainties (k = 2) for the Bru¨el & Kjær type 8305 (a) and ENDEVCO type 2270 (b).
Utilizing a recursive digital filter derived from the considered differential equation, the accelerometer’s behaviour upon shock-shaped (transient) excitations was predicted. For the calculation of associated uncertainties a Monte Carlo simulation technique was employed. The proposed modelling approach was applied to sinusoidal calibration measurements and the derived model was validated by means of additional calibration measurements using shock excitations. As a result, predicted and measured shock sensitivities were consistent with most differences being below 1%. These results thus confirm the proposed modelling approach and encourage its use for determining accelerometer output signals also for input accelerations differing from those used during the calibration measurements.
For the number of Monte Carlo trials we used MC = 106 which was found large enough such that the results remained essentially unchanged when repeating the whole analysis. A Gaussian distribution was used due to lack of distributional knowledge. Since no information was available on a possible correlation between the measurements Sm = S(xm), um = u(xm), m = 1, . . . , L, no correlation has been considered. Note that knowledge of correlation can immediately be taken into account. In a similar way uncertainties associated with the estimated model were derived based on the covari1 ance matrix V l^ ¼ ðH T V 1 (8) associated with y HÞ ^. For instance, uncertainties (and the covariances) l associated with the resulting physical parameter ^ 0; q ^ were calculated as follows: estimates ^d; x ^k ¼ l ^ þ ek where ek 1. For k = 1:MC: Calculate l denotes a three-dimensional random vector drawn from a multivariate normal distribution with zero mean and covariance matrix 1 V l^ ¼ ðH T V 1 For each k, calculate y HÞ . ^dk ; x ^ k0 ; q ^k from l ^k according to (4). ^ 0; q ^ is 2. The covariance matrix associated with ^d; x then determined as the covariance matrix of the ^ k0 ; q ^k ÞT ; k ¼ 1; . . . ; MC, with the vectors ð^dk ; x diagonal elements of the covariance matrix being
A. Link et al. / Measurement 40 (2007) 928–935
the squared standard uncertainties. (Note that the ^ k0 ; q ^k ÞT ; k ¼ 1; . . . ; MC mean of the vectors ð^ dk ; x ^ 0; q ^.) essentially coincided with ^ d; x Similarly, the uncertainties associated with estimates derived from the model such as shock sensitivities were calculated. Appendix B. Approximation of a linear continuoustime system by a discrete-time system Application of the Laplace transformation to the differential equation (1) provides the continuoustime system function G(s). From G(s), a corresponding discrete-time system function D(z) can be obtained by substituting s according to 2 1 z1 s¼ ðB:1Þ T 1 þ z1 that is 2 1 z1 DðzÞ ¼ G : T 1 þ z1
ðB:2Þ
The system function D(z) has the algebraic form PM bk zk ðB:3Þ DðzÞ ¼ Pk¼0 N k k¼0 ck z satisfying a difference equation of the form N X k¼0
ck yðn kÞ ¼
M X k¼0
bk xðn kÞ:
ðB:4Þ
935
Hence, it is straightforward to obtain the corresponding difference equation (B.4) from the system function D(z). For further details see [9]. References [1] ISO, International Standard 16063-11 Methods for the calibration of vibration and shock transducers – part 11: Primary vibration calibration by laser interferometry, International Organization for Standardization, Geneva, 1999. [2] ISO, International Standard 16063-13 Methods for the calibration of vibration and shock transducers – part 13: Primary shock calibration by laser interferometry, International Organization for Standardization, Geneva, 2001. [3] A. Link, H.J. von Martens, Accelerometer identification using shock excitation, Measurement 35 (2004) 191–199. [4] A. Link, A. Ta¨ubner, W. Wabinski, T. Bruns, C. Elster, Calibration of accelerometers: determination of amplitude and phase response upon shock excitation, Measurement Science and Technology 17 (2006) 1888–1894. [5] Bru¨el & Kjær, Instructions and Applications, Naerum, Denmark, 1975. [6] Calibration and measurement capabilities, Acoustics, Ultrasound and Vibration, PTB, Germany. Available from: