ACTA IMEKO
ISSN: 2221‐870X
June 2015, Volume 4, Number 2, 52‐56

ACTA IMEKO | www.imeko.org June 2015 | Volume 4 | Number 2 | 52

Accelerometer transverse sensitivity calibration; validation
and uncertainty estimation

Christiaan S Veldman

National Metrology Laboratory of South Africa, Acoustics, Ultrasound and Vibration Laboratory, South Africa Private Bag X34, Lynnwood
Ridge, 0004, South Africa

Section: RESEARCH PAPER

Keywords: accelerometer; calibration; transverse sensitivity; validation; uncertainty of measurement; ISO 16063‐31

Citation: Christiaan S Veldman: Accelerometer transverse sensitivity calibration; validation and uncertainty estimation, Acta IMEKO, vol. 4, no. 2, article 9,
June 2015, identifier: IMEKO‐ACTA‐04 (2015)‐02‐09

Editor: Paolo Carbone, University of Perugia, Italy

Received September 23
rd
, 2014; In final form February 14

th
, 2015; Published June 2015

Copyright: © 2015 IMEKO. This is an open‐access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited

Funding: This work was supported by the Department of Trade and Industry, South Africa

Corresponding author: Ian Veldman, e‐mail: CSVeldman@NMISA.org

1. INTRODUCTION

Due to their ease of use and low cost, accelerometers are
widely considered as the vibration sensor of choice. A variety of
different models are required to cover the wide range of
vibration measurement applications. To select the
accelerometer best suited for a specific application, the user will
typically scrutinize the manufacturer’s specifications. Apart
from the general (usually the most relevant) specifications such
as size, sensitivity, frequency- and acceleration ranges, the
manufacturer also specifies the relative transverse sensitivity
(RTS) of an accelerometer.

For specialized application, the transverse sensitivity is of
importance. For some applications, a more accurately known
value of the transverse sensitivity might be required [1], [2], [3].
In addition, knowledge of the angle (mechanical orientation) of
the transverse sensitivity is also required.

NMISA developed a capability to accurately measure the
transverse sensitivity of accelerometers as part of its research in
vibration metrology. NMISA modified its existing low
frequency accelerometer calibration system to facilitate the

measurement of accelerometer transverse sensitivity and will
offer this calibration service to industry in the near future.

In section 2, the author defined “transverse sensitivity”. The
system hardware and specifications are described in section 3.
In section 4, the approach followed to estimate the uncertainty
of measurement as well as the major uncertainty of
measurement contributors is discussed. The method followed
for validating the system, with the validation results are
discussed in section 5. Finally, in the concluding section the
findings are summarized.

2. TRANSVERSE SENSITIVITY

The transverse sensitivity of an accelerometer is defined as
the sensitivity to acceleration applied perpendicular to its
sensitive axis [4]. The axis of maximum sensitivity of the
transducer is not necessarily aligned with the sensitive axis, as
shown in Figure 1. As a result, any motion not in line with the
sensitive axis will produce an output.

If the transducer is placed in a rectangular co-ordinate
system, as shown in Figure 1, the vector, Smax, representing the
maximum transducer sensitivity can be resolved into two

ABSTRACT
The National Metrology Institute of South Africa (NMISA) has implemented a system to measure the transverse sensitivity of vibration
transducers. As a mechanical device, the principle sensing axis of an accelerometer is not 100 % perpendicular to the mounting axis
(surface). This gives rise to the effect that the accelerometer will produce an electrical output even when a mechanical input
perpendicular to the principle measurement axis is applied. The quantification of this “defect” parameter is of importance when high
accuracy acceleration measurements are performed using accelerometers. This paper gives a brief overview of the system developed
by the NMISA to measure the transverse sensitivity of vibration transducers. The paper then explores the validation of the system
along with the uncertainty of measurement associated with the calibration system.

ACTA IMEKO | www.imeko.org June 2015 | Volume 4 | Number 2 | 53

components: the sensitive axis sensitivity, SN, (sensitivity) and
the maximum transverse sensitivity, ST,max.

The theoretical transverse sensitivity curve is shown in
Figure 2. The transverse sensitivity, expressed as a percentage
of the sensitivity, is referred to as the Relative Transverse
Sensitivity (RTS). The RTS is dependent on the excitation
angle.

For high quality accelerometers, manufacturers supply
devices with low RTS, typically ≤ 1 %, and with the direction of
the lowest transverse sensitivity, βTMin, indicated with a red dot
on the accelerometer. The manufacturer supplies these low
transverse sensitivity devices through selection. That is, they
physically measure the transverse sensitivity and select the units

that meet the required RTS specification.

3. SYSTEM DESCRIPTION

The transverse sensitivity calibration system of NMISA [5]
was developed in compliance with ISO 16063-31 [6]. The
transverse sensitivity capability was developed as an extension
of the existing primary low frequency accelerometer calibration
system. A schematic diagram of the system configuration is
shown in Figure 3. The system utilizes the existing long stroke
(152 mm peak to peak) electro-dynamic exciter, connected to
an air bearing linear translation stage (ABT). A stepper motor
controlled turntable is mounted on top of the ABT (Figure 4).
Table 1 provides the system parameters.

For the vibration generator with turntable system
implemented by NMISA, once the unit under test (UUT) is
mounted on the turntable and all the hardware and cable
connections are completed, the in-house developed software is
executed.

The software performs a set of procedural steps as part of
each transverse sensitivity measurement per turntable angular
position as follows:

 Move the turntable to the angular position of interest,
(from 0° to 360° in 5° steps);

 Ramp the exciter to the selected vibration level, in
frequency and amplitude;

 Sample two analogue inputs simultaneously,
streaming the data (time series data) directly to
computer storage;

 Apply the three parameter sine fit algorithm (3PSF)
[7] to the REF and UUT output voltages time series
to calculate acceleration amplitude and the UUT
voltage output;

 Calculate the relative transverse sensitivity (RTS)
using (1) and (2);

 Record the RTS in the result sheet;
 Plot the RTS on a polar diagram.

Table 1. Parameters of NMISA transverse sensitivity calibration system.

Vibration frequency range 5 Hz to 20 Hz

Transverse acceleration range 5 m/s
2
to 50 m/s

Analogue inputs Four simultaneously sampled 12
bit channels

Sampling frequency 500 kHz

Turntable rotation angle 0° to 360°

Turntable rotation resolution 1°

Reference Polytec OFV‐505 Heterodyne
laser interferometer system

Figure 2. Polar plot indicating a theoretical transverse sensitivity curve for
an accelerometer.

Figure 1. Graphical illustration of transverse sensitivity.

Figure 3. Diagram of the system configuration.

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These steps are executed for each angular position from 0°
to 360° using the selected step size, without the need of any
intervention by the metrologist. For the calculation for
transverse sensitivity, ST, the following formula was used:

out
T ˆ

u
S  (1)

where ST is the transverse sensitivity, ûout is the amplitude of the
output signal of the transducer vibrating perpendicularly to its
sensitivity axis, and âT is the acceleration level in the test
direction.

The acceleration level is measured by means of a reference
transducer and calculated as:

Ref

Ref
T

ˆ
ˆ

u
a  (2)

where ûRef is the voltage output of the transducer; and SRef is the
sensitivity of the reference transducer.

The relative transverse sensitivity, *TS , is calculated from

T*
T

S
S

S  (3)

where SN is the transducer sensitivity.

Once all the measurements have been completed, the

results are saved using an Excel template file. The result sheet
also displays the RTS in graphical format, similar to Figure 2.

4. UNCERTAINTY OF MEASUREMENT

A conservative approach was followed w.r.t. the
consideration of uncertainty contributors. The “worst case”
(but still scientifically valid) values were used for the uncertainty
contribution. This allows for a single uncertainty calculation
that is valid for a wider range of measurement conditions. It
also reduces the need to re-calculate the uncertainty budget for
each calibration. However, it does not relieve the metrologist of
the responsibility of having to consider the uncertainty for each
calibration performed. By using the uncertainty values
estimated for a specific calibration, instead of the generalized
values, an RTS calibration with a smaller uncertainty might be
possible.

The uncertainty of measurement (UoM) was estimated in
accordance with the GUM [8]. The root uncertainty
contributors were identified from the mathematical model in
(1), which was expanded to (4) by inserting (1) and (2) into (3):

Ref

out

Ref*
T ˆ

u
S  (4)

Through consideration of the mathematical model (4) and
the measurement procedure, a detailed set of uncertainty
contributors was identified. This full set was reduced to a sub-
set containing the uncertainty contributors with a significant
contribution. This process produced the list of “dominant”
uncertainty contributors listed in Table 2. A summary of the
uncertainty budget is reported in Table 3.
The voltages for both the reference signal as well as the
accelerometer output signal were captured using an A to D
converter and applying the 3PSF. The Sine Approximation
Method is a well-established and adopted time domain signal
processing technique [9]. It requires for the samples to be
equidistance sampled, that is that the time between t0, t1, t2…be
constant, and if the phase difference between signals is
required, that the sampling of the two signals be performed
simultaneously. For the system implemented, both these criteria
were achieved using a four channel analogue to digital converter
(ADC) (four individual channels, not multiplexed channels),
with a single sample timing clock, thus synchronising the
sampling done by the four ADCs.

It has been established that the 3PSF method is influenced
by various factors [10-13] for instance:

 the sampling frequency;
 number of ADC bits;
 number of samples per period to be an integer number;
 number of periods to be a prime number;
 signal to noise ratio (SNR).
Of particular relevance to this application is the SNR [10] as

the accelerometer output is a very small signal due to the low
transverse sensitivity. As a result, for this application, the SNR
is measured. However, for simplicity, a minimum SNR limit of

Figure 4. Turntable mounted on top of air‐bearing translation stage.

Table 2. Significant uncertainty contributors

Uncertainty Contributor Source of Uncertainty

Reference transducer sensitivity Calibration certificate

Acceleration Level Reference voltage measurement

Accelerometer Output Voltage Accelerometer voltage
measurement

Sensitivity (sensitive axis sensitivity) Calibration certificate

Type A Uncertainties Statistical means (Standard
Deviation)

ACTA IMEKO | www.imeko.org June 2015 | Volume 4 | Number 2 | 55

15 dB is set for the purpose of calculating the 3PSF uncertainty
contribution. For a SNR of 15 dB, the uncertainty in the
magnitude determination, σA, is about 0.3 %. σA is calculated
using

(5)

where σA is the 3PSF amplitude precision, σ is the zero mean
white Gaussian noise, Q is the quantization error [10, 11]; N is
the number of samples.

Due to the inherent small transverse sensitivity of
accelerometers, additional steps were required to achieve the
SNR limit requirement. This was accomplished by filtering the
time signals using a 4th order Butterworth (Infinite Impulse
Response (IIR)) digital bandpass filter. The forward-reverse
filtering model was applied to accomplish zero phase response.
The lower and upper cut-off frequencies were selected to be
0.8909·fv and 1.225·fv respectively, where fv is the vibration
frequency.

4.1. Reference Sensitivity Uncertainty

For this transverse sensitivity calibration system, the
metrologist may choose to use either an accelerometer as the
reference device or a laser interferometer (vibrometer). For
both options, the sensitivity of the reference device is known
from a prior calibration. The uncertainty associated with the
sensitivity, along with its coverage factor, is obtained from the
calibration of the reference.

4.2. Acceleration Level Uncertainty

The voltage amplitude of the reference signal is determined
using the 3PSF algorithm [12]. From [13], the uncertainty
associated with the amplitude is obtained from (5).

4.3. Accelerometer Output Voltage Uncertainty

The uncertainty associated with the accelerometer output
voltage is estimated in the same manner as described in
section 4.2 since the amplitude is also determined using the
3PSF. However, the calculated uncertainty for this parameter is
expected to be larger, due to the smaller SNR, in light of the
smaller output voltage.

4.4. Sensitive Axis Sensitivity

The uncertainty contribution of the sensitivity will depend
on the source of the value of sensitivity. Generally, this
sensitivity will be obtained from a valid calibration certificate.
In such an instance, the uncertainty will be taken from the
certificate. It is possible to determine the sensitivity of the
sensitive axis using the calibration system, prior to performing
the RTS measurements. In this instance, the uncertainty
contribution needs to be calculated separately.

4.5. Type A Uncertainty

At each measurement point, 0° to 355° in 5° steps, the
system captures an equidistant sampled time series, containing
100 vibration cycles for both the reference- as well as the
accelerometer channels. To eliminate the undesired effects
introduced by the bandpass filtering, the 3PSF is applied to the
centre 50 cycles only. The final voltage amplitude is calculated
as the mean and standard deviation of these 50 voltage
amplitudes (per channel).

The largest Type A uncertainty was determined by
calibrating an accelerometer with a low RTS (≈ 0.1 %). Using

this accelerometer the largest standard deviation calculated,
considering all the measurement points through the complete
360° rotation was 3.5 %. As was to be expected, the Type A
uncertainty reached a peak value at an angular position with the
lowest RTS.

5. VALIDATION

The performance of the transverse sensitivity calibration
system and procedure was validated by performing a bilateral
interlaboratory comparison (ILC). The ILC partner was the
Deutsche Akkreditierungsstelle (DakkS) accredited laboratory,
SPEKTRA GmbH, Dresden, Germany.

The purpose of the ILC was to evaluate and validate the
metrological operation of the two participating laboratory’s
accelerometer transverse sensitivity calibration systems and
relevant procedures. The differences in the RTS measurements
obtained between the two laboratories would support (or
disprove) each laboratory’s measurement capability, within their
stated UoM.

The parameter covered by the ILC was the measurement of
the relative transverse sensitivity (RTS) of an accelerometer at
16 Hz. Three accelerometers were used as the ILC transfer
devices. The accelerometers that were used are listed in Table 4.

5.1. Evaluation Criteria

For each laboratory i, the data where xi,S, is the maximum
relative transverse sensitivity ST, reported and u(xi,S), is the
reported standard uncertainty associated with the RTS. For
each of the comparison artefacts (transfer devices) an ILC
reference value xR,S was determined as the weighted mean of
the results of n laboratories (for this comparison, n = 2)
according to

∑
xi,S

u2 xi,S

n
i 1

∑
,

(6)

, ∑
,

(7)

The degree of equivalence, DLAB-WM, and ULAB-WM, was

Table 4. Comparison transfer devices.

No Accelerometer Serial number

1 Endevco 2270M8 16194

2 PCB 3701G2FA3G 8353

3 PCB J353B01 47794

Table 3. Summary uncertainty budget.

Source of Uncertainty Estimated
Uncertainty

Probability
Distribution

(k)

Reference transducer sensitivity 0.5 % 2

Voltage measurement accuracy
(Sine Fitting)

0.2 % 1

Uncertainty in the UUT
sensitivity value

0.8 % 2

Voltage measurement accuracy
(Sine Fitting)

0.3 % 1

Rotation angle accuracy 0.5° √3
Vibration table horizontal
alignment

0.1° √3

Type A Evaluation 3.5 % 1

ACTA IMEKO | www.imeko.org June 2015 | Volume 4 | Number 2 | 56

determined for the RTS measurements using

(8)

(9)

where xLab represents the measurement results obtained by the
laboratory for each RTS and xWM represents the ILC reference
value calculated as the weighted mean (WM) using (8). ULab-WM
is the uncertainty of measurement associated with the calculated
DLab-WM for k = 2, calculated using (9).

5.2. Comparison Results

The RTSs for the three individual accelerometers as
reported by each laboratory (with the associated uncertainties)
are reported in Table 5. The calculated comparison reference
values (weighted mean values) are reported in Table 6, while the
degrees of equivalence (DoE) for each participant is reported in
Table 7. The DoE for the measurement results reported by
NMISA are shown in graphical format in Figure 5. The graph
clearly indicates uncertainties which overlap the reference
values indicating NMISA equivalence with SPEKTRA.

6. CONCLUSIONS

A transverse sensitivity calibration system was
implemented in compliance with ISO 16063-31 by NMISA. The
transverse motion is generated using an electro-dynamic
vibration exciter with a stepper motor controlled turntable for
the angular positioning control.
The requirement for a relatively high SNR (≥ 15 dB) was
highlighted. This minimum level of SNR is maintained through
the use of digital narrow band band-pass filtering.

Four major sources of uncertainty were identified; the

reference transducer sensitivity, the acceleration level, the
accelerometer output voltage and the sensitive axis sensitivity.
For this system, the upper limit for the Type A uncertainty was
calculated to be 3.5 %.

The system was validated through a bi-lateral ILC. The
results for the three different accelerometers used, support the
UoM estimated by NMISA. The ILC results further indicate
that the UoM estimated by NMISA could be considered as
fairly conservative.

REFERENCES

[1] T. Petzsche, “Determination of the transverse sensitivity using a
mechanical vibration generator with turntable,” ISO TC 108/SC
3/WG 6 Doc. N153, 2007.

[2] J. Dosch and M. Lally, “Automated testing of accelerometer
transverse sensitivity,” Proceedings of the International Modal Analysis
Conference (IMAC), Kissimee, Florida, USA, pp. 1-4, 2003.

[3] R. Sill and E. Seller, “Accelerometer transverse sensitivity
measurement using planar orbital motion,” Proceedings of the 77th
Shock and Vibration Symposium, Monterey, California, USA, pp. 8-
12, November 2006.

[4] ISO, “Mechanical vibration, shock and condition monitoring –
Vocabulary” ISO 2041, 2009.

[5] C.S. Veldman, “Implementation of an Accelerometer Transverse
Sensitivity Measurement System”, NCSL International, June
2013.

[6] ISO, “Methods for the calibration of vibration and shock
transducers -- Part 31: Testing of transverse vibration sensitivity”
ISO 16063-31.

[7] Peter Händle, “Amplitude estimation using IEEE-STD-1057
three-parameter sine wave fit: Statistical distribution, bias and
variance”, Measurement, vol. 43, pp. 766–770, 2010.

[8] BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML
2008 Evaluation of Measurement Data—Guide to the Expression of
Uncertainty in Measurement Joint Committee for Guides in
Metrology, JCGM 100:2008.

[9] ISO, “Methods for the calibration of vibration and shock
transducers -- Part 11: Primary vibration calibration by laser
interferometry” ISO 16063-11.

[10] M. Bertocco, C. Narduzi, P. Paglierani, D. Petri, “Accuracy of
Effective Bits Estimation Methods”, IEEE Instrumentation and
Technology Conference, Brussels, Belgium, June 4-6, 1996.

[11] Konrad Heijn, Andrzej Pacut, “Generalized Model of the
Quantization Error – A Unified Approach”, IEEE Transactions
on Instrumentation and Measurement, vol. 34, n. 1, Feb. 1996.

[12] F.Corrêa Alegria, “Bias of amplitude estimation using three-
parameter sine fitting in the presence of additive noise”,
Measurement, 42, pp.748 – 756, 2009.

[13] F. Corrêa Alegria, A. Cruz Serra, “Uncertainty of the Estimates
of Sine Wave Fitting of Digital Data in the Presence of Additive
Noise”, IEEE Instrumentation and Technology Conference,
Sorrento, Italy, April 24-27, 2006.

Table 5. Reported ILC Results.

Accelerometer

NMISA SPEKTRA

RTS
(%)

Uc
(%)

RTS
(%)

Uc
(%)

Endevco 2270M8 0.8 3 0.95 0.3

PCB 3701G2FA3G 0.24 3 0.25 0.3

PCB J353B01 0.9 3 0.81 0.3

Figure 5. NMISA Degrees of Equivalence.

Table 6. Calculated Weighted Mean Values.

Accelerometer

Weighted Mean

RTSWM UWM

(%) (%)

ENDEVCO 2270M8 0.95 0.3

PCB 3701G2FA3G 0.25 0.3

PCB J353B01 0.81 0.3

Table 7. Calculated Degrees of Equivalence.

Accelerometer

Degrees of Equivalence

DNMISA‐WM UNMISA‐WM DSPEKTRA‐WM USPEKTRA‐WM

(%) (%) (%) (%)

ENDEVCO
2270M8

‐0.15 3.15 0.001 0.42

PCB
3701G2FA3G

‐0.01 3.01 0.0 0.42

PCB
J353B01

0.09 3.11 ‐0.001 0.42

‐5,0

‐2,5

0,0

2,5

5,0

0 1 2 3 4D
e
v
ia
ti
o
n
f
ro
m
W

M
(
%
)

Accelerometer Number

NMISA DoE