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An analysis of Instantaneous Angular Speed measurement errors
Quentin Lecl`
ere1, Franc¸ois Girardin1and Didier R´
emond2
1INSA-Lyon, LVA, F-69621, France
2Universit´
e de Lyon, CNRS, INSA-Lyon, LaMCoS UMR5259, F-69621, France
{quentin.leclere, francois.girardin, didier.remond}@insa-lyon.fr
Abstract
The instantaneous angular speed (IAS) of a rotating machine is a crucial information to understand the machine
operation and to diagnose potential faults. There are several difficulties related to the assessment of the IAS,
which is generally estimated using an angle encoder. First of all the IAS is never really instantaneous data,
it is always averaged between two increments of the encoder, leading to a spectral aliasing error. The second
potential source of estimation error is linked to the manner the time delays between angle increments are
measured. A first possibility is to use a high frequency counting approach: a high frequency pulse signal is
used as reference, typically several tens of MHz, and an electronic device is used to count the number of pulses
of the high frequency clock between two events of the angle encoder signal. A second possibility is to use
a standard analog to digital converter with an anti-aliasing filter to acquire the angle encoder signal, using a
lower sampling rate (typically several tens of kHz), and to determine event’s times of the encoder by numerical
processing like upsampling or interpolation. For both approaches, the counting error is related to the uncertainty
of the estimated time between two pulses of the angle encoder. A third potential cause of IAS estimation error
is related to the angle encoder itself. The encoder is made of a rotating device linked to the shaft, on which
a given number of marks are distributed on the whole circumference (the angle between two marks being the
resolution of the encoder). These marks can be either teeth or holes, or alterning sectors, depending on the
technology (magnetic, optical, ...). In all cases, the angle between consecutive marks is never perfectly the
same all around the encoder. These imperfections are the cause of geometrical errors on the estimated IAS.
The aim of this paper is to give some guidelines to handle globally these estimation errors, and to propose a
calibration procedure to correct the geometrical error. Some examples will be given on both numerical and
experimental illustrations.
1 Introduction
The instantaneous angular speed (IAS) of a rotating machine can be of major interest for real-time mon-
itoring or diagnosis applications. The rotation of the shaft is indeed the result of the global operation of the
machine, thus carrying information about all mechanical processes contributing to the rotation. Any fault, fail-
ure or defect has theoretically an effect on the instantaneous speed. IAS has widely been used in the monitoring
of reciprocating engines [1, 2, 3], gear transmissions [4, 3], rolling bearings [5], or machining [6].
The IAS of rotating machines is generally assessed using a angle encoder, which is basically made of a
rotating wheel with marks, and a fixed sensor delivering a signal carrying the mark passing frequency. Several
methods can be used to acquire such a signal, either using a high frequency counting approach [7, 8] or post
processing the signal acquired with a standard ADC DAQ [9]. The analysis of errors is of course an important
issue when dealing with such signals, associated by either measurement noise [10], counting errors [11] or
more recently by geometric error [12].
The aim of this paper is to give an overview of potential causes of IAS estimation errors, associated to two
different acquisition methods: the high frequency counting approach and the standard DAQ system approach.
The first method uses a high frequency counting electronic device, counting the number of elementary periods
between two events of the encoder. The second method is based on the use of a standard DAQ to digitalize
1
the encoder signal, using a sampling frequency much lower than the one used for the counting approach. The
time durations between consecutive encoder events are then estimated using interpolation. The first section of
the paper is dedicated to errors due to spectral aliasing, induced by the fact the estimated IAS is average over
angular periods defined by the encoder resolution. The second part focuses on counting errors, for which a
model is established for both methods. In the third section, the causes of geometric error are discussed, and a
calibration method is proposed to attenuate it. The last section is devoted to experimental illustrations.
2 Errors due to spectral aliasing
The IAS (Instantaneous Angular Speed) assessed using an angle encoder is never strictly instantaneous
data [11]. The speed is estimated from a measure of the time duration between two events of the encoder. The
resulting speed is in fact averaged data, over an angle length equal to the encoder resolution
θ
. This averaging
operation can generate differences between the real IAS and the estimated IAS, because of spectral aliasing.
This effect is not easy to formulate because the time duration between two angle steps is not constant. However,
the averaging process can be formulated in angle to simplify the analysis:
¯
ω
(
θ
) =
θ
t=
θ
θ
+
θ
/2
θ
θ
/2
d
θ
ω
(
θ
)1
1
θ
θ
+
θ
/2
θ
θ
/2
ω
(
θ
)d
θ
(1)
the approximation in equation (1) being reasonable for small speed fluctuations. The local moving averaging
process in angle can be expressed in the Fourier domain using a simple multiplication:
¯
ω
(
ν
) =
ω
(
ν
)sinc(
πν
θ
)(2)
A second intrinsic characteristic of IAS data is that it is sampled at a sampling frequency 1/
θ
. This angular
domain sampling is equivalent to a convolution of the IAS spectrum by a comb filter with a period of 1/
θ
. This
operation generates spectral aliasing if the IAS spectrum is not null above 1/(2
θ
)(the Nyquist frequency),
which is not guaranteed at all by the low pass filtering expressed in Eq. (1), as illustrated in figure 1.
1/
θ
useful
frequency range
|sinc(
π
ν
θ
)|
|sinc(πν∆θ)| * comb1/∆θ
ν
Figure 1: Spectrum of the local averaging operator, and spectrum of the sampled local averaging operator.
2.1 Numerical illustration
A numerical illustration is conducted for the IAS given by the following function:
ω
(
θ
) = 100 +sign(sin(25
θ
)) (3)
The real IAS is simulated with an angular resolution of d
θ
=2
π
/Zradian, with Z=1e5. The corresponding
time history is estimated by numerical integration. The measured IAS is simulated for an encoder resolution of
θ
=2
π
/Nradian, N=720 corresponding to the number of pulses by encoder revolution. The angular spectra
of the measured and real IAS are given in figure 2, for the useful frequency range of the simulated measured
IAS. The energy of the real IAS spectrum is distributed on odd harmonics of order 25, which is characteristic
of the square nature of the generated signal. The measured IAS spectrum on odd harmonics of order 25 is
slightly underestimated because of the spectral weighting by the cardinal sine (cf. equation 2), and several
ghost harmonics are observed between real harmonics because of spectral aliasing.
2
Figure 2: Angular spectra of the real (solid black) and measured (dotted red) IAS. Numerical simulations.
3 Counting error
3.1 High frequency counting approach
The counting error results from the manner the time delays between angle increments are measured. The
first possibility is to use a high frequency clock delivering pulses at a high frequency. An electronic device
is used to count the number of pulses of the high frequency clock between two angle increments. Of course,
this number of pulses is an integer, which generates a rounding error on the estimated time delays, also called
quantification error [11].
time
τ
j
τ
j+1
Tj
j
T
~
=Nj/F = Tj + τj - τj+1
1/F
Angle encoder increments
High frequency pulses
Nj pulses
Figure 3: Counting error when using a high frequency pulse counting approach.
The error made on the determination of time occurrences of angle increments is considered here as indepen-
dent random variables, noted
τ
jfor increment j, uniformly distributed between 1/(2F)and 1/(2F), where
F is the frequency of the high frequency clock. Note that taujis here defined as the difference between the
estimated time occurrence of the encoder increment and the real one (see Fig. 3).
The estimated IAS with counting error is thus expressed as follows:
˜
ω
j=
θ
Tj+
τ
j+1
τ
j
¯
ω
j
θ
T2
j
(
τ
j+1
τ
j)(4)
3
The spectrum of the IAS is then given by:
ˆ
˜
ω
k=ˆ
¯
ω
k
θ
N
N1
j=0
τ
j+1
τ
j
T2
j
ei2
π
jk/N(5)
Let us now assume that the IAS variation effects are negligible, or that the speed is simply constant: Tj=
T=
θ
/. After few rearrangements it gives, for k>0:
ˆ
˜
ω
k=2
N
θ
τ
Nei2
π
k(N1)/N
τ
0+2isin(
π
k/N)ei
π
k/N
N1
j=1
τ
jei2
π
jk/N(6)
Of course, the spectrum of the real IAS is in this case equal to zero for k>0, the spectrum given in equation
6 can thus be considered as the contribution of the counting error. It is interesting to give the expected value of
the energy of the counting error, as a function of k:
E|ˆ
˜
ω
k|2=Vˆ
˜
ω
k=4
N2
θ
24(N1)sin2(
π
k/N) + 2V[
τ
],(7)
with, remembering that
τ
is uniformly distributed between 1/(2F)and 1/(2F),
V[
τ
] = 1
12F2(8)
3.2 Numerical illustration
We consider the following law of instantaneous speed:
ω
(
θ
) = +|sin(
θ
/2)|+1
10
θ
2
π
M(9)
where Mcorresponds to the number of cycles. The first term () corresponds to a constant component, set
successively to 10 rad/s and 100 rad/s. The second term in |sin(
θ
/2)|introduces a periodicity of the speed
equal to 2
π
. The last term introduces a very small acceleration to avoid numerical artifacts encountered when
considering a pure constant speed: the speed increases of 0.1 rad/s during the whole simulation, from to +
0.1 rad/s.
ω
(
θ
)is computed with a relatively fine resolution of d
θ
=2
π
/Z, with Z=1e5. The corresponding
time history is obtain by simple numerical integration:
t[
θ
p] =
p
i=1
d
θ
w[
θ
i](10)
The angle encoder used in the simulation has N=2048 pulses per revolutions:
θ
=2
π
/2048. The time
corresponding to the angle encoder tjfor j[1...NM]increments are determined by interpolation of t[
θ
k]. The
IAS without the counting error (but with aliasing, though negligible for this illustration) is calculated from these
data:
¯
ω
j=
θ
tj+1tj
(11)
The counting frequency is adjusted to F=80MHz. The number of events of the high frequency clock that
are elapsed between two consecutive encoder increments are then obtained as follows:
nj=floor(F.tj+1)floor(F.tj)(12)
where floor(x)stands for the largest integer less than x.njwith j[1...NM]corresponds typically to the output
of a counter connected to an angle encoder. The IAS with counting error is finally:
˜
ω
j=F
θ
nj
(13)
The Discrete Fourier Transform of the IAS with and without counting error is computed at each cycle, for
=10 rad/s and 100 rad/s. The squared modulus of the DFT is then averaged over cycles. The result are
4
drawn in Fig. 4, together with the expected value of the contribution of the counting error given in equation
7. Note that the only difference between IAS spectra for =10 rad/s and 100 rad/s is at k=0. It can be
seen in Fig. 4 that the contribution of the counting error is largely dominant over a wide high frequency range,
and in good agreement qualitatively and quantitatively with the prediction. The amplification of the counting
error between =10 rad/s and 100 rad/s is conform to equation (7): a factor 10 on the average rotation speed
induces a factor 100 on the counting error.
0 200 400 600 800 1000
10−7
10−6
10−5
10−4
10−3
10−2
10−1
order k
IAS spectrum amplitude
IAS spectrum
IAS with counting error
Theoretical error spectrum
=100 rad/s
=10 rad/s
Figure 4: IAS spectrum with and without counting error. Numerical illustration.
3.3 Standard DAQ system approach
Another way to assess the IAS is to post process the signal of the encoder recorded using a standard DAQ
system, with a frequency sampling much smaller than the one used in the high frequency counting approach.
The idea is to interpolate zero-crossings of the encoder from its low-pass filtered signal. The error made on
time
Analog coder signal
LP Filtered Analog coder signal
Sampled points
interpolation
Figure 5: Zero-crossing interpolation using a standard DAQ system.
zero crossing (and thus on the angle increment times) is more difficult to handle than for the high frequency
approach. However, the same formalism can be used, just by changing the distribution of
τ
, the error made on
the estimated zero crossing times. The distribution used in the previous section was uniform. For the approach
described in this section, a Gaussian centered distribution will be assumed, all the difficulty being to determine
its variance.
4 Geometric error
4.1 Effect of the geometric error on the IAS spectrum
A third potential cause of IAS estimation error is related to the angle encoder itself. The encoder is made of a
rotating device linked to the shaft, on which a given number of marks are distributed on the whole circumference
(the angle between two marks constituting the resolution of the encoder). These marks can be either teeth or
5
holes, or alterning sectors, depending on the technology (magnetic, optical, ...). In all cases, the angle between
consecutive marks is never perfectly the same all around the encoder. These imperfections are the cause of
geometrical errors on the estimated IAS. Let us note
δ
jthe positioning error of the encoder mark j. The effect
of the geometric error on the estimated IAS is thus simply as follows:
˜
ω
j=
θ
Tj
=
θ
j+
δ
j
δ
j+1
Tj
=¯
ω
j
δ
j+1
δ
j
Tj
(14)
which is quite similar to equation 4, replacing the zero-crossing time error by the geometric error. The spectrum
of the estimated IAS with geometric errors is thus:
ˆ
˜
ω
k=ˆ
¯
ω
k1
N
N1
j=0
δ
j+1
δ
j
Tj
ei2
π
jk/N(15)
Let us now focus on the spectrum of the IAS acquired during one revolution of the encoder. In this case, N
corresponds to the number of marks on the encoder’s disk. The angular reference can be arbitrarily set to the
first mark j=0. In this case,
δ
0=
δ
N=0. Then, assuming that the speed fluctuation is negligible, the spectrum
of the IAS for k>0 is:
ˆ
˜
ω
k=
N
θ
2isin(
π
k/N)ei
π
k/N
N1
j=1
δ
jei2
π
jk/N(16)
The shape of the spectrum of the geometric error is then given as follows:
E|ˆ
˜
ω
k|2=Vˆ
˜
ω
k=2
N2
θ
24(N1)sin2(
π
k/N)V[
δ
](17)
where V[
δ
]characterizes the precision of the encoder. It is interesting to note that the shape of the geometric
error spectrum is the same than the shape of the counting error spectrum, with a weighting in sin2(
π
k/N)for
order k.
There is however a major difference between the counting and geometrical errors. The former is a purely
random process linked to the measurement uncertainty, while the latter is due to the fact that the exact angle
steps of the encoder are not known exactly, and wrongly assumed to be constant. The consequence is that the
geometric error, that is repeated cycle to cycle, will exhibit a strong first order cyclostationarity (periodicity),
while the counting error will be purely random. In other words, an averaging over cycles of the squared modulus
of the IAS will converge to its expected value for the counting error only. A convergence of the geometric error
would have been encountered for an averaging over different angle encoders instead of an averaging over cycles.
4.2 Correction of the geometric error
The geometric error can be efficiently reduced by taking into account the real angle steps of the encoder,
noted
θ
j. These steps have to be assessed during a calibration procedure. The encoder has to be mounted on a
rotating machine operating ideally constant rotation speed: the first order cyclostationarity of the real IAS has
to be as small as possible. Then the time intervals Tbetween angle steps have to be acquired during several
cycles, in order to average out the counting error by synchronous averaging:
Tj=1
M
M
c=1
Tjc =1
Fnj=1
FM
M
c=1
njc (18)
where jstands for the jt h angle step, cdenotes the cth cycle, Mthe total number of cycles, and njc the number
of events of the high frequency clock, if the high frequency counting approach is used. Then, assuming that the
speed is constant, angle steps can be estimated simply by:
θ
j=
θ
+
δ
j+1
δ
j=Tj=2
π
Tj
N
j=1Tj=2
π
nj
N
j=1nj(19)
The main hypothesis when assessing the geometric error with equation (19) is that the speed should be
constant. In fact, this condition is not strictly necessary, a less restrictive condition would be that the expected
6
value of the IAS over cycles at each step of the encoder should be the same, or in other words that the first
order cyclostationarity part of the IAS should be constant. This means that all periodic fluctuations of the
IAS (at a period equal to 2
π
or any entire multiple) should be avoided. But unfortunately, this condition is
very difficult - not to say impossible - to satisfy: the rotating device is never perfectly balanced, the load is
potentially periodic... Anyhow, it is possible to assess the geometric error for different operating conditions
(speed, load) of for different encoder/shaft coupling angle, or even for different rotating machines, in order to
verify the stability of the result.
5 Experimental illustrations
5.1 Experimental setup
In order to approach the perfect rotating machine, a NC lathe Optimab 350 has been used. A special
workpiece is first machined on the lathe, and then encoders are mounted according to Fig. 6. The rotation of
the encoder’s stator is blocked using a support on a drill, hold on the tool turret. Two optical encoders are studied
in this paper, a Heidenhain ERN 120 with a resolution of 2048 pulses/revolution (Fig. 6, center) and an AVL
364 with 720 pulses/revolution (Fig. 6, left). The driving system of Optimab 350 is composed of small pulley
Figure 6: Left : AVL 364 encoder mounted on the chuck. Center : Heidenhain encoder mounted on the chuck.
Right : driving mechanism of the lathe.
directly mounted on the motor, linked through a Poly-V belt to a bigger pulley which is part of the chuck shaft.
The motor is a three-phase four-pole (pairs) squirrel-cage induction motor (ref. 1PH7107), driven by numerical
servo-driver. The speed is thus numerically controlled and is supposed to be - very close to - constant. Angular
speed can be driven from 60 rpm to 5000 rpm, in both directions. As ratios between the chuck pulley perimeter
Pc and belt length Lb or motor pulley perimeter Pm are not integers (see table 1), the first order cyclostationarity
due to rotating system is reduced to coaxiality and run-out precision. Typical coaxiality precision is about 0.05
mm for the assembly [chuck+workpiece]. The IAS measured with the AVL 364 encoder is assessed using
Chuck pulley Motor pulley Belt
Perimeter /length (mm) 625 350 2335
Ratio to Pc 1 1.78 0.27
Table 1: Optimab 350 driving system parameters
the standard DAQ system approach, the angle encoder signal is digitalized with an OROS OR38 system, with
a frequency sampling at 102400Hz (24 bits). The IAS measured with the Heidenhain ERN 120 encoder is
obtained using the high frequency counting approach, with a NI Compact RIO [9178+9401], with intergrated
counter based on 80 MHz high frequency clock. The IAS is estimated at two rotation speeds for the AVL
encoder (100 and 500 rpm) and five rotation speeds for the Heidenhain encoder (100, 500, 1000, 2000, 3000).
7
5.2 Calibration (correction of the geometric error)
The geometric error is assessed for both angle encoders, using equation (19), and for several rotation speeds.
The measured angle steps are drawn in Fig. 7, and their empiric standard deviations are given in table 2, for
the lowest rotation speed (10.5 rad/s). It can be seen in Fig. 7 that the identified angle steps are very similar
for different rotation speeds, which validates the hypothesis discussed in section 4.2. The identified standard
deviation of the geometric error is very different for the two encoders, about 20 times lower for the Heidenhain
than for the AVL. Another notable fact is that the geometric error signals exhibit strong transient components,
reflecting some localized default on the encoder’s disks. These defaults, as well as the whole geometric error
data, can be seen as a kind of signature of the encoder. It is noteworthy that this signature can be used to get
an absolute angle reference, avoiding the necessity to use an additional revolution trigger signal [12, 13]. The
Encoder Nstd(
θ
) std(
θ
)/mean(
θ
)
AVL 364 720 1.4e-5 rad 0.16 %
Heidenhain ERN 120 2048 7.3e-7 rad 0.024 %
Table 2: Measured standard deviation of the geometric error for the two encoders
500 1000 1500 2000
3.066
3.067
3.068
3.069
3.07
x 10−3 heidenhain N=2048
angle steps (rad)
480 500 520 540 560 580
3.066
3.067
3.068
3.069
3.07
x 10−3 heidenhain − zoom
200 400 600
8.65
8.7
8.75
8.8
8.85 x 10−3 AVL N=720
angle steps (rad)
encoder increment 300 350 400
8.6
8.7
8.8
8.9 x 10−3 AVL − zoom
encoder increment
Figure 7: Measured angular steps of the 2 angle encoder and several rotation speeds. Top: Heidenhain encoder,
=10.5,52,105,210,314 rad/s. Bottom: AVL encoder, =10.5,52 rad/s.
IAS spectrum obtained at low speed with both encoders are drawn in Fig. 8, with and without correcting the
geometric error. Note that the constant component of the spectrum (at 10.5 rad/s) is largely out of the figure
axes, and that the order is limited to k=200 for both encoders, even if their Nyquist orders are different (360 for
the AVL and 1024 for the Heidenhain). The two encoders are giving equivalent results up to order 30, and the
correction is not significant. Concerning the Heidenhain encoder, the correction is actually not significant on
the whole spectrum, due to the very low standard deviation of
θ
(cf table 2). The correction has a substantial
effect for the AVL encoder above order 30: the IAS spectrum obtained with correction is significantly lowered,
reaching the spectrum obtained using the Heidenhain encoder between orders 30 and 80. Above order 80, the
two encoders are giving different results, which could mean that the assembly of the device on the lathe affects
the measured IAS.
Some peaks are clearly visible on the IAS spectrum at orders 7, 11, 21 and 43, illustrating clearly that the
hypothesis of constant speed is not totally fulfilled. However, these peaks are not suppressed by the correction
of the geometric error, which means that they do not contribute to the periodic part of the IAS spectrum. The
true orders of the involved phenomena are thus not integers. Remembering that ratios to chuck pulley perimeter
is 0.27 for belt and 1.78 for the motor pulley, these two parts are not the cause of speed variations. Nevertheless,
8
0 50 100 150 200
10−4
10−3
10−2
order k
IAS spectrum amplitude
AVL
AVL − geometric error correction
Heindenhain
Heindenhain − geometric error correction
Figure 8: IAS spectrum measured at =10.5 rad/s using both encoders, with and without the correction of
the geometric error.
the motor is a three-phase induction motor, with four pole pairs, that corresponds to 3x4x2=24 windings. Thus,
orders 7, 11, 21, 43 corresponds to motor specific orders through belt pulley system (ratio 1.78), that is to say
respectively 4 poles, 6, 12, and 24 windings. Orders 6 and 12 are perhaps not linked to stator windings, but the
constitution of the rotor is not known. These orders could be due to number of elements in the cage.
5.3 Estimation of the counting error
5.3.1 Using the high frequency counting approach
The IAS spectrum obtained using the Heidenhain encoder, and with the high frequency counting approach,
is shown in Fig. 9 (left) for 2 rotation speeds, up to order 1024. Some peaks are oberved at low orders, but the
shape of the spectrum at higher orders (above about 300) is conform to the shape of the theoretical counting
error spectrum (weighting in sin2(
π
k/N)of equation 7). However, the amplitude is largely underestimated.
The ratio between theoretical and experimental error spectra is estimated for each rotation speed, and the scaled
theoretical counting error spectrum is also drawn in figure 9 (left). The scaling factors are given in table 3. The
level of the counting error spectrum is determined by the variance of
τ
(eq. 7). The theoretical and measured
variances of
τ
are also given in table 3. At low speed (10.5 rad/s), the measured standard deviation of
τ
is
about 7 times higher than the theoretical one, which means that the detection of events of the angle encoders
is about 7 times less accurate than expected. This ratio decreases when the speed increases: at 314 rad/s, the
measured variance almost equals the theoretical one. This unexpected measurement uncertainty, depending
on the rotation speed, is not fully understood for the time being, but the electronic device generating the TTL
signal from the physical information of the encoder is strongly suspected.
Rotation speed (rad/s) theoretical std(
τ
)= (12F)1/2experimental std(
τ
) scaling ratio
10.5 3.6e-9s 27e-9s 7.4
52 3.6e-9s 9e-9s 2.4
105 3.6e-9s 6e-9s 1.70
210 3.6e-9s 5e-9s 1.26
314 3.6e-9s 4e-9s 1.18
Table 3: Theoretical and measured standard deviation of
τ
for the high frequency counting method.
9
0 200 400 600 800 1000
10−7
10−6
10−5
10−4
10−3
10−2
IAS spectrum amplitude
order k
IAS spectrum
Theoretical counting error spectrum
Scaled counting error spectrum
= 10.5 rad/s
= 105 rad/s
x7.5
x1.7
0 50 100 150 200 250 300 350
10−7
10−6
10−5
10−4
10−3
10−2
order k
IAS spectrum amplitude
IAS spectrum
Scaled counting error spectrum
= 10.5 rad/s
= 52 rad/s
Figure 9: Left: Heidenhain angle encoder (N=2048). IAS spectrum and (scaled) theoretical counting error
contribution, using the high frequency counting approach. Right: AVL angle encoder (N=720). IAS spectrum
and scaled theoretical counting error contribution, using the standard DAQ system approach.
5.3.2 Using the standard DAQ system approach
The IAS spectrum obtained using the AVL encoder with the standard DAQ system approach (with correc-
tion of the geometric error) is shown in figure 9 (right) for 2 rotation speeds, up to order 360. The global shape
of the spectra are identical to those obtained using the Heidenhain encoder with the high frequency counting
method. The high frequency part of the spectrum has the shape of a counting error spectrum. However, the
level of the counting error with the standard DAQ system approach is more difficult to predict theoretically than
for the high frequency counting method. Thus, the theoretical error spectra have been scaled so as to fit the
measured IAS spectra at high frequencies. The standard deviation of the detection error
τ
obtained experimen-
tally from the scaling are given in table 4 for two rotation speeds. The ratio of the standard deviations to the
sampling period are also given in table 4, giving a idea of the precision improvement due to the interpolation
of the zero crossings of the angle encoder signal. At 52 rad/s (500rpm), the detection error standard deviation
is about 500 times lower than the sampling period, which illustrate the possibility to reach a good precision
with the standard DAQ system approach. At this rotation speed, the detection uncertainty is indeed only 5
times higher than the uncertainty obtained using the high frequency counting approach, while the frequency
sampling is about 800 times lower (80 MHz over 102400 Hz). This remarkable result can be explained by the
fact that the encoder signal is acquired with a high dynamic (24 bits) with the standard DAQ system, allowing
a fine interpolation of zero crossings, while the dynamic of the encoder signal that could be reproduced from
the high frequency counting method is only 1 bit, making impossible any interpolation between two samples.
Rotation speed (rad/s) std(
τ
) ratio fe/std(
τ
)
10.5 50e-9s 220
52 20e-9s 500
Table 4: Measured standard deviation of
τ
for the standard DAQ system counting method
6 Conclusion
An overview of different types of IAS measurement errors has been given in this paper: the spectral aliasing
error, the counting error and the geometric error. The counting error has been theoretically studied for two
counting methods, one method using a high frequency clock and a counter, and one method based on the
post processing of the encoder signal recorded using a standard DAQ system. A correction of the geometrical
error based on calibration measurements has been proposed. The theoretical results have been illustrated on
10
experiments realized on a lathe. The geometric error correction has been significantly attenuated for one of
the two tested encoders. One important results concerns the counting error, that has been found to be higher
experimentally than theoretically using the high frequency counting approach, particularly at low speed. The
error caused by the electronic device, that is not considered in the theoretical expression of the counting error
in this work, is thus certainly important to consider.
References
[1] P. Charles, J. K. Sinha, F. Gu, L. Lidstone, and A.D. Ball. Detecting the crankshaft torsional vibration of
diesel engines for combustion related diagnosis. Journal of Sound and Vibration, 321:1171–1185, 2009.
[2] M. Desbazeille, R.B. Randall, F. Guillet, M. El Badaoui, and C. Hoisnard. Model-based diagnosis of
large diesel engines based on angular speed variations of the crankshaft. Mechanical Systems and Signal
Processing, 24(5):1529–1541, 2010. Special Issue: Operational Modal Analysis.
[3] Quentin Lecl`
ere and Nacer Hamzaoui. Using the moving synchronous average to analyze fuzzy cyclosta-
tionary signals. Mechanical Systems and Signal Processing, 2013. In Press, Corrected Proof, Available
online.
[4] Didier Remond. Practical performances of high-speed measurement of gear transmission error or torsional
vibrations with optical encoders. Measurement Science and Technology, 9(3):347, 1998.
[5] L. Renaudin, F. Bonnardot, O. Musy, J.B. Doray, and D. R´
emond. Natural roller bearing fault detection
by angular measurement of true instantaneous angular speed. Mechanical Systems and Signal Processing,
24(7):1998 – 2011, 2010. Special Issue: ISMA 2010.
[6] M. Lamraoui, M. Thomas, M. El Badaoui, and F. Girardin. Indicators for monitoring chatter in milling
based on instantaneous angular speeds. Mechanical Systems and Signal Processing, 2013. In Press,
Corrected Proof, Available online.
[7] X. Kong. Gear train monitoring by transmission error method. PhD thesis, University of Wisconsin–
Madison, 1987.
[8] Yuhua Li, Fengshou Gu, Georgina Harris, Andrew Ball, Nick Bennett, and Ken Travis. The measurement
of instantaneous angular speed. Mechanical Systems and Signal Processing, 19(4):786–805, 2005.
[9] K.R. Fyfe and E.D.S. Munck. Analysis of computed order tracking. Mechanical Systems and Signal
Processing, 11(2):187–205, 1997.
[10] Fengshou Gu, Isa Yesilyurt, Yuhua Li, Georgina Harris, and Andrew Ball. An investigation of the effects
of measurement noise in the use of instantaneous angular speed for machine diagnosis. Mechanical
Systems and Signal Processing, 20(6):1444–1460, 2006. Special Issue: Laser Doppler Vibrometry.
[11] H. Andr´
e, F. Girardin, A. Bourdon, J. Antoni, and D. R´
emond. Precision of the ias monitoring system
based on the elapsed time method in the spectral domain. Mechanical Systems and Signal Processing,
2013. In Press, Corrected Proof, Available online.
[12] Alessandro Rivola and Marco Troncossi. Zebra tape identification for the instantaneous angular speed
computation and angular resampling of motorbike valve train measurements. Mechanical Systems and
Signal Processing, 2013. In Press, Corrected Proof, Available online.
[13] Matthieu Boucaud, C´
eline Sandier, Nicolas Totaro, Quentin Leclere, Didier Remond, and Jo¨
el Perret-
Liaudet. Improved acoustical sources location in gear power transmissions. In Proceedings of the 18th
Symposium Vibrations, Shocks and Noise, page paper N ˚ 40, Clamart, France, July 2012.
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... Several methods can be employed to obtain the IAS: the elapsed time method uses a high frequency clock while the ADC (Analog to Digital Conversion)-based method exploits the analog encoder signal [12]. Some recent works analyse the IAS measurement errors such as the geometric error, the filtering effect due to elapsed time measurement or the counting error [13,14]. From these results IAS can be considered as a good and precise way to reach dynamic behaviour of gears when rattle noise occurs. ...
... The relative velocity is calculated from the IAS of both gears. The assessment of the IAS with optical encoders is subject to several difficulties recently described in references [13,14]. Rotating encoders are composed of a rotating device containing a number of marks which defines the resolution of the encoder. ...
... η Dωω f θ À Á ¼ À lnðαÞ 3F 2 sin ðπf θ ΔθÞ 2 ω 4 0 Δθ 2 (13) with f θ being the "angle" frequency, Δθ ¼ 2π=R the angular distance between two marks of the encoder and ω 0 the mean operating speed. In other words, a spectral component of the power spectral density (PSD) Dωω ðf θ Þ which is higher than the quantization threshold η Dωω has a probability α to be induced by the counting error. ...
... Several methods can be used to obtain the IAS: the elapsed time method uses a high frequency clock while the ADC-based method exploits the analog encoder signal [3]. Some recent works analyse the IAS measurement errors such as the geometric error, the filtering effect due to elapsed time measurement or the counting error [4] [5]. ...
... The relative velocity is calculated from the IAS of both gears. The assessment of the IAS with optical encoders is subject to several difficulties recently described in the references [4] and [5]. We focus here on counting error. ...
... Generally, direct IAS estimation can be performed using two major groups of measurement methods [13]: timer based methods, which consist of counting the Elapsed Time (ET) between two consecutive edges of the TTL signal using a high frequency counter [2,4,5,[13][14][15][16][17]. The other group of methods is ADC-based methods which consist of estimating the Instantaneous Frequency (IF) of the temporally sampled signal (by the means of a classical ADC acquisition board) [6][7][8][18][19][20]. ...
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This paper treats the spectral precision of frequency demodulation method based on Gabor’s analytic signal applied to Instantaneous Angular Speed (IAS) estimation from speed sensor under stationary speed conditions. The quadrature decomposition combined with geometric complex representation are used to approach the spectral modeling of the influence of wide-sense stationary additive noise on IAS spectral estimate using the periodogram. A method is proposed for IAS noise spectral estimation from signal’s envelope. Moreover, the effect of the carrier power and encoder resolution is demonstrated through real life signals. Finally, a statistical approach is proposed in order to set thresholds on IAS spectral observation allowing a good separation of additive random perturbations from real information under a certain probability. The proposed tools are applied on both synthetic and experimental signals and are shown to be effective in estimating IAS noise level from the input signal properties.
... Analogously, IAS have been applied in the diagnosis for reciprocating compressor valve leakage faults caused by discharge plates or suction valves [33]. Leclè re [34] extracted the IAS signal from the multi-order probability density function of the gearbox acceleration instantaneous spectrum to monitor the operating conditions of the wind turbine. These studies show the effectiveness of applying IAS based high-performance diagnostics to many different machines, while it has not been evaluated upon a PG system. ...
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Planetary Gearbox (PG) is widely used in many vital transmission systems with the advantages such as high power-weight ratio and high load capacity. To maintain the performance of these system, extensive attentions are paid in the vibration signal analysis for early fault detection and diagnosis of PGs. However, vibration signals are less accurate as they severely influenced by multi-interfaces energy dissipation and varying transmission paths. In contrast, Instantaneous Angular Speed (IAS) is directly related to the gear dynamics and free from the path varying effect. This paper evaluates the performance of an improved IAS measurement system to diagnose PG faults, in which a dual detector setup is adopted to suppress IAS errors caused by the setting parameters and encoder errors. Firstly, a PG dynamic model is established to simulate IAS responses from a PG with local faults on sun and planet gears. Secondly, the noise level of the demodulated IAS is studied under different system parameter configurations. Subsequently, a dual detector self-denoising method is developed to suppress the false IAS caused by encoder errors. Finally, the performance of proposed IAS measurement system is evaluated based on an industrial PG with sun and planet faults. Experimental results show PG faults can be effectively diagnosed from the simple IAS order spectra, indicating that the proposed IAS measurement system is a more promising and yet lower costing approach to monitor rotating machines.
... Badaoui and Bonnardot [10] showed that, using a counting system, angular sampling is a non-uniform time sampling introducing a noise which power is proportional to the jitter variance. André et al. [11] studied the accuracy of the counting system used to estimate the IAS , and Leclere et al. [12] made a deeper analysis of the measurement errors and their influence on the speed estimation, taking the irregularity of the magnetic coder and the jitter of the clock into account. Palermo et al. [13] proposed a calibration tool to minimize the geometrical error when estimating the angular speed with zebra-tapes. ...
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The use of Instantaneous Angular Speed (IAS) in condition monitoring of rotating machines is an appealing alternative to traditional approaches such as those based on accelerometers: the direct angular sampling characteristic of IAS measurements has proven to be a favoured framework to study mechanical phenomena involved in rotating machines. Physical models have been established to link the variations of IAS to torque disturbances in case of mechanical fault such as bearing failure. In this paper an original point of view is given on the IAS measurement by linking IAS variations and shaft vibrations. First the orbit of a rotating shaft is studied to show that it contains useful information about the health state of the bearing. Then angle position sensors traditionally used in IAS measurements are combined to reconstruct this orbit. It is shown that IAS measurements are sensitive to the shaft vibrations, which may be used advantageously to enhance the diagnosis possibilities. Finally the links between various defect sizes and the reconstructed orbits are investigated to propose a fault severity indicator.
... Cylinder leakiness was simulated by changing the rate of the decompressor valve opening with the simultaneous recording of signals from the sensor of angle marks, current, voltage and actual pressure inside the cylinder. Binding to the rotation angle of the crankshaft was done with the help of processing data from the angle marks sensor, that is, from the inductive sensor positioned opposite the flywheel ring gear [4,12,13]. In this case software processing of oscilloscope charts was used (Figure 2). ...
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Vibration analysis of rotating machinery is an important part of industrial predictive maintenance programmes, so that wear and defects in moving parts can be discovered and repaired before the machine breaks down, thus reducing operating and maintenance costs. One method of vibration analysis is known as order tracking. This is a frequency analysis method that uses multiples of the running speed (orders), instead of absolute frequencies (Hz), as the frequency base. Order tracking is useful for machine condition monitoring because it can easily identify speed-related vibrations such as shaft defects and bearing wear. To use order tracking analysis, the vibration signal must be sampled at constant increments of shaft angle. Conventional order tracking data acquisition uses special analog hardware to sample at a rate proportional to the shaft speed. A computed order tracking method samples at a constant rate (i.e. uniform Δt), and then uses software to resample the data at constant angular increments. This study examines which factors and assumptions, inherent in this computed order tracking method, have the greatest effect on its accuracy. Both classical and computed methods were evaluated and compared using a digitial simulation. It was found that the method is extremely sensitive to the timing accuracy of the keyphasor pulses and that great improvements in the spectral accuracy were observed when making use of higher-order interpolation functions.
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We evaluated in this paper an improved technique for measuring gear transmission error (GTE) at high speed, by using low pulse per revolution optical encoders. The originality of this technique lies in the fact that highly precise, completely digital measurements of torsional vibration or transmission error (TE) at high speed are achievable by the use of low-price, basic optical components. The lengths of encoder pulses are estimated with a high-frequency timer (100 MHz): thus, it appears that the theoretical precision of this device depends only on the angular speed of shafts, not on the number of pulses per revolution of the encoder. In practice, the intrinsic encoder accuracy (namely the grating or electronic signal processing precision) directly affects precision measurements. Alternatively, the number of pulses per revolution of the encoder specifies the resolution. We examined the possibility of calibrating encoders through using a specific test rig. The determination of corrective data assigned to each grating leads to an insignificant improvement of the precision measurement. The coherence from one revolution to another does not present any significant deterministic component. The overall precision achieved is less than 0.03 second of arc for each frequency of the power spectral density. This calibration device only gives a good assessment of eccentricity induced by mechanical mounting of optical discs on a shaft, compared with the direct measurement on grating discs. The correlation between the two measurements is less than 3% of the magnitude of the relative eccentricity. Thus, the encoder technique seems to be a cheap and easy way to implement transmission error measurement on real mechanical systems with high precision and sufficient reliability.
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Early fault detection and diagnosis for medium-speed diesel engines is important to ensure reliable operation throughout the course of their service. This work presents an investigation of the diesel engine combustion related fault detection capability of crankshaft torsional vibration. The encoder signal, often used for shaft speed measurement, has been used to construct the instantaneous angular speed (IAS) waveform, which actually represents the signature of the torsional vibration. Earlier studies have shown that the IAS signal and its fast Fourier transform (FFT) analysis are effective for monitoring engines with less than eight cylinders. The applicability to medium-speed engines, however, is strongly contested due to the high number of cylinders and large moment of inertia. Therefore the effectiveness of the FFT-based approach has further been enhanced by improving the signal processing to determine the IAS signal and subsequently tested on a 16-cylinder engine. In addition, a novel method of presentation, based on the polar coordinate system of the IAS signal, has also been introduced; to improve the discrimination features of the faults compared to the FFT-based approach of the IAS signal. The paper discusses two typical experimental studies on 16- and 20-cylinder engines, with and without faults, and the diagnosis results by the proposed polar presentation method. The results were also compared with the earlier FFT-based method of the IAS signal.
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This work aims at monitoring large diesel engines by analyzing the crankshaft angular speed variations. It focuses on a powerful 20-cylinder diesel engine with crankshaft natural frequencies within the operating speed range. First, the angular speed variations are modeled at the crankshaft free end. This includes modeling both the crankshaft dynamical behavior and the excitation torques. As the engine is very large, the first crankshaft torsional modes are in the low frequency range. A model with the assumption of a flexible crankshaft is required. The excitation torques depend on the in-cylinder pressure curve. The latter is modeled with a phenomenological model. Mechanical and combustion parameters of the model are optimized with the help of actual data. Then, an automated diagnosis based on an artificially intelligent system is proposed. Neural networks are used for pattern recognition of the angular speed waveforms in normal and faulty conditions. Reference patterns required in the training phase are computed with the model, calibrated using a small number of actual measurements. Promising results are obtained. An experimental fuel leakage fault is successfully diagnosed, including detection and localization of the faulty cylinder, as well as the approximation of the fault severity.
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Many different methods have been developed for the measurement of angular speed. Each successive method has attempted to improve measurement performance using a different strategy to process encoder signals based on two basic principles: counting the number of pulses in a given time duration and measuring the elapsed time for a single cycle of encoder signal. A review of existing methods reveals that little effort has been made in the measurement of instantaneous, multi-channel, wide-range angular speed. As a result, this paper aims to develop a method that provides instantaneous speed information in the form of angular displacement. It addresses the general process and considerations that ensure effective measurement of instantaneous angular speed (IAS). The paper then presents two different techniques: one is based on a general-purpose data acquisition system, another uses pure software implementation. Both methods aim to maximise the use of hardware resources without incurring additional costs in the form of upgrades to the measurement system. Finally, an optimisation of measurement system parameters for a reliable IAS measurement has been carried out based on measurement error analysis.