Reduction of pseudo vibrations in rotational measurements, using
sensor arrays.
M. L. Jakobsen and S. G. Hanson
Department of Optics and Plasma Research.
Risoe National Laboratory, DK-4000 Roskilde, Denmark.
[email protected] and [email protected]
ABSTRACT
Laser speckle related pseudo vibrations, appearing in laser vibrometry, applied to rotating targets, are difficult to
distinguish from angular velocity fluctuations, appearing at the cycle frequency of the rotating target and its harmonics.
This paper will discuss reduction of pseudo vibrations from the point of the view of using an array of low cost laser
vibrometer sensors.
INTRODUCTION
Angular velocity measurements obtained with optical technologies, using coherent sources, such as laser Doppler
velocimetry (LDV) [1] and speckle shearing velocimetry [2], are often contaminated by a noise phenomenon called pseudo
vibrations [3]. Pseudo vibrations are minor fluctuations appearing in the angular velocity measurements, and are caused
by laser speckles and their influence up through the sensor and the signal processing. In essence, the speckle pattern
repeats itself for each revolution, and the processing algorithm will repeat the same “errors” Therefore, they appear as a
noise contribution, which strongly correlate with the angular position of a rotating target, through the fine structure of its
surface. For the same reason the pseudo vibrations are strongly represented in e.g. the power spectrum of angular
velocity time series, where it will appear as a comb function with harmonics of the fundamental frequency of the rotating
target. Therefore, pseudo vibrations will interfere particularly with any measurements of fluctuations in angular velocity,
which are located spectrally near the cycle frequency of the rotating target or its harmonics. The pseudo vibrations are not
connected directly to a real variation in the rotational speed, but they are difficult to distinguish from the real torsional
vibrations.
Efforts have been put into this subject partly describing the phenomenon [3,4], partly finding ways to overcome it as a
problem for laser vibrometry. The pseudo vibrations have been reduced/removed by e.g. recording a reference
measurement ideally without torsional vibrations, then obtaining experimental measurements synchronized to the
reference in terms of target angle, and finally obtaining the torsional vibrations by subtracting the reference measurement
form the experimental measurements [5]. This method relies on the possibility to obtain a reference measurement without
torsional vibration, and no decorrelation of the speckle noise during the experiment.
In the light of the option of using compact, high-performance sensors [6] at low-cost prices, the prospect of implementing
sensor with multiple laser sources or even sensor arrays appears. Therefore, this paper will study these new practical
opportunities to reduce speckle-related noise.
THEORY
It is the purpose here to analyze the spectrum observed from a torsional vibrometer. Specifically, it is the aim to consider
the influence of the so-called pseudo-vibrations, and how their influence might be reduced with special means. We will
here assume that the instantaneous rotational speed is presented as a signal, the strength of which is proportional to the
instantaneous rotation. This gives:
ω(t ) = ω 0 (t ) + α1 sin (ω1t + ϕ1 (t )) + n (t )
(0)
where ω0(t) is the mean (DC) rotational speed, α1 is the modulation depth of the torsional vibration, with a constant
torsional vibration frequency ω1 and a stochastic phase function ϕ1(t), and n(t) is an additive noise arising from speckle
repetition and electronic signal processing. We have deliberately only assumed one torsional frequency to be probed in
the mechanical structure. We will later discuss the influence of having a multitude of torsional vibrations; but for the sake
of clarity, we will in the following only include a single torsional mode.
To find the power spectrum of the signal, we need to find the autocorrelation of the time-dependent rotational speed, i.e.
AΣ (τ ) ≡ ω(t ) ω(τ − t ) .
(1)
We will assume that the statistics of the process is static, which means that the autocorrelation function has no absolute
time-dependence, but only depends on the time-delay (τ). The above cross products will in the following be discussed
individually:
ω 0 (t )ω 0 (τ − t ) : This is the autocorrelation of the basic rotational speed. We assume this rotational speed to be
constant but with a slight decorrelation due to a slow variation in the rotational speed, with an assumed long decorrelation
time of τ0. We get:
⎡ τ2 ⎤
A0 (τ ) ≡ ω 0 (t )ω 0 (τ − t ) = ω 0 exp ⎢− 2 ⎥ .
⎢⎣ τ 0 ⎥⎦
2
(2)
α1 sin (ω1t + ϕ1 (t )) sin (ω1 (τ − t ) + ϕ1 (τ − t )) : This is the autocorrelation of the torsional vibration. Assuming the
2
stochastic phase perturbation to adhere to a Gaussian random process, this may be written:
2
A1 (τ ) ≡
α1
2
cos (ω1 τ ) exp [−σ1 (1 − b1 (τ ))] .
2
(3)
Here
2
σ1 ≡ ϕ1
2
and b1 (τ ) ≡
ϕ1 (t )ϕ1 (τ − t )
ϕ1
(4)
2
which are the phase variance and the normalized phase covariance, respectively. The phase variance thus reflects how
much the torsional vibrations vary from a deterministic sinusoidal variation, and the normalized phase covariance tells on
which time scale this occurs. Assuming a Gaussian form with a decorrelation time τ1 for the normalized phase covariance,
we get
⎡
2
A1 (τ ) =
α1
⎛
⎡
2
⎤⎞⎤
τ
2
cos (ω1 τ ) exp ⎢−σ1 ⎜⎜1 − exp ⎢− 2 ⎥ ⎟⎟⎟⎥ .
⎢⎣
⎢⎣ τ1 ⎥⎦ ⎠⎥⎦
⎝⎜
2
(5)
Assuming τ << τ 1 , the inner exponential can safely be series expanded to first order, which gives:
2
A1 (τ ) =
α1
2
2
⎡
⎤
τ
⎥.
2
2
⎢⎣ τ1 / σ1 ⎥⎦
cos (ω1 τ ) exp ⎢−
(6)
This means that the effective decorrelation time of the torsional frequency is given by τ 1 / σ 1 .
n (t ) n (τ − t ) : This is the correlation function for the noise, which may be due to the cyclic speckle pattern and due to
the electronic noise. We can thus write this correlation function as:
An (τ ) ≡ n (t ) n (τ − t ) = nsp (t ) n sp (τ − t ) + nel (t ) nel (τ − t ) ,
(7)
where we have divided this contribution up into the two statistically independent contributions.
The speckle term is repetitive for each revolution with a decorrelation time τsp. We can write
⎡ τ2 ⎤ ∞ ⎛
⎞
⎥ ∑ ℜ ⎜⎜τ − n 2π ⎟⎟,
2
⎢
⎥
⎜
ω 0 ⎠⎟
⎣ τsp ⎦ n =−∞ ⎝
Asp (τ ) ≡ nsp (t ) nsp (τ − t ) = exp ⎢−
(8)
where τsp is the decorrelation time for repeating the speckles, i.e. the time it takes for the probed speckle pattern to
decorrelate. Usually, we have ω0τ sp >> 1 , which means that the shaft has to rotate several revolution before the probed
speckle pattern is changes, either by translation of the shaft or displacement of the probing unit, or by contamination of the
surface. The preceding function ℜ (τ ) is a system-specific function that depends on the processing unit and describes
the deviation of the measurement of the instantaneous value of ω ( t ) from the correct one due to the repetition of the
same speckle pattern. Therefore ℜ (τ ) = 0 for τ < 0 or τ ≥ 2π / ω0 . The exact form of ℜ (τ ) depicts to what extent the
higher harmonics of the pseudo-vibrations will be present in the vibrational spectrum, while the exponential terms in Eq. 9.
In reality, the form the expression for ℜ (τ ) is based on the ability of the signal processing scheme to follow the speckle
velocity, and thus depends on the processing scheme and the characteristics of the speckle pattern. What may be
controlled by the user is to some extent τsp, i.e. the decorrelation time for repeating the speckles.
The last term of importance is the autocorrelation of the electronic noise here assumed to be band limited with an upper
2
frequency B, where 1/B is a time usually larger than any of the cyclic periods and decay times and σel is the noise power.
We get:
⎡ (B τ )2 ⎤
⎥.
Ael (τ ) ≡ nel (t ) nel (τ − t ) = σel exp ⎢−
⎢⎣ (2π)2 ⎥⎦
2
(9)
None of the cross terms for the autocorrelation function in Eq. 2 will contribute in case the averaging time is sufficiently
large, and in case there are no hidden correlations between the contributions.
In case we have had not only one mechanical torsional vibration, but a multitude of vibrations, all cross terms between
these that will appear in Eq. 2 will vanish in case the averaging time exceeds the period defined by their difference in
frequency.
el (ω) etc.
The spectral contributions are given by the Fourier transform of the above correlation functions, here denoted A
We get for the four terms:
A% 0 (ω ) =
⎡ ω 2τ 02 ⎤
⎥,
⎣ 4 ⎦
π ω0 τ 0 exp ⎢ −
2
ατ ⎛
⎡ (ω − ω1 ) τ 1
A%1 (ω ) = π 1 1 ⎜ exp ⎢ −
2
4σ 1 ⎝
4σ 1
⎣
2
2
2
⎤
⎡ (ω + ω1 ) 2 τ 12 ⎤ ⎞
⎥ + exp ⎢ − 4σ 2
⎥⎟,
⎦
⎣
⎦⎠
1
(10)
⎛
⎡ ω 2τ sp2 ⎤ ∞ %
⎞
⎥ ∗ ∑ ℜ(ω − nω0 ) ⎟ ,
⎣ 4 ⎦ n = −∞
⎠
A% sp (ω ) = πτ sp ⎜ exp ⎢ −
⎝
and
2π σ el
⎡ πω
exp ⎢ − 2
A% el (ω ) =
B
⎣ B
3/ 2
2
2
2
⎤
⎥,
⎦
where ∗ stands for convolution. It is seen that the spectra for pseudo vibrations and mechanical vibrations will overlap, in
case the mechanical vibrations have a periodicity of the average rotational frequency ω0 (i.e. ω1 = nω0 for n = 1,2,3,..).
Next, we will analyze how to reduce the influence of pseudo-vibrations from the mechanical “real” torsional vibrations.
Here, we will assume these two contributions to be positioned at the same frequency, being it the fundamental frequency,
ω0, or at one of the higher harmonics. It is seen from Eq. 11 that a reduction in the correlation time τsp for the pseudovibrations will reduce its peak value but not reduce the power in the total peak. The power in the pseudo-vibrations for a
certain harmonics can only be reduced through the processing scheme, i.e. by reducing the influence of ℜ (τ ) . A
reduction of the correlation time may be achieved either by moving the object along the rotor axis (at least the
decorrelation length for translation per revolution), or by contaminating its surface during measurement. Both schemes will
have the effect of decorrelating the speckle pattern during measurement, and thus avoiding that the analyzer repeats the
same error from one revolution to the next.
But realizing that the mechanical- and the pseudo-vibrations both have amplitude and phase, the possibility for a further
reduction of the spurious vibrations is present by considering the two contributions as phasors. If we have several
independent systems probing the same rotating object, we may assume that they present the same phasor for the
mechanical vibration, but have statistically independent phasors for the pseudo-vibrations. We can without loss of
generality assume the phasor for the mechanical vibration to have amplitude Amech with zero phase. The phasor for the
noise i’th system can be written An , i exp [ −iϕ i ] . If we have N¸ independent measurement systems, each providing the
measurement
Ai = Amech + An , i exp [ −iϕ i ] , the wanted mechanical vibrations can be found as the vectorial average value, i.e.:
Amech ≅
1
N
∑A
N
i
=
i =1
1
N
∑( A
N
mech
i =1
+ An , i exp [ −iϕ i ])
(11)
where Amech is the estimated value of the mechanical vibration.
Of interest in this context is the standard variation of the estimated value for the vibrational amplitude and its phase, based
on N independent measurements. We may here take advantage of the mathematical derivation by J.W. Goodman [7] in a
slightly different field, namely speckle statistics. Here, statistics for a constant phasor plus a random phasor sum is
analyzed. The derivation will not be repeated here, merely will his findings be used for our derivation, and the results given
with respect to this investigation.
The joint probability for measuring a phasor A is given by
p ( A, ϕ ) =
2
− 2 A Amech cos [ϕ ] ⎤
⎡ A2 + Amech
2
⎥
2σ
⎣
⎦
A
exp ⎢ −
2πσ
(12)
where σ ≡ An / N . The individual probabilities can then be found to give:
2
2
p A ( A) =
and
A
σ
2
2
⎡ A2 + Amech
⎤ ⎡ A Amech ⎤
I0
'
2
2
⎥
2σ
⎣
⎦ ⎢⎣ σ ⎥⎦
exp ⎢ −
2
ϕ
⎡
⎤
pϕ (ϕ ) ≅
exp ⎢ −
.
2 ⎥
2πσ / Amech
⎣ 2(σ / Amech ) ⎦
1
(13)
In the above equations, the modified Bessel function of the first kind is denoted I0. The probability density function for the
phase is approximated, assuming that Amech / σ is large. Finally, we can find the standard variation for the amplitude and
for the phase. The second order moment of the amplitude becomes:
A
2
= Amech + 2σ .
2
2
(14)
And the standard variation of the phase is σ / Amech . The average value of the phase is zero, whereas the average value of
the amplitude is slightly larger than Amech .
Therefore, by averaging N measurements, obtained with independent speckle noise sources, the contribution from the
speckle noise to the standard deviation of both the amplitude and the phase of the measured torsional vibrations will be
reduced with a factor of N-1/2. This result describes both the situation of having N laser vibrometers to monitor individual
paths on the rotating target for an arbitrary number of revolutions, and the situation of having one sensor to monitor the
rotating target for N revolutions with a mechanism, decorrelating the speckles for each revolution.
A low-cost laser vibrometer
A compact, high-performance laser vibrometer could be based on optical spatial-filtering velocimetry [8] of speckle
dynamics, observed at the focal plane of a lens. A schematic drawing of the sensor is illustrated in Figure 1. The speckles
are produced when illuminating a non-specular target surface with coherent light. The scattered light propagates through a
Fourier-transforming optical (L4) arrangement to the entrance plane of the spatial filter. When the target rotates, then the
speckle pattern, observed at the entrance plane of the spatial filter, will translate [9]. The relation between target rotation
and speckle translation at the lenslet array is given by:
Δ p x = 2 f 4 ⋅ Δθ = 2 f 4 ⋅ ω0 ⋅ Δτ ,
(15)
where Δθ equals the angular displacement of the target within a time lag of Δτ at a constant angular speed of ω0. The
given relation for the speckle translation, Δpx, can be derived via the time-lagged cross covariance function of the intensity
distribution in the Fourier plane [9]. A similar analysis shows that in case of a translation of the target [10], the
decorrelation length of at the speckle patterns will correspond to the mean speckle size, and that means that the speckle
pattern will boil, rather than translate. Thus, target translation due to e.g. vibrations will not bias the measurements of
angular velocity. Furthermore, this fact is used as the criteria for correct alignment of the Fourier plane by replacing the
rotating target with a target undergoing a linear translation.
The spatial filter is based on a lenticular array (L1) and the spherical lens (L2), and implements narrow spatial band-pass
filtering of the intensity distribution of the observed speckle pattern. The spatial filter extracts a spatial frequency, defined
by the pitch (Λ) of the lenticular array, and at the detector plane the corresponding quasi-sinusoidal spatial intensity
distribution is monitored at various phase steps by an arrangement of photodetector pairs. When the target rotates, the
intensity distribution at the detector plane translates, and accordingly, a photodetector pair will provide a quasi-sinusoidal
photocurrent, with a phase varying proportionally with the target angle. The quasi-sinusoidal signal has a stochastic
amplitude and phase, but it will allow for simple processing schemes based on e.g. zero-crossing detection. Zero-crossing
detection of the frequency of the signals provides real-time measurements of the angular velocity of a rotating target.
Figure 1. A schematic of the sensor design
The calibration of the sensor depends on the focal length of the Fourier transforming lens (f4), and the spatial frequency,
specified by the spatial filter, thus the robustness of the system can be high, and the fundamental frequency (fsignal) of the
quasi-sinusoidal signal becomes:
f signal =
2 f4
Λ
⋅ ω0 ,
(16)
The sensor, used for the application below, uses the following optical parameters. The lenticular array (L1) has a pitch of
15 μm, and a mean focal length of 7-8 μm. The lens, L2, has a focal length of 3.9 mm, and the detector pairs have an
internal centre spacing of 3.9 mm. The lens, L4, has a focal length of 20 mm, and the working distance is approximately 20
mm as well. The VCSEL emits coherent light at 850 nm and is approximately collimated. The position of the focal plane of
the lens, L4, is tuned to provide a sensor with the minimum decorrelation length, which means that it is comparable in size
with the speckle size – i.e. few microns.
An application
An application of our interest is the torsional vibrations of the main rotor in a wind turbine, where aerodynamic interaction
between the blades (three blades) and the tower generates e.g. a torsional vibration with its fundamental frequency
starting at the third harmonic of the cycle frequency of the axis. The sensor is mounted to probe the rotor between the
main bearings, carrying the weight of the blades and rotor, and the gearbox. The sensor probes the raw surface on the
rotor, and accordingly it provides a quasi-sinusoidal photocurrent for a zero-crossing-detection algorithm. As described
earlier [11] the algorithm actually measures the averaged time steps during sets of zero crossings as a measure for the
angular velocity. The algorithm provides angular velocity measurements continuously to a PC, connected to an ether
network.
The wind turbine, used for our tests, has three blades and can produce 500kW. The wind turbine is locked to the grid, and
rotates with a constant angular velocity of 2.7 rad/s (26 rpm). The sensor acquires an angular velocity with a sample rate
of 10 measurements per s or 22 measurements per revolution. The wind speed was 8 m/s during the tests.
Figure 2 illustrates the power spectra of two time sequence of angular velocity measurements obtained from the sensor
mounted in the wind turbine. Clearly, the power spectrum of the pseudo vibrations appears as a comp function as it is
described in Eq. 10. The lines are very sharp and actually provide an excellent calibration for the mean cycle frequency of
the rotor, being entirely independent of the optical sensor. However, the information we seek is located at the third
harmonic of the cycle frequency, and here we expect the third harmonic of the pseudo vibrations to contribute as well. A
clear indication of this is obtained by poring oil onto the target surface. The film of oil will evaporate slightly and flow
continuously due to the gravitational pull as the rotor rotates, and consequently the speckle pattern will decorrelate from
one revolution to the other. Therefore, when coating the surface with a thin film of oil, the speckle patterns do not repeat
themselves for each cycle, and in the power spectrum, obtained over hundreds of rotor revolutions, the speckle noise is
dramatically reduced. The conclusion is that the signal obtained at the third harmonics on a dry target surface was pseudo
vibrations, and within the noise floor still present, when measuring with a thin oil film on the target surface, we cannot
detect any torsional vibrations at the third harmonic of the cycle frequency of the rotor. Figure 3 illustrates the power
spectra of two time sequence of angular velocity measurements obtained from the sensor, with an oil film attached to the
target surface. The data in the left power spectrum is obtained 30 minutes after the oil has been pored onto the target
surface, while the data for the right power spectrum has been obtained 120 minutes later. Clearly, the speckledecorrelating effect of the oil film has worn off by then.
0.02
0.02
0.015
0.015
0.01
0.01
0.005
0.005
1
2
3
4
5
1
2
3
4
5
Figure 2. The power spectra of two angular velocity time series illustrate the pseudo vibrations. These measurements
have been obtained from a dry target surface.
0.02
0.02
0.015
0.015
0.01
0.01
0.005
0.005
1
2
3
4
5
1
2
3
4
5
Figure 3. The power spectra of two angular velocity time series illustrate a temporal reduction in pseudo vibrations due to
having attached an oil film to the target surface. The left power spectrum is obtained 30 minutes after poring oil on the
target, while the right power spectrum is obtained 120 minutes later.
Of course an oil film on the target surface is not an option for applications in general, and as mentioned above an
alternative option to introduce sufficient speckle decorrelation could be achieved by translating the laser vibrometer along
the axis of rotation. As mentioned above the decorrelation length for a perfectly tuned laser vibrometer is comparable with
the mean speckle size, which for the given sensor means few microns. Thus, even, acquiring data through a high number
of target revolutions, still a fairly short translation length will be required. However, a mechanical active device will always
reduce the robustness of the complete sensor.
Prospect for the near future
We have illustrated that the speckle related noise will appear in the power spectrum at the fundamental cycle frequency of
the target and its harmonics. Therefore, in case we intend to study torsional vibrations of a rotating target, at a frequency
which is identical to the cycle frequency of the target or its harmonics we encounter a severe noise floor. With the
possibility of averaging several independent measurements, either by decorrelating the speckles from revolution to
revolution or by using an array of independent vibrometer sensors we can use the fact that the torsional vibrations of the
target will be repeated through each individual measurement, while the speckle noise will be stochastic from one
measurement to the other. Therefore, according to Eq. 14, the speckle noise can be reduced by averaging, and the noise
equivalent signal decreases with N-1/2. E.g. in case of having an array of 25 vibrometer sensors the standard deviation
related to the speckle noise will drop by a factor of 5.
At the moment the described spatial-filtering-velocimetry technology is being developed for mass production by OPDI
technology Aps., DK. The concept is based on reproduction of the optical parts in polymers, and integration of
optoelectronics onto an ASIC, containing analogue electronics and digital signal processing. The spatial-filteringvelocimetry sensor will become a single chip, which combined with a Fourier transforming lens (L4), again produced in a
polymer, will provide a compact low-cost laser vibrometer. The prototype of the spatial-filtering-velocimetry sensor is
illustrated in Figure 4. The spatial filter velocimetry sensor is implemented with a single piece of polyetherimide (ULTEM)
or TOPAS (top item). The photodetectors are implemented directly with the technology the ASIC, while the VCSEL is
mounted onto a platform on the ASIC (bottom item). The optic element and the ASIC are positioned mutually in a plastic
housing, e.g. using optical feedback. In the near future we propose to use such sensors chips, arranged in an array of N
-1/2
laser vibrometers, to reduce pseudo vibrations (N ), and thereby gain more information of torsional vibrations in rotating
solid structures, particularly vibrations coinciding with the fundamental cycle frequency of the target rotation or its
harmonics.
Figure 4. The photograph illustrates the chip solution for a spatial filtering velocimetry sensor, which can be combined
with a Fourier transforming lens to become a compact low-cost vibrometer. At the top the optical element is seen from
both sides. The optical element is the mounted on top of the ASIC, illustrated below.
ACKNOWLEDGEMENTS
We thank OPDI Technology Aps. (DK) for their contributions to this work.
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