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Determining Transfer Functions from Measured Data
APPENDIX C Determining Transfer Functions from Measured Data
C.1 INTRODUCTION We have discussed transfer functions throughout this book as though they are known exactly, by calculation from first principles, for example. But much of the time, our information is based on experimental data and is therefore imprecise. The purpose of this appendix is to indicate how noise affects measured quantities and how we can evaluate noise contamination of data. C.2 CONSTRUCTION OF INVERSE FILTERS BASED ON MEASURED DATA Let us now consider the situation in Figure C.l(a), where a transfer mobility Y relates an input force L to an output velocity V. The mobility is defined by the ratio of the velocity to the force, Y = V/L, but if we calculate the mobility by using the Fourier transforms of the velocity and the force directly in this way we would get a very poor representation of the transfer mobility. This is because truncation effects in the evaluation of the velocity and force transforms cause relatively large variations from one computation to the next. Measurement errors and noise will also affect the calculations. Let us call the combination of these noise and error components SV„ and (5L„, so in the presence of noise the velocity and force are νη-*ν\
+ δνη9
L„ = L< + <5L„,
(C.l)
where Vxn and Un are the "true" values of the force and the velocity in the absence of
£(t)
v(t) or Vn
or e
U
(a)
Figure C.l
im |—
-9 V(t) 'm(t)
n(t)
(b)
A linear two-port system with and without additive noise at input and output. 287
288
Machinery Noise and Diagnostics
noise. With these introduced, we would get
™-(£ -t -t or δγ γ,
= (γτ-ΤΓ)-
^
Since the relative errors in measuring the velocity and force may be considered independent, the relative variance in the transfer mobility equals the sum of the relative variances in the measured velocity and the input force: T2
T2
σγ 2
σν 2 + 771Γ· GL
\Y\
\V\
„2
(Q3)
\L\
Let us now multiply the numerator and denominator of the transfer mobility by the complex conjugate of velocity and then multiply mobility by the complex conjugate of force. If we now make N measurements and estimate Y by averaging over N, we get <ν^:>Ν
=SOl(œ)
( ^ = 7«L7nΓ )2)N^ = S9Ή n(œ) <\K\2>N =Sw(œ) SUc»)
(="ιλ (C.4)
(=H2\
where the two possibilities generate two estimates for the transfer functions. In this equation, Svl is the cross spectrum between the velocity and force, Sn is the auto (or power) spectrum of the force, and Svv is the auto spectrum of the velocity. Since noise at the system input where the force is applied will affect the auto spectrum of the force more than it affects the cross spectrum between input force and output velocity, we would use H2 when the measurement is expected to be contaminated by input noise; by a similar argument, we would use Hi if noise in the output were expected. This situation can change frequency by frequency. At system resonances, the drive point force will become smaller, so noise at the input will be more important at frequencies near system resonances. At the same frequencies, the output velocity will tend to be large. At or near zero in the transfer function, however, the output velocity is small, even if the input force is large, so the measurement error of the output velocity will tend to be larger near transfer function minima. In such regions, we would tend to use Hl. Thus, the choice between the use of Hx and H2 tends to be a frequency-by-frequency matter, but it is possible to write a simple computer program that will take the measured data and use the
Determining Transfer Functions from Measured Data
289
more appropriate value of Hl and H2, depending on the spectral magnitudes of the input force and the output velocity. A quantity that is closely related to the functions Hl and H2 is the coherence. Whenever we measure a transfer function, the coherence is also calculated to test the quality of the data as they are taken. Consider the situation as shown in Fig ure C.l(b), where noise n(t) is added to the input and noise m(t) is present at the output. The definition of the coherence y2 is
?2=ff=w-
(C5)
The measured input force and the output velocity are then v(t) = vt + m(i),
l(t) = lt + n(t).
(C.6)
If there were no noise in either the input or output, we would have
y2 =
\su^=ί^ννλ = (ΗΛ = L
SuSvv
\SnJ\SvvJ
\H2J
(C7)
Since the input and output noise are uncorrelated, the cross spectrum is not affected by the presence of input or output noise, but the auto spectra for input and output are, so the coherence with noise present is 7
2
_ \svl\2
isy
=
i + s,Vi + sm + sjmj-1 < L $it J\
SVt
(C8)
SVt
The result in Equation C.8 shows how input and output noise or both reduce coherence. Another way of reducing coherence is to have frequency components at the output that are generated within the mechanical system and are not present in the input. This usually results from some sort of nonlinear behavior of the system, so a reduction of coherence can also be a good indicator of nonlinearity in the system. Note, however, that coherence does not discriminate against correlated noises present at both the input and output or noises that are individually correlated with either the input or the output. Thus, we cannot use correlation to distinguish between different sources in a machine that have phase coherence. Since many machines have a periodic operating cycle, all the sources of vibration and excitation are likely to be coherent. Coherence analysis cannot easily discriminate against or determine the contribution of various noise sources in such machines. Let us explore this last part further by referring to Figure C.2. Here we have two sources in a machine: a valve impact and a combustion pressure, both producing vibration pulses at an accelerometer. We have sketched the time
290
Machinery Noise and Diagnostics jitter in waveform valve impact => "["combustion φ \ I force
Λ-
\Λ·
► «
^ 7 \ ^
^/^-^
_» ** T
accelerometer
Figure C.2 Vibrations due to different sources that have afixedtime relation to each other will be coherent. A temporal uncertainty (jitter) in one of the sources produces loss in coherence. waveforms of the two signals in the figure, and we note that since these two signals occur every cycle at the same time relative to each other they are perfectly coherent. Therefore, any combustion pressure signal that arrives at the accelerometer cannot be distinguished from a valve impact signature on the basis of coherence. Let us suppose, however, that there is a jitter in the time of valve impact due to uncontrolled irregularities and that this jitter is about 0.1 msec. Since 0.1 msec corresponds to a period of oscillation of 10 kHz, we might expect that at frequencies above 2.5 kHz, where the 0.1-msec jitter is a quarter of a period, this jitter will cause a loss in coherence between the valve impact and the combustion pressure. Thus for frequencies above 2.5 kHz, coherence analysis would allow an identification and separation of the sources, but below this frequency the signals will be coherent and cannot be separated by coherence analysis.
C.1 INTRODUCTION We have discussed transfer functions throughout this book as though they are known exactly, by calculation from first principles, for example. But much of the time, our information is based on experimental data and is therefore imprecise. The purpose of this appendix is to indicate how noise affects measured quantities and how we can evaluate noise contamination of data. C.2 CONSTRUCTION OF INVERSE FILTERS BASED ON MEASURED DATA Let us now consider the situation in Figure C.l(a), where a transfer mobility Y relates an input force L to an output velocity V. The mobility is defined by the ratio of the velocity to the force, Y = V/L, but if we calculate the mobility by using the Fourier transforms of the velocity and the force directly in this way we would get a very poor representation of the transfer mobility. This is because truncation effects in the evaluation of the velocity and force transforms cause relatively large variations from one computation to the next. Measurement errors and noise will also affect the calculations. Let us call the combination of these noise and error components SV„ and (5L„, so in the presence of noise the velocity and force are νη-*ν\
+ δνη9
L„ = L< + <5L„,
(C.l)
where Vxn and Un are the "true" values of the force and the velocity in the absence of
£(t)
v(t) or Vn
or e
U
(a)
Figure C.l
im |—
-9 V(t) 'm(t)
n(t)
(b)
A linear two-port system with and without additive noise at input and output. 287
288
Machinery Noise and Diagnostics
noise. With these introduced, we would get
™-(£ -t -t or δγ γ,
= (γτ-ΤΓ)-
^
Since the relative errors in measuring the velocity and force may be considered independent, the relative variance in the transfer mobility equals the sum of the relative variances in the measured velocity and the input force: T2
T2
σγ 2
σν 2 + 771Γ· GL
\Y\
\V\
„2
(Q3)
\L\
Let us now multiply the numerator and denominator of the transfer mobility by the complex conjugate of velocity and then multiply mobility by the complex conjugate of force. If we now make N measurements and estimate Y by averaging over N, we get <ν^:>Ν
=SOl(œ)
( ^ = 7«L7nΓ )2)N^ = S9Ή n(œ) <\K\2>N =Sw(œ)
(="ιλ (C.4)
(=H2\
where the two possibilities generate two estimates for the transfer functions. In this equation, Svl is the cross spectrum between the velocity and force, Sn is the auto (or power) spectrum of the force, and Svv is the auto spectrum of the velocity. Since noise at the system input where the force is applied will affect the auto spectrum of the force more than it affects the cross spectrum between input force and output velocity, we would use H2 when the measurement is expected to be contaminated by input noise; by a similar argument, we would use Hi if noise in the output were expected. This situation can change frequency by frequency. At system resonances, the drive point force will become smaller, so noise at the input will be more important at frequencies near system resonances. At the same frequencies, the output velocity will tend to be large. At or near zero in the transfer function, however, the output velocity is small, even if the input force is large, so the measurement error of the output velocity will tend to be larger near transfer function minima. In such regions, we would tend to use Hl. Thus, the choice between the use of Hx and H2 tends to be a frequency-by-frequency matter, but it is possible to write a simple computer program that will take the measured data and use the
Determining Transfer Functions from Measured Data
289
more appropriate value of Hl and H2, depending on the spectral magnitudes of the input force and the output velocity. A quantity that is closely related to the functions Hl and H2 is the coherence. Whenever we measure a transfer function, the coherence is also calculated to test the quality of the data as they are taken. Consider the situation as shown in Fig ure C.l(b), where noise n(t) is added to the input and noise m(t) is present at the output. The definition of the coherence y2 is
?2=ff=w-
(C5)
The measured input force and the output velocity are then v(t) = vt + m(i),
l(t) = lt + n(t).
(C.6)
If there were no noise in either the input or output, we would have
y2 =
\su^=ί^ννλ = (ΗΛ = L
SuSvv
\SnJ\SvvJ
\H2J
(C7)
Since the input and output noise are uncorrelated, the cross spectrum is not affected by the presence of input or output noise, but the auto spectra for input and output are, so the coherence with noise present is 7
2
_ \svl\2
isy
=
i + s,Vi + sm + sjmj-1 < L $it J\
SVt
(C8)
SVt
The result in Equation C.8 shows how input and output noise or both reduce coherence. Another way of reducing coherence is to have frequency components at the output that are generated within the mechanical system and are not present in the input. This usually results from some sort of nonlinear behavior of the system, so a reduction of coherence can also be a good indicator of nonlinearity in the system. Note, however, that coherence does not discriminate against correlated noises present at both the input and output or noises that are individually correlated with either the input or the output. Thus, we cannot use correlation to distinguish between different sources in a machine that have phase coherence. Since many machines have a periodic operating cycle, all the sources of vibration and excitation are likely to be coherent. Coherence analysis cannot easily discriminate against or determine the contribution of various noise sources in such machines. Let us explore this last part further by referring to Figure C.2. Here we have two sources in a machine: a valve impact and a combustion pressure, both producing vibration pulses at an accelerometer. We have sketched the time
290
Machinery Noise and Diagnostics jitter in waveform valve impact => "["combustion φ \ I force
Λ-
\Λ·
► «
^ 7 \ ^
^/^-^
_» ** T
accelerometer
Figure C.2 Vibrations due to different sources that have afixedtime relation to each other will be coherent. A temporal uncertainty (jitter) in one of the sources produces loss in coherence. waveforms of the two signals in the figure, and we note that since these two signals occur every cycle at the same time relative to each other they are perfectly coherent. Therefore, any combustion pressure signal that arrives at the accelerometer cannot be distinguished from a valve impact signature on the basis of coherence. Let us suppose, however, that there is a jitter in the time of valve impact due to uncontrolled irregularities and that this jitter is about 0.1 msec. Since 0.1 msec corresponds to a period of oscillation of 10 kHz, we might expect that at frequencies above 2.5 kHz, where the 0.1-msec jitter is a quarter of a period, this jitter will cause a loss in coherence between the valve impact and the combustion pressure. Thus for frequencies above 2.5 kHz, coherence analysis would allow an identification and separation of the sources, but below this frequency the signals will be coherent and cannot be separated by coherence analysis.