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Analysis of uncertainties of TPA with tonal excitation
S. Mohamady 1, M. Vorländer
1
1 Institute of Technical Acoustics, RWTH Aachen University
Kopernikusstraße 5, 52074 Aachen, Germany
e-mail: [email protected]
Abstract In the analysis of acoustic systems the focus is mainly on the ultimate result of the analysis such as
spectra, levels or other post-processing results for acoustic evaluation and sound design. The aim of this
research is to derive uncertainty models and sensitivity analyses in order to facilitate applications in the
field of condition monitoring and sound design. As one of the applications, post-processing of the
uncertainty data and characterization of the engine noise during a run-up gives specific information on
condition of an electric engine. However the output uncertainty and its relation to the uncertainty of input
variables are usually unknown. In this regard it is designed a case study as a rectangular enclosure with
interior sound source and receiver. To conduct the uncertainty analysis it is estimated two uncertain input
variables: the reproducibility of sensor positions, and the temperature in the system. In this observation it
is defined the random error of sensor positions and random errors of temperature. By using an analytic
model of a modal response the output variable – the sound pressure show a variance, too. By using the
uncertainty propagation method described in GUM, at first the specific standard deviations in the transfer
functions are determined, and then discussed by applying specific excitation signals such as tonal sound
and machine run-ups. The results show a specific dependence between the output variances from the input
variances but interestingly differently for the low-frequency and the high-frequency range. Moreover, with
post-processing analysis the audibility of changes in transfer path analysis can be examined to find the
psychoacoustic significance of variations with respect to input uncertainties. Loudness, roughness and
tonality are typical sound quality parameters that can be investigated in future.
1 Introduction
Uncertainty is of increasing interest in transfer path analysis (TPA). Since this phenomenon is non-
preventable, an uncertainty evaluation of vibro-acoustic systems becomes more significant. In the
traditional transfer path analysis, broadband input excitations are likely to be found. Large uncertainties of
the output amplitude in the modal response occur mainly at steep slopes of the transfer functions. In a
broadband excitation by combustion noise, for example, the mean level of the transfer function is rather
robust. Especially in systems with tonal excitation as occurs in electric drives, the uncertainties in the
amplitudes and phases of the output signals were not studied in detail yet. It can be expected that sound
pressure amplitudes in modal systems with tonal excitations are much more sensitive to variations of the
excitation or to uncertainties in the system itself. One cause for uncertainties is temperature changes and
another cause is reproducibility of sensor positioning. The main objective of this research is to assess the
range of tonal excitation uncertainty in TPA to establish an uncertainty model which links the uncertainty
of output variations to the input variations.
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2 Case study
The case study is a rectangular enclosure with hard boundaries and dimensions of . Sound source and receiver are placed inside the geometry shown in figure 1 below.
The relation between sound pressure at the receiver and positioning of the receiver is obtained using
modal synthesis approach as follow (Kuttruff, 2007):
∑
(1)
where is the eigenfunction in the room at the sender and receiver positions, with as volume
velocity for harmonic excitation and modal damping constant. The Schroeder frequency of the
proposed case study is estimated to be around 2 kHz (Kuttruff, 2007).
3 Source of uncertainties
It is defined two sources of uncertainties which propagate through the acoustic system: sensor positioning
and temperature, both cases are assumed to be normally distributed.
3.1 Sensor positioning
The uncertainty of sensor positioning is defined with a spherical area with radius of R centered in the
initial position of the receiver. Then 100 sample points are normally distributed in this area and the
transfer path of the system for each point is calculated. The size of the sphere is increased three times with
steps of 0.015 m and the same samples numbers are translated by shifting their position. In Figure 2 the
sensor points in the upper corner of the box with a standard deviation radius of 3.5 cm are shown.
Figure 1:Normally distributed sample of source-to receiver positions with standard deviation of the
distance, r, to the nominal receiver position, rr. Example for r = 0.035 m
3.2 Temperature variation
The speed of sound changes with temperature, which also has an influence on the modal superposition of
the sound field. The relation between sound speed and temperature is as follows:
(2)
where is temperature in centigrade ( ), and c the sound speed. In this case 100 samples with a mean
temperature of 20 °C is selected and sound pressure with respect to each variation is obtained. With
variation of temperature the modal superposition of the transfer function varies.
Sound source
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3.3 Combined uncertainties
In this work the two input variation sources are assumed to be uncorrelated. For studying combined effect
of sensor positioning and temperature changes the third setup of the receiver positioning (R = 0.035 m) is
combined with temperature variation described above.
4 Uncertainty analysis of transfer paths
In this research Mont Carlo simulation was used as uncertainty modeling tool with a sample size of 100.
The uncertainty analysis procedure is described in the block diagram depicted in Figure 2. First, the
influence of uncertainty sources on transfer path of the system is studied then the model is expanded for
the case with tonal excitations. The propagation of uncertainties is started with analyzing the influence of
sensor positioning and temperature individually and then in their combination just for the transfer path as
such.
Figure 2:Block diagram of uncertainty analysis
A detailed uncertainty analysis of transfer function is extensively investigated by (Dietrich, 2013) to
assess the independent uncertainty contribution in transfer path analysis (TPA). In this work the attempt is
to obtain the relation between the input uncertainty parameter and the output uncertainty of data such as
total level and loudness.
4.1 Influence of sensor positioning
The probability distribution function (PDF) of sensor positioning around center position of (0.7, 0.4, 0.2)
is depicted in Figure 3 (a).
The input PDF is used in the Monte-Carlo Simulation as variation of inputs to the system. The distribution
of the resulting sound pressure levels at the first eigenfrequency (215 Hz) is calculated and depicted in
Figure 3 (b). The probability density function at that frequency seems to be normally distributed.
For a general conclusion, however, the resulting sound pressures must be discussed frequency
dependent. This result cannot be shown in a single diagram. Instead, it is used the standard
deviation expressed p in decibels
(3)
Sound pressure
level
Sensor
positioning
Excitation
signal
Temperature
Transfer path of DUT
(TPA)
Uncertainty
variation
relation
Uncertainty sources
TRANSFER PATH ANALYSIS AND INVERSE METHODS 3959
The fact that the standard deviation expressed in decibels distorts the symmetry of the normal distribution,
is neglected here, and only the positive deviation is considered. The sound spectrum of the transfer path of
the system with respect to each set of the input uncertainties is calculated and the result of uncertainty
propagation in transfer path of the system is shown as a waterfall plot in Figure (4). In this plot the x, y
and z axis represent the standard deviations of sensor positioning in meter, the frequency in Hz and the
relative standard deviation of the sound pressure amplitude in dB, respectively.
(a) (b)
Figure 3:a) PDF of sensor positions with radius of 3.5 cm, b) PDF of sound pressure levels at the first
eigenfrequency (215 Hz)
Considering each set of position uncertainties, the uncertainty propagation increases with the frequency up
to the Schroeder frequency. Then it reaches a relatively constant value of 8 dB. The maxima are found at
the eigenfrequencies of the system. Towards larger input variation the sound pressure deviations increase
as well but only below the Schroeder frequency. Above the Schroeder frequency the constant value of 8
dB seems to be independent of the input standard deviation.
Figure 4:Spectral relative deviation of the sound pressure amplitude as a function of the input standard
deviations in frequency domain in dB
The pattern of relative standard deviations will be later visualized with higher resolution by adding more
input uncertainty positions. For now the next step of the analysis will be introduced which is to integrate
the variances in each one-third octave band:
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√∑
(4)
where and
are standard deviations and mean values of the sound pressure levels at each frequency,
respectively.
Figure 5 shows the result of integrated variance over one-third octave bands. In this figure each line
belongs to pointed uncertainty of sensor positioning.
Figure 5:Level differences resulting from added variances in one-third octave bands
Figure 5 confirms that there is a tendency for a saturation of the output level deviation above the
Schroeder frequency. Only for the smallest input deviation the value of 8 dB is not reached. The next step
in future of the analysis should be to identify a general quantitative relation between the input standard
deviation of sensor positioning and the resulting level uncertainty in the modal overlap region.
4.2 Influence of temperature
In this part the influence of temperature on the modal behavior of the acoustic system is reported. Figure 6
(a) shows the PDF of temperature around 20°C and (b) the PDF of the resulting sound pressure at 215 Hz.
Figure 6: a) PDF of temperature with mean temperature of 20°C, b) PDF of sound pressure at 216 Hz due
to temperature variation
2.5 kHz
TRANSFER PATH ANALYSIS AND INVERSE METHODS 3961
The ensemble of transfer functions is calculated for each of the given standard deviations. The standard
deviation of the temperature is 1 K. The spectral standard deviations, p, of the resulting transfer
functions, p(), are calculated and expressed in terms of the level difference between the actual result and
the mean spectrum at the nominal receiver position, as shown in Figure 6 (b).
Figure 7: Example: Relative standard deviation of sound pressure in dB caused by temperature
variations with STD of 1K
The method of integrating the variance in each one-third octave is used again to show the trend of
uncertainties propagation in the system with temperature uncertainties, see Figure 8. The consequence of
this effect could be that a reference transfer function is measured at one temperature conditions and others
for the purpose of comparison or monitoring at another time at another temperature. The spectral details
may differ as shown in Figure 7 and 8.
Figure 8: Added variances in third octave band (example: temperature STD of 1 K)
4.3 Influence of combined uncertainty analysis
The two sources of uncertainties discussed above are uncorrelated. Thus the following equation can be
used to estimate the combined uncorrelated uncertainties:
(5)
In this case the relative standard deviation of sound pressure is again achieved by using eq. (3).
As mentioned above, the third set of input positioning is selected and combined with temperature
variation. Figure (9) and (10) show the influence of combined uncertainty as relative standard deviation of
the sound pressure level.
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Figure 9: Relative standard deviation of sound pressure in dB (combined)
Figure 10: Added variances in one-third octave bands of the relative standard deviation of sound
pressure in dB
Up to now variations of the transfer function due to input uncertainty parameters were studied. This
directly corresponds to the expectation values of an arbitrary tonal input on the one hand, in case of the
specific spectral uncertainties. On the other hand, the band averages reflect the expectation of output
deviations with broadband excitations. However, the study can be expanded for the system with specific
tonal input excitation signals. In the next sub-sections this analysis are presented.
5 Uncertainty analysis with tonal excitation signals
In present study the source of uncertainty is mainly on transfer path of the system. With an excitation
signal just a part of the frequency range of the transfer path will be excited. The purpose of such studies is
mainly an application-dependent analysis like condition monitoring and sound design with tonal sources
such as electric machines.
For the uncertainty analysis in the next step the excitation signals will be considered according to their
feature to pick specific frequencies, where in the transfer function uncertainties will occur simultaneously.
The fact again illustrates that broadband excitation and band averages are less affected by uncertainties
than tone complexes picking possibly large deviations.
To start the analysis first tonal noise is used to excite the system. A harmonic series of pure tones with the
fundamental frequency f0 up to N tones is given by
TRANSFER PATH ANALYSIS AND INVERSE METHODS 3963
∑
(6)
For calculation of the output signal spectrum, Stone(f) is multiplied with p(f) from eq. (1). Due to the
statistical independence of the events the total error of tone excitation is to be calculated by summing the
variances, 2, of the spectral uncertainties involved (GUM, 2008):
(7)
At first it is studied one tonal complex according to eq. (7), with a fundamental frequency of fo = 200 Hz.
The relative STD of the output in decibels as a function of the input STD is obtained and depicted in
Figure (11).
Figure 11: Relative STD of power excited with tonal noise excitation (receiver positioning)
Obviously a large increase can be detected due to increasing input uncertainties. In second step simulated
engine noise during run-up is used to excite the system. A run-up can be created by sweeping the
fundamental frequency, f0, in eq. (8) and choosing appropriate excitation amplitudes according to a model
of an electric machine. A simulated electric machine noise during run-up (v.d. Giet, 2011) with using a
linear sweep with frequency order of and time duration of in time domain:
(
),
(8)
where , , and are start speed, end speed, frequency order and run-up time, respectively.
The excitation amplitudes can be derived from processing the linear sweep with machine interior impulse
response of the electric engine, which leads to:
∑ ( ∑ ( )
∑ ( )
)
(9)
where and are impulse response of transfer function and from radial force with
mode r to acoustic pressure and azimuthal case. Consequently and are radiation efficiency and
electromagnetic torque. In the same way the standard deviation of sound pressure excited with broadband
engine noise can be calculated using the integral over the variances:
∫
(10)
Since the run-up excitation is a time-dependent process, the relation between input and output uncertainty
can be shown in each time segments. Figure 12 shows the results.
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In each time segment an increasing output STD with an increase of input STD can be seen in this example.
The influence of the temperature on the system with engine noise excitation and the combined uncertainty
with sensor positioning can be studied in the same way as presented above.
Figure 12: Relative STD of output pressure due to sensor positioning variation as a function of time during run-
up
6 Loudness Analysis
As an example, an application of the uncertainty model for a task in sound design is discussed. The
parameter loudness is analyzed. The resulting loudness of the transfer function due to an input position
uncertainty is depicted in Figure 13 with the mean loudness result surrounded by two curves expressing
the mean plus and minus the obtained deviations. This illustrates the mean inside a confidence interval
with a probability that an arbitrary result is found within this interval with a probability of 68%.
Figure 13: Loudness analysis of transfer function due to receiver positioning uncertainty
Up to 20 sone from the critical band of 3 Bark there is a little influence of sensor positioning on loudness.
Whether or not the loudness deviations are audible will depend on the excitation signal and on masking
effects. In the example of Figure 13, the largest deviation occurs at the critical band of 6 Bark.
Bark
TRANSFER PATH ANALYSIS AND INVERSE METHODS 3965
7 Conclusion
Considering uncertainty analysis, in this study a detailed uncertainty parameter evaluation is proposed. A
simple case study is introduced as an example, and two sources of uncertainties are defined. The
propagation of uncertainties was studied with the goal of the GUM framework. An interesting result is that
the band-average variance above the Schroeder frequency seems to saturate at 8 dB. So far no analytical or
statistical explanation can be given.
The system is then excited with two samples of tonal excitation: tonal noise and engine noise during run-
up. The relation between input and output uncertainties is further evaluated to facilitate further
applications in the field of condition monitoring and sound design.
8 Future Work
At first, the variances in the transfer function of the modal enclosure must be studies in more details and
explained in a more general way. The result from the case study suggests that there is a certain
independence of the statistics of the variances in the modal superposition range. For this, the statistics of
room transfer functions must be re-visited. Then, the case studies must be extended towards coupled
airborne and vibration modal systems.
One of the main objectives of the proposed study is to use the uncertainty results as condition monitoring
tool for electric engines. The main problem of noise and vibration monitoring of electric engines is the
combination of source effect and transmission path. This work is founded to overcome this problem and
extract influence of transfer path from the main signature of electric engine operation. The results of
condition monitoring are a part of future study.
References
[1] Kuttruff, H. (2007). Acoustics – An Introduction, Taylor and Francis.
[2] v. d. Giet, M. (2011). Analysis of electromagnetic acoustic noise excitations. Doctoral dissertation
RWTH Aachen University
[3] Dietrich, P. (2013). Uncertainties in Acoustical Transfer Functions. Doctoral dissertation RWTH
Aachen University
[4] ISO GUM (2008) Guide to the Uncertainty in Measurement
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