analysis of uncertainties of tpa with tonal...

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

3958 PROCEEDINGS OF ISMA2014 INCLUDING USD2014

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:


√∑

(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


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.


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


∑

(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.


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


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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