Assessing the vibro-acoustic radiation characteristics of acompact consumer appliance

Daniel TAYLOR1 and John LAMB2

Dyson Ltd, Malmesbury, England

ABSTRACTCustomer acceptance and regulatory requirements require manufacturers to develop quieter, morepleasant sounding consumer appliances. In developing and deploying noise mitigation strategies, itis important to understand the transfer functions from a source of noise or vibration to the externalenvironment.

This study looks to investigate the low frequency vibro-acoustic transfer function of a curvedshell, loaded with a spatially varying forced vibration. To better understand the vibro-acousticsof the compact system of sub-wavelength dimensions, the radiation characteristics of the curvedshell under vibration loading is investigated. By then correlating the average surface velocity of thecurved shell with far field acoustic measurements the vibro-acoustic characteristics can be obtained.This is then compared to a three dimensional scan performed with a pressure/velocity (PU) intensityprobe to interrogate the developing sound field at different distances.Keywords: vibro-acoustics, radiation efficiency, consumer appliance

1. INTRODUCTION

The drive from consumers and regulatory bodies for quieter domestic appliances as well as the generaltrend towards compact, lightweight construction, presents challenges for noise control engineering.To inform the correct engineering decisions it is becoming increasingly important for the structuralvibration and radiation characteristics of a realistic geometry to be understood at the design stage,to aid in the reduction of radiated noise.

Although prototypes provide a platform upon which accurate measurements can be performedthe further these components diverge from classical problems (e.g. baffled plates), the less intuitivethey become to understand. This can be particularly problematic when rapid development cyclesrequire time efficient measurement procedures that are still sufficiently accurate to direct the designprocess. For example, without performing in-depth analysis it is difficult to know in advance wherebest to perform a coarse set of vibration measurements in order to obtain representative vibrationprofiles.

In this regard, the radiation efficiency is a useful construct to collapse the effects of spatiallyvarying vibration profiles and modal behaviour into a single correction factor for a compact sourcemodel. One limitation of this approach is that a radiation efficiency is specific to a single operatingpoint of the machinery under test. As a result it cannot be applied generally to different operat-ing points of the device, especially if the vibration profile varies strongly between these operatingconditions. Obtaining analytic expressions for the radiation efficiency of classical structures suchas baffled, flat plates has been studied in depth [1–3]. Work has also been conducted on curvedshells to obtain suitable approximations for the radiation efficiency of these structures. This workhas mainly been focused on applications in aerospace where the radius of curvature is large com-pared to the dimensions of the panel under consideration [4–7]. For more complex geometries theradiation efficiency must usually be determined experimentally. The ISO standard ISO 7849:2 [8]provides guidelines for determining the radiation efficiency (radiation factor) experimentally as well

1email: Daniel.Taylor@dyson.com2email: John.Lamb@dyson.com

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as outlining procedures for using it to estimate the radiated sound power level from a coarse set ofvibration measurements.

Revel and Rossi [9] proposed an approach for predicting the pressure field at a given positionwhich involves a linear superposition of small radiating ‘elements’ distributed over the surface of theenclosure. As this method accounts for spatial variations in the velocity profile over the radiatingsurface, it offers a more generally applicable technique. Acquiring the data with sufficient resolutioncan be a time consuming process. In their work Revel and Rossi compared the superposition methodto the ISO 7849:2 standard approach and found that they agreed well for a simplistic test case.

In this work the ISO 7849:2 standard approach and the piecewise linear summation approach arecompared in their ability to predict the sound power radiated from the compact curved shell of aDyson vacuum cleaner head. During operation the beater bar within the cleaner head (Figure 1) issubjected to high levels of spatially varying vibration as a result of the interaction of the beater bar’sbristles with the floor. This vibration excites the compact curved shell of the cleaner head assemblythrough the mounting points and is subsequently radiated as low frequency sound. As the enclosuredimensions are smaller than the acoustic wavelength in air, simple transfer function expressions suchas the compact source model [10] with a suitable radiation efficiency term should provide reasonableengineering approximations. However, the strongly varying spatial vibration response may makethis approach sensitive to measurement position.

Figure 1: Typical Dyson cleaner head with beater bar.

A pressure/velocity PU intensity probe was used to measure the intensity field radiated by thecurved shell. The visualization provides qualitative insight to the developing sound field.

In Section 2 the experimental procedure for obtaining the high resolution laser Doppler vibro-metry scan required for the linear patch summation method is discussed along with the methodologyfor measuring the acoustic response of the system. The methodologies used to determine the lin-ear summation method and the radiation efficiency are presented in Sections 3 and 4 respectively.The accuracy of these methods are compared in Section 5 along with a brief investigation into thesensitivity of the ISO standard approach with respect to measurement position. The results for theintensity scan are also presented followed by a discussion of the accuracy of the two methods.

2. EXPERIMENTAL PROCEDURE

The experimental procedures used to capture the vibration of the light weight curved shell usinga laser Doppler vibrometer is described in this section, as well as the configuration used in thehemi-anechoic chamber to accurately capture the radiating sound field of the appliance.

2.1 Laser Doppler-vibrometry

To replicate the intended operating conditions of the appliance, the system was placed on carpetand air flow introduced by a slave vacuum pump to generate the equivalent suction force. Powerwas supplied to the beater bar motor to provide the in-use vibration loading. A photograph of theexperimental configuration is shown in Figure 2.

To determine an appropriate spatial sampling frequency, the expression for the transverse wavelengthon a flat plate [11] was used to calculate an approximate value for the bending wavelength at the

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Figure 2: The vibration measurement setup.

highest frequency of interest, fmax = 1000 Hz,

λb,max = 1fmax

4

√Bω2

m, (1)

where B is the bending stiffness of the material and m the mass per unit area of the casing.Using the material parameters of the shell, a wavelength (λb,max) of 70 mm was calculated. Aspatial sampling rate of 20 mm was deemed adequate to capture at least three measurements perwavelength of the highest frequency of interest. Following this, the surfaces of the enclosure werediscretised into 159 measurement points.

The vibration of the shell was captured using a Polytech 100 portable digital vibrometer [12].The rig was mounted onto an X-Y traverse system which was used to accurately position the LDVover the surface of the curved shell. Normal incidence of the laser with the surface of the enclosurewas achieved using an adjustable angled mount and measurements were captured for each of the159 positions. The error in positioning the vibrometer was less than ±2 mm. An accelerometer wasplaced on the upper surface of the shell to provide a phase reference. Data acquisition was performedusing National Instruments hardware and a Labview measurement framework.

2.2 Acoustic testing

The experimental configuration as shown in Figure 2 was transferred to a hemi-anechoic chamberto acquire the acoustic response of the system. The acoustic response of the system could notbe measured simultaneously with the laser vibrometry scan as the X-Y traverse system was notportable and could not be located within the hemi-anechoic facility. A 10 microphone setup ona 2 m hemisphere was used to determine the sound power level of the system in accordance withIEC 60704 [13].

Experimental determination of the radiation efficiency requires the measured sound power tobe a function solely of the forced vibration of the structure. Contamination from airborne noisesources, such as aero-acoustic noise or that produced by a motor would reduce the accuracy of thederived radiation efficiency and limit its efficacy as a predictive tool. It is therefore paramount thatall efforts are made to isolate the structure borne noise from the airborne sources during testing.To minimize unwanted noise, the slave vacuum was situated outside of the chamber and an inletsilencer used to reduce the airborne noise in the airflow.

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While the airflow through the cleaner head generates noise, at low frequencies the predominantsource of noise is due to the vibro-acoustic response of the enclosure. With the airflow on, acousticmeasurements were made with and without the beater bar running. Figure 3 presents the soundpower level in Constant Percentage Bandwidth (CPB) for these two cases. With the beater barengaged, the sound power level is >10 dB above that for the flow noise over much of the frequencyrange of interest. The airflow noise therefore contributes a negligible amount to the total soundpower of the system at lower frequencies, however above 600 Hz its contribution increases.

Figure 3: Comparison of CPB for the beater bar on and off.

3. LINEAR SUMMATION METHOD

As shown by Revel and Rossi [9], sound pressure level estimates of the sound radiated by a vibratingsource can be obtained through discretization of the vibrating surface and treating each ‘patch’or element as an independent radiator. The sound field at a given position may be calculated byperforming a linear superposition over each of these independent sources. Although similar to aboundary element approach, the transfer function from each patch to the field position does notsatisfy the true boundary conditions of the enclosure. Therefore diffraction and shadowing by partsof the structure are not taken into account. The consequence is that in general, unlike a boundaryelement approach, as the ‘patch’ size becomes infinitesimal the result will not converge to an exactsolution.

Using the naming system defined in Figure 4 the sound pressure level at a given position may beestimated by calculating the linear summation

SPLmic,n = jρock

2π

I∑i=1

SiVi,rms|ri,n|

ej(−k|ri,n|−φi), (2)

where Vi,rms is the RMS velocity magnitude at each discrete patch, k is the wavenumber in air,φi is the relative phase, I is the total number of elements and |ri,n| is the magnitude of the positionvector between the ith element and the field location

|ri,n| =√

((xmic,n − xi)2 + (ymic,n − yi)2 + (zmic,n − zi)2). (3)

To replicate the acoustic sound power level measurement, the geometry of the experimentalarrangement was simulated and the sound pressure level calculated at the equivalent locations ofeach of the 10 microphones in the hemisphere. To calculate the pressure at a single microphoneposition SPLmic,n the measured velocity and phase at each of the 159 LDV measurement positionswas used. The measured response using the LDV at each position was assumed to be representativeof the vibration response across the surface of each element and each element was given an equalarea weighting Si with

∑159i=1 Si = Stotal. No interpolation or smoothing was performed between

measurement positions.

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Figure 4: Discretisation of the curved shell.

4. RADIATION EFFICIENCY

The radiation efficiency of a structure provides an engineering solution to assess the radiated soundpower of a system with a low resolution set of vibration data. As used in this work, the radiationefficiency matches the experimental acoustic measurements with the average surface velocity over theradiating system, condensing contributions of the modal behavior and different radiating surfacesinto a frequency dependent transfer function. In the strictest sense, it is valid only when consideringsimilar sources under identical measurement and operating conditions. However, in circumstanceswhere changes in operating conditions are unlikely to dramatically effect the radiation characteristicsof the system the radiation efficiency provides a convenient framework for coarse prediction anddiagnostics.

The radiation efficiency is calculated from experimental data using

σrad = Wrad

ρocS〈v2〉, (4)

where Wrad is the radiated sound power of the structure, 〈v2〉 is the space averaged normalsurface velocity of the structure, ρo is the density of air, c the equilibrium speed of sound and S theradiating surface area of the structure [14].

The spatial average of the RMS velocity 〈v2〉 was calculated using all 159 measurement positionsover the total surface of the shell S = Stotal. This quantity, along with Wrad, the sound powerdetermined from the 10 microphone hemisphere was used to calculate the radiation efficiency forthe curved shell.

5. RESULTS

This section outlines the initial observations from the surface vibration measurements as well asa comparison with measured SWL for the linear summation method and the radiation efficiencyprocedures.

5.1 Shell vibration response

A MATLAB routine was used to post-process the vibration signals for each of the 159 measurementpositions. Figure 5 compares the velocity magnitude for the upper curved face of the enclosure at100, 200 and 1000 Hz. It can be seen that the response has high spatial dependence even at lowfrequencies. Figure 5a shows the vibration response at the lowest frequency of interest, 100 Hz. Thestrongest vibration can be seen at the edges of the upper surface corresponding to the mountingpoints of the beater bar. In Figure 5b, the uneven vibration loading is shown to introduce strong

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asymmetry into the modal response. At 1000 Hz, Figure 5c demonstrates that the complex modalresponse of the shell is captured.

(a) 100 Hz. (b) 200 Hz.

(c) 1000 Hz.

Figure 5: Spatial distribution of velocity magnitude across the top surface of the curvedstructure at different frequencies.

5.2 Linear summation method

The estimated sound power level of the linear summation approach was determined by first calculat-ing the sound pressure level at each of the 10 simulated microphone positions. These estimates wereaveraged and the sound power level determined in accordance with IEC 60704. The difference in dBcompared to the actual measured value is shown in Table 1. The comparison between the measuredand estimated sound power level in CPB is shown in Figure 6. The agreement of the total soundpower level calculated between 100 and 1000 Hz is very good, although it can be seen in Figure 6for frequencies above 600 Hz, the fit between the estimate and actual radiated sound power beginsto diverge.

5.3 Radiation efficiency

The radiation efficiency determined for the curved shell is shown in Figure 7, along with a quadraticcurve fit to this data. From baffled plate theory, the radiation efficiency tends to unity at frequenciesabove the coincidence frequency. This is when the bending wave speed matches the speed of soundin air. From the known material properties of the curved shell this was approximated to be above20 kHz, well away from the frequency region of interest. The quadratic fit is in good agreement withthe 20 dB/decade increase expected for the subsonic bending wave regime [14].

To assess the accuracy of the approximated radiation efficiency the sound power level was calcu-lated using the surface averaged RMS velocity for all 159 LDV measurement points and the radiationefficiency derived using the quadratic best fit. The radiated SWL predicted using the approximatedradiation efficiency with the full LDV data set gives good accuracy for the measurement at <0.5 dB,

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Figure 6: Comparison of sound power level pre-diction for the linear summation method with themeasured sound power level in CPB for the curvedshell.

Figure 7: The calculated radiation efficiency of thecurved shell.

see Table 1. This demonstrates that the approximated radiation efficiency shown by the purple lineon Figure 7 is a good description of the radiation characteristics.

The ISO 7849:2 standard recommends that for sound power level estimation at least 10 meas-urement positions are used for applications where the surface area is <1 m2. A subset of 10 evenlydistributed measurement points i.e every 16th element, was selected from the 159 LDV measure-ments. This was used to generate a ‘new’ surface averaged vibration magnitude from which a soundpower level calculation could be performed. The predicted sound power level in CPB, made usingthis measurement subset and the approximated radiation efficiency is shown in Figure 8 alongsidethe measured data. The difference between the measured and predicted total sound power levelintegrated over 100 to 1000 Hz is shown in Table 1, showing good agreement.

Table 1: Comparison between the measured SWL, and two prediction methods.

Method Predicted SWL (∆dB)Linear summation -1.1Radiation efficiency (σrad,approx), with 159 points +0.3Radiation efficiency (σrad,approx), with 10 points +1.3

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Figure 8: Comparison between the predicted result using the ISO 7849:2 procedure withthe quadratic fit radiation efficiency and the measured SWL. A subset of 10 evenly distibutedmeasurements from the 159 LDV data set was used to provide a surface averaged velocityprofile.

5.3.1 Sensitivity to measurement position

As shown in Section 5.1 the vibration profile of the enclosure is spatially dependent. The ISO stand-ard suggests that the number of vibration measurements should increase until the surface averagedvelocity profile 〈v2〉 converges. It is not always possible to evaluate 〈v2〉 during a measurementsession so it is of interest to evaluate the sensitivity of the approach to the measurement positionwhen using a reduced data set. To evaluate the sensitivity as a function of measurement positionthe sound power level was calculated using a subset of 10 measurements selected randomly fromthe 159 LDV data points. This process was first performed for the exact radiation efficiency calcu-lated with all 159 LDV data points and the 10 randomly selected measurements, see Figure 9 (redmarker). This was also undertaken with the 10 random measurement subset using the approxim-ated radiation efficiency (purple markers). Figure 9 presents the mean and spread of the 10 repeatsusing each procedure. The sound power level calculation can be seen to be more accurate whenusing the approximated radiation efficiency when a reduced measurement set is selected, howeverboth efficiency terms provide very reasonable predictions. Extending the sensitivity investigationfurther, the number of data points in each subset was reduced to 5 and 3 and the approximatedradiation efficiency was used in the calculation. Reducing the number of data points increases theaverage error and spread of results, although the predictions remain suitably accurate even with 3data points, for many engineering applications.

6. INTENSITY FIELD SCAN

A PU intensity probe was used to measure the intensity field around the shell up to a radius of 0.5 mfrom the structure. The resulting visualisation allows a qualitative analysis of the developing soundfield to be made. Contour plots of the intensity fields at 100, 200 and 1000 Hz are shown in Figure10. In all three cases strong asymmetry is observed close to the shell, however as the sound fielddevelops the intensity begins to homogenise.

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Figure 9: Sensitivity study results using randomly sampled elements. The impact of usingthe measured radiation efficiency and the approximated radiation efficiency is shown.

(a) 100 Hz. (b) 200 Hz.

(c) 1000 Hz.

Figure 10: Sound intensity scan of the developing sound field at different frequencies.

7. DISCUSSION

Both the linear summation method and the radiation efficiency approach were able to provide verygood estimates of the sound power despite a spatial vibration profile that varied strongly across thesurface of the shell.

The patch summation method provided a very good agreement in terms of total sound powerlevel and reasonable agreement in certain CPB center frequency bands. As the vibration data wasrecorded during a separate session to the acoustic measurement, it is possible that some of thedifferences might be as a result of variations in the operating conditions during the two experiments.Figure 6 shows that the agreement between the prediction and measurement diverges for frequenciesabove 600 Hz. Referring to Figure 3 the aero-acoustic noise due to flow through the structure wasseen to contribute increasingly to the SWL in this frequency region which may account for thisdiscrepancy.

The radiation efficiency appears to provide a robust and accurate method to predict the sound

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power level from a coarse set of measurements. The 10 measurement positions recommended by theISO standard appears to provide suitable resolution. If a greater trade off in accuracy is acceptable,reasonable agreement can be achieved with even lower sample sizes, however the sensitivity withrespect to the choice of measurement position increases. The radiation efficiency shown in Figure 7is highly oscillatory, this suggests that this quantity would be sensitive to changes in the system.One would expect that the implicitly smooth gradient of the quadratic fit, while less accurate inspecific cases, would offer a more robust quantity for general use.

The intensity field visualisation highlights that the vibration profile results in an asymmetricalsound intensity in the near field, however this does not persist strongly into the far field. Qual-itatively this helps to explain why the two methods provide similarly accurate predictions. For acompact enclosure with sub-wavelength dimensions the spatial dependence of the vibration profiledoes not translate into the far field. The simple compact source transfer function is therefore a gooddescription of the far field pressure.

8. CONCLUSIONS

Two methods for determining the sound power level from vibration measurements were comparedfor a compact curved shell with high levels of spatial variation in the vibration profile. Both thelinear superposition method and the ISO 7849:2 standard approach were found to provide accuratesound power level estimates. Approximating the radiation efficiency to provide a function withsmooth frequency variation was shown to be more robust to choice of measurement position thanthe exact expression. It is reasonable to assume that approximated radiation efficiency terms thatvary smoothly with frequency will be more generally applicable when system operating conditionsvary, however more work would be needed to validate this. The linear superposition method takesthe spatial variation of the vibration profile into account. The prediction made using this approachwas accurate, providing a closer match across the frequency range, however the data collection andpost processing requirements are more intensive than the radiation efficiency approach. An intensityprobe was used to measure and visualise the intensity field radiated by the vibrating curved shell. Asthe radiating structure is much smaller than the acoustic wavelengths of interest in air, the spatialdependence of the vibration profile does not persist into the far field.

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[8] DD ISO/TS 7849-2:2009. Acoustics- determination of airborne sound power levels emitted bymachinery using vibration measurement - Part 2: Engingeeing method including the determ-ination of the adequate radiation factor. ISO Standard, 2009.

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