A Model for Three-Dimensional Simulation of Acoustic Emissions from Rotating Machine Vibration

A Model for Three-Dimensional Simulation of Acoustic Emissions fromRotating Machine Vibration.

Magalhaes,R. S.,a) Almeida,L. A. L., Embirucu,M., and Fontes,C. H. O.Programa de Pos-Graduacao em Engenharia Industrial - PEI, Escola Politecnica, Universidade Federal da Bahia -UFBA, Rua Aristides Novis, 2 - Federacao, Salvador-BA, Brasil, CEP 40.210-630.

Santos,J.M.C.Faculdade de Engenharia Mecanica, Universidade Estadual de Campinas - UNICAMP, Campinas - SP, Brasil,CEP 13.083-970.

(Dated: June 16, 2009)

Industrial noise generated by equipment such as compressors, vacuum pumps, and many others canbe successfully mitigated with the combined use of passive and active noise control strategies. Ina noisy area where equipment access for the human operator is necessary, a practical solution fornoise attenuation may include both the use of baes and Active Noise Control (ANC ). When theoperator is required to stay in movement in a delimited spatial area, conventional ANC is usuallynot able to adequately cancel the noise over the whole area. New control strategies need to bedevised to achieve acceptable spatial coverage. In the pursuit of this goal, a three-dimensionalvibration-acoustic model is proposed in this paper. The signal of an accelerometer attached to thebulk of a centrifugal pump installed in an empty room was used as the input of this model. Thesignal of a microphone that changes its position in a spatial grid inside this room is the output. Theprocedure proposed comprises the vibration-acoustic modeling of the machine-room transfer functionusing ARX (Auto-Regressive with eXogenous input) models. In the first stage, the ARX modelswere used to describe a SISO (Single-Input Single-Output) system where the input is the machinevibration and the output is the noise level measured at a certain point. A least square method wasused to estimate the ARXmodel parameters. In the second stage, spatial interpolation was usedto estimate the model parameters for any given point of the room with Cartesian coordinates X,Y and Z. These new parameters, calculated for each point, may be used to map noise level in thesystem output over the whole room. Results show good agreement between experimental data andmodel predictions, indicating the potential of using the developed model for the design of ANC.

PACS numbers: 43.60.Gk

I. INTRODUCTION

In an industrial environment, the noise emitted byrotating equipment housed in rooms is very noxious andespecially harmful to operational personnel. Appropriateattenuation for this noise may be obtained by associatinga simulation model for the acoustic emission caused bymachine vibrations to an Active Noise Control (ANC )system. ANC requires the introduction, in an acousticarrangement, of a controlled secondary acoustic sourcedriven in such way that the acoustic field generated bythis source interferes destructively over the field causedby the original primary acoustic source1.

The waveform found in the acoustic field produced by arotating machine is almost periodic and the fundamentalfrequency and noise level can be estimated by anappropriate model. Therefore, previous knowledge ofthe acoustic field behavior of the primary source in avibrate-acoustic system is very useful for effective noise

a)The author is also with SENAI (Servico Nacional deAprendizagem Industrial) - CIMATEC (Centro Integrado deManufatura e Tecnologia), Av. Orlando Gomes,1845- Piata, CEP:41650-010, Salvador-BA. Author to whom correspondence shouldbe addressed. Electronic mail: [email protected]

level control. Through the adjustment of the amplitudeand phase of the output signal predicted by a model,the secondary source must be driven so that the fieldoriginated by the primary acoustic source is cancelledout. Information about the pressure and the acousticpower of the vibrate-acoustic system is therefore veryuseful in the early stage of effective noise control, eitherby passive or active means.The modeling of the phenomenon involved is not

simple and different numerical methods of varyingcomplexity have been developed. Many theoretical andexperimental studies have been performed to identify theappropriate model for simulation of acoustic emissionsin vibrate-acoustic systems in three dimensions. Somemethods require boundaries or domain division in a largenumber of elements or sections where very fine meshes areneeded to solve excitations at high frequencies, such asthe Infinite Element Method - IEM 2 and the BoundaryElement Method - BEM 35. These methods have notbeen widely used to compute the propagation of sounddue to the high computation effort involved, hamperingreal-time applications and making their use unfeasiblefor ANC. Excessive calculation efforts and computermemory requirements are the main limitations imposedon the shape of the room to be modeled.Methods based on geometric acoustics are also widely

used in room acoustic prediction. Among these methods,

A Model for Three-Dimensional Simulation 1

the Image Source Method - ISM 68 requires a largeamount of virtual sources which can limit its application.The Ray Tracing Technique - RTT 9 is valid in highfrequency ranges and includes a certain degree ofuncertainty, since it is not assured that all the necessaryrays will be included in the output signal response.The Method of Fundamental Solutions (MFS ) ) is

applicable when a fundamental solution of the differentialequation that describes the sound propagation in theacoustic arrangement analyzed is available10.The Room Transfer Function (RTF ) method, which

describes the sound transmission characteristics betweena source primary and a receiver in a room11, plays avery important role in acoustic signal processing andsound field control, especially when an ANC uses inversefilters based on RTF s to reduce noise12. A multi-inputmulti-output sound control system has recently beeninvestigated using this method13. In such a system,multiple RTFs between the sources and receivers wereused. An efficient modeling method called Common-Acoustical-Pole-Zero (CAPZ) was proposed for multipleRTF s11. The common acoustical poles correspondto the resonance frequencies (eigen-frequencies) of theroom. The zeros correspond to the time delay andanti-resonances. This model requires fewer variableparameters (zeros) than the conventional all-zero andpole-zero models to express the RTF s because thecommon acoustical poles are common to all RTF s in theroom. However, even when the CAPZ model is used, theRTF has to be measured for every source-receiver due tothe dependence on the zeros from the source and receiverpositions.This paper proposes the Machine-Room Transfer

Function (MRTF ), a method based on the theory ofthe impulse response that includes the machine vibration(primary source) in the dynamic modeling of RTF s. TheMRTFmethod models the vibrate-acoustic transmissionbetween a primary source and a receiver in a room. Theprediction of the acoustic field inside the enclosed spaceis the main objective. As well as the RTF in the CAPZmodel, the MRTF has to be measured for every source-receiver setting. Given the difficulty and feasibility of thistask, this work also proposes an interpolation procedureto estimate an unknown MRTF at an arbitrary positionbetween known MRTF at an arbitrary position betweenknown MRTF s. The interpolations are applied over themodel parameters of the known MRTF s and can provideintermediary models (or intermediary parameters) foreach one of the arbitrary source-receiver positions.

II. MODELLING AND METHODOLOGY

A. Model structure and parameter estimation procedures

Consider a vibrate-acoustic system that comprises acentrifugal pump housed in a room (Figure 1). Thecentrifugal pump is the primary noise source in thissystem.Placing the accelerometer in the pump and a

microphone at any given point in the room (identified by

FIG. 1. Acoustic field mapping generated by a rotatingmachine operating in a closed room identified by coordinatesX, Y , Z: X = 1, 2, , 7(0.44m), Y = 1, 2, , 10(0.44m)and Z = 1, 2, , 5(0.44m). Microphone displacement(passive sensor): experimental setup (left), mesh of the 350collected data (7 10 5 positions assumed by the passivesensor)(right).

FIG. 2. Idealized system response to an idealized step input:phenomenon depiction (top); gauge pressure (bottom-left);sound or dynamic pressure (bottom-right).

its coordinates X, Y , Z), the Machine-Room TransferFunction of the vibrate-acoustic signal-transmissionchannel between the accelerometer signal u(t) andthe microphone signal y(t), denoted by H(s),can beidentified. This model represents the vibrate-acousticsystem between the pump and any given point in theroom.A key feature for identifying parsimonious and reliable

empirical models is to establish representative modelstructures. Theoretically driven model structures arethe best way to establish such structures. If we appliedan idealized unitary step in the accelerometer, one ofthe simplest idealized responses the system could bringout would present a profile that resembles the one de-picted in Figure 2 (bottom-right). Since the mass ofthe pump shell is constant, a step signal measured bythe accelerometer is equivalent to a step function force


applied on the pump shell itself. Let us consider airtransportation across a square cross section duct, inwhich one of the faces (face A) is adjacent to the pumpshell and the sound pressure is measured on the otherface (face B). A step change in the force applied to theshell will cause a motion of face A according to Figure 2(top). Assume that the sides of the box are completelyfrictionless, i.e. any viscous drag between gas particlesinside the box and those outside are negligible. Thus theonly forces acting on the enclosed gas are due to the pres-sures at the faces of the box. The difference between theforces acting on the two sides (A and B) of our tiny boxof gas is equal to the rate at which the force changes withdistance multiplied by the incremental length of the box(X). Considering the equilibrium conditions, with nullgauge pressure in the inner box, a step change in the forceapplied on the face A produces a behavior in the gaugepressure inside the box according to Figure 2 (bottom-left). The sound pressure, i.e. the dynamic pressure, willbehave as described by Figure 2 (bottom-right) with a

static gain equal to zero. If the duct is not hermeticallysealed, not only would the sound pressure but also thegauge pressure (static pressure) have a null static gain.This idealized behavior is reasonable and can be

parsimoniously represented by an under-damped secondorder system. This is the lowest order or thesimplest model structure capable to describe thedynamic behavior of the system presented in Figure1 and in agreement with what has been proposed byFang14. Therefore, the starting transfer function, H(s),considered to describe the experimental data is thefollowing second order one:

H (s) = k 2 s

s2 + 2 s+ 2 etds (1)

where k, , , and td are model gain, characteristicfrequency, damping factor and time delay, respectively.A discrete-time model equivalent to Equation (1) is:

y (n) = a1 y (n 1) a2 y (n 2) + b0 [u (n d) u (n d 1)] , d 0 (2)

where u(n) is the system input signal sample at instantn, y(n) is the system output signal sample at instant n, dis the delay (dead time) of the system output with regardto input u, q is the forward shift operator and nb na .Or, using Equation (3):

A (q) y (n) = qd B (q) u (n) + e (n) (3)

A (q) = 1 + a1 q1 + a2 q2B (q) = b0

(1 q1) (4)

where e(n) is white noise. In this work, the ARXmodel is applied to predict the output in a simulationfashion (or long step ahead prediction), and a least squarerecursive procedure was used to estimate the parameters.Although the structure defined by Equation (4) is enoughto cope with the identification problem (a reasonableidentification was obtained with such a structure), thereal phenomenon behavior is quite more complex and aslightly increased order model was assumed in order toimprove model predictions. Moreover, Equation (1) is aminimum-order model capable of describing the behaviordepicted in Figure 2, but not the only one. Therefore,the use of Equation (5) is fully justified.

A (q) = 1 + a1 q1 + a2 q2B (q) = (b0 + b1 q1 + b2 q2)

(1 q1) (5)

As well as model structure, time delays werederived on a theoretical basis. The time delay

of each estimated model considers sound velocity(340m/s) and longitudinal distance (distance along Yaxis, since contributions of X and Z coordinates arequite negligible) between primary source and the gridmeasurement point (Figure 1) where the model is beingestimated, that is:

d =tdTs

=dl/vsTs

(6)

where dl is the longitudinal distance, vs is sound velocityand Ts is sample time. The nearest X Zplaneis 1.32m distant from primary source, and the soundemitted by the primary source reaches this point witha 3.9ms delay. Similarly, the furthest X Zplane is5.30m distant from primary source, with a 15.6ms delay.Time delays of intermediary points are determined inan analogous fashion. Considering time delays providesmuch better results than when they are neglected, as willbe shown later. Moreover, determining their values usinga theoretical approach furnishes much more consistentmodels than using parameter estimation procedures.The remaining model parameters were determined

using a standard least square algorithm, resulting in1750 parameters defining 350 MRTF s which describe thedynamics and spatial behavior of the acoustic pressurein the room. As will be discussed later, in order toallow model reduction and to enable the location of theMRTF at arbitrary positions, an interpolation procedurewas employed. When choosing an interpolation method,it is worth remembering that some techniques require


TABLE I. Model parameter spatial distribution. Plane Z = 1(0.44m).

Plane Z = 0.44m

Identified Reduced InterpolatedParameter Grid Grid Grid

b0

more memory or longer computation time than others.However, it may be necessary to trade this off to achievethe desired smoothness in the result. Nearest neighborinterpolation is the fastest method. However, it providesthe worst results in terms of smoothness, althoughthis did not pose a problem here. Nearest neighborinterpolation (also known as proximal interpolation or,in some contexts, point sampling interpolation) is asimple method of multivariate interpolation in 1 or moredimensions. The nearest neighbor algorithm simplyselects the value of the nearest point and does notconsider the values of other neighboring points at all,yielding a piecewise-constant or zero-order interpolation.In a squared-grid, an interior point (0) will have eightpossible nearest neighbors (a point located at a face willhave four possible nearest neighbor and a point located atan edge will have two). Therefore, the following equationapplies:

p0 = pmin dist (7)

where pmin dist is the parameter value of the point withthe minimum distance from point 0 among the eightvertex that surround point 0, and the distance betweenpoint 0 and vertex i (d0,i) is given by:

d0,i =(X0 Xi)2 + (Y0 Yi)2 + (Z0 Zi)2 (8)

The interpolation procedure follows a given directionfrom the origin toward the last point. If the point to beinterpolated falls at an equal-spaced position regardingtwo or more vertex the first one is chosen, consideringthe interpolation path.

B. Methodology for data gathering

The environment in which the experiment was carriedout is a room 7.5 3.5 meters and 3.2 meters high,

composing the acoustic mesh presented in Figure 1.The room houses a centrifugal pump powered by anelectrical asynchronous motor, two ICP sensors, anaccelerometer which receives the dynamic generated bythe primary source and a microphone that receives thesound pressure at each point in the mesh previouslydefined. A piezoelectric accelerometer of 100mV/g wasadopted.During each measurement the passive sensor was

positioned with its axis parallel to the wall (length ofthe room) and to the floors plane in front of the primarysource. The data was collected by a CMXA50 Micrologcollector (SKF ) which relies upon a compact collectingdata device. The signal treatment is composed of anICP integrated font linked with a pass-band filter (101000Hz), adjusted to a sample frequency of 2560Hz witha collect span time of 1.6 seconds. A collection of 4096points per channel in each mesh of sampling was carriedout for each variable data. The nominal rotation of thecentrifugal pump (primary source) is 29Hz. Becausethe highest level of power is below the 200Hz band, thecollected signal underwent a tenth power reduction priorto the identification of each MRTF.

III. RESULTS

By applying the least square algorithm using thedata mesh, 350 MRTF s were identified describing thedynamics and spatial behavior of the acoustic pressurein the room through its relationship with the vibrationsignal from the pump. Each machine-room transferfunction (MRTF ) comprised an ARX model accordingto Equations (3) and (5).Considering 350 models and 5 parameters (a1, a2, b0, b1

and b2) for each one, the acoustic pressure mapping inthe entire room uses a total of 1750 1750 parameters.This number of parameters is too high for real-timeapplications and a reduction in the data mesh was made


FIG. 3. Worst (left) and best (right) adjustments for time response of identified and interpolated models.

by interpolating model parameters which also enabledthe location of the MRTFat arbitrary positions amongknown (measured) MRTF s.In the interpolation procedure, the measurements at

the following coordinates were considered: X = 1, 4, 7(0.44m), Y = 1, 5, 10 (0.44m) and Z = 1, 3, 5 (0.44m).Therefore, the reduced data mesh comprises only 27MRTF s and the acoustic pressure mapping in thiscase comprises a total of 135 parameters (a reductionof 93% in the measured mesh), which represents anotable reduction in computational cost facilitating itsimplementation in real-time control systems.The spatial distribution of the estimated parameters

can be represented through surfaces, which are presentedin Table I. This table presents the spatial behavior ofmodel parameter (b0) in a specific Z plane (Z = 1).Therefore, each surface shows parameter variations inboth X and Y directions. The parameter values at aspecific point are related to the physical-acoustic featuresof this point, such as the distance from the primaryacoustic source or the wave sound reflection measuredfrom the point. The first column of Table I presentsan example of the surface obtained using the parametervalues of the identified models using all input-outputdata (350 in the whole space considered and 70 for eachZ plane) according to Figure 1. The second columnpresents the surface obtained through the grid reductionprocedure that considers only 9 input-output pairs foreach Z plane. Finally, the third column presents theresults obtained by the spatial interpolation procedure,applied on the reduced grid (second column), providing350 interpolated models.Considering also the similar spatial distribution of

other parameters, likewise the parameter b0 shown inTable I, two main conclusions can be drawn regardingthe spatial distribution of model parameters. First,in all cases the interpolated models (third column)present good agreement with respect to the original

0 20 40 60 80 100 120

108

104

100

freq [Hz]

PSD

Measured output Estimated output identified models with delaysEstimated output interpolated models with delaysEstimated output identified models without delays

FIG. 4. Average PSD for the output.

identified models (first column), attesting the potentialand efficiency of the interpolation procedure proposedin this work. Second, a supposed symmetric behaviorexpected with respect to the central point of X axis wasnot found. This is associated to the physical features ofthe room that does not assure uniform conditions in thewhole space and mainly due to the pump displacementas its driving shaft is not aligned with Y direction, butwith X direction.The resulting models were used to predict the dynamic

and spatial behavior of the microphone output signal.Figure 3 shows the worst and best mean squareerror models for the measurements at the followingcoordinates: X = 2 (0.44m), Y = 7 (0.44m), Z = 5(0.44m) and X = 2 (0.44m), Y = 1 (0.44m), Z = 4(0.44m). It can be seen that even the worst model


FIG. 5. Dynamic and three-dimensional prediction of system behavior in response to input from a sine wave (the dark ellipsoidrepresents pump location): output signal as a function of X and t (left), Y and t (center) and Z and t (right).

results provide a suitable description of the experimentaldata, capturing the main trends of system behavior, whenusing either identified or interpolated models.Figure 4 presents the average PSD (Power Spectral

Density) of the 350 mesh points. The interpolatedmodel provides a good description of the systemdominant dynamic (PSD peaks, where most of the energysignal is concentrated) without degradation of outputprediction (both amplitude and frequency features ofthe interpolated model fit those of the identified modelswell). Further reductions in the interpolated mesh ledto significant degradation in the output prediction of theinterpolated model.As commented earlier, when considering sound

propagation time delays in model identification betterresults are achieved, as shown by Figure 4. When usingtime delays the summed square error is 2491, however,this rises to 2586 (4% greater) when identifications arecarried out without time delays.As stated earlier, one of the benefits of the

interpolating procedure is the ability to locate theMRTFat arbitrary positions. Figure 5 shows this to predict thetransient behavior of the system over the whole spatialdomain, in response to an arbitrary input (in this specificcase a 2 amplitude and 5Hz frequency sine wave). Itmust be pointed out that this modeling approach allowspredicting system behavior with any desired detail, withuniform or non-uniform grids. Figure 5 exemplifies a 700uniform grid points simulation.Figure 5-left shows an expected symmetry of the

picture regarding the middle X. In Figure 5-centercoordinates X and Z are kept constant in the graphand Y varies along room length, and so this coordinateis responsible for determining the distance between thepump and a given point. One can observe an expectedphase variation of wave y(t) along Y axis. This isphysically meaningful since the response delay of outputy(t) is directly proportional to the distance between thepump and the analyzed X Y Z point. Figure 5-right shows a central position regarding room width(X coordinate) at the nearest position regarding pumplocation (Y coordinate), and one can see that the greatestwave amplitude falls near the floor.

IV. CONCLUSIONS

This paper presents the development of a Machine-Room Transfer Function (MRTF ) to describe thevibrate-acoustic transmission between a primary sourceand a receiver in a room. Identified modelsperform satisfactorily in describing system behavior.Furthermore, in order to provide model reduction andto describe the whole spatial behavior, an interpolationprocedure was applied to the identified models. Thisprocedure resulted in a significant model reductionof up to 93%, maintaining a good description ofsystem dominant dynamics without degradation ofoutput prediction and allowing for real-time modelimplementation in control systems. Model consistencyand usefulness for analysis purposes were demonstratedthrough simulations that describe the dynamic and full-spatial behavior of sound pressure. The inclusion of timedelays derived from a theoretical approach have improvedmodel performance and also its consistency.

NOMENCLATURE

a, b, d discrete model parametersA,B,D discrete model polynomialsd time delay (time instants)d distancee white noiseH machine-room transfer functionk model gainn time instantna, nb, nd model ordersp parameterq forward shift operators Laplace operatort, T timeu system input signalv model prediction errorv velocityX,Y, Z Cartesian coordinatesy system output signal


GREEK LETTERS

difference operator model characteristic frequency model damping factor

SUBSCRIPTS

0 interpolatedd time delay (time)F filteredi vertexl longitudinalmin dist minimum distancep estimateds sample times sound

ABBREVIATIONS

ANC Active Noise ControlARX Auto-Regressive with eXogenous inputBEM Boundary Element MethodCAPZ Common-Acoustical-Pole-ZeroIEM Infinite Element MethodISM Image Source MethodMFS Method of Fund amental SolutionsMRTF Machine-Room Transfer FunctionPSD Power Spectral DensityRTF Room Transfer FunctionRTT Ray Tracing TechniqueSISO Single-Input Single-Output

Acknowledgments

The authors acknowledge SENAI (Servico Nacional deAprendizagem Industrial), CAPES and CNPq (Brazilianfederal research agencies) for their financial support.

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A Model for Three-Dimensional Simulation of Acoustic Emissions from Rotating Machine Vibration