ACOUSTIC EXCITATION OF MECHATRONIC SYSTEMS

Transcription

1 Twelfth Intenational Congess on Sound and Vibation ACOUSTIC EXCITATION OF MECHATRONIC SYSTEMS N.B. Roozen, B.T. Vehaa and R.M.G. Rijs Royal Philips Electonics, Philips Applied Technologies, P.O. Box 18, 56MD Eindhoven, the Nethelands, Abstact Within the specialty mechatonics a pluality of disciplines such as mechanics, electonics, softwae and contol ae combined to develop extemely accuate pecision machiney. Examples ae ulta-pecise measuing equipment with nanomete accuacy, stages fo lithogaphy applications and stages fo electon micoscopes. The accuacy of mechatonic systems is apidly inceasing. Key in the development of such highly pecise machiney is to contol the distubances affecting the accuacy of the machine. A systematic way to do so, is to define a dynamic eo budget which is divided amongst the diffeent distubances. Many diffeent distubances need to be consideed. To mention a few: floo vibations, vibations geneated intenally by the machine, acoustic excitation due to flow and/o cleanoom ai-conditioning systems, etc. The latte distubance, acoustic excitation, claims a significant pat to the eo budget, especially fo extemely accuate pecision machiney. In ode to estimate the contibution to the dynamic eo budget aleady in the design phase of the machine, it is necessay to pedict the esponse of the system to acoustic excitation. In the design phase of a machine only appoximate dimensions ae available, which calls fo appoximate estimates of the machines sensitivity to acoustic excitation. The pape discusses such an appoximate method, which consides igid body motion of the machine only, excited by plane acoustic waves. The method uses an analytical model. The analytical model is deived, and the theoy is validated by means of expeiments. 1

2 INTRODUCTION Within the specialty mechatonics a pluality of disciplines such as mechanics, electonics, softwae and contol ae combined to develop extemely accuate pecision machiney. Examples ae ulta-pecise measuing equipment with nanomete accuacy, stages fo lithogaphy applications and stages fo electon micoscopes. The accuacy of mechatonic systems is apidly inceasing. Key in the development of such highly pecise machiney is to contol the distubances affecting the accuacy of the machine. A systematic way to do so, is to define a dynamic eo budget which is divided amongst the diffeent distubances (ef [1]). Many diffeent distubances need to be consideed. To mention a few: vibations geneated intenally by the machine like setpoint-motion elated vibations, flow noise, pneumatic noise etc. Othe souces that usually claim a significant pat of the eo budget ae extenal distubances like floo vibations and envionmental acoustical noise. The site dependent behaviou of these distubance souces makes it had to quantify them pecisely. Commonly, depending on the application, numbe of machines and the numbe of applicable envionments, an envelope distubance specification is equied. The envelope distubance specification specifies the wost-case conditions of floo vibations and envionmental acoustical noise at which the machine still pefoms well. Duing the conceptual design stage of highly pecise machiney an essential step is to evaluate diffeent conceptual design studies with espect to extenal distubances. Despite the lack of details in the design it is possible to make calculations that deal with floo vibations and acoustics. Pedicting the influence of floo vibations is quite accuately possible by using simple 1D models consisting of just a few igid bodies. The main compliances in these models usually include the floo vibation isolation system and the fist esonance fequency. The powe of these calculations is that they povide insight and that they ae vey time efficient. The science of pedicting the influence of acoustical envionmental noise at the ealy machine design stage is less developed than fo floo vibations. The explanation fo that is twofold: Fistly, the elation between floo vibations and machine vibations is staightfowad, whee the elation between acoustics and machine vibations appeas moe complicated. In this pape is explained that also fo the acoustic excitation of machiney staightfowad elations exist. Secondly, floo vibations usually dominate in the 1-5 Hz fequency egion whee highly pecise machines behave like igid bodies. Acoustic excitation usually dominates in the low to medium fequency egion. Fo instance, the acoustic noise spectum of clean oom ai-conditioning systems usually dominates at fequencies aound the 15 Hz 1/3 d octave band (ef. []). Fo many mechatonic applications this implies that both the igid body motion and the fist intenal esonance of a machine ae excited. This pape focuses upon the igid body behavio of machines subjected to acoustic excitation. Refeence is made to the liteatue to deal with esonant behavio. In the next section, an analytical deivation of the sensitivity of a igid body to acoustic excitation is given. Next, the expeimental tests that validate the analytical model ae discussed.

3 ACOUSTIC SENSITIVITY OF NON-RESONANT RIGID BODIES Basically, two fequency egimes of pactical impotance can be consideed. The fist fequency egime is above the suspension esonance fequency and below the fist intenal esonance of the stuctue, whee the stuctual esponse is govened by inetia. This situation will be teated analytically in this section. A second fequency egime of pactical impotance is at stuctual esonance, whee damping contols the stuctual esponse. The deivation given hee applies to the fist fequency egime. It follows the wok of Fahy (ef. [3]), whee Fahy gives a theoy that applies to the esonant esponse of stuctues. The theoy deived in this pape applies to a non-esonant igid body motion, fo which inetia effects goven its esponse. This is the case fo fequencies above the suspension esonance fequency and below the fist intenal esonance of the stuctue. In many high-pecision mechatonic applications this fequency ange, which is typically between 1 and a few hunded Hetz, is most impotant with espect to acoustic excitation. In situations that the esonant stuctual esponse needs to be detemined (the second fequency egime), the eade is efeed to the wok of Fahy (ef. [3]). The ecipocal poblem; acoustic adiation of a vibating stuctue Fist, conside the ecipocal poblem, i.e. a vibating igid body that adiates acoustic p,θ,φ, noise. Following Fahy (ef. [3]) the mean squae acoustic pessue ( ) measued at distance and at uniquely defined angles ( θ, φ ) fom the vibating object, is as follows elated to the adiated acoustic sound powe P ad : ρ cp D( θ, (,, p D is the souce diectivity facto and ρ c is the chaacteistic acoustic whee ( θ, impedance. The souce diectivity facto D( θ, whee ( ) θ,φ ad θ = (1) is defined as I ( ) ( θ, φ ) D θ, φ = () I I is the sound intensity in the fa field, unde the angles ( φ ) θ, with the vibating igid body, and I is the sound intensity poduced at the same distance by a unifomly diectional souce of the same powe, whee P ad = I. Defining the acoustic adiation efficiencyσ as Pad σ = (3) ρ cs v n with v n the spatially aveaged squae nomal velocity of the igid body, Eq. (1) can be ewitten as 3

4 n ( θ, ρ c σs v D p (, θ, = (4) Assume that the igid body vibates due to an extenal foce F ( ) at point of the igid body. Assuming igid body motion, and using the nd law of Newton, F ( ) = jωvm, whee m is the mass of the igid body and using equation (4) and whee v is the velocity of the igid body in the diection of the exciting foce F ( ), the sound pessue adiated by the igid body can be witten as F ( ) σsd ( ) ( θ, φ ) vn p, θ, φ = ρ c. (5) ωm v v n The facto, which accounts fo the suface mean squaed nomal velocity, given v a velocity of the body as a whole, will be denoted by ψ : v n v ψ (6) ψ is a constant, depending upon the shape of the body and the diection of vibation. Using definition (6), Equation (5) can be ewitten as F ( ) σsd ( ) ( θ, p, θ, φ = ρc ψ. (7) ωm In Figue 1 a schematic dawing is given. The left of Figue 1 epesents the ecipocal poblem which is consideed now, and the ight of Figue 1 epesents the diect poblem of inteest, which is teated next. p' Q F' v Figue 1 On the left the ecipocal poblem (a foce F' exciting the stuctue mechanically, causing an acoustic pessue p') and on the ight the diect poblem (a point souce Q exciting the stuctue acoustically, causing the stuctue to vibate with a velocity v). 4

5 The diect poblem; acoustic sensitivity of stuctue to acoustic excitation Now, conside the diect poblem of inteest, i.e. a igid body that is exposed to acoustic sound pessue field. Following Veheij (ef. [4]), the following ecipocity elationship holds between the,θ,φ θ, φ fom the pessue p ( ) at a distance and at uniquely defined angles ( ) vibating stuctue, due to a foce F ( ) acting on the stuctue at position (ou peviously dicussed ecipocal poblem, see the left pat of Figue 1), and the velocity esponse of the stuctue v( ) at point of the stuctue due to souce Q (,θ, at a distance and at uniquely defined angles ( θ, φ ) fom the vibating stuctue (the diect poblem of inteest, see the ight pat of Figue 1): v( ) p(, θ, = (8) Q(, θ, F ( ) Using the ecipocity elation (8), the sensitivity of the stuctue to acoustic excitation can be detemined fom Eq. (7): v( ) ρc σsd( θ, = ψ (9) Q, θ, φ ωm ( ) Although this equation is valid fo any position of the point souce (,θ, Q elative to that of the stuctue, and fo any acoustic envionment, it is convenient to wite the point souce Q (,θ, in tems of the pessue at the location of the stuctue, but in its absence. Let s denote this pessue by p ( ), whee the apostophe denotes the fact that the pessue is meant in the absence of the igid body. Also assume that the point souce Q (,θ, is located in a homogenous, fee field, without flow, elatively fa away fom the igid body (i.e. >>L and >>λ, whee L is a typical dimension of the igid body), in which case the acoustic pessue at position can be witten as ρ ck ( ) Q( θ, p =, (1) Using this elation, Eq (9) can be witten as v( ) c = p ω m 4πσSD( θ, ψ (11) ( ) Note that by assuming that the point souce (,θ, Q is located fa away fom the stuctue, the stuctue will be excited by plane acoustic waves. Thus the elationship deived is valid fo plane acoustic waves only. This end esult diffes fom that of Fahy (ef. [3]) at one main point. As we ae consideing a igid body motion fo which the inetia of the igid body dominates, we have a tem ω m in the denominato of Eq. (11), esulting in 1 db/oct fo fequencies above the citical adiation fequency, while in Fahy s esult the denominato is educed to a tem ω, esulting in 6 db/oct fo fequencies above the citical adiation fequency. This is because Fahy assumes a modal esponse, fo which intetia effects cancel the stiffness effects and fo which the esponse of the 5

6 stuctue is detemined by damping of the stuctue only. This means that in the end expession of Fahy, the mass of the stuctue is not found, and the dependence on 1 fequency is one ode diffeent (i.e. ω instead ofω ). EXPERIMENTAL VALIDATION In this section the analytical model is validated by means of expeiments. In designing the expeiment, a body was chosen fo which analytical solutions of the souce diectivity facto D ( θ, is available; a igid sphee. The fa-field diectivity of an oscillating igid sphee is given by D ( θ, φ = ) = 3cos ( θ ) (1) Fo an oscillating igid sphee the facto ψ, which accounts fo the vaying nomal velocity given a velocity of the body as a whole, as defined by Eq. (6), equals ψ = 1/ 3 (13) The acoustic adiation efficiency σ of an oscillating igid sphee is given by 4 ( ω / ω ) fo ω < ω σ = (14) 1 fo ω ω wheeω is the citical angula fequency, which is given by c ω = (15) a and whee c is the speed of sound and whee a is the adius of the sphee. In the expeimental tests, a football with a adius of.1 m was used and a weight of.361 kg. Using Equations (11) though (15) then gives 3 c πa 4π ω < ω ( ) a = fo v = ω m cm (16) p ( ) c 4πa fo ω ω ω m This elationship shows some typical popeties of a igid body excited by plane acoustic waves: ω The citical adiation fequency f =, which is commonly used to indicate the π fequency above which a vibating stuctue adiates acoustic noise efficiently, plays an impotant ole fo acoustic excitation also. Below the citical adiation fequency, the sensitivity to acoustic excitation in tems of velocity ove pessue, is fequency independent. Above the citical adiation fequency, the sensitivity to acoustic excitation in tems of velocity ove pessue, falls off with -1 db/oct. Expeiments wee pefomed using a football with a adius of.1 m and a weight of.361 kg. The expeiments wee conducted in an anechoic oom. A loudspeake is used to excite the football. The distance between the loudspeake and the football was 6

7 lage enough to ceate plane waves. The football was hanging on the ceiling by means of a sting. The velocity of the football was measued by means of a lase vibo mete, which was located outside the anechoic oom to avoid acoustic excitation of the lase vibo mete equipment. See Figue. sting footbal p' lase vibo Figue Expeimental test set-up in an anechoic oom. The theoetical sensitivity of the football can be deived fom Equation (16), giving: v p ( ) ( ) 3 π.1 5 = 5 1 = 85dB ef = c 4πa ω m 1 m spa fo fo f f < f f = 8Hz = 8Hz (17) The measuement esults ae shown in left subfigue of Figue 3. This figue also shows the theoetical sensitivity of the football. Figue 3 Stuctual esponse of the football to acoustic excitation, expeiment & theoy. On the left: football, a=.1m, 361 gam, Right subfigue: football filled with PUR, a=.875 m, 58 gam. Below the citical adiation fequency of 8 Hz the coespondence between theoy and expeiment is sufficient. Above the citical adiation fequency the esults ae 'polluted' by an anti-esonance and a esonance of the acoustic sensitivity. This is caused by intenal acoustic esonances of the football. Unfotunately (but logically) 7

8 these esonances ae of the same ode as the citical adiation fequency f, thus making it difficult to identify the -1 db/oct fall-off. To solve this poblem, anothe expeiment was conducted with anothe football that was filled with polyuethane foam (PUR), so to avoid acoustic esonances inside the football. The esults ae shown in the ight subfigue of Figue 3. The esonant behavio of the football almost completely disappeaed. Though thee is some decease of the acoustic sensitivity of the football above the citical adiation fequency, it is difficult to daw fim conclusions egading the 1 db/octave decease as pedicted by the theoy. CONCLUSIONS An analytical model is deived to descibe the acoustic excitation of a igid body above its suspension esonance fequency. The model is based on the elation with the ecipocal case, i.e. the acoustic pessue esulting fom a vaying foce exciting the igid body. The model is applied to the acoustic excitation of a sphee. The esulting expession is expeimentally veified in an anechoic chambe, using a speake and a lase vibo measue to detemine the sphee's vibation amplitude. While the model pedicts a 1 db/octave decease of the acoustic excitation sensitivity above the citical fequency of the ball, this dop-off is not well confimed by the expeiments. Unfotunately, the esult is 'polluted' by an anti-esonance and a esonance of the acoustic sensitivity just above the citical adiation fequency of the igid sphee due to intenal acoustic esonances. Additional expeiments by filling the sphee with foam suppessed the esonant behavio of the sphee easonably, and the measuements seems to give a decease of acoustic sensitivity with fequency, but vey fim conclusions can not be dawn fom this. Below the citical fequency howeve, the model gives a athe accuate pediction of the acoustic sensitivity of the sphee. AKNOWLEDGEMENT Ilan Vink, student at Inholland, is acknowledged fo pefoming the measuements. REFERENCES [1] L. Jabben, W. Monkhost, R. Tousain, J. van Eijk, Dynamic eo budgetting a Design tool fo mechatonic systems, nd benelux meeting on systems and contol, Lommel, Belgium (3). [] C.G. Godon, M.Q.Wu, Noise and Vibation Chaacteistics of Cleanoom Fan-Filte Units, Poc. of the Annual Meeting of the Institute of Envionmental Sciences and Technology, Phoenix, AZ (1998). [3] F.J. Fahy, Sound and stuctual vibation : adiation, tansmission and esponse, (Academic Pess, 1994) [4] J.W. Veheij, Invese and ecipocity methods fo machiney noise souce chaacteization and sound path quantification, Int. Jounal of Acoustics and Vibation, Vol., No. 3, pp (1997). 8