CHARACTERISTICS OF A DYNAMIC PRESSURE GENERATOR BASED ON LOUDSPEAKERS. Jože Kutin *, Ivan Bajsić

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1 Sensors an Actuators A: Physical 168 (211) oi: 1.116/j.sna Elsevier B.V. CHARACTERISTICS OF A DYNAMIC PRESSURE GENERATOR BASED ON LOUDSPEAKERS Jože Kutin *, Ivan Bajsić Laboratory of Measurements in Process Engineering (LMPS), Faculty of Mechanical Engineering, University of Ljubljana, Aškerčeva 6, SI-1 Ljubljana, Slovenia * Corresponing author: T: , F: , E: joze.kutin@fs.uni-lj.si ABSTRACT The ynamic pressure generator uner iscussion consists of two, face-to-face-orientate electroynamic louspeakers an the air chamber between. The aim of this paper is to investigate the static an ynamic characteristics of this pressure generator. Physical moelling an an experimental analysis were employe to emonstrate its capabilities an limitations. The generator's static sensitivity, which was efine by the ratio between the generate pressure an the excitation electric current, mainly epens on the ratio between the force factor an the effective area of the louspeakers. The presente system has relatively goo precision an stability, an a small sensitivity to changes in the internal volume. Its ynamic characteristics are efine by the properties of the louspeaker iaphragm, the air chamber between the louspeakers an the connection of the evice uner test to the pressure generator. Keywors: pressure generator; pressure ynamics; louspeaker 1. INTRODUCTION Pressure is an important process variable in a wie variety of inustrial an scientific applications. Whenever measurements an/or control of the changing pressure conitions are require, the installe sensors an other equipment shoul have suitable ynamic characteristics. As a result, the research an evelopment of ynamic pressure generators are closely linke to the growing nees for ynamic testing an calibration of such equipment. It is possible to choose from a variety of operating principles an configurations when it comes to ynamic pressure generators [1,2]. One of these operating principles relates to pressure generators base on louspeakers, which are generally use for low-pressure amplitues an acoustic frequencies. The accessible literature escribes some of their potential areas of application: the ynamic calibration of low-range pressure sensors [,4], the testing of the ynamic response of pneumatic transmission line systems [5-7], an the testing of hyrostatic pressure switches [8]. This paper eals with the louspeaker-base pressure generator shown in Fig. 1. A similar configuration was introuce at the conference in 2 [9], since when it has been successfully employe as a ynamic pressure generator in a variety of applications [6,8]. This pressure generator consists of two, face-to-face-orientate electroynamic louspeakers. On the circumference of the chamber between, there are evenly istribute points for connecting pressure sensors or other equipment uner test. The employe louspeakers were chosen on the basis of the large ratio between the force factor an the effective area, which etermines the attainable magnitue of the generate pressure. 1

2 Sensorss an Actuators A: Physical 168 (211) oi: 1.116/j.sna Elsevier B.V. Fig. 1. Pressure generator with two louspeakers. The aim of this paper is to t investigate the static an ynamic characteristicss of the louspeaker-base pressure generator (see Section 2 an, respectively). Physical moelling an an experimental analysis were employe to emonstrate its i capabilities an limitations. The physical moel of the pressure generator, presente in Sections 2.1 an.1, is base on the linear lumpe moels off mechanical an acoustical elements [1,11]. The employe acoustical moel assumes that the wavelength of the generate pressure oscillations is large compare to the longest linearr imensionn of the air chamber, which w assures that the generate pressuree insie is essentially uniform. Similar assumptions are often use in basic moelling of a louspeaker mounte in a (meium size) close boxx [12,1]. In Section 2.1, the acoustical loa of the louspeaker iaphragm ue to the gas stiffness s is briefly reerive to properly consier a simultaneous motion of both louspeakers. The experimental work, presente in Sections 2.2 an.2, was esigne to iscuss the main characteristics of the pressuree generator,, such as the static sensitivity, the precision an a stability, an the frequency range. A particularr attention is focuse on the influence of changes in thee pressure-generator internal volume, which can occur in some applications. The finings off this paperr will be useful for the proper application an the further evelopment off such evices. 2. STATIC CHARACTERISTICS OF THE PRESSURE GENERATOR 2.1 Static physical moel Both the louspeakers of the pressure generator are moelle ass being ientical. The effective area S of their iaphragms is i assume to be constant within the t range of the isplacements x x max, x max. The x isplacement of both iaphragms with respect to one another results r in a change of the internal volume V V 2Sx, where V is thee initial internal volume. In the close system, such a change in the volume of a gas leas to t an increase in the absolute pressure, from an initial value P to P P p. The gas volume iss consiere as an aiabatically close system in which PVV is constant, with being the aiabatic inex [11]. For relatively small changes the following relation between the pressure change an the iaphragm isplacement can be erive: 2

3 Sensors an Actuators A: Physical 168 (211) oi: 1.116/j.sna Elsevier B.V. p SP 2 x. (1) V (If the gas volume were to change relatively slowly over time with respect to heat transfer to the surrounings, it might be more reasonable to consier the system as isothermal, in which PV is constant; substitute the erive expressions with 1.) When the iaphragm is move, ue to the pressure change the internal gas acts on the iaphragm as an ae stiffness force, which can be expresse as follows using Eq. (1): ps k x, k 2 S P 2, (2) V where k is the gas stiffness. The movement of the louspeaker iaphragm results from the magnetic force BLi acting on the voice coil, where L is the effective length of the voice coil wire in the magnetic fiel B an i is the excitation voice coil current [1]. Within the framework of the static moel there shoul be a balance between the excitation force an the combine influence of the gas stiffness force (2) an the suspension stiffness force k susp x. This results in: BLi x. () k k susp Substituting Eq. () into Eq. (2) leas to the linear static characteristic of the pressure generator. This can be expresse in terms of the static sensitivity K PG, which is efine by the ratio between the generate pressure change an the excitation electric current: p BL 1 KPG. (4) i S 1 k / k susp The generator s static sensitivity epens linearly on the ratio between the force factor BL an the effective area S, but the last fraction represents some ecreasing effect of the ratio between the suspension an the gas-relate stiffness. 2.2 Static measurements Fig. 2 presents a scheme of the measurement system where the experiments on the pressure generator were carrie out. In the case of static measurements, the louspeakers (Beyma, 5"MP-6/N) were electrically excite from a voltage amplifier connecte to the analog output of a DAQ boar (National Instruments, DAQPa-62E, max. input range ± 1 V, resolution 12 bit, max. sampling frequency 1 khz). The excitation current was etermine by measuring the voltage rop across a known resistor. The generate pressure was measure by a single variable-reluctance pressure transmitter (Valiyne, P855, measuring range ± 14 Pa, output voltage ± 5 V, accuracy.15 % of upper range limit, frequency range to 25 Hz ( B)). The signal acquisition an processing were realize in the LabVIEW programming environment. In some experiments an aitional volume V ADD of.5 m or 1 m was connecte to the pressure generator with the volume volume was increase to V V VADD. V.26 m, so the total internal

4 Sensorss an Actuators A: Physical 168 (211) oi: 1.116/j.sna Elsevier B.V. Fig. 2. Block iagram of the measurement system. The generator s static characteristic wass measure at nine points, equally ivie over the chosen testing range, incluing both increasing an ecreasing pressure.. Fig. shows one examplee of the time variations of the excitation current an thee generate pressure. For each testing point, the excitation voltage was set to some constant value v for a stabilization time of about 1 s. The variation of the excitation current (an proportionallyy of the generate pressure) uring the stabilization time, which is more evientt at higher excitation voltages, results from the changing temperature of the louspeaker coils. An increasing coil temperature leas to an increasing coil resistance an, consequently, to a ecreasing current at a constant voltage supply. 1 5 Pressure Current Pressure p [Pa] Current i [A] Time [s] Fig.. Time variation of the excitation current an the generate pressure for measurements of the generator static characteristic. The resulting static characteristic that relates to the last measure values at each testing point (after the stabilization time) iss presente in Fig. 4. The measurements are approximate by a linear regression line throughh the origin, the slope of which represents r the static sensitivity 4

5 Sensors an Actuators A: Physical 168 (211) oi: 1.116/j.sna Elsevier B.V. K PG. Such estimate sensitivities are collecte in Table 1 for three ifferent values of the ae volume V ADD an three repetitions in each configuration. The pressure generator s sensitivity is about 8 Pa/A. As expecte from the physical moel in Eqs. (2) an (4), the sensitivity shows some ecrease with the increasing internal volume. An approximation of the results in Table 1 with the physical moel shows that k susp in the pressure generator uner iscussion is relatively small in terms of k : ksusp / k,1 without the ae volume an ksusp / k,4 for the V ADD of 1 m. Pressure p [Pa] 1 5 Measurement Linear fit Current i [A] Fig. 4. Static characteristic of the pressure generator. Table 1. Measurement results of the pressure generator sensitivity for three values of the ae volume Number of Sensitivity of the pressure generator K PG [Pa/A] measurement V V V V.5 m V V 1 m Maximal eviations of any measure static characteristic of the pressure generator from its best straight line o not excee 1 Pa, or about 1 % of the generate-pressure upper range limit. These estimate eviations combine the effects of nonlinearity an hysteresis an the remaining temperature ynamic effects after the stabilization time of 1 s. Other possible contributions may be also relate to non-ieal performances of the employe measurement instruments. Three repetitions of measurements in each configuration of the system can be use for estimating the repeatability (the short-term stability) of the pressure generator; a maximal relative eviation of the repeate estimations of the static sensitivity in Table 1 is about.15 %. The long-term stability of the static characteristic may be expecte to epen on the uration an the magnitues of the generator's loaing an its effect on the properties of the rubber iaphragms. We performe the following test: the pressure generator was operating twelve hours generating the sinusoial pressure with the amplitue of 5 Pa an the frequency of 1 Hz (more than 4 million of vibration perios in total). After each hour of operation the static sensitivity was measure. The results showe relative eviations smaller than.5 %, without any characteristic tren. That means the observe loaing i not cause any noticeable systematic effects in the presente pressure generator. 5

6 Sensors an Actuators A: Physical 168 (211) oi: 1.116/j.sna Elsevier B.V. Fig. 5 presents the time variation of the generate pressure in the case of a step change of the internal volume. A han-operate valve was built in between the pressure generator an the ae volume of.5 m ; this valve was opene as quickly as possible at some moment uring the pressure generation an the internal volume was increase in the ratio of about 1:. As woul be expecte, the generate pressure shows a ecrease ue to the smaller static sensitivity with the larger internal volume. However, it is important to realize that the pressure ecrease woul be much greater without the electroynamic operation of the pressure generator, as is also preicte for the comparison in Fig Pressure p [Pa] V ADD =.5 m With the pressure generator Without the pressure generator (preicte) Time [s] Fig. 5. Time variation of the pressure showing the effect of the suen increase of the pressure generator s volume. Small influences of the volume changes on the generate pressure can be avantageous for applications where volume changes occur uring the testing, e.g., when testing the switching pressure of hyrostats. Fig. 6 presents the time variation of the pressure uring the switching action of the hyrostat that was teste on the linear pressure input. The measure pressure in the hyrostat cavity changes ue to progressive eflections of the hyrostat iaphragm an the final switch of the hyrostat mechanism. On the other han, the measure pressure in the pressure generator shows the ability of generating the linear pressure input without observable influences of the hyrostat volume changes. If one pressure generator is use for simultaneous testing of more hyrostats, such characteristic woul be of high importance to prevent their mutual interaction Pressure generator Hyrostat Pressure p [Pa] Time [s] Fig. 6. Time variation of the pressure uring the switching action of the teste hyrostat. 6

7 Sensors an Actuators A: Physical 168 (211) oi: 1.116/j.sna Elsevier B.V. In some publishe references, see, e.g., [,5], the close-volume pressure generators with one louspeaker were use in contrast to the iscusse configuration with two louspeakers. If the static physical moel presente in Section 2.1 was moifie for such configuration with one louspeaker, the only change woul be twice smaller gas stiffness k. Therefore, their static sensitivity woul become about twice more epenent on the internal gas volume changes. Such properties can represent an important benefit of the pressure generator with two louspeakers when employe in applications with the varying volume.. DYNAMIC CHARACTERISTICS OF THE PRESSURE GENERATOR.1 Dynamic physical moel The simplifie ynamic moel of the louspeaker-base pressure generator is set up within the framework of iscrete, lumpe-parameter oscillators. The pressure generator is assume to consist of an internal volume V between the iaphragms, to which the evice uner test with the internal volume V is connecte using a tube with an internal cross-section S t. Other assumptions follow Section 2.1. The natural frequency of the iaphragms moving with respect to one another can be estimate as: f 1 2 k k susp, (5) m where m is the effective mass of the vibrating iaphragm, but its stiffness k ksusp sums the contributions of the gas an the suspension springs, where k is efine by Eq. (2) using V. For the pressure generator uner iscussion, ksusp k (see Section 2.2), so the natural frequency of the iaphragms epens strongly on the properties of the internal gas volume. If the close-volume pressure generator use only one louspeaker, the effective gas stiffness k in Eq. (5) woul become twice smaller (as alreay iscusse at the en of the previous section). Therefore, when ksusp k, the pressure generator with one louspeaker woul have about the 2 smaller iaphragm's natural frequency compare to the iscusse configuration with two louspeakers. If the imensions of the internal gas volume between the iaphragms are relatively small, in comparison with the wavelength of the generate pressure oscillations, this flui egree of freeom can be neglecte in the ynamic moel. Thus, the pressure change insie can be relate to the iaphragm isplacement by Eq. (1) using V. The wavelength is efine as c/ f, where c is the spee of soun (about 4 m/s for ry air at 2 C) an f is the oscillation frequency [1]. For an internal iameter of about 12 cm, which represents the longest linear imension in the pressure generator uner iscussion, such an assumption is appropriate for pressure oscillation frequencies of a few hunres of Hz,. However, the natural frequency of the iaphragm still oes not represent the only ynamic limitation. The connecting tube of the evice uner test, together with the sie volumes, forms 7

8 Sensors an Actuators A: Physical 168 (211) oi: 1.116/j.sna Elsevier B.V. the flui oscillator calle a Helmholtz resonator [11]. The natural frequency of this resonator can be written as: f f 1 2 k k f, t f, t, (6) m f where m f is the effective mass of the flui in the connecting tube an k f, t k f, t is the combine influence of the gas springs of both sie internal volumes V an V : k 2 2 St P St P k. (7) V V f, t f, t If the pressure generator volume is much larger than the internal volume of the evice uner test, V V, the natural frequency of the flui oscillator only epens on the evice uner test an its connection to the pressure generator. In that case both ientifie oscillators can be treate as uncouple..2 Dynamic measurements In the case of ynamic measurements, the generate pressure was measure by two piezoelectric pressure-measurement systems, incluing the piezoelectric pressure sensors (Kistler, 7261, sensitivity 22 pc/kpa, cal. measuring range ± 4 Pa, accuracy (best straight line).25 % of upper range limit, resonant frequency without connecting tube 25 Hz (see the measure resonant frequencies for ifferent connecting tubes in [6])) an the charge amplifiers (Dewetron, DAQ-Charge, sensitivity.1 V/pC, output voltage ± 5 V, accuracy (best straight line).5 % of upper range limit, frequency range. Hz to 5 khz ( B)). Pressure sensors have an internal volume of 1.5 cm an were connecte to the chamber of the pressure generator with short plastic tubes (length about 5 cm, internal iameter about 4 mm). The sinusoial excitation of the louspeakers was performe with a function generator (Golstar, FG-82) using a voltage-controlle frequency from the DAQ boar. The sampling frequency of the DAQ boar for the excitation current an the pressure signals was set to 2 khz. The other etails about the measurement system are escribe in Section 2.2 an schematically presente in Fig. 2. The frequency characteristic was etermine at excitation frequencies up to 1 Hz, in steps of about 1 Hz. At each point the amplitues an phases of the excitation current an the pressure signals were calculate after a stabilization time of about 5 s. Fig. 7 shows the amplitue an phase frequency characteristics between the generate pressure an the excitation current for both the oppositely connecte pressure sensors (P1 in P2). Two resonant frequencies are evient the resonance at about 4 Hz correspons to the natural frequency of the iaphragm, but the resonance at about 6 Hz correspons to the natural frequency of the flui oscillator forme by the connecting tube an the pressure sensor volume (see the physical moel in Section.1). 8

9 Sensors an Actuators A: Physical 168 (211) oi: 1.116/j.sna Elsevier B.V. Amplitue ratio A p / (A i K PG ) [-] P1 P Frequency f [Hz] Phase ifference p - i [ o ] P1 P Frequency f [Hz] Fig. 7. Amplitue an phase frequency characteristics between the generate pressure an the excitation current for two pressure sensors (P1 an P2). In spite of the fact that the amplitues an the phases of the generate pressure, in comparison with the excitation current, vary with the excitation frequency, there are relatively small eviations of the pressure oscillations at both connection points P1 an P2 on the circumference of the pressure generator. Fig. 8 shows the corresponing amplitue an phase frequency characteristics between the generate pressures up to 5 Hz. Although the louspeaker s natural frequency is in the observe frequency range, the amplitue an phase eviations o not excee 2 % an 1, respectively. Such results inicate a relatively homogeneous istribution of the pressure in the chamber of the pressure generator up to 5 Hz. 9

10 Sensors an Actuators A: Physical 168 (211) oi: 1.116/j.sna Elsevier B.V. 1.2 Amplitue ratio A p2 / A p1 [-] Frequency f [Hz] Phase ifference p2 - p1 [ o ] Frequency f [Hz] Fig. 8. Amplitue an phase frequency characteristics between the generate pressures measure by two pressure sensors (P1 an P2). 4. CONCLUSIONS The physical moelling an experimental analysis presente in this paper offer an insight into the capabilities an limitations of a pressure generator base on electroynamic louspeakers. The generator's static sensitivity, which was efine by the ratio between the generate pressure an the excitation electric current, mainly epens on the ratio between the force factor an the effective area of the louspeakers. The attainable magnitue of the statically generate pressure by the presente system is of the orer of 1 Pa. It is limite by the allowe electric current through the louspeaker coil an the resulting temperature loa. The estimate precision an stability of the pressure generator is of the orer of 1 % of the upper range limit. The generator proves a small sensitivity to changes in the internal volume, showing only about 1.5 % ecrease of the static sensitivity when the internal volume was increase three times. The pressure generator is applicable in the frequency range of hunres of Hz. Its ynamic characteristics are efine by the properties of the louspeaker iaphragm, the air chamber between the louspeakers an the connection of the evice uner test to the pressure generator. The measurement results for the generate pressures at the raially opposite connection points show the amplitue an phase eviations, which o not excee 2 % an 1, respectively, in the frequency range up to 5 Hz. 1

11 Sensors an Actuators A: Physical 168 (211) oi: 1.116/j.sna Elsevier B.V. The presente physical moels show that the pressure generator uner iscussion, which is realize with two louspeakers, has some avantages over an alternative configuration with one louspeaker. The iaphragm gas-relate stiffness is expecte to be twice as high with two louspeakers. Therefore, the effect of the internal gas volume on the generator s static sensitivity is smaller an the iaphragm s natural frequency is higher. REFERENCES [1] V. E. Bean, Dynamic pressure metrology, Metrologia (1994) [2] J. Hjelmgren, Dynamic Measurement of Pressure A Literature Survey, SP Sweish National Testing an Research Institute, Boras, 22. [] J. Zakrzewski, K. Wrobel, Dynamic calibration of low-range silicon pressure sensor, IEEE Transactions on Instrumentation an Measurement 51 (22) [4] H. Urzeniczok, J. Zakrzewski, Pressure pulse generating system for ynamic calibration of silicon low range pressure sensors, Proceeings of the 17th IMEKO Worl Congress (2) [5] A. Yoshia, Y. Tamura, T. Kurita, Effects of bens in a tubing system for pressure measurement, Journal of Win Engineering an Inustrial Aeroynamics 89 (21) [6] I. Bajsić, J. Kutin, T. Žagar, Response time of a pressure measurement system with a connecting tube, Instrumentation Science & Technology 5 (27) [7] C. Antonini, G. Persico, A. L. Rowe, Preiction of the ynamic response of complex transmission line systems for unsteay pressure measurements, Measurement Science an Technology 19 (28) [8] J. Kutin, A. Smrečnik, G. Bobovnik, I. Bajsić, Analysis of the system for testing a switching pressure of hyrostatic pressure switches, Project Report, Faculty of Mechanical Engineering, University of Ljubljana, 24 (in Slovene). [9] T. Lokar, A. Smrečnik, I. Bajsić, Generating ynamic pressure with louspeakers, Proceeings of the 16th IMEKO Worl Congress (2) [1] T. D. Burton, Introuction to Dynamic System Analysis, McGraw-Hill, New York, [11] R. D. Blevins, Formulas for Natural Frequency an Moe Shape, Krieger, Malabar, [12] H. Schurer, Linearization of electroacoustic transucers, Ph.D. Thesis, University of Twente, Enschee, [1] R. E. Apfel, Acoustic lumpe elements from first principles, Hanbook of Acoustics, e. M. J. Crocker, Wiley, New York,