HYDRODYNAMIC RESONATION THROUGHFLOW SYSTEM AS ACOUSTIC CIRCUIT

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1 KOMISJA BUDOWY MASZYN PAN ODDZIAŁ W POZNANIU Vol. 8 nr 4 Archiwum Technologii Maszyn i Automatyzacji 008 MILENA KUŠNEROVÁ, JAN VALÍČEK, SERGEJ HLOCH MAREK SADÍLEK, ROBERT ČEP HYDRODYNAMIC RESONATION THROUGHFLOW SYSTEM AS ACOUSTIC CIRCUIT The paper proposes a derivation of general characteristic acoustic parameters of the hydrodynamic resonation system elements. The acoustic parameters of the system are the acoustic compliance and acoustic mass as primary parameters for calculation of the theoretical resonation fundamental frequency for turbulent waterflow. Frequency model (5 khz to 5 khz) refers to the interpolation for the liquid pressure range from 5 MPa to 5 MPa. The analysis travels in a direction of a value comparison of concrete acoustic characteristic parameters for any element of the system and has been contrasted with theoretical conditions about an acoustic mass for tubes and an acoustic compliance for chambers. This system generates a modulate liquid jet as a result of resonation chamber implantation. The theoretical model was built on the basis of knowledge of primarily geometric dimensions of the system and the pressures of the liquid to predict values of fundamental frequencies of oscillations. Predicted oscillation frequencies were verified by laboratory measurements. Key words: hydrodynamic system, acoustic mass, acoustic compliance, fundamental frequency, analytical model. INTRODUCTION Our aim is to achieve the better disintegration of surface of material being machined than in the case of continuous jet machining. Technology of the material surface treatment by the highspeed water flow is relatively effective method for material cutting. The process efficiency and the possibility of its regulation can be enhanced by the resonance effect influence, i.e. by the flow modulation in consequence of the tuned chambers introduction into a system. As RNDr., PhD. Institute of Physics, Technical University of Ostrava, Ostrava Czech Ing., PhD. Republic. Ing., PhD. Department of Technology Systems Operation, Technical University of Košice with the seat in Prešov, Slovakia. Ing., PhD. Department of Technology Systems Operation, Technical University of Košice with the seat in Prešov, Slovakia. Ing., PhD. Department of Machining and Assembly, Technical University of Ostrava, Ostrava, Czech Republic.
2 44 M. Kušnierová, J. Valiček, S. Hloch, M. Sadílek, R. Čep a real acceptable system proper for this technical and physical application, the hydrodynamic acoustic throughflow system can be regarded. Theoretically the system can be represented by analytical model with continuously distributed characteristic acoustic parameters, above all by acoustic mass m a and acoustic compliance c a, eventually by acoustic impedance Z a [3]. The elements of the system are tubes and chambers. Tubes (Fig., : input tube, connecting tube 4, output tube and nozzle 6,7) are generally characterized by acoustic parameters m a, their parameter c a is relatively insignificant. Chambers (Fig., : capacious cavities, 3, 5) are defined by parameter c a, their acoustic parameters m a have minor effect on system functionality. Fig.. Photography of the hydrodynamic throughflow resonation system (device of Institute of Physics, Faculty of Mining and Geology, Technical University of Ostrava) Rys.. Zdjęcie urządzenia do badań rezonansu przepływu hydrodynamicznego (Instytut Fizyki, Wydział Górnictwa i Geologii, Uniwersytet Techniczny w Ostrawie) Fig.. Scheme of the hydrodynamic throughflow resonation system Rys.. Schemat systemu do badań rezonansu przepływu hydrodynamicznego
3 Hydrodynamic resonation throughflow system 45 The aim of acoustic characteristic parameters derivation is the subsequent fundamental frequency, amplitude and system oscillation energy derivation, namely in dependence on primary, directly measured parameters of the system (liquid pressure and geometrical dimensions of the system). The above mentioned analytical models of liquid flow parameters have been created upon the analogy of the functionality of acoustic circuits with the air medium and upon the analogy of electromagnetic RLC circuits. For bearing hydraulic environment we have derived analogically the acoustic mass as well as we have m a approximated, whereas functional values of both models were comparable []. Acoustic compliance has been derived as well by analogy, but theoretical presumptions of model functional values c a had to be approximated upon the experimentally acquired data, because the fluid properties are right in such case hardly comparable with gas properties (above all with respect on relatively low water compressibility in comparison with air compressibility).. MATHEMATICAL AND PHYSICAL CONDITIONS OF SYSTEM FUNCTIONALITY Hydrodynamic resonation system generates (i.e. amplify) sound wave by analogy as classical acoustic system with gaseous bearing environment. Longitudes of system elements would have been shorter then it is half wave of stationary waves, to reacted functionally as elements of vibrating system. If the element of this system is longer then acoustic halfwave, the stationary wave can come into being, or resonance does not have to realize in general. So frequency active are (at in the concrete discussed setting of geometrical proportions of elements) only connecting tube, resonance chamber and output tube (nozzle) with reference to geometric dimension of system []). Ray tracheid condition for averages of system elements is executed in specimen case of setting, because near of all elements is medium regular insignificant in comparison with longitude of acoustic halfwave (acoustic we regard as planar). We consider longitudinal and transverse conditions as necessary conditions non competent. Relevant capacity of liquid so entertains as solid and has function only one acoustic element yclept acoustic mass m a. Contrariwise the capacities embody compliance (elasticity) and is them possible to assign parameter acoustic compliance c a. Which with efficient settingmovement input parameters elements system is concerned, it is possible differentiate invariable values of parameters for given to system (e.g. averages elements) and regularly setting of parameters for ordinary operation. As optimalization of function given to system then we understand either protoxylem made system of all mathematically and physicaly conditions, or already made system and after it tuned car on maximum disposable frequence
4 46 M. Kušnierová, J. Valiček, S. Hloch, M. Sadílek, R. Čep (amplitude, energy) of oscillations, i.e. additionally. In technical practice we consider as well adjustable input parameter  change of fluid pressure and tuning on longitude of element. Analysis of dependencies of longitude of element on longitude acoustic halfwave (Table ) is related to indirect measurement of phase speed, through which we're evaluate longitude of acoustic halfwave. Table Analysis of dependencies of element longitude h on acoustic halfwave longitude λ/ Analiza wpływu długości elementu h na długość połówki fali akustycznej λ/ conditions λ h conditions λ h < λ stationary wave h = k ; k =,, 3... rises λ break in series h k ; k =,, 3... elements acoustic mass acoustic acoustic oscillations rise compliance oscillations with frequence of stationary waves oscillations with frequence other elements of series oscillations with frequence of elements in series 3. RESULTS OF HARMONIC OSCILLATIONS ACOUSTIC MASS m a For acoustic mass derivation at processes running in hydrodynamic acoustic flow oscillating system, the notation, terminology and physical relations are as follows: Y = S y () Y volume dislocation; S crosssection of an element; y acoustic displacement; dy u = y = u () u acoustic oscillations velocity; du du U = S u = S (3) U acoustic flow velocity; du du p = m p = m S (4) a a a a du du pa S = ma S Fa = pas Fa = ma S (5) p a acoustic pressure; m a acoustic mass; F a acoustic force expressed by acoustic pressure;
5 F ma acoustic Newton force; Hydrodynamic resonation throughflow system 47 du F ma = m (6) m F a = F ma m a = S m inertial mass of oscillating elements; m a l element length; ρ liquid density; k = k rkk : m S l l ρ = S ρ = S a kg 4 m (7) (8) krkk l ρ = (9) S k conversion coefficient (k rkk relative correction correlation coefficient), which expresses material and geometrical properties of a concrete system (e.g. sharpness of edges at transition between neighbouring elements), analytical and experimental []. We show the demonstration of concrete values of acoustic parameters of hydrodynamic acoustic system (Table ). Indexes at symbols of acoustic parameters are: P input tube, H connecting tube, T output tube (nozzle), VK input chamber, RK resonant chamber. By other say: we are awaiting, that the values of acoustic mass of tubes will be higher than the values of acoustic mass of chambers and opposite: the values of acoustic plasticity of chasubles will be higher than the values of tubes acoustic compliance: m > m c > c (0) H RK Above all, input chamber and the nozzle do not behave according to hypothesis of optimal acoustic mass (Table ). Chamber appreciates m a relatively high and nozzle appreciates m a relatively low. RK H The values of acoustic mass of system at different pressure of liquid Wartości mas akustycznych systemu dla różnych wartości ciśnienia płynu Table m a [kg m 4 ] 5 MPa 0 MPa 5 MPa 0 MPa 5 MPa m P m H m T m VK m RK
6 48 M. Kušnierová, J. Valiček, S. Hloch, M. Sadílek, R. Čep 4. RESULTS OF HARMONIC OSCILLATIONS ACOUSTIC COMPLIANCE c a For acoustic compliance derivation in processes running in hydrodynamic acoustic flow vibrating system, the notation, terminology and physical relations are as follows: dfa dfa = dy = dy () c S c S dpa = dy S dpa = dv () c S c S df a elementary acoustic force expressed by acoustic pressure; dp a elementary acoustic pressure; c mechanical compliance (inverse value of rigidity); V element volume (cylindrical) V = S l ; dpca = dv (3) c c a acoustic compliance; dp a m 4 s dp c a = c S (4) kg dv dp c = K (5) V ca = a K module of volume elasticity; v ϕ K = K = vϕ ρ ρ K = (6) γ γ liquid compressibility; dp ca = dp c γ = γ 0 p 0 a = 4 V K c a = γ V (7) ρ0 ρ = (8) γ p p flow pressure; γ 0 liquid compressibility at room temperature; ρ liquid density; ρ 0 liquid density at room temperature. We show the demonstration of concrete values of acoustic parameters of hydrodynamic acoustic system (Table 3).
7 Hydrodynamic resonation throughflow system 49 The values of acoustic plasticity of system at different pressure of liquid Wartości akustycznej plastyczności systemu dla różnych wartości ciśnienia płynu Table 3 c a [m 4 s kg ] 5 MPa 0 MPa 5 MPa 0 MPa 5 MPa c P c H c T c VK c RK Above all input tube according to hypothesis of optimal acoustic compliance. Does not behave its value c a is comparable with plasticity of resonance chamber. Acoustic parameters of these elements would have been adjusted, i.e. optimized. 5. RESULTS OF HARMONIC OSCILLATIONS FUNDAMENTAL FREQUENCY c a Method of solving [] is based on the evaluation of acoustic impedance of oscillating system under resonance conditions. As a basis we use the analogy with the classical evaluation of impedance of oscillating electromagnetic circuits by means of their apparent resistances (inductance and capacitance). Then, we shall determine the reactance of acoustic mass X ma and the reactance of acoustic compliance X ca as acoustic apparent resistances, and the total acoustic resistance then as a vector of acoustic impedance Z a with the real component of acoustic resistance r a and with the imaginary component of apparent resistance X a = X ma X ca. Furthermore, we analogically assume that the acoustic impedance is minimal just under resonance conditions, so that the eigenfrequency f 0 of oscillations of the system corresponds to the formal equivalent of Thomson relation: Z = + a = ra + i X a = ra + i ω 0ma ra i X ma ω0ca X ca X ma = X ca f0 = [ Hz] (9) π m a c a In the oscillating electromagnetic circuits it is true that the total impedance is equal to the sum of partial impedances. We assume again analogically that this is true even for the given hydrodynamic oscillating system composed of elements connected in series and that its total acoustic mass m a corresponds to the sum of
8 50 M. Kušnierová, J. Valiček, S. Hloch, M. Sadílek, R. Čep partial acoustic masses, and the reciprocal value of total acoustic compliance c a is equal to the sum of reciprocal values of partial acoustic compliances. In our specific case, the system consists of three tubes (supply tube, socket and exit nozzle, i = 3) and two chambers (entry chamber and resonant chamber itself, j = ). 3 m a = mai ; =. (0) c c i= We took altogether 86 direct measurements of frequency at setting various values of input parameters, i.e. liquid pressure and geometric dimensions of the system. The submitted example of experiment corresponds to the setting of liquid pressure at 5 MPa. We processed the timehistory record of the force using the FFT method. The fundamental resonant frequency is 8.3 khz (Fig. 3). The amplitude of the force has the value of about 5.6% of static component of the force; the values of components of higher harmonic frequencies of spectrum are relatively high as well (the first higher harmonic component corresponds to.4% of static component of the force, Fig. 3). a j= aj 3,5 Amplitude [N],5 0, Frequency [khz] Fig. 3. A record of measurements of force amplitude versus frequency Rys. 3. Zapis pomiarów amplitudy siły w funkcji częstotliwości The source of data for the creation and the verification of the model of fundamental frequency of eigenoscillations of the system was the direct measurements of tunable geometric dimensions of elements, the accurate setting of liquid pressures, and the indirect measurements of flow rate and of acoustic oscillation rate derived from them, the direct measurements of frequency and various other indirect measurements by means of tabulated values, regressions and calculations with the Excel program.
9 Hydrodynamic resonation throughflow system 5 6. CONCLUSIONS In objective problems we have noted satisfactory agreement of theoretical results with experimental results, namely for 86 direct measurements (at setting different pressures of liquid and geometrical proportions of system) []. By analysis of acoustic parameters of hydrodynamical acoustic system we have found out, that acoustic elements connected in series: connecting tube, resonance chamber and output tube (nozzle) are the resonant bearing elements. Conclusion of discussion of mathematically physical conditions of optimal functionality of system contributes to this conclusion. By confrontation of material flow and geometrical conversion (9) coefficients of all elements we have realized, that their values agree according to the theoretical expectation, but the punctuality of evaluation is relatively differ. Punctuality of evaluation in hundrehs embodies: input chamber, input tube, punctuality of evaluation in tenths embodies: nozzle, resonance chamber and connecting tube. It is possible to come to a conclusion, that in acoustic most active elements of system rises relatively biggest problem with interpretation of effect of acoustic resistance. By comparison of relative differences between measured and theoretically predicated values the concordance rate of results of model experiment is relatively high (it varies only in tenths %), at frequency bearing elements connecting tube  resonance chamber nozzle it is still higher. We have evaluated sensitivity to changes input parameters as well, whereas the pressure liquid was intended the as primary parameter. Sensitivity of acoustic mass on relatively small changes pressure of fluid is minimum in comparison with sensitivity of acoustic plasticity, whereas with increasing pressure acoustic mass goes up and acoustic plasticity contrariwise declines (and on the contrary). ACKNOWLEDGEMENTS The authors would like to acknowledge the support of the project MŠMT No. MSM REFERENCES [] Foldyna J., Sitek L., Měření stagnační síly. Helmholtzova komora, Research report, Ostrava, Ústav geoniky AV Ostrava 003. [] Kušnerová M., Hlaváč L., Self vibrating chambers with continuous passage of liquid, in: Sborník vědeckých prací VŠBTUO. Řada strojní č./006; ročník LII. VŠBTUO: Ostrava, 006, p [3] Kuttruff H., Room acoustics. Applied Science, London, Publishers Ltd Praca wpłynęła do Redakcji Recenzent: prof. dr hab. inż. Jan Kołodziej
10 5 M. Kušnierová, J. Valiček, S. Hloch, M. Sadílek, R. Čep REZONANS HYDRODYNAMICZNY Z WYKORZYSTANIEM SYSTEMU PRZEPŁYWOWEGO JAKO OBWODU AKUSTYCZNEGO S t r e s z c z e n i e W artykule zaproponowano ogólną charakterystykę parametrów akustycznych hydrodynamicznego systemu rezonansowego. Podstawowymi parametrami akustycznymi systemu, wykorzystywanymi do obliczeń podstawowej częstotliwości teoretycznego rezonansu przepływu turbulentnego wody, są odkształcalność akustyczna i masa akustyczna. Model częstotliwościowy (od 5 khz do 5 khz) dotyczy interpolacji dla ciśnienia płynu od 5 MPa do 5 MPa. Analiza zmierza w kierunku porównania wartości konkretnych parametrów charakterystyk akustycznych każdego elementu systemu i została zestawiona z warunkami teoretycznymi dotyczącymi masy akustycznej rur i akustycznej odkształcalności komór. System ten tworzy modulowany strumień płynu jako wynik zastosowania rezonansu komory. Model teoretyczny był zbudowany na podstawie wiedzy o pierwotnych wymiarach geometrycznych systemu i ciśnień płynu po to, aby przewidzieć wartości podstawowych częstotliwości drgań. Przewidywane częstotliwości drgań były sprawdzane przez pomiary laboratoryjne. Słowa kluczowe: system hydrodynamiczny, masa akustyczna, odkształcalność akustyczna, częstotliwość podstawowa, model analityczny
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