Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 53, 3, pp. 557-567, Warsaw 2015 DOI: 10.15632/jtam-pl.53.3.557 REFINED MODEL OF PASSIVE BRANCH DAMPER OF PRESSURE FLUCTUATIONS Sylwester Kudźma West Pomeranian University of Technology Szczecin, Faculty of Mechanical Engineering and Mechatronics, Szczecin, Poland; e-mail: sylwester.kudzma@zut.edu.pl Zygmunt Kudźma Wrocław University of Technology, Faculty of Mechanical Engineering, Wrocław, Poland e-mail: zygmunt.kudzma@pwr.edu.pl This paper presents an analysis of the influence of the kind of a friction model on the di- mensioning of a branchpressure fluctuation damper.Themathematicalmodel of the branch damper is defined by determining the damper input impedance and finding its minimum corresponding to the maximum effectiveness in reducing pressure fluctuations. Three kinds of friction for the oscillatory flow in the damper, i.e. a lossless line, steady friction and a nonstationary friction model, are considered. Experimental studies confirmed that the use of the nonstationary friction model in the calculation of branch damper length ensures the highest effectiveness in reducing the amplitude of pressure fluctuations characterized by a given frequency. Keywords: pressure fluctuations, damper, noise 1. Introduction Hydrostatic drive systems have their well-known advantages, but their main disadvantage is that they generate much noise which may disqualify such kind of drive when the increasingly more stringent noise emission standards dictated by ergonomic considerations are exceeded. For this reason, a properly designed hydrostatic drive system should not only have the expected static and dynamic properties, but also emit as little noise as possible. Directive 98/37/WE includes a general recommendation that a machine should be designed in such a way that the hazard arising from the emission of the noise generated by it is reduced (preferably at the noise generation source) to the lowest level possible owing to the technological progress and the available means of noise reduction. The directive also requires that information about the noise at the operator workstation be included in the machine documentation. The following should be specified: the equivalent sound pressure level, the instantaneous peak acoustic pressure and the acoustic power level. Noise in a hydraulic system can be generated in two ways: • directly – the noise source (e.g. the impeller of a fan in the electric motor driving the pump) produces changes in pressure in the surrounding air, • indirectly – time-variable forces make the components of a hydraulic system vibrate. The vibration of the surfaces of the components results in noise emission. Indirectly generated noise is the principal noise in hydraulic systems. Changeable forces acting on the hydraulic system components arise as a result of: • pressure fluctuations (Tijsseling, 1996), 558 S. Kudźma, Z. Kudźma • mechanical connection of the system components through conduits and the common mo- unting. A single component (e.g. a valve) may vibrate as a result of the action of the fluid, causing the vibration of the components connected with it. One of the major noise sources in a hydraulic system is the unstable operation of the pressu- re relief valves due to, among other things, external excitations produced by vibration of the machine frame or the feeder cover to which the pressure relief valves are often mounted. One should note that vibrations may arise in the resonant region of the element which controls the valve. Therefore, the coincidence of the frequency range of, e.g., the foundation and that of the hydraulic valve controlling element should be avoided (Stosiak, 2012). Experimental studies aimed at locating and identifying vibration and noise sources must be carried out in order to effectively eliminate annoying noise. Energymeasuringmethods are par- ticularly suitable for locating noise sourceswhen diagnosing the acoustic condition of hydraulic machines and equipment (Kollek et al., 2001). For locating noise sources, Osiński and Kollek (2013) recommend using the acoustic intensity method (AIM) with a two-microphone acoustic probewhereby amap of noise intensity around the investigated equipment can be obtained and the loudest places can be indicated. The causes of noisiness in the hydraulic system can be divided into mechanical causes and hydraulic causes. The group of mechanical causes includes workmanship and assembly faults, excessive clearances inallmoving joints, unbalanced rotatingpartsandsoon.Themainhydraulic causes are: cavitation phenomena (Kollek et al., 2007), forcing pressurefluctuations andworking liquid pressure surges in the pump or displacement motor chambers. In a properly designed hydraulic system, cavitation should not occur while the occurrence of working liquid pressure surges largely depends on the type of the pump. Axial multiplunger pumps are the noisiest while vane pumps and internal gear pumps are the most silent-running. The research so far has identified pressure fluctuations and the resulting vibrations as the principal causes of noise generation in hydraulic systems, seeMikota andManhartsgruber (2003), Kudźma (2001, 2006, 2012), Wacker (1985). Thus by reducing pressure fluctuations, one can reduce the noisiness of the individual system components and thereby prolong their service life. One of the effective ways of reducing pressure fluctuations, and so the hydrostatic noise of the drive system, is the use of pressure fluctuation dampers. This paper presents a refined passive branch dampermodel taking into account nonstationary flow resistance. 2. The supply conduit as a long hydraulic line Simplifying assumptions, in detail described by Kudźma (2012), are commonly made when deriving equations for thenonstationaryflowof a liquid in closed conduits.On suchassumptions, the (laminar and turbulent)motion of the liquid is described by the following equation,Kudźma (2012), Zarzycki (1994): — the equation of motion towards the axis z ∂vz ∂t =− 1 ρo ∂p ∂z +ν 1 r ∂ ∂r ( r ∂vz ∂r ) + 1 r ∂ ∂r ( rνt ∂vz ∂r ) (2.1) — the equation of continuity ∂p ∂t +ρoc 2 o (∂vz ∂z + ∂vr ∂r + vr r ) =0 (2.2) where: vz is the instantaneous velocity of the liquid in the conduit in the axial direction, vr – instantaneous velocity of the liquid in the radial direction, p – instantaneous pressure of the liquid, ν – kinematic coefficient of molecular viscosity, νt – kinematic coefficient of turbulent Refined model of passive branch damper of pressure fluctuations 559 viscosity, ρo – steady density of the liquid, z – axial coordinate of the conduit, r – radial coordinate of the conduit, co – velocity of pressure wave propagation, t – time. In the case of turbulent motion, vz and p are quantities averaged in accordance with the Reynolds rules. By integrating equations (2.1) and (2.2) over the conduit cross section, one gets a system of equations which can be presented as follows, see Kudźma (2012), Zarzycki (1994), Zarzycki et al. (2007) ρo ∂v(z,t) ∂t + ∂p(z,t) ∂z + 2 R τw =0 ∂p(z,t) ∂t +ρoc 2 o ∂v(z,t) ∂z =0 (2.3) where: v = v(z,t) is the average in the conduit cross section velocity of the liquid, p = p(z,t) – average pressure in the conduit cross section, R – inside radius of the conduit, τw – shear stress on the conduit wall. The expression (2/R)τw in equation (2.3)1 represents pressure drop due to friction, per unit length. In the case of a laminar flow, the most accurate model of hydraulic resistance is the model with variable resistance, which takes into account pressure losses as a function of frequency. Using this model one gets the following expression for impedance Z0 of a long hydraulic line, see Zielke (1968) Z0(s)= ρos πR2 1− 2J1 ( jR √ s ν ) jR √ s ν J0 ( jR √ s ν ) (2.4) where: s is theLaplace transformation operator,J0,J1 are respectively zero-order andfirst-order first-type Bessel functions, j – imaginary unit. By applying the inverseLaplace transformation, Zielke (1968) obtained the following relation for the instantaneous shear stress on the conduit wall τw(t)= 4µ R v+ 2µ R t∫ 0 w(t−u) ∂v ∂t (u) du (2.5) where: w(t) is the weighting function and u is the time in the convolution integral. The second term in equation (2.5) describes the influence of flownonstationarity on the shear stress. It is a convolution integral of the instantaneous acceleration of the liquid and weighting function w(t). The instantaneous conduit wall shear stress τw can be presented as the sum of quasi-steady quantity τwq and time-variable quantity τwn, see Vardy and Brown (2003) τw = τwq+ τwn (2.6) If only the quasi-steady frictionmodel is to be taken into account, the second term in equations (2.5) and (2.6) should be omitted, whereby only τw = τwq remains. It should be noted that the quasi-steady frictional resistance models used in calculations of nonstationary states are valid only in the case of slow velocity changes, which applies to low excitation frequencies or small accelerations of the liquid. 2.1. Weighting function The so-calledweighting function,which depends on, among other things, the character of the flow, features significantly in above relation (2.5). For the laminar flow, the relation presented by Zielke (1968) is commonly used, whereas for the turbulent flow there are twomainmodels (also 560 S. Kudźma, Z. Kudźma dependent on the Reynolds number) proposed by Vardy and Brown (2003, 2004) and Zarzycki (1994), Zarzycki et al. (2007). Since the weighting functions presented by the above authors, especially the ones for the turbulent flow, are complicated and difficult to handle in numerical computations, their approximations are used in practice. From among the weighting function approximating relations proposed in the literature, the relation presented by Urbanowicz and Zarzycki (2012), which through the appropriate scaling of the coefficients can be used for both laminar and turbulent flows, deserves special attention w(t̂)= 26∑ i=1 mie −nit̂ (2.7) where: t̂ = νt/R2 is dimensionless time, and the coefficients mi, ni assume the following values (Urbanowicz and Zarzycki, 2012): m1 =1; m2 =1; m3 =1; m4 =1; m5 =1; m6 =2.141; m7 =4.544; m8 =7.566; m9 =11.299; m10 = 16.531; m11 = 24.794; m12 = 36.229; m13 = 52.576; m14 = 78.150; m15 = 113.873; m16 =165.353; m17 =247.915; m18 =369.561; m19 =546.456; m20 =818.871; m21 =1209.771; m22 =1770.756; m23 =2651.257; m24 =3968.686; m25 =5789.566; m26 =8949.468; n1 = 26.3744; n2 = 70.8493; n3 = 135.0198; n4 = 218.9216; n5 = 322.5544; n6 = 499.148; n7 = 1072.543; n8 = 2663.013; n9 = 6566.001; n10 = 15410.459; n11 = 35414.779; n12 = 80188.189; n13 = 177078.960; n14 = 388697.936; n15 = 850530.325; n16 = 1835847.582; n17 = 3977177.832; n18 = 8721494.927; n19 = 19120835.527; n20 = 42098544.558; n21 = 92940512.285; n22 = 203458923.000; n23 = 445270063.893; n24 = 985067938.878; n25 =2166385706.058; n26 =4766167206.672. The function can be easily transformed to the Laplace variable domain. Then it assumes the form L[w] = 26∑ i=1 mi ŝ+ni (2.8) where ŝ is dimensionless operator of the Laplace transformation ŝ =(R2/ν)s. The values of the universal coefficients are determined in the following way, see Urbanowicz and Zarzycki (2012) n1u = n1−B ∗; n2u = n2−B ∗; . . . ; n26u = n26−B ∗ m1u = m1 A∗ ; m2u = m2 A∗ ; . . . ; m26u = m26 A∗ where A∗ = √ 1 4π B∗ = Reκ 12.86 = 2320κ 12.86 κ = log 15.29 Re0.0567 = log 15.29 23200.0567 Thevalues of the universal coefficients of the laminar-turbulentweighting function are necessary to determine the current shape of the weighting function used in numerical computations. Thus the function defined by the universal coefficients L[w] = 26∑ i=1 miu ŝ+niu (2.9) sufficiently well represents the nonstationary frictional loss model for both the laminar flow and the turbulent flow, provided that the Reynolds number is determined earlier. Performing theLaplace transformation on equations (2.1) and (2.2) for zero initial conditions and then integrating the equations relative to thevariable z with the limits of 0-L (L is the length Refined model of passive branch damper of pressure fluctuations 561 of the hydraulic line) one gets a matrix transition function for a long hydraulic line. Assuming a harmonic excitation, the function can be presented in the following form, see Kudźma et al. (2002), Zarzycki (1994), Zarzycki et al. (2007) [ p1 q1 ] = H(jω) [ p2 q2 ] (2.10) where:H(jω) is thematrix transition function, p1, p2, q1, q2 are harmonically variable deviations from the mean value of the pressure and the rate of flow respectively. When at the harmonic excitation a quasi-steady state is considered using the model with distributed parameters, the transmittance matrix assumes the following form, see Kudźma (2012) and Zarzycki (1994) H(jω)= [ h11 h12 h21 h22 ] (2.11) where the particular matrix terms are expressed by the relations h11 =cosh(Tψzjω) h12 = Zcψz sinh(Tψzjω) h21 = 1 Zcψz sinh(Tψzjω) h22 =cosh(Tψzjω) (2.12) where:Zc = ρco/(πR 2) is the characteristic impedance of the conduit,T = L/co – time constant. ψz – operator defining the influence of viscosity (a viscosity function) expressed by ψz = ψ jΩ ψ = ε+jδ (2.13) ε – coefficient of sinusoidal pressure wave amplitude damping, δ relates to wave phase velocity, j is the imaginary unit ε = √ −(Ω2+2b2Ω)+ √ (Ω2+2b2Ω)2+(2b1Ω)2 2 δ = √ (Ω2+2b2Ω)+ √ (Ω2+2b2Ω)2+(2b1Ω)2 2 (2.14) and b1 =ℜ (1 2 Ro πR4 µ +2jΩL[w] ) b2 =ℑ (1 2 Ro πR4 µ +2jΩL[w] ) (2.15) where: L[w] is simple Laplace transformation of the weighting function, Ω = ωR2/ν – dimen- sionless frequency,ω – angular frequency of excitations, Ro – constant resistance calculated from the Darcy-Weisbach formula Ro = λReµ 8πR4 (2.16) λ – dimensionless coefficient of linear frictional losses, Re – Reynolds number, µ – dynamic vi- scosity of the liquid.Whenonly the quasi-steadyfrictional losses are taken into account, relations (2.14) are reduced to the form, see Zarzycki (1994) ε = √ 1 2 Ω √√√√ −1+ √ 1+ (Ro Ω )2 δ = √ 1 2 Ω √√√√ 1+ √ 1+ (Ro Ω )2 (2.17) 562 S. Kudźma, Z. Kudźma 3. Branch damper dimensioning based on long hydraulic line equations Abranch damper is a conduit of proper length inserted at the right angle into themain conduit and stoppered at its end. The principle of operation of the branch damper is based on the in- terference of the pressure wave generated by an excitation with the pressure wave bounced off the damper and propagating in the opposite direction. Thus the branch damper dimensioning problem comes down to determining its length L0 depending on the frequency of the excitations which are to be damped. There is a prevailing view that this damper is a narrow-band damper and its effectiveness in reducing pressure fluctuations is limited to one frequency – the dam- per resonance frequency, see Kudźma (2001, 2006),Wacker (1985), Kollek andKudźma (1997), Mikota andManhartsgruber (2003). It is assumed that thedamping effectiveness sharplydimini- shes already at slight deviations from the resonance frequency.However, the above analyseswere based on the ideal liquidmodel (not representative of the real conditions) and their conclusions have not always been corroborated in operational practice (Kudźma, 2001, 2006). A schematic of the branch damper hydraulic system is shown in Fig. 1. Fig. 1. Schematic of a branch damper hydraulic system A branch damper mathematical model is defined by determining the damper input impe- dance and finding its minimum corresponding to the maximum pressure fluctuation reduction effectiveness. Three cases: lossless oscillatory flow, flow with quasi-steady losses and the case with the nonstationary friction model are considered. In order to carry out an analysis of the hydraulic system incorporating a branch damper onemust first determine the operational impe- dance ZT(s)= pT1(s)/QT1(s) in the supply station TT , where pT(s) and QT(s) are the Laplace transforms of the deviations of the pressure pT and flow rate QT . Thus one should select an im- pedance valuewhich ensures theminimumvariation of pressurepT . Treating the branch damper with length L0 as a long hydraulic line, for a harmonic excitation one can write (consistently with relations (2.12) and (2.13)) the following [ pT1 QT1 ] =   cosh(TΨzjω) ZcΨz sinh(TΨzjω) 1 ZcΨz sinh(TΨzjω) cosh(TΨzjω)   [ pT2 QT2 ] (3.1) Since the flow is blocked at the damper end, QT2 =0, the impedance Zd at the place where the branch is connected, according to equation (3.1), has the form Zd = ρocoΨz πR2 tanh L0Ψzjω co (3.2) When the lossless model is adopted, the viscosity function Ψz = 1 should be used in equation (3.2), whereas for the quasi-steady frictional losses one should use relations (2.17). The nonsta- tionary frictionmodel is taken into account through equations (2.13)-(2.17) and substituting jΩ for ŝ (the dimensionless operator of the Laplace transformation) in relation (2.9). Refined model of passive branch damper of pressure fluctuations 563 Figure 2 shows an example of how the geometric parameters of the branch pressure fluctu- ation damper are determined for the basic harmonic of pump 2110 (this type of pumpwas used for experimental verification) manufactured by the Warsaw Waryński Construction Machine- ry Plant. The dominant frequency in the pump delivery fluctuation spectrum follows from the relation fi = npztK 60 (3.3) where: np is the rotational speed of the pump shaft [rpm], zt – number of teeth, K – next number of the harmonic component, f1 = 250Hz, zt = 10 teeth and pump shaft rotational speed np = 1500rpm −1. From formula (3.3), after transformations, it follows that the initial damper impedance modulus |Zd| for the lossless model will be minimal when the following condition is satisfied ωw co L0 = Kπ+ π 2 (3.4) whereK =0,1,2, . . ..Using thedependencebetween theangular frequencyωw and frequencyfw, and the following expression for pressure wavelength λf (Kudźma, 2012) λf = co fw (3.5) one can determine (fromcondition (3.5)) lengthL0 of thebranchdamper ensuring themaximum pressure fluctuation amplitude damping for a given frequency fw as a function of length λf of the pressure wave in the pipeline L0 = λf 4 (3.6) Fig. 2. Modulus of the initial impedance |Zd| of the branch damper as a function of its length L0 for different frictionmodels at viscosity µ =30 ·10−3Ns/m2, pressure wave propagation velocity coszt =1288m/s (acc. to Kudźma (2012)) and R =4.5mm Numerous numerical studies, corroborated by experiments, indicate that in order to obtain the maximum pressure fluctuation damping for a given excitation frequency fw, the damper length calculated for the ideal liquid should be shortened by the value of correction ∆L0 deter- minedassuming thenonstationary frictionmodel and introducing the notion of relative change χ in damper length χ = ∆L0 L0 (3.7) 564 S. Kudźma, Z. Kudźma Fig. 3. Relative change χ in the branch damper length versus viscosity υ of the workingmedium The numerically determined value of coefficient χ depending on the kinematic viscosity of the oil is shown in Fig. 3. In real conditions, theoptimal lengthof thebranchdamper shouldbecalculated fromrelation L0opt = λ 4 (1−χ) (3.8) 4. Experimental verification Branch damper effectiveness tests and acoustic tests were carried out in the real loader Ł-200 boom lifting gear system incorporating P2C2110C5B26A gear pump made by the War- sawWaryńskiConstructionMachineryPlant (themanufacturer-installed pumpmodel). Figure 4 shows a schematic of the hydraulic system of the Ł-200 loader boom lifting gear which was pla- ced in a sound chamber (the drivemotor and the supply systemwere outside the chamber) with an insulating power of 50dB. Fig. 4. Schematic of Ł-200 loader boom lifting gear hydraulic systemwith throttle valves and dampers: 1 – pump, 2 – distributor R1011VF1V, 3 – throttle valve, 4 – branch damper, 5 – branch damper λ/8, 6, 7 – pressure sensors A retunable damper, with adjustable length Lo (and so with adjustable natural frequency) was designed in order to experimentally verify the method of determining (selecting a friction model) the optimal length. Figure 5 shows an axial cross section of the investigated damper. The pressure fluctuation amplitude levels pT1 in the hydraulic system versus branch damper length L0 are shown in Fig. 6. The damper whose length L0opt =1.24mwas determined on the basis of the nonstationary friction model was found to bemost effective. For the lossless model the damper length was 1.28m, but the effectiveness of the damper was by 4dB lower than that of the damper with the nonstationary friction model. Refined model of passive branch damper of pressure fluctuations 565 Fig. 5. Branch damper with retunable natural frequency: 1 – branch damper, 2 – piston, 3 – bleeder screw, 4 – copper washer, 5 – cotter pin, 6 – sealing ring, 7 – connector shell, 8 – connecting nut, 9 – cutting ring, 10 – coupling shell, 11 – nut, 12 – cutting ring Fig. 6. Pressure fluctuation amplitude levels pT1 in the hydraulic system versus the branch damper length L0 – pump delivery fluctuation first harmonic f1 =250Hz, average pressure pT =10MPa 5. Conclusion The branch damperwhose dimensioning comes down to determining its length is effective in re- ducing amplitudes only for specific frequencies. If in the first approximation the optimal length is assumed in accordance with L0 = λf1/4 (leaving out oscillatory flow resistances in the damper), the pressure fluctuation amplitude damping is obtained for the basic harmonic f1and harmo- nic 3f1, i.e. generally fw = 2K − 1, K = 1,2,3, . . .. In order to suppress even harmonics, one should assume damper length L0 = λf1/8 (λf1 – the wavelength for the basic harmonic). In terms of pressure fluctuation effectiveness, the most advantageous solution is to use a double branch damper.When the flow resistances are left out and the optimum length is adopted, the particular pressure fluctuation components are suppressed completely. In real conditions, when the nonstationary frictionmodel is assumed, the optimal length of the branchdamper for a given excitation frequency is calculated from relation (3.2). One can determine the optimal branch damper length using simplified relation (3.8) and the data shown in Fig. 3. In this case, the following regularity is observed: the higher the coefficient of viscosity of the working liquid, the shorter (by 2-15%) the optimal length in comparisonwith the length definedby formula (3.6) for 566 S. Kudźma, Z. Kudźma the ideal liquid.When the double branch damper is installed in the outlet port of the pump in theŁ-200 loader boom lifting system, the pressurefluctuation amplitude is reduced several times in the whole range of the excitation frequencies, whereby the total noise (the measure of which is the sound pressure level subject to correction) is reduced by a few to about twenty dB(A), depending on the system load, see Fig. 7. Fig. 7. Corrected sound pressure level LAdB(A) of Ł-200 loader boom lifting gear hydraulic system with the double branched damper and without damper versus the forcing pressure pT References 1. Directive 98/37/WE (in Polish) 2. Kollek W., Kudźma Z., 1997, Passive und active Metoden der Druckpulsation und Larminde- rung Hydrostatischen Systemen, II Deutsch-Polnisches Seminar Innovation und Fortschritt in der Fluidtechnik, Warsaw 3. KollekW.,KudźmaZ.,OsińskiP., 2001,Theuse of acousticholographyto locatenoise sources in hydrostatic drive systems (in Polish),Problemy Maszyn Roboczych, 17, 93-102 4. Kollek W., Kudźma Z., Stosiak M., 2008, Propagation of vibrations of the bearing elements of a heavy engineeringmachine (in Polish),Transport Przemysłowy i Maszyny Robocze, 2, 50-53 5. Kollek W., Kudźma Z., Stosiak M., Mackiewicz J., 2007,Possibilities of diagnosing cavita- tion in hydraulic systems,Archives of Civil and Mechanical Engineering, 7, 1, 61-73 6. Kudźma Z., 2001, Pressure fluctuation damper with retunable natural frequency (in Polish), Czynniki stymulujące rozwój maszyn i systemów hydraulicznych, Konferencja naukowo techniczna, Wrocław-Szklarska Poręba, 3-6.X.2001, OficynaWydawnicza PolitechnikiWrocławskiej,Wrocław 7. Kudźma Z., 2006, Passive branch pressure fluctuation damper (in Polish),Hydraulika i Pneuma- tyka, 6 8. Kudźma Z., 2012, Pressure Fluctuation and Noise Damping in Hydraulic Systems in Transient and Steady States (in Polish), OficynaWydawnicza PolitechnikiWrocławskiej,Wrocław 9. KudźmaZ.,Kudźma S., 2002,Wave phenomena in proportionally controlled hydrostatic systems (in Polish),Hydraulika i Pneumatyka, 6, 15-17 10. Mikota J., Manhartsgruber B., 2003, Transientresponse dynamics of dynamic vibration ab- sorbers for the attenuation of fluid-flow pulsations in hydraulic systems, International Journal of Fluid Power, 4, 1 11. Ohmi M., Iguchi M., 1980, Flow pattern and frictional losses in pulsating pipe flow. Part 3: General representation of turbulent flowpattern,Bulletin of JSME, December, 23, 186, 2029-2036 Refined model of passive branch damper of pressure fluctuations 567 12. Ohmi M., Kyonen S., Usui T., 1985, Numerical analysis of transient turbulent flow in a liquid line,Bulletin of JSME, May, 28, 239, 799-806 13. Osiński P., Kollek W., 2013, Assessment of energetistic measuring techniques and their appli- cation to diagnosis of acoustic condition of hydraulic machinery and equipment, Archives of Civil and Mechanical Engineering, 13, 3 14. StosiakM., 2012,Themodelling of hydraulic distributor slide-sleeve interaction,Archives of Civil and Mechanical Engineering, 12, 2, 192-197 15. Tijsseling A.S., 1996, Fluid-structure interaction in liquid-filled pipe systems: a review, Journal of Fluids and Structures, 10, 109-146 16. UrbanowiczK., ZarzyckiZ., 2012,Newefficient approximationofweighting functions for simu- lations of unsteady friction losses in liquid pipe flow,Journal of Theoretical andAppliedMechanics, 50, 2, 487-508 17. Vardy A.E., Brown J.M.B., 2003, Transient turbulent friction in smooth pipe flows, Journal of Sound and Vibration, 259, 5, 1011-1036 18. VardyA.E., Brown J.M.B., 2004,Transient turbulent friction in fully roughpipe flows, Journal of Sound and Vibration, 270, 233-257 19. Vardy A.E., Brown J.M.B., 2007, Approximation of turbulent wall shear stress in highly tran- sient pipe flows, Journal of Hydraulic Engineering, ASCE, November, 1219-1228 20. Wacker K., 1985, Schalldampfer auslagen zum Vermindern des Larmes von Hydraulikanlagen, Maschinenmarkt 21. Zarzycki Z., 1994,Resistances of nonstationary motion of liquid in closed conduits (in Polish), Prace Naukowe Politechniki Szczecińskiej, 516 22. Zarzycki Z., Kudźma S., Kudźma Z., Stosiak M., 2007, Simulation of transient flows in hydraulic system with a long liquid line, Journal of Theoretical and Applied Mechanics, 45, 4, 853-871 23. ZielkeW., 1968, Frequency-dependent friction in transient pipe flow,Transactions of the ASME, Journal of Basic Engineering, 90, 1, 109-115 Manuscript received June 13, 2014; accepted for print November 21, 2014