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Journal of Fluids and Structures

Journal of Fluids and Structures 42 (2013) 456–465

0889-97http://d

n CorrE-m

journal homepage: www.elsevier.com/locate/jfs

An experimental study on the stability of a direct springloaded poppet relief valve

C. Bazsó n, C.J. Hős n

Department of Hydrodynamic Systems, Budapest University of Technology and Economics, P.O. Box 91, 1521 Budapest, Hungary

a r t i c l e i n f o

Article history:Received 13 September 2012Accepted 16 August 2013Available online 20 September 2013

Keywords:Poppet relief valveValve chatterGrazing bifurcation

46/$ - see front matter & 2013 Elsevier Ltd.x.doi.org/10.1016/j.jfluidstructs.2013.08.008

esponding authors. Tel.: þ36 1 463 1680; faail addresses: csaba.bazso@hds.bme.hu (C. B

a b s t r a c t

This paper presents detailed experimental results on the static and dynamic behaviour ofa hydraulic pressure relief valve with poppet valve body, with a special emphasis on theparameters influencing the valve instability. The first part of the paper presents the staticmeasurements; sonic velocity in the hydraulic hose, discharge coefficient and fluid flowforces. The results are compared to the data found in the literature and a reasonableagreement was found. The second part presents dynamic measurements of valve chatter.While varying the feed flow rate, pressure and displacement time histories were recordedfor a wide range of set pressure. It is shown that the spectra of both signals have similarfrequency content, moreover, the frequency of chatter is fairly constant for a wideparameter range, both in terms of flow rate and set pressure. The regimes of qualitativelydifferent motion forms (stable operation, free, impacting and chaotic oscillations) areshown in the flow rate–set pressure parameter plane.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Direct loaded pressure relief valves are widely used in the process industry to protect the system against excessiveoverpressure, however, as it is well known, they have a tendency to become unstable and self-oscillate, see e.g. Misra et al.(2002), Botros et al. (1997), Chabane et al. (2009). This motivated a significant effort to better understand the nature of valveinstability, often referred to as valve chatter. These studies mostly deal with theoretical model development and analysisand – up to the authors best knowledge – there is a lack of systematic experimental studies.

Among the few experimental work on relief valve chatter, maybe the study of Kasai (1968) was the first one, in which theauthor developed a mathematical model considering the inlet piping and analysed the stability of the valve theoretically inthe presence of harmonic disturbance. The author obtained conditions to determine the disturbance frequency range inwhich instability occurs. Some experiments were also performed and good agreement with the theory was found. An in situexample is given by Misra et al. (2002), where a few recorded time histories are shown, mainly for model validationpurposes. Moussou et al. (2010) focused on the valve dynamics itself (without upstream or downstream piping), yet wasable to demonstrate that upon increasing the supply pressure dynamic instability occurs, especially for small lifts. Thisagrees well with the explanation often encountered in the literature (Hayashi, 2001) that essentially there are two origins ofinstability in the system, one being the valve plus reservoir (valve mode) as a stand-alone system that might be unstable byitself, while the other one is the ‘pipe mode’, i.e. the interaction of the valve and pipeline dynamics. These two modes werealso experimentally observed, see Botros et al. (1997). Typically for lower pipe lengths (up to approx. L/D¼20) the oscillation

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x: þ36 1 463 3091.azsó), csaba.hos@hds.bme.hu (C.J. Hős).

C. Bazsó, C.J. Hős / Journal of Fluids and Structures 42 (2013) 456–465 457

frequency reflects pipe's one-quarter-wave resonance frequency, which is consistent with the theoretical results by e.g.Hayashi et al. (1997). Frommann and Friedel (1998) present experimental data to validate his improved pressure surgecriterion for the compressible case. The authors also deduce that long pipelines are more dangerous from the stability pointof view. Finally, Chabane et al. (2009) studied the effect of built-up back pressure on valve chatter and found that more than10% back pressure influence the stability margin significantly.

From the theoretical modelling and analysis point of view, a large number of studies can be found. MacLeod (1985)analysed analytically the dynamic stability of a relief valve by means of linear stability analysis and found that the parameterchanges that improve the valve stability (e.g. increasing the spring stiffness or the damping) result in slower valve opening.The usage of direction-sensitive damping was proposed that dampens only the closing motion. Hayashi et al. (1997)performed a detailed numerical analysis on the chaotic vibrations with the help of the toolbox of nonlinear dynamicalsystems. An interesting outcome of this study was that self-excited oscillations may be present even below the crackingpressure. Dasgupta and Karmakar (2002) investigated the sensitivity of the system parameters on the transient responseand found that the behaviour of the system is critically influenced by the pre-compression of the main spring and thegeometry of the damper. More recently, Licskó et al. (2009) developed an analytical model of a simple hydraulic relief valveand demonstrated the richness of the dynamics: free, impacting and chaotic oscillations. Moreover, he proposed to employthe recently developed techniques of non-smooth dynamical systems given in e.g. Di Bernardo et al. (2008). Hős andChampneys (2011) presented a detailed study on the non-smooth dynamics of the system described by Licskó et al. (2009)and gave a qualitative explanation on the birth and fate of the different motion forms. Moreover, it was demonstrated thatqualitative bifurcation analysis can be used to efficiently investigate the influence of the system parameters.

Due to the recent development of CFD techniques several studies have been published which employ state-of-the-artCFD solvers (notably with deforming/moving mesh). Srikanth and Bhasker (2009) present 2-D flow analysis of a circuitbreaker valve with moving grid, which was then extended with FSI simulation by Song et al. (2010). Viel and Imagine (2011)reported the methodology of co-simulation between AMESim and CFD software for transient simulation of hydrauliccomponents in their surrounding environment. Unfortunately neither experimental nor analytical validation was given inthese studies. Yonezawa et al. (2012) in their study differentiated amplified vibration and self-excited vibration in a controlvalve by means of FSI simulations. CFD techniques also provide the ability of determination of such parameters whose valueis cumbersome to measure, for example the value of damping coefficient (see e.g. Khalak and Williamson, 1997; Vandiver,2012) or added mass (see e.g. Vandiver, 2012) due to flow induced vibration.

This paper aims to present the results of a systematic series of measurements, which focused on the influence of feedflow rate and set pressure on the dynamic behaviour of the valve. One of our primary aims was to experimentally verify themotion types described by Hős and Champneys (2011) (e.g. non-impacting and impacting oscillations) and the influence ofthe system parameters. For the sake of simplicity and to avoid additional gas dynamical effects, slightly compressible fluid(hydraulic oil) was chosen to perform the measurements with. The results will be interpreted in terms of bifurcationdiagrams, similar to those ones given by Hős and Champneys (2011).

The rest of the paper is organized as follows. The next section (Section 2) is devoted to the description of the test facilityand measurements accuracy. Then Section 3 presents the results of the steady-state measurements on discharge coefficient,fluid forces and sonic velocity. As already mentioned, of central interest is the dynamic behaviour of the relief valve with anemphasis on unstable operation, chattering. These results are presented in Section 4. First, in Section 4.1, the set pressure isfixed and the dynamic response under the variation of flow rate is shown. Then, in Section 4.2, the effect of the set pressureis also studied. The results are concluded in Section 5.

2. Experimental set-up

The sketch of the hydraulic test rig is shown in Fig. 1. A screw pump (2) driven by a variable-speed AC electric motor wasused to provide the desired flow rate. The test valve and the pump were connected by hydraulic rubber hose (3). Pressuretransducers (4, type: HBM P8-AP) were mounted at two locations along the pipeline: one close to the test valve, whileanother one close to the pressure side of the screw pump. The test valve was a direct spring-loaded one with conical valvebody and sharp seat, see the right panel of Fig. 1. The valve shaft (8) was led out of the chamber to provide direct access formeasuring the displacement of the valve body and the forces acting on the valve body. The exhaust port (10) was placed atthe top of the chamber to ensure oil lubrication between the shaft and the pre-compression thread. The diameter of thechamber was Dch¼100 mm, which, compared to the seat diameter (Ds¼15 mm), was believed to be sufficiently large not toinfluence the flow pattern at the downstream part of the valve (for details on the effect of the geometry of the downstreamchamber on the fluid force and discharge coefficient, see McCloy and McGuigan (1964) or Vaughan et al. (1992)). The springwas a linear helical spring with s¼15 kN/m stiffness.

The displacement was measured directly with an inductive displacement transducer denoted by 12 in Fig. 1 (type: BalluffBAW). The fluid force measurement was performed with the help of a load cell (type Megatron Kraftaufnehmer KM300).A metering tank was connected to the discharge of the valve to determine the flow rate, however, it was only used toestablish a calibrated correlation between the revolution number of the screw pump n, the mean system pressure Δp andthe flow rate, i.e. Q ðn;ΔpÞ. Once this relationship was measured, the flow rate was computed merely by means of therevolution number and the system pressure. The data acquisition and the signal processing were preformed with the help ofa HBM QuantumX device, that allowed 9.6 kHz sampling frequency in the case of the static measurements and 19.2 kHz for

Fig. 1. Sketch of the experimental test-rig. 1 – hydraulic oil tank, 2 – screw pump, 3 – transmission line, 4 – pressure-transducers, 5 – signal processingdevices, 6 – inlet, 7 – valve body, 8 – valve shaft, 9 – spring, 10 – outlet, 11 – pre-compression thread, 12 – inductive displacement transducer.

Table 1Parameters of the test-rig.

Quantity Symbol Value Units

Valve mass mv 0.458 kgSpring mass ms 0.117 kgSpring stiffness s 15 kN/mHalf cone angle of poppet α 30 deg.Pipe diameter Dp 10 mmPipe length Lp 4.5 mSeat diameter Ds 15 mmDensity of fluid ρ 870 kg/m3

Kinematic viscosity of fluid ν 20 mm2/s

C. Bazsó, C.J. Hős / Journal of Fluids and Structures 42 (2013) 456–465458

dynamic ones. The measurements were carried out at temperatures between 28 and 321C. The parameters of the test-rigand the fluid are presented in Table 1.

Force, pressure, and pump revolution number were measured directly, the flow rate was interpolated based on thepressure and revolution number using previous, calibrated data sets corresponding to the pump performance curve. Theoverall (absolute) accuracy of the force measurement was 3.1 N, while the relative error of the revolution number andpressure measurement was below 1%.

3. Static measurements

3.1. Discharge coefficient

It is conventional in the corresponding literature (e.g. Borutzky et al., 2002; Urata, 1969; Takenaka, 1964) to assume fullydeveloped turbulent flow through the valve and use the orifice equation for calculating the flow rate, that is

Q ¼ CdA xð Þffiffiffiffiffiffiffiffiffi2Δpρ

s; ð1Þ

where Δp is the pressure drop, ρ is the fluid density, A(x) is the flow-through area and Cd is the discharge coefficient. Theflow-through area for sharp seat, conical valve body and small lifts can be calculated as (see Urata, 1969)

AðxÞ ¼Dsπh¼Dsπ sin ðαÞx; ð2Þwhere Ds is the diameter of the inlet and h¼ sin ðαÞx is the gap width. Although the discharge coefficient Cd depends on twoparameters primarily, which are the valve lift and system pressure (see e.g. Bergada and Watton, 2004; Merritt, 1967;Takenaka, 1964), it is usually assumed in the literature that it can be described by the (gap) Reynolds number as a singleparameter (see Borutzky et al., 2002; Merritt, 1967; Takenaka, 1964). By considering the mean velocity v in the gap withflow-through area A(x), the clearance h between the seat and the valve as characteristic length and the kinematic viscosity ν,the Reynolds number takes the form:

Re¼ vhν¼ QDsπν

: ð3Þ

0 100 200 300 4000

0.2

0.4

0.6

0.8

1

Re

Cd

x = 0.1 mmx = 0.15 mmx = 0.2 mmx = 0.25 mmx = 0.3 mmx = 0.35 mmCd = B(Re)C

Fig. 2. Discharge coefficient as a function of Reynolds number.

Fig. 3. Measurement of fluid force.

C. Bazsó, C.J. Hős / Journal of Fluids and Structures 42 (2013) 456–465 459

Simultaneous measurement of displacement, pressure and flow rate allowed us to estimate the value of dischargecoefficient. The measurement was carried out as follows. The spring was replaced by a positioner ring to set fixed valvedisplacements of x¼0.1, 0.15, 0.2, 0.25, 0.3, and 0.35 mm. Then, for each displacement, the flow rate was increasedsystematically up to the maximum flow rate of the screw pump (Q ¼ 1:5…24 l=min).

The measured discharge coefficients are shown in Fig. 2 as a function of the Reynolds number (3).Merritt (1967) and Takenaka (1964) approximated Cd as the square root function of the Reynolds number Cd ¼ f ðRe1=2Þ.

However, in our case, a curve fit assuming square root dependence did not give satisfactory accuracy, hence the exponent ofthe Reynolds number was also considered as a free parameter, i.e. Cd ¼ BReC . By simple logarithmic curve fit it was foundthat B¼0.2028 and C¼0.2358 (goodness of fit: R2 ¼ 0:9090), which are also shown in Fig. 2. By performing errorpropagation analysis it was found that the maximum absolute error of the Reynolds number was ERe;max ¼ 71:3� 10�4

while maximum absolute error of the discharge coefficient was ECd ;max ¼ 70:012. The dominant parameter influencing theaccuracy of Reynolds number and of the discharge coefficient was the valve positioning accuracy.

3.2. Fluid force

In order to measure the fluid force acting on the valve body, the spring was again removed and a load cell was mountedto the valve shaft, see Fig. 3.

The theoretical force is given by Bergada and Watton (2004):

F ¼ FpþFmom ¼ΔpAs�Q2ρ1As� cos ðαÞ

AðxÞ

� �; ð4Þ

100

200

300

F [N

] pset = 5.3 bar

−5

0

5

erro

r [%

]

100

200

300F

[N] pset = 7.6 bar

−5

0

5

erro

r [%

]

200

300

400

F [N

] pset = 12.0 bar

−5

0

5

erro

r [%

]200

300

400

F [N

]

pset = 13.9 bar

−5

0

5

erro

r [%

]

0 5 10 15 20300

400

500

Q [l/min]

F [N

] pset = 17.5 bar

0 5 10 15 20−5

0

5

Q [l/min]

erro

r [%

]

Fig. 4. Fluid force on the valve body. Left panel: absolute measured values. Dots represent the measurements, plus signs and crosses stand for thetheoretical fluid force given by (4) without and with the momentum term, respectively. Right panel: Relative differences between (a) plus signs: themeasured force and the pressure term in (4) and (b) crosses: the measured force and both terms in (4). Solid lines represent the error boundaries (see textfor details). The rows correspond to different spring pre-compression.

C. Bazsó, C.J. Hős / Journal of Fluids and Structures 42 (2013) 456–465460

where Fp stands for the pressure force, Fmom stands for the momentum force and α is the half-cone angle. We employ theusual assumption that the flow angle through the gap equals the half cone angle, see Urata (1969) and McCloy andMcGuigan (1964). Fig. 4 compares the measured results and the analytical prediction; the left column shows the measuredvalues (dots) and the theoretical fluid force given by (4) without (plus sign) and with (crosses) the momentum term,respectively. The right column depicts the discrepancy between them, defined by

error %½ � ¼ Fmeasured�FFmeasured

: ð5Þ

It is clearly seen that the higher the flow rate is, the more significant the momentum term in (4) becomes, which isconsistent with the results of Urata (1969). Note that for our particular geometry and parameters, by neglecting themomentum term in (4), the relative difference between the measurements and the pressure force is mostly below 3%. Eq. (4)tends to underestimate the force in the case of small flow rates, yet the error is below 5% in the whole parameter range. Theerror boundaries of the force measurement are depicted by the solid lines.

3.3. Hydraulic capacity and sonic velocity

The hydraulic capacity of the transmission essentially measures the ‘stiffness’ of the system and thus plays an importantrole in the dynamic of the system, see Vašina and Hružík (2009), Keramat et al. (2012). It is given by

CH ¼ Vρa2

; with a¼ffiffiffiffiffiffiffiffiEredρ

s; ð6Þ

where V is the oil volume in the hose, a is the sonic velocity in the fluid, and Ered is the reduced bulk modulus including boththe fluid and pipe wall elasticity. Although there are formulae available (e.g. Rabie, 2009; Wiley and Streeter, 1978; Keramatet al., 2012) to compute Ered, as the hydraulic hose is neither thin-walled nor linearly elastic, it was found to be morestraightforward to measure it with the help of two pressure transducers as follows. After the revolution number was set, the

C. Bazsó, C.J. Hős / Journal of Fluids and Structures 42 (2013) 456–465 461

system was excited by hitting the valve body with a hammer. By correlating the two pressure signals and finding the timedelay between them, the sonic velocity could be simply computed. The average value of sonic velocity was found to bea¼1062 m/s but its characteristic shows a slight increase as pressure rises, which coincides with the results of Vašina andHružík (2009) (Fig. 5).

4. Dynamic measurements

The main focus of this study is on valve chatter measurements, notably on the parametric stability boundaries and thedominant frequency of chatter. Thus dynamic measurements were performed for different set pressure values (i.e. springpre-compression values, x0), in which, after starting from low flow rates, the revolution number of the screw pump (thus theflow rate) was increased with small steps up to the maximum value (approx. 23 l=min) and then decreased backwards to seeif hysteresis occurs. At each flow rate, pressure and displacement time histories were recorded for 4 s with 19.2 kHzsampling frequency. This resulted in a total number of 60 measurements per set pressure.

0 5 10 15 20 25 300

500

1000

1500

p [bar]

a [m

/s]

Fig. 5. Sonic velocity as a function of system pressure.

0 5 10 15 20 250

0.2

0.4

0.6

Q [l/min]

x [m

m]

(b)(c)(d)(e)

0 0.1 0.2 0.30

0.2

0.4

0.6

t [s]

0 0.02 0.04 0.06 0.080

0.10.2

0 0.05 0.10

0.2

0.4

0.6

t [s]

x [m

m]

0 0.05 0.10

0.2

0.4

0.6

0 0.05 0.10

0.2

0.4

0.6

x [m

m]

Fig. 6. Panel (a): Measured bifurcation diagram showing the behaviour of the system while (slowly) varying the flow rate. Panels (b)–(e): Displacementtime histories at flow rates indicated in panel (a). Spring pre-compression: x0¼17 mm.

C. Bazsó, C.J. Hős / Journal of Fluids and Structures 42 (2013) 456–465462

4.1. One-parameter study: effect of flow rate at fixed set pressure

We start by describing the typical behaviour of the system while varying the flow rate, for a fixed set pressure. Theresults will be presented by means of bifurcation diagrams as in panel (a) of Fig. 6, where those displacement values areshown which correspond to zero velocity or impact with the seat. Panels (b)–(e) of the same figure show the four typicalmotion types, that is (b) stable equilibrium, (c) oscillation without impact with the seat, (d) regular impacting oscillation and(e) chaos-like impacting oscillation. Note that this experimentally obtained diagram is qualitatively similar to that onepresented by Licskó et al. (2009) and Hős and Champneys (2011), which was obtained by numerical modelling.

As it can be observed in Fig. 6, the valve motion is stable for high flow rates and, upon decreasing the flow, a critical valueis reached at 20:4 l=min where the valve looses its stability and chatter appears. As described by Hős and Champneys (2011),the oscillation is born via a Hopf bifurcation, i.e. there is a pair of purely complex eigenvalues of the linearized system. Byfurther decreasing the flow rate, the amplitude of the oscillation grows and, at approx. 14:7 l=min, it reaches the seat and

0 200 400 6000

5

10

15

20

25

f [Hz]

Q [l

/min

]

0 200 400 6000

5

10

15

20

25

f [Hz]

Fig. 7. Frequency content of the self-excited vibration for x0¼17 mm (left: frequency map of the displacement, right: frequency map of the pressure beforethe valve). Cross, asterisk, and rectangle represents the point of primary Hopf bifurcation, grazing bifurcation, and chaotic chaterring, respectively.

0 5 10 15 20 250

0.5

1

x [m

m]

x0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mmx0 = 13.5 mm

0 5 10 15 20 250

0.5

1x0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mmx0 = 15 mm

0 5 10 15 20 250

0.5

1

x [m

m]

x0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mmx0 = 17 mm

0 5 10 15 20 250

0.5

1x0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mmx0 = 19 mm

0 5 10 15 20 250

0.5

1

Q [l/min]

x [m

m]

x0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mmx0 = 21 mm

0 5 10 15 20 250

0.5

1

Q [l/min]

x0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mmx0 = 23 mm

Fig. 8. Measured bifurcation diagrams for different spring pre-compression values. Cross, plus sign, asterisk, and rectangle represents the point of primaryHopf bifurcation, secondary Hopf bifurcation, grazing bifurcation, and chaotic chaterring, respectively.

C. Bazsó, C.J. Hős / Journal of Fluids and Structures 42 (2013) 456–465 463

the impacting oscillation regime starts. The point where the valve body first ‘grazes’ the seat is called a grazing bifurcationand is a unique feature of non-smooth dynamical systems, for details see e.g. Di Bernardo et al. (2008). After the grazingbifurcation point the oscillation amplitude decreases with decreasing flow rate. For low flow rates (below 4:15 l=min) weexperience highly complicated, chaos-like motion form. Finally, at 2:07 l=min the system becomes stable again.

Fig. 7 shows the dominant frequency content of the measured signals for (left) displacement and (right) pressure signals.Note that the two signals contain essentially the same frequency components. In the free-oscillation regime, the dominantfrequency is 238 Hz, which is clearly the pipe eigenfrequency fpipe¼a/L¼1064/4.5¼236.4 Hz. Once the impacting oscillationappears (at approx. 10 l=min.), there is a slight increase in the frequency, which is caused by the fact that – roughly speaking– the impact interrupts and ‘cuts off’ a portion of the free oscillation. Finally, in the chaotic regime, we observe a widerfrequency content, which is due to the many random-like impacts with the seat.

4.2. Two-parameter study: effect of set pressure

Next, we present the effect of set pressure by analysing the bifurcation diagrams obtained by setting different spring pre-compression, see Fig. 8.

First, notice that upon increasing the set pressure, the point of the initial instability denoted by cross is also increasing,i.e. the unstable region increases. For low set pressures (x0¼13.5 and 15 mm) we observe a second critical flow rate, belowwhich the motion stabilizes. Although this point was not captured by the model of Hős and Champneys (2011), we speculatethat it is a secondary Hopf bifurcation. For higher set pressures, this point becomes hard to identify, and it is also unclear if itpersists or is destroyed.

The same mechanism applies for the rest of the regimes: the chaotic and the impacting regimes also become wider. Inthe case of the last three set pressures, no stable valve motion was experienced neither at high nor at low flow rates.

The appearing frequency at the primary stability loss is listed in Table 2. For a given set pressure value, this frequencyremained constant up to the occurrence of impacting oscillations, as already shown in Fig. 7. We emphasize again that thesefrequencies are very close to the pipe eigenfrequency, which is 236 Hz.

Finally, we present the essence of the measurements in Fig. 9, where all the previously described special points aredepicted on the flow rate–set pressure plane. Previous studies (Licskó et al., 2009; Hős and Champneys, 2011) dealing with

Table 2Critical flow rate and oscillation frequency at the initial stability loss.

Spring pre-compression (mm) Set pressure (bar) Flow rate (l/min) Frequency (Hz)

13.5 11.46 8.033 244.914 11.88 10.69 243.214.5 12.31 12.44 241.415 12.73 14.15 239.715.5 13.16 15.36 239.716 13.58 17.28 237.917 14.43 20.42 237.9

0

0.2

0.4

0.6

0.8

1

q [−]

0 5 10 15 20 pset [bar], δ [−]5 10 15 20 25

5

10

15

20

25

x0 [mm]

Q [l/s]

Stable Untable

Boundary of loss ofstabilityPrimary Hopf bifurcationSecondary Hopf bifurcationGrazing bifurcationChaotic chattering

Fig. 9. Boundary of loss of stability and the experienced types of stability losses.

C. Bazsó, C.J. Hős / Journal of Fluids and Structures 42 (2013) 456–465464

such pressure relief valve system analysed the dynamics in terms of dimensionless parameters such as compressibilityparameter β, pre-compression parameter δ, and dimensionless flow rate q that are calculated for the particular set-up asfollows:

β¼ EredV

Qref

p0ffiffiffiffiffiffiffiffiffis=m

p ¼ 57:4; ð7Þ

δ¼ sx0A p0

¼ psetp0

¼ 0:8491

mm

� �� x0 mm½ �; ð8Þ

q¼ Q ½l=m�Qref ½l=m�; with Qref ¼ 21:5 l=m

� �: ð9Þ

Notice that the δ spring precompression parameter's numerical value coincides with the set pressure given in bars. Thesenormalized values are also depicted in Fig. 9 giving more generality to the results. It also worth mentioning that therelatively high value of β is typical for liquid systems. In the case of highly compressible fluids (gases), this value is wellbelow 1.

5. Conclusion

A systematic experimental study was presented on relief valve instability for slightly compressible fluid (hydraulic oil).The experimental system consisted of a positive displacement pump, a simple direct spring loaded valve and a hydraulichose connecting them. Pressure and displacement time histories were recorded for a large number of flow rates and setpressures.

We have experimentally validated the qualitative bifurcation diagram given by Hős and Champneys (2011). Theexperiments show that for high flow rates, the valve equilibrium is stable, which, upon decreasing the flow rate, looses itsstability via a Hopf bifurcation. A free, non-impacting oscillation is born, whose amplitude increases with decreasing theflow rate and once the valve body reaches the seat impacting periodic orbit is born, whose amplitude decreases withdecreasing flow rate. At low flow rates, highly nontrivial, chaos-like impacting motions were observed. Finally, for very lowflow rates, the valve stabilizes again, but only for low set pressures. Upon increasing the set pressure the unstable regimeexpands quickly and vice versa: a critical set pressure can be found, below which the valve is stable for all flow rates.

An interesting outcome of our study was that although the experimental results qualitatively agreed with the ‘no-pipe-model’ presented by Hős and Champneys (2011), the oscillation frequency remained constant for a wide parameter range(both in terms of flow rate and set pressure). Moreover, this frequency coincides with the pipe eigenfrequency a/L, whichsuggests that the initial valve stability loss (Hopf bifurcation) immediately couples with the pipe's internal dynamics and thelatter dominates the behaviour of the system. Future theoretical studies should also take into account its internal dynamics.

From a more practical point of view it was clearly seen that there is a critical spring pre-compression below which thevalve is unconditionally stable. Qualitatively explaining, the higher the spring pre-compression is, the smaller the valveopenings are and the more intense the acoustical feedback inside the pipe is (see Misra et al., 2002). This critical spring pre-compression can be found by simple linear stability analysis during the design phase. It is also important to emphasize thatthe valve itself is neither stable nor unstable and one cannot design a ‘stable valve’without taking into account the system itis connected to, notably the connecting pipeline.

Acknowledgements

This research was supported by the János Bolyai research grant of the Hungarian Academy of sciences of C. Hős and hasbeen developed in the framework of the project “Talent care and cultivation in the scientific workshops of BME” project.This project was supported by the Grant TÁMOP-4.2.2.B-10/1-2010-0009.

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