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8/13/2019 PH-LecNotes-HH4 (AY2013-V1.0) http://slidepdf.com/reader/full/ph-lecnotes-hh4-ay2013-v10 1/26 PAGE 1 Power Hydraulics Module PH-LecNotes-HH4 (AY2013-V1.0) A valve is a device which, upon input of a mechanical, pressure or electrical signal, changes the direction , pressure or flow of the fluid passing through it. In a hydraulic circuit, the symbol of a valve consists of squares, representing its positions. As in pneumatic and electro-pneumatic circuits, valves are shown in their initial positions. 1. VALVE POSITION 1.1 Finite Position Finite-position valves have fixed or finite number of positions represented by squares. E.g. in the case of a 5/3 way valve (figure 1), there are three positions that the valve can assume and therefore there are three squares. In the left-hand position, full flow occurs from P to A. But in the right-hand position, full flow occurs from P to B. In the mid-position, there is no flow at all. 1.2 Infinite Position Flow through an infinite-position valve varies steplessly from no flow to full flow. It is represented by one square with an arrow showing the direction of flow (figure 2). If the arrow is in line with the port connection the valve is normally open . If it is offset from the port connection the valve is normally closed . C HAPTER H4 H YDRAULIC VA LV E S Figure 1 Figure 2 Normally Open Normally Closed

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PA G E 1

P o w e r H y d r a u l i c s M o d u l e P H - L e c N o t e s - H H 4 ( AY 2 0 1 3 - V 1 . 0 )

A valve is a device which, upon input of a mechanical, pressure or electrical signal, changes thedirection , pressure or flow of the fluid passing through it. In a hydraulic circuit, the symbol of a valveconsists of squares, representing its positions. As in pneumatic and electro-pneumatic circuits, valvesare shown in their initial positions.

1. VALVE POSITION

1.1 Finite PositionFinite-position valves have fixed or finite number of positions represented by squares. E.g. in thecase of a 5/3 way valve (figure 1), there are three positions that the valve can assume andtherefore there are three squares. In the left-hand position, full flow occurs from P to A. But in theright-hand position, full flow occurs from P to B. In the mid-position, there is no flow at all.

1.2 Infinite Position

Flow through an infinite-position valve varies steplessly from no flow to full flow. It is representedby one square with an arrow showing the direction of flow (figure 2). If the arrow is in line with the

port connection the valve is normally open . If it is offset from the port connection the valve isnormally closed .

C HAPTER H4

H YDRAULIC V ALV ES

Figure 1

Figure 2

Normally Open Normally Closed

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2. TYPES OF VALVES

In general, valves can be classified into three main categories according to their functions:

3. DIRECTIONAL CONTROL VALVES

Directional Valves

Function and ClassificationDirectional valves control the directions of flow. In hydraulic systems they are used to: Control direction of actuator motion Select alternative control circuits Perform logic functions

Directional valves are classified according to their design: Type of valve element e.g. Ball poppet, disc poppet, rotary spool, sliding spool, etc. Method of actuation e.g. Solenoid, fluid pilot, manual, mechanical, etc. Number of flow paths and positions, e.g. 4/2, 5/3, etc. Size e.g. l/min, gpm, etc. Types of connections e.g. straight threads, pipe threads, flanged mounting, sub-plate

mounting, etc.

Sliding-Spool ValvesMany directional valves are of the sliding spool construction. A sliding spool in the valve housingdirects fluid to different ports or stops the flow. The grooves on the spool allow flow while its landblocks flow (figure 3). The valve directs fluid from inlet port P to either port A or port B. Port Tpermits fluid to return to the reservoir. Figure 3 shows a 5-way, 2-position valve. In the case of a 3-

position valve there is a centre or neutral position.

Figure 3

Directional cont rol valve

Directs flow of fluid to therequired line.

e.g. Directional valve,check valve

Pressure control valve

Regulates maximumpressure in a circuit.

e.g. Pressure relief valve,sequence valve, counter-balance valve, pressurereducing valve

Flow control valve

Changes speed ofactuator by varying thefluid flowrate .

e.g. Non-pressurecompensated valve,pressure compensatedvalve, pressure-cum-temperature compensated

A B

T P T

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Leakage across Sliding-spool Valves

In a sliding-spool valve, fluid will leak from a pressurized port to one of lower pressure.Pressurized cavities are sealed from non-pressurized cavities by virtue of an oil film at the smallclearance between the land and the bore. The leakage past this clearance depends on theclearance fit, fluid viscosity, overlapping length of the land and bore and the pressure differencebetween the high and low-pressure sides.

Valve Centre Positions and their Applications

Three-position spool valves have a variety of interchangeable spools. Figure 4 shows four typesof commonly used spool centers. These valves have identical flow patterns in the shiftedpositions but different center conditions for different applications. Spools may be held in theircentre positions by centering springs or spring-loaded detents.

Figure 4

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The following explains the properties of the four types of spool centre :

Open-Centre Spool

An open centre connects all ports. The pump flow is directed totank at low pressure. The piston is free to move with the load.

While the spool is in the centre position, obviously otheractuators supplied by the same pump will not be able to do worksince the flow is always at atmospheric pressure.

Piston floats with loadmovement

Low resistancepath back to tank

Figure 4(a)

A B

P T

A B

P T

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Closed-Centre Spool

In a closed centre all ports are blocked. The cylinder piston willbe locked. However the locking will be possible for a shortperiod only, otherwise pressure that is building up in bothactuator lines due to leakage will cause the piston to drift

gradually. In the centre position, flow is re-directed to tankthrough the relief valve at high pressure, and therefore muchenergy is wasted and converted into heat.

When the piston is locked, energy could be re-channeled to independent operation of otheractuators supplied by the same pump. The circuit below shows how closed-centre valves canbe connected in parallel controlling three actuators.

Piston locks

Pump flow re-direct ed totank via relief valve- Energy Inefficient

Figure 4(b)

A B

P T

A B

P T

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Tandem-Centre Spool

A tandem centre allows the actuator to be held in anintermediate position for only a short time (for the same reasonas the closed-centre spool) while directing the pump flow to thetank. Tandem-centre creates high pressure drop of about 300

kN/m2

or 3 bar.

Figure below shows that two directional valves with tandem centre can be arranged in a seriesswitching circuit.

Piston locks

Low resistancepath back to tank

Figure 4(c)

Allows for operationof another actuator

A B

P T

A B

P T

A B

P T

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Float-Centre Spool

Since both actuator lines can return to tank, the pressure doesnot build up in these lines, unlike the closed centre and thetandem centre. A float centre hence allows the piston to floatwhile the pump flow is unloaded at maximum operating to the

tank via the system relief valve.To save energy, the pump flow could be re-directed to anotheractuator for independent operation (parallel connection). When itis used in conjunction with pilot check valves (see Section 3.2),the load can be stopped in an intermediate position for a longperiod.

Other Centre Valve Positions

Also available are valves made in other configurations. An example is shown in figure 5, inwhich switching of the valve to the left-hand position results in a regenerative circuit, and theright-hand position a normal circuit.

Normal Positionfor retractionstroke

Figure 5

RegenerativePosition forextension stroke

Pump flow can bere-directed tooperate anotheractuator

Piston floats

Figure 4(d)

A B

P T

A B

P T

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3.2 Check Valves

Check valves are one-way directional valves. A check valve permits free flow in one direction andblocks flow in the reverse direction.

(a) Check Valve or Non-Return Valve

The simplest type of directional control valve is the non-return valve or check valve, which allowsfree flow in one direction and blocks flow in the other. The symbol and characteristic of the valve isshown in Figure 6.

Valve operation In the free flow direction, flow usually begins to occur at about 0.34 bar pressure drop. This

is known as the cracking pressure of the valve. Full flow pressure occurs at a higherpressure.

In the reverse flow direction, the poppet or ball is pushed against its seat blocking all flow. If a specific cracking pressure is essential to the functioning of a circuit, it is usually required

to show a spring on the check-valve symbol.

(b) Pilot-Operated Check Valve

A pilot-operated check valve, whose symbol is shown in figure 7, permits free flow in onedirection and blocks flow in the other direction, until opened by a pilot-pressure signal at its pilotport (as indicated by the dotted line). The poppet is loaded against its seat by a light, non-adjustable spring when pilot pressure is not present. It operates like an ordinary check valve.

Appl ication of Pilot-Operated Check Valves

Pilot-operated check valves are commonly used in numerous circuit applications. We shall discussits application in a circuit for locking a cylinder (figure 8),

Free flow No flow

Figure 6

Pilot signal Figure 7

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The use of closed-centre or tandem-centre directional valve does not ensure that the cylinder willremain in an intermediate position for an indefinite length of time because pressurized oil will leakthrough ports A and B into the cylinder. The use of a pilot-operated check valve can solve theproblem.

The circuit operates in the following manner: When no signal is applied to the directional valve, the piston remains stationary as the pilot

check valve prevents the pressurized fluid in the cylinder from leaking through the directionalvalve thus locking the piston in position.

When the piston is signaled to extend, the pilot line to the right-hand check valve ispressurized and the fluid in the rod end can return to tank.

The same principle applies when the piston is signaled to retract.

4. PRESSURE CONTROL VALVES

Pressure control valves perform functions such as limiting maximum system pressure or regulatingreduced pressures in certain parts of a circuit. They are named for their primary function, such as reliefvalve, sequence valve, pressure reducing valve, etc.

4.1 General Principle of Operation

The operation is based on a balance between pressure and spring force. Most are infinitepositioning, i.e. they can assume any position between fully open and fully closed, depending onflowrate and pressure differential. The symbolic representation of a simple pressure control valveis shown in figure 9.

Figure 8

Figure 9

in

drain

out

pilot

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4.2 Pressure Relief Valve

Function

A simple relief valve limits the maximum pressure in a system to prevent damage to the hydrauliccomponents and lines by diverting flow to the tank when pressure rises to a set value. The pumpdraws maximum power when the relief valve operates as it will be pumping the full flow againstthe maximum pressure.

Appl ication of Pressure Relief Valve

In this simple hydraulic circuit, a pump sucks fluid from the tank and pushes it into the system.

The circuit operates in the following manner: When the directional valve is at the center position, the hydraulic fluid circulates almost

without pressure (no resistance to flow) from the pump back to the tank. When 1SOL is energised, cylinder extends and the extension speed depends on the pump

flowrate and the cylinder area. When the piston is fully extended or when it engages the load, flow from pump encounters

resistance and pressure builds up in the system. When the cracking pressure is reached, some fluid will be diverted to tank via the relief

valve. As the pressure increases, more fluid will be diverted until full-flow pressure is reached,

during which all pump flow is diverted back to tank.

External and Internal Drainage

As fluid tends to leak past the lands of the spools, the leakage to the spring chamber will causepressure build-up. In order to shift the spools, the pilot pressure acting on the spool will not onlyhave to overcome the spring force, but also the oil pressure in the spring chamber. But as thepressure in the spring chamber continues to build up, eventually the pilot pressure will not beable to shift the spool. It is therefore important to drain the spring chamber.

For some pressure valves with the valve’s outlet leading to the tank, a drain passage can be

provided within the valve itself to lead the leakage flow to the outlet of the valve. This is knownas internal drainage . If the outlet is pressurized, a separate external drain to the tank is required.

Figure10

2SOL1SOL

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4.3 Sequence Valve

Function A sequence valve is used to direct fluid to a secondary line while maintaining a set pressure inthe primary (inlet) line.

Appl ication A typical application of the valve is in sequencing the cylinder action in a drilling workstation as

shown in figure 12.

When 1SOL is actuated, cylinder A extends, while sequence valve S1 prevents flow fromentering cylinder B. When cylinder A extends fully and clamps the workpiece, resistance isencountered and system pressure increases. The system pressure will build up until it reachesthe setting of S1 whereupon it switches. Flow is then directed to cylinder B causing it to extend.The pressure at this time is the setting of sequence valve S1. When cylinder B encounters theload, system pressure continues to rise. When the system pressure reaches the relief valvesetting, the relief valve opens and direct the flow back to the tank.

Note that in the circuit, the sequence valve, which is pressure-controlled, is used to controlsequencing of cylinders. The sequence valve, however, does not ensure that cylinder B willextend only after cylinder A completes its stroke during extension.

Figure11

Direct Control Remote Control

S1SequenceValve

CYL B

CYL A

Drill Head

ClampingCylinder

2SOL1SOL

Figure12

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4.4 Pressure reducing valve

Function Pressure reducing valve is a normally open valve used to maintain reduced pressure in the primarycircuit. It is actuated by pressure sensed in the secondary circuit and tends to close as it reachesthe valve setting, thus preventing further pressure build-up in the primary circuit.

Appl ication in a Pressure-Cont rol Clamping Circui tIn Figure 12, it can be seen that after the drilling cylinder encountered the load, the systempressure would rise and hence the clamping pressure would also increase. If it is desired to

maintain a certain clamping pressure that is less than the maximum system pressure, a pressure-reducing valve is required as shown in Figure 14.

Note:

The sequence valve ensures that the clamp pressure is maintained at a pre-determinedminimum during the extending stroke of the drill cylinder.

Pressure reducing valve limits pressure to a safe maximum when higher pressure is required fordrilling operation at the drill cylinder.

Figure13

Figure14

SequenceValve

Dri ll Cyl inderworkhead

Pressure ReducingValve

CYL B

CYL A

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Worked Example 1

A drilling fixture uses the hydraulic circuit, which is shown in the figure. Because of the shape of theworkpiece, clamp cylinder A has to be placed vertically while drill cylinder B is positionedhorizontally. The clamping force required is to be at least 50 kN before drilling operation can begin.The clamping device weighs 400 kg. The system relief valve is set at 200 bar and the sequencevalve 100 bar. Assume that the frictional resistance is negligible.

a) Determine the required bore diameter of cylinder A (note that the preferred cylinder diameterwould be 63mm, 78mm, 100mm or 110mm). (Ans. 100mm)

b) Determine the maximum and minimum output forces of drill cylinder B, if its bore diameter is 12mm. (Ans. 2262N, 1131N)

c) In the circuit, when the directional valve is switched, clamp cylinder A will retract either ahead of,or simultaneously, with drill cylinder B. This is not satisfactory. Modify the circuit to overcomethis deficiency.

Solution:

a) Cyl B begins drilling operation whenPressure reaches 100 bar.

Forces on Cylinder A at this pressure are:

mmor m Dbore

m A

N A

9.820829.0.min

103924.510200392450000

50000392410200

235

5

The selected bore diameter for cylinder A is 100 mm.

b) min drill pressure is at 100 bar and max drill pressure is at 200 bar.

Cyl B Bore Area = 2424 10131.1012.0 m

Min. output force from the drill cylinder = N 113110131.110100 45

Max. output force from the drill cylinder = N 226210131.110200 45

Cyl B

Cyl A

400 kgor3924 N

50 kN net force

200 bar

3924 N

100 bar x A

Net50 kN

=

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c) The clamp cylinder A will have a sequence valve at its rod end. The system pressure needto overcome the pressure set by this sequence valve before cylinder A can retract.

Cyl A

* Add a Sequence Valve

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4.5 Counterbalance Valve

FunctionThe counterbalance valve is used to create a back pressure or cushioning pressure on theunderside of a vertically moving piston to prevent the load from "running away" because of

gravity as it is being lowered.

The primary port of the valve is connected to the lower cylinder port and the secondary port to

the directional control valve. The pressure setting is slightly higher (usually about 30%) than isrequired to hold the load to prevent it from falling (figure 16).

Appl icationIn Figure 16, the load being lowered is prevented from falling freely due to gravity by thecounterbalance valve as shown.

Figure15

Figure16

Counter-balancingValve

Load

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During Raising

Since the valve is normally closed, a reverse free-flow check valve is provided. The hydraulic fluidflows freely past the check valve to raise the

cylinder.

During Suspension

The spring-controlled back pressure is greaterthan the load pressure and the load is heldsuspended. However, because of leakage pastthe spool valves, a pilot-operated check valve isrequired if the suspension is required for aprolonged period.

During Lowering

The spring tends to close the valve thatprevents the load from falling freely.

To force open the counterbalance valveagainst the spring, the cylinder piston must beforced down to lower the load. This causes thepressure in the bore end of the cylinder to rise,switching the counterbalance valve, thusproviding a flow path.

The return flow from the cylinder thendischarges to tank via the directional valve.

Internal DrainagesThe counterbalance valve can be internally drained since the secondary port is connected totank during lowering and the check valve bypasses the spool during raising. In construction, thecounterbalance valve is similar to the sequence valve except that the top cap is bolted in adifferent manner to convert the external drain of one into the internal drain of the other.

Resistance fromback pressure

B - ass

Load

Resistance fromback pressure

Load + P x A

The counter-balancingvalve opens

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Worked Example 2

The counterbalance valve used in the circuit creates a back pressure to prevent the load fromrunning away during lowering. The usual pressure setting is 1.3 times the load-induced pressure.

(a) With a load of 10 kN and a cylinder bore area of 0.002 m 2, determine the counterbalance valvesetting.

(b) Modify the circuit such that the cylinder is able to suspend the load at standstill for a prolongedduration.

Solution:

(a) At the suspended position

Load Induced pressure

bar

m N

50

002.010000

2

Counter-balancing valve setting

= 1.3 times x Load Induced pressure

bar 65

503.1

(b) The back pressure 65 bar is greaterthan the load pressure and the load isheld suspended. However, because ofleakage past the spool valves, a pilot-operated check valve is required if thesuspension is required for a prolongedperiod.

10 kN Load

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SUMMARY OF PRESSURE CONTROL VALVES

Name SymbolNormal

PositionSource of

Pilot Signal Drain

PressureRelief Valve Closed Inlet Internal

SequenceValve Closed

Inlet(Direct

Control)External

Circuit(RemoteControl) External

CounterbalanceValve

Closed Inlet Internal

PressureReducing Valve

Open Outlet External

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5. FLOW CONTROL VALVES

Flow control valves are used to control the flowrate of hydraulic fluid into the actuators in order toregulate their speed. This is simply because the speed of an actuator is dependent on the amount offluid that is pushed into it per unit time. It is also possible to regulate the speed with a variable-displacement pump, but in many hydraulic drive applications, a fixed displacement pump is preferred andthe flow is then regulated using flow control valves.

Figure 17 shows the symbolic representation of a flow control valve. Generally a check valve isincorporated to allow free flow in the reverse direction while flow is regulated in the forward direction.

Figure 17

5.1 Flow Control Methods

There are three ways that flow control valves can be used in a hydraulic circuit.

Meter-In Circuit (Figure 18)

In meter-in operation, the flow control valve is placed between the pump and the actuator. In thisway, it controls the amount of fluid going into the actuator. Pump delivery in excess of the meteredamount is diverted to tank through the relief valve.

A check valve must be included in parallel to the flow control valve to allow for return free flow. Formeter-in speed control in both directions, the flow control valve can be placed in the pump outletline prior to the directional valve. For separately adjustable flowrates in both directions, two flowcontrol valves must be used between the directional valve and the actuator.

Meter-in is highly accurate and used for resistive loads only. Because of the relief valve operation,it is not energy efficient.

Meter-Out Circui t (Figure 19)In meter-out circuit, the flow control valve is located where it restricts the return flow from theactuator and the piston is "cushioned." This restriction will also cause the system pressure toincrease to the point where the relief valve will open to divert part of the flow to tank.

To regulate speed in both directions, the valve can be installed in the tank line from the directionalcontrol valve. For separately adjustable flowrates in both directions, two flow control valves mustbe used between the directional valve and the actuator. For free flow of the supply, a by-passcheck valve is used.

This circuit can be used for both resistive and "run-away" loads. It is not energy efficient. There isalso a risk of pressure amplification in the cylinder circuit.

Throttled Flow

Free Flow

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Bleed-Off Circuit (Figure 20)

In a bleed-off arrangement, the flow is bled off the supply line from the pump and the actuatorspeed is controlled by metering a portion of the pump delivery to tank. The advantage is that thepump always operates at the pressure required by the work since excess fluid returns to tankthrough the flow control valve instead of the relief valve.

Its disadvantage is some loss of accuracy because the measured flow is to the tank rather thaninto the cylinder, making the latter subject to variations in the pump delivery due to changingwork load.

Bleed-off circuits should not be used for "run-away" load as the return flow in the tank line is notcontrolled directly by bleed-off.

Comparison of the Different Flow-Control Circuit s

MethodRun-away load Efficiency in

Energy use Accurate

control

Meter-in X X (better than meter-out)

Meter-out X

Bleed-off X X

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Figure 18: Meter-InConfiguration

Control ofextensionSpeed only

Control ofretractionSpeed only

Control ofretractionSpeed

Control ofextensionSpeed

Control ofextension andretraction speed atthe same time

From Pump

Control ofextensionSpeed only

Control ofretractionSpeed only

Control ofextensionSpeed

Control ofretractionSpeed

Control ofextension andretraction speed atthe same time

From Pump

Figure 19: Meter-OutConfiguration

Pressure = pressure at reliefvalve less

pressure dropeven at no load

Pressure at cylrod-end greaterthan Pressure atbore-end due to

pressureintensification

0 bar PressurePressure =

pressure at reliefvalve less

pressure dropeven at no load

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Control ofextensionSpeed only

Control ofretractionSpeed only

Control ofextensionSpeed

Control ofretractionSpeed

Control ofextension andretraction speed atthe same time

From Pump

Figure 20: Bleed-offConfiguration

0 PressurePressure = pressurerequired at work

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5.2 Types of Flow Control Valves

Flow control valves fall into two basic categories : pressure compensated and non-pressurecompensated types. Also, for the pressure-compensated type, we can have the pressure-cum-temperature compensated type.

Non-Pressure Compensated Flow Control Valve

A symbolic representation of a typical non-pressure compensated flow control valve is shownbelow and they are sometimes referred to as throttle valves. These valves are used wherepressures remain relatively constant and actuator speeds are not critical.

Pressure-Compensated Flow-Control Valve

The figure below shows the symbolic representation of a pressure-compensated flow controlvalve (or simply known as flow control valve). This valve maintains a constant pressure dropacross an adjustable throttle. The constant pressure drop across the throttle causes a constantflow rate even though the overall pressure differential across the valve may change due to loadchanges.

Pressure-Cum-Temperature Compensated Flow Control Valve

Flowrate changes with temperature and oil viscosity even if the pressure drop can remainconstant. Hence for more accurate flow control we should use a valve with a temperaturecompensator as well as a pressure compensator as shown by the following symbolicrepresentation.

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Worked Example 3

A speed-controlled hydraulic circuit is used in a grinding machine. The cylinder has a bore areaof 0.0075 m 2 and rod area of 0.00385 m 2. The pump flow flowrate is 20 /min and the FCV

setting is at 3.93 /min.

(a) Determine the extension and retraction speeds of the cylinder if the cylinder is connected inthe Meter-in conf iguration as shown Figure 21(a).

The extension speed is throttled

min/524.0

0075.0

min/1093.32

33

m

m

m Aq

The retraction speed

min/480.500385.00075.0

1020

)(3

m

a AQ p

(b) Determine the extension and retraction speeds of the cylinder if the cylinder is connected inthe Meter-out conf iguration as shown Figure 21(a).

The extension speed is throttled

min/077.1

)00385.00075.0(

min/1093.3)( 2

33

m

m

ma A

q

The retraction speed

min/480.500385.00075.0

1020

)(3

m

a A

Q p

Figure 21(a)

Figure 21(a)

q3.93 /min

Q20 /min

Q20 /min

q3.93 /min

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5.3 Deceleration Valve

Function A deceleration valve is used to control piston speed during some parts of the extension stroke.

Appl ication : Rapid-Advance-to-Feed ci rcui t

Rapid-advance-to-feed circuit is employed in applications where the cycle time of operatingsequence is to be reduced. The operating cycle involved rapid advance feed of tool to theworkpiece, change over to a slower feed, then rapid retraction to the starting position.

In Figure 23, a normally-closed deceleration valve is connected with the flow control valve inmeter-out configuration.

During the rapid advance, the return flow of the cylinder passes freely through the decelerationvalve. When the cam closes the deceleration valve, the oil flows through the flow control valveand the flow is throttled. During the retraction stroke, oil flows freely into the cylinder via the

check valve in the deceleration valve.

Figure22

Figure 23

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Worked Example 4

A horizontal feed table of a milling machine is driven by a hydraulic cylinder with the rapid-advance slow-feed circuit as shown in Figure 20. Milling operating only begins just after slowfeed movement has started.

The cylinder has a bore area of 0.0113 m 2 and rod area of 0.0031 m 2. The pump flowrate is 90/min and the FCV setting is at 5 /min.

Determine(a) Initial fast forward velocity in m/min

(b) Flowrate in litres/min at the 2/2 way valve during the fast forward stroke(c) Final slow feed velocity in m/min

Solution:(a)

The fast forward extension speed

min/96.7

0113.0min/090.0

2

3

m

mm

AQ

(b)The return flowrate

min/3.65

)0031.00113.0(min/96.7

)(2mm

a Av

q

E

(c)The slow extension speed

min/61.0

)0031.00113.0(min/105

)( 2

33

m

mm

a Aq