With a clear idea of the basic electro-mechanical building blocks andtheir relationship to one another (Part1), the use of simulation modellingtools makes it possible to derive the basic mathematical models of thehydraulic components, and more, importantly simulate and test them in atypical implementation and extract meaningful results that will beuseful in coming up with a final design.
Tools such as the Matlab Simulinkallow the developer to firstderive a top level system design as shown in Figure A-1 below and then proceed tolower levels and derive models for each of the important elements inthe design.
|FigureA-1: Top level system diagram of a typical electro hydraulic system.|
Modelling the Flow Control ServoValve
The two stage nozzle-flapperservo-valve consists of three main parts: an electrical torque motor,hydraulic amplifier, and valve spool assembly and is shown in detail inFigure 8, below .
|Figure8. Valve Torque Motor Assembly (illustration courtesy of Moog)|
The torque motor consists of an armature mounted on a thin-walledsleeve pivot and suspended in the air gap of a magnetic field producedby a pair of permanent magnets.
When current is made to flow in the two armature coils, the armatureends become polarized and are attracted to one magnet pole piece andrepelled by the other. This sets up a torque on the flapper assembly,which rotates about the fixture sleeve and changes the flow balancethrough a pair of opposing nozzles, shown in Figure 9, below . The resultingchange in throttle flow alters the differential pressure between thetwo ends of the spool, which begins to move inside the valve sleeve.
|Figure9. Valve Responding to Change in Electric Input (illustration courtesyof Moog)|
Lateral movement of the spool forces the ball end of a feedbackspring to one side and sets up a restoring torque on thearmature/flapper assembly.
When the feedback torque on the flapper spring becomes equal to themagnetic forces on the armature the system reaches an equilibriumstate, with the armature and flapper centred and the spool stationarybut deflected to one side. The offset position of the spool opens flowpaths between the pressure and tank ports (P s and T ), andthe two control ports (A and B ), allowing oil to flow to andfrom theactuator.
Modelling the Torque Motor
For simplicity, the electrical characteristics of the servo-valvetorque motor may be modelled as a series L-R circuit, neglecting forthe time being any back-EMF effects generated by the load. The transferfunction of a series L-R circuit is:
where L c i sthe inductance of the motor coil, and R c the combined resistance of the motor coil and the current senseresistor of the servo amplifier. Values of inductance and resistancefor series and parallel winding configurations of the motor are usuallypublished in the manufacturer's data sheet.
The lateral force on the valve spool is proportional to torque motorcurrent, but oil flow rate at the control ports also depends upon thepressure drop across the load.
Modelling the Valve Spool Dynamics
A servo-valve is a complex device which exhibits a high-ordernon-linear response, and knowledge of a large number of internal valveparameters is required to formulate an accurate mathematical model.Indeed, many parameters such as nozzle and orifice sizes, spring rates,spool geometry and so on, are adjusted by the manufacturer to tune thevalve response and are not normally available to the user.
Practically all physical systems exhibit some non-linearity: in thesimplest case this may be a physical limit of movement, or it may arisefrom the effects of friction, hysteresis, mechanical wear or backlash.When modelling complex servo-valves, it is sometimes possible to ignoreany inherent non-linearities and employ a small perturbation analysisto derive a linear model which approximates the physical system.
Such models are often based on classical first or second orderdifferential equations, the coefficients of which are chosen to matchthe response of the valve based on frequency plots taken from the datasheet.
A simple first or second order model yields only an approximation toactual behavior, however the servo-valve is not the primary dynamicelement in a typical hydraulic servo system and is generally selectedsuch that the frequency of the 90 degree phase point is a factor of atleast three higher than thatof the actuator.
For this reason it is usually only necessary toaccurately model valve response through a relatively low range offrequencies, and the servo-valve dynamics may be approximated by asecond order transfer function without serious loss of accuracy.
|Figure10. Typical Servo-valve Frequency Response Curve (illustration courtesyof Moog)|
A typical performance graph for a high-responsive servo-valve isshown in Figure 10, above. Assuming a second order approximation is tobe used, suitable values for natural frequency and damping ratio willneed to be determined from the graph.
Natural frequency ( v ) can be read fairly accurately fromthe -3dB or 90 degree phase point of the 40% curve. Damping can bedetermined from an estimate of the magnitude of the peaking present.For an under-damped second order system, the damping factor (S v ) can be shown to be related to peak amplitude ratio (M v )bythe formula
In this example, a reasonable estimate of peaking based on the 40%response curve would be about 1.5 dB, which corresponds to an amplituderatio of about 1.189. A suitable value of damping determinediteratively from Equation 2 is about 0.48. Armed with these values, asimplified model of the servo-valvespool dynamics may be constructed.
The input to the model will be the torque motor current derived fromEquation 1 in Part 1 normalized to the saturation current obtained fromthedatasheet, and the output will be the normalized spool position. Shownin Figure A-2 below i s theSimulink modelof the servo valve.
|FigureA-2. A Simulink model of the servo-valve|
Modelling Valve Flow-Pressure
The servo-valve delivers a control flow proportional to the spooldisplacement for a constant load. For varying loads, fluid flow is alsoproportional to the square root of the pressure drop across the valve.Control flow, input current, and valve pressure drop are related by thefollowing simplified equation:
In the above equation, Q L , is the hydraulic flowdelivered through the load actuator, Q R the rated valveflow at a specified pressure drop P R , and i* v is normalized input current. P R is the pressuredrop across the valve given by P V = P S +P T + P L ,where P S , P T and P L are system pressure, return line (tank) pressure, and load pressurerespectively.
Maximum power is transferred to the load when P L = 2/3 PS ,and since the most widely used supply pressure is 3,000 psi, it iscommon practice to specify rated valve flow at P = 1,000 psi(approximately 70 bar). The static relationship between valve pressuredrop and load flow is often presented in manufacturer's datasheets as afamily of curves of normalized control flow against normalized loadpressure drop for different values of valve input current as shown inFigure 11, below .
|Figure11. Servo-valve Flow-pressure curves (illustration courtesy of Moog)|
The horizontal axis is the load pressure drop across the valve,normalized to 2/3 of the supply pressure. The vertical axis is outputflow expressed as a percentage of the rated flow, Q R .Thevalve orifice equationis applied separately for the two control ports to obtain expressionsfor oil flow into each of the two actuator chambers. Since load flow isdefined as the flow through the load: Q L = Q A =-Q B
|FigureA-3: Simulink model of the Hydraulic Actuator|
As shown in the Simulink model in FigureA-3, above , the inputs arecommand voltage from the amplifier, supply and return oil pressuresfrom the hydraulic power supply (P S andP T ),andload pressures from the actuator chambers (P A and P B ).Outputs are the flows to each side of the piston (Q A and Q B ),and the load flow (Q L ).
Modelling the Linear Actuator
Cylinder ChamberPressure. The relationship between valve control flow andactuator chamber pressure is important because the compressibility ofthe oil creates a “spring” effect in the cylinder chambers whichinteracts with the piston mass to give a low frequency resonance. Thisis present in all hydraulic systems and in many cases this abruptlylimits the usable bandwidth. The effect can be modelled using the flowcontinuity equation from fluid mechanics which relates the net flowinto a container to the internal fluid volume and pressure.
The left hand side of the equation is the net flow delivered to thechamber by the servo valve. The firstterm on the right hand side isthe flow consumed by the changing volume caused by motion of thepiston, and the second term accounts for any compliance present in thesystem. This is usually dominated by thecompressibility of the hydraulic fluid and it is common to assume thatthemechanical structure is perfectly rigid and use the bulk modulus of theoil as a value for .
Mineral oils used in hydraulic control systems have a bulk modulusin the region of 1.4 x 109 N/m. The aboveprevious equation can be re-arranged to findthe instantaneous pressure in chamber A as follows:
PistonDynamics. Once the two chamber pressures are known, the netforce acting on the piston (F P )can be computed by multiplying by the area of the piston annulus (A P ) by the differentialpressure across it.
F P = (P A – P B )A P
An equation of forces for piston motion can now be established byapplyingNewton's second law. For the purposes of this analysis, itwill be assumed that the piston delivers a force to a linear springload with stiffness K L ,which will allow us to investigate the load capacity of the actuatorlater. The effects of friction (F f )between the piston and the oil seals at the annulus and end caps willalso be included. The resulting force equation for the piston is shownbelow and may be modelled in Simulink using two integrator blocks.
The total frictional force depends on piston velocity, driving force(F P ), oil temperature and possiblypiston position. One method of modelling friction is as a function ofvelocity, in which the total frictional force is divided into staticfriction (a transient term present asthe actuator begins to move ),Coulomb friction (a constant force dependent only on the direction ofmovement), and viscous friction (aterm proportional to velocity ).Assuming that viscous and Coulomb friction components dominate,frictional force (F f ) can be modelled as
where viscous and Coulomb friction coefficients are denoted by F v0 and F c0 respectively. Frictional effects are notoriously difficult to measureand accurate values of these coefficients are unlikely to be known, butorder of magnitude estimates can sometimes be made from relativelysimple empirical tests.
One test which can yield useful information is to subject the systemto a low frequency, low amplitude sinusoidal input signal, and plot theoutput displacement over one or two cycles. A low friction systemshouldreproduce the input signal, but the presence of friction will tend toflatten the tops of the sine wave as the velocity falls to a level blowthat necessary to overcome any inherent Coulomb friction.
In actuators fitted with conventional PolyTetraFluoroEthylene(PTFE)-basedbearings, frictionis related fairly linearly to supply pressure and oil temperature andcare should be taken to conduct testing under representativeconditions.
In a first analysis, leakage effects in the actuator are sometimesneglected, however this is an important factor which can have asignificant damping influence on actuator response. Leakage occurs atthe oil seals across the annulus between the two chambers and at eachend cap, and is roughly proportional to the pressure difference acrossof the seal. Including leakage effects, the flow continuity equationfor chamber A
is similar with appropriate changes of sign. It is a relativelysimple matter to modify the Simulink model to compute the instantaneouschamber leakages and subtract them from the total input flow.
Modelling the hydraulic power supply
The behavior of the hydraulic power supply described in Part 1 of this series may bemodelled in the same way as the chamber volumes: by applying the flowcontinuity equation to the volume of trapped oil between the pump andservo-valve. In this case, the input flow is held constant by thesteady speed of the pump motor, and the volume does not change. Thetransformed equation is
This equation takes into account the load flow (Q L )drawn from the supply by the servo-valve, and accurately models thecase of a high actuator slew rate resulting in a load flow whichexceeds the flow capacity of the pump. In such cases the supplypressure (P S ) falls, leading to a correspondingreduction in control flow and loss of performance. The action of thepressure relief valve may be modelled using a limited integrator toclamp the system pressure to the nominal value.
Next in Part 3: Modelling a typicalservo actuator
To read Part 1 go to : “Thebasics of electro-hydraulic servo actuator systems.”
Richard Poleyis Field Application Engineer at TexasInstrumentswith focus on digital control systems .
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