Hydraulic (Fluid) Systems

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Hydraulic (Fluid) Systems Basic Modeling Elements Resistance apacitance Inertance Pressure and Flow Sources Interconnection Relationships

[N/m 2 ] ( ) v : volume [m 3 ] ( ) The analogy between a hydraulic system and an electrical system will be used often.

The pressure, p,, used in this chapter is the absolute pressure.

1 Hydraulic (Fluid) Systems Basic Modeling Elements Resistance apacitance Inertance Pressure and Flow Sources Interconnection Relationships ompatibility Law ontinuity Law Derive Input/Output Models ME375 Hydraulic - 1 Key oncepts : volumetric flow rate [m 3 /sec] ( ) p : pressure [N/m 2 ] ( ) v : volume [m 3 ] ( ) The analogy between a hydraulic system and an electrical system will be used often. Just as in electrical systems, the flow rate (current) is defined to be the time rate of change (derivative) of volume (charge): d = dt v = v The pressure, p,, used in this chapter is the absolute pressure.. You need to be careful in determining whether the pressure is the absolute pressure sure or gage pressure, p *. Gage pressure is the difference between the absolute pressure and the atmospheric pressure, i.e. p * = p p atmospheric ME375 Hydraulic - 2 1

2 Basic Modeling Elements Fluid Resistance Describes any physical element with Ex: The flow that goes through an orifice or a valve and turbulent flow that goes through the characteristic that the pressure a pipe is related to the pressure drop by drop, Δp, across the element is = k p12 proportional to the flow rate. Find the effective flow resistance of the p 1 + Δp p 2 element at certain operating point (, p 12 ). p 1 + Δp p 2 R R Δp = p p = p = R Δ R p 1 = = R p 12 Orifices, valves, nozzles and friction in pipes can be modeled as fluid resistors. p 12 1 R = d dp = k 12 e 2 p, p12j 2 p12 2 R = = 2 k k 12 p 12 ME375 Hydraulic - 3 Basic Modeling Elements Fluid apacitance Describes any physical element with the characteristic that the rate of change in pressure p in the element is proportional to the difference between the input flow rate, IN, and the output flow rate, OUT. IN ef + p r p OUT IN OUT d ( p pref ) = p r = IN OUT dt pr Hydraulic cylinder chambers, tanks, and accumulators are examples of fluid capacitors. Ex: onsider an open tank with a constant cross-sectional sectional area, A: IN IN p = = OUT p 1 p r = ( IN OUT ) = = = h OUT ME375 Hydraulic - 4 2

3 apacitance Examples Ex: alculate the euivalent fluid capacitance for a hydraulic chamber with only an inlet port. IN p chamber volume v Recall the bulk modulus ( (β ) of a fluid is defined by: β = v dp r dv Apply the chain rule: F dpr H I dtk β = v = ddv dt i F = H G vi β K J 4 d dt p r Ex: Will the effective capacitance change if in the previous open tank example, a load mass M is floating on top of the tank? M h p IN OUT ME375 Hydraulic - 5 Basic Modeling Elements Fluid Inertance (Inductance) Describes any physical element with the characteristic that the pressure drop, Δp, across the element is proportional to the rate of change (derivative) of the flow rate,. p 1 + Δp p 2 I Δp = p = ( p p ) = I d dt = I Δp p 1 p 2 I Ex: onsider a section of pipe with cross- sectional area A and length L, filled with fluid whose density is ρ : p 1 + Δp p 2 Start with force balance: F = ma L A Long pipes are examples of fluid inertances. Q:What will happen if you suddenly shut off one end of a long tube? --- (Water Water Hammer effect) I = ρ L A ME375 Hydraulic - 6 3

4 Basic Modeling Elements Pressure Source (Pump) An ideal pressure source of a hydraulic system is capable of maintaining the desired pressure, regardless of the flow reuired for what it is driving. p 1 p S + p 2 p S p 21 = p 2 p 1 = p S Flow Source (Pump) An ideal flow source is capable of delivering the desired flow rate, regardless of the pressure reuired to drive the load. p 1 p 2 S = S ME375 Hydraulic - 7 Interconnection Laws ompatibility Law The sum of the pressure drops around a loop must be zero. Similar to the Kirchhoff voltage law. Δp = p = 0 losed Loop j losed Loop p 1 p 2 B ij ontinuity Law The algebraic sum of the flow rates at any junction in the loop is zero. This is the conseuence of the conservation of mass. Similar to the Kirchhoff current law. or Any Node IN j = = 0 OUT A 1 2 p + p + p = r1 12 2r = o o ME375 Hydraulic - 8 4

5 Modeling Steps Understand System Function and Identify Input/Output Variables Draw Simplified Schematics Using Basic Elements Develop Mathematical Model Label Each Element and the orresponding Pressures. Label Each Node and the orresponding Flow Rates. Write Down the Element Euations for Each Element. Apply Interconnection Laws. heck that the Number of Unknown Variables euals the Number of Euations. Eliminate Intermediate Variables to Obtain Standard Forms: Laplace Transform Block Diagrams ME375 Hydraulic - 9 In lass Exercise Derive the input/output model for the following fluid system. The T pump supplies a constant pressure p S to the system and we are interested in finding out the volumetric flow rate through the nozzle at the end of the pipe. Valve p S Label the pressures at nodes and flow rates Write down element euations: ME375 Hydraulic

6 In lass Exercise No. of unknowns and euations: Interconnection laws: Eliminate intermediate variables and obtain I/O model: Q: an you draw an euivalent electrical circuit of this hydraulic system? Note that pressure is analogous to voltage and flow rate is analogues to electric current. ME375 Hydraulic - 11 In lass Exercise Electrical Analogy: ME375 Hydraulic

7 Motion ontrol of Hydraulic ylinders Hydraulic actuation is attractive for applications when large power is needed while maintaining a reasonable weight. Not counting the weight of the pump and reservoir, hydraulic actuation has the edge in power-to to-weight ratio compared with other cost effective actuation sources. Earth moving applications (wheel loaders, excavators, mining euipment,...) are typical examples where hydraulic actuators are used extensively. A typical motion application involves a hydraulic cylinder connected to certain mechanical linkages (inertia load). The motion of the cylinder is regulated via a valve that is used to regulate the flow rate to the cylinder. It is well known that such systems chatter during sudden stops and starts. an you analyze the cause and propose solutions? M R V p S R V ME375 Hydraulic - 13 Motion ontrol of Hydraulic ylinders Let s s look at a simplified problem: The input in the system to the right is the input flow rate IN and the output is the velocity of the mass, V. A: : ylinder bore area : : ylinder chamber capacitance B: : Viscous friction coefficient between piston head and cylinder wall. Derive the input/output model and transfer function between IN and V. Draw the block diagram of the system. an this model explain the vibration when we suddenly close the valve? M V B p L A p S p Sr IN R V ME375 Hydraulic

8 Motion ontrol of Hydraulic ylinders Element euations and interconnection euations: Take Laplace transforms: Block diagram representation: ME375 Hydraulic - 15 Motion ontrol of Hydraulic ylinders Transfer function between IN and V: How would the velocity response look if we suddenly open the valve to reach constant input flow rate Q for some time T and suddenly close the valve to stop the flow? Analyze the transfer function: Natural Freuency Damping Ratio Steady State Gain ME375 Hydraulic

Lecture Fluid system elements

Lecture Fluid system elements Lecture 8.1 Fluid system elements volumetric flowrate pressure drop Detailed distributed models of fluids, such as the Navier-Stokes equations, are necessary for understanding many aspects of fluid systems