Hydraulic System - ME375 Handouts Hydraulic (Fluid) Systems...

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Unformatted text preview: ME375 Handouts Hydraulic (Fluid) Systems • Fundamental Principles – Pascal’s Law: Pressure applied to the pp fluid is transmitted equally in all directions – Transmit forces (incompressible) • Applications – High force – Heavy loads – Precise motion School of Mechanical Engineering Purdue University ME375 Hydraulic - 1 Hydraulic System Modeling • Basic Modeling Elements – – – – Resistance Capacitance Inertance Pressure and Flow Sources • Interconnection Relationships – Compatibility Law Law – Continuity Law • Derive Input/Output Models School of Mechanical Engineering Purdue University ME375 Hydraulic - 2 1 ME375 Handouts Key Concepts • q : volumetric flow rate [m3/sec] [m 2] [N/ [N/m • p : pressure [N/m • v : volume [m3] [m ( ( ( ) ) ) The analogy between a hydraulic system and an electrical system will be used often. often. Just as in electrical systems, the flow rate (current) is defined to be the time rate of change (derivative) of volume (charge): (charge): q= d & v=v dt 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 or gage pressure, p*. Gage pressure is the difference between the absolute pressure and the atmospheric pressure, i.e. p * = p − patmospheric School of Mechanical Engineering Purdue University ME375 Hydraulic - 3 Basic Modeling Elements • Fluid Resistance • Fluid Capacitance Describes any physical element with Describes any physical element with the the characteristic that the pressure characteristic that the rate of change in drop, Δp, across the element is pressure p in the element is proportional proportional to the flow rate q. to the difference between the input flow rate, qIN , and the output flow rate, qOUT . p1 + Δp − p2 + Δp − p1 q R p2 R Δp = p1 − p2 = p12 = R ⋅ q 1 1 q = Δp = p12 R R pC qIN ( – Orifices, valves, nozzles and friction in pipes can be modeled as fluid resistors. resistors. + pCr − pref q C qOUT qIN − qOUT C ) d & C pC − pref = C ⋅ pCr = q IN − qOUT dt 14243 pCr – Hydraulic cylinder chambers, tanks, and accumulators are examples of fluid capacitors. capacitors. School of Mechanical Engineering Purdue University ME375 Hydraulic - 4 2 ME375 Handouts Basic Modeling Elements Ex: Consider an open tank with a constant crosscross-sectional area, A: Ex: Will the effective capacitance change if in the previous open tank example, a load mass M is floating on top of the tank? pr pr h M pC qIN qOUT h pC qIN pC = qOUT qIN − qOUT = & pCr = 1 (qIN − qOUT ) = C = ⇒ C= School of Mechanical Engineering Purdue University ME375 Hydraulic - 5 Basic Modeling Elements • Fluid Inertance (Inductance) Ex: Consider a section of pipe with crosscrosssectional area A and length L, filled Describes any physical element with the with fluid whose density is ρ : characteristic that the pressure drop, Δp , p1 + Δp − p2 across the element is proportional to the rate of change (derivative) of the flow q L A rate, q. p1 + Δp − p2 p1 q I Δ p = p12 = ( p1 − p2 ) = I Start with force balance: F = ma ma + Δp − I p2 q d & q = I ⋅q dt – Long pipes are examples of fluid inertances. inertances. Q: What will happen if you suddenly shut off one end of a long tube ? --- (Water Hammer effect) School of Mechanical Engineering Purdue University ⇒ I= ρL A ME375 Hydraulic - 6 3 ME375 Handouts 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 required for what it is driving. driving. p1 − pS + p2 pS q p21 = p2 − p1 = pS • Flow Source (Pump) – An ideal flow source is capable of delivering the desired flow rate, regardless of the pressure required to drive the load. load. p1 p2 qS q q = qS School of Mechanical Engineering Purdue University ME375 Hydraulic - 7 Interconnection Laws • Continuity Law • Compatibility Law – The algebraic sum of the flow rates – The sum of the pressure drops at any junction in the loop is zero. around a loop must be zero. – This is the consequence of the – Similar to the Kirchhoff voltage conservation of mass. mass. law. law. ∑ Δp j = Closedpij = 0 ∑ – Similar to the Kirchhoff current law. Closed Loop p1 ∑ Loop B A qj = 0 Any Node p2 or C ∑q IN = pr School of Mechanical Engineering Purdue University OUT q2 q1 p r 1 + p1 2 + p 2 r = 0 ∑q q1 + q 2 = q o qo ME375 Hydraulic - 8 4 ME375 Handouts Modeling Steps • Understand System Function and Identify Input/Output Variables • Draw Simplified Schematics Using Basic Elements Si El • Develop Mathematical Model – – – – – Label Each Element and the Corresponding Pressures. Label Each Node and the Corresponding Flow Rates. Write Down the Element Equations for Each Element. Apply Interconnection Laws. Check that the Number of Unknown Variables equals the Number of Equations. Equations. – Eliminate Intermediate Variables to Obtain Standard Forms: • Laplace Transform • Block Diagrams School of Mechanical Engineering Purdue University ME375 Hydraulic - 9 Example Derive the input/output model for the following fluid system. The pump supplies a constant system. pressure pS to the system and we are interested in finding out the volumetric flow rate through the nozzle at the end of the pipe. pipe. pr Valve pr pS pr • Label the pressures at nodes and flow rates • Write down element equations: School of Mechanical Engineering Purdue University ME375 Hydraulic - 10 5 ME375 Handouts Example • No. of unknowns and equations: laws: • Interconnection laws: • Eliminate intermediate variables and obtain I/O model: Q: Can you draw an equivalent electrical circuit of this hydraulic system ? Note that pressure is analogous to voltage and flow rate is analogues to electric current. School of Mechanical Engineering Purdue University ME375 Hydraulic - 11 School of Mechanical Engineering Purdue University ME375 Hydraulic - 12 Example Electrical Analogy: 6 ME375 Handouts Motion Control of Hydraulic Cylinders Hydraulic actuation is attractive for applications when large power is needed while maintaining a reasonable weight. Not counting the weight of weight. the pump and reservoir, hydraulic actuation has the edge in power-to-weight ratio compared with power-toother cost effective actuation sources. Earth sources. moving applications (wheel loaders, excavators, mining equipment, ...) are typical examples ...) where hydraulic actuators are used extensively. 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 cylinder. is well known that such systems chatter during sudden stops and starts. Can you analyze the starts. cause and propose solutions? M RV RV pS pr School of Mechanical Engineering Purdue University ME375 Hydraulic - 13 Motion Control of Hydraulic Cylinders Let’s look at a simplified problem: The input in the system to the right is the input flow rate qIN and the output is the velocity of the mass, V. A: Cylinder bore area C: Cylinder chamber capacitance B: Viscous friction coefficient between piston head and cylinder wall. wall. • Derive the input/output model and transfer function between qIN and V. • Draw the block diagram of the system. the block diagram of the system • Can this model explain the vibration when we suddenly close the valve? V A C M School of Mechanical Engineering Purdue University pL pr B qIN RV pS pSr pr ME375 Hydraulic - 14 7 ME375 Handouts Motion Control of Hydraulic Cylinders Element equations and interconnection equations: Take Laplace transforms: Block diagram representation: School of Mechanical Engineering Purdue University ME375 Hydraulic - 15 Motion Control of Hydraulic Cylinders Transfer function between qIN 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 Frequency Damping Ratio Steady State Gain School of Mechanical Engineering Purdue University ME375 Hydraulic - 16 8 ...
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This note was uploaded on 12/22/2011 for the course ME 375 taught by Professor Meckle during the Fall '10 term at Purdue University-West Lafayette.

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