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Unformatted text preview: ME375 Handouts Hydraulic (Fluid) Systems
• Basic Modeling Elements
– – – – Resistance Capacitance Inertance Pressure and Flow Sources • Interconnection Relationships
– Compatibility Law – Continuity Law • Derive Input/Output Models
School of Mechanical Engineering Purdue University ME375 Hydraulic - 1 Key Concepts
• q : volumetric flow rate [m3/sec] [m 2] • p : pressure [N/m [N/m • v : volume [m3] [m ( ( ( ) ) ) 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 is time rate of change (derivative) of volume (charge):
q= d v=v dt The pressure, p, used in this chapter is the absolute pressure. You need to be absolute careful in determining whether the pressure is the absolute pressure or gage pressure gage pressure, p*. Gage pressure is the difference between the absolute pressure and and the atmospheric pressure, i.e. * p = p − patmospheric
School of Mechanical Engineering Purdue University ME375 Hydraulic - 2 1 ME375 Handouts Basic Modeling Elements
• Fluid Resistance Ex: The flow that goes through an orifice or a valve and turbulent flow that goes through Describes any physical element with a pipe is related to the pressure drop by the characteristic that the pressure q = k p12 drop, Δp, across the element is proportional to the flow rate q. Find the effective flow resistance of the
p1 + Δp − q R p2 p1 + Δp − R element at certain operating point ( q , p12 ). ).
p2 q q
q Δp = p1 − p2 = p12 = R ⋅ q q= 1 1 Δp = p12 R R
p12 p12 – Orifices, valves, nozzles and friction in pipes can be modeled as fluid resistors. dq k 1 = = R d p12 e q , p j 2 p1 2 12 R= 2 p12 2q =2 k k
ME375 Hydraulic - 3 School of Mechanical Engineering Purdue University Basic Modeling Elements
• Fluid Capacitance Ex: Consider an open tank with a constant cross-sectional area, A: crossDescribes any physical element with the characteristic that the rate of change in pr pressure p in the element is proportional to the difference between the input flow h rate, qIN , and the output flow rate, qOUT . pC
pref + pCr − C qIN qOUT qIN pC C qOUT qIN − qOUT pC = qIN − qOUT = pCr = = ⇒ C=
ME375 Hydraulic - 4 C d pC − pref = C ⋅ pCr = q IN − qOUT dt pCr ( ) 1 (qIN − qOUT ) = C – Hydraulic cylinder chambers, tanks, and accumulators are examples of fluid capacitors.
School of Mechanical Engineering Purdue University 2 ME375 Handouts Capacitance Examples
Ex: Calculate the equivalent fluid capacitance Ex: Will the effective capacitance change if in for a hydraulic chamber with only an inlet the previous open tank example, a load port. mass M is floating on top of the tank? pr
qIN pC C chamber volume v pr M h qIN pC qOUT Recall the bulk modulus (β ) of a fluid is defined by: dpCr
dv Apply the chain rule:
Cr FH dp dt IK β=v = d dv dt i F vI d p ⇒ q=G J β H4K dt C Cr School of Mechanical Engineering Purdue University ME375 Hydraulic - 5 Basic Modeling Elements
• Fluid Inertance (Inductance) Ex: Consider a section of pipe with crossEx crosssectional 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 p2 + Δp − across the element is proportional to the rate of change (derivative) of the flow q L A rate, q.
p1 q I
+ Δp − p2 p1 + Δp − p2 q Start with force balance: F = ma ma I Δ p = p12 = ( p1 − p2 ) = I 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
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. • Flow Source (Pump) – An ideal flow source is capable of delivering the desired flow rate, regardless of the pressure required to drive the load.
p1 p2 p1 − pS + p2 q pS p21 = p2 − p1 = pS 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. law. ∑ Δp j = Closedpij = 0 ∑ – Similar to the Kirchhoff current law. Closed
Loop Loop ∑ qj = 0 = p1
A B p2
C Any Node ∑q IN ∑q OUT q1 q2 pr p r 1 + p1 2 + p 2 r = 0
School of Mechanical Engineering Purdue University q1 + q 2 = q o
ME375 Hydraulic - 8 4 ME375 Handouts Modeling Steps
• Understand System Function and Identify Input/Output Variables • Draw Simplified Schematics Using Basic Elements • 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. – Eliminate Intermediate Variables to Obtain Standard Forms: • Laplace Transform • Block Diagrams
School of Mechanical Engineering Purdue University ME375 Hydraulic - 9 In Class Exercise
Derive the input/output model for the following fluid system. The pump supplies a constant The pressure pS to the system and we are interested in finding out the volumetric flow rate volumetric through the nozzle at the end of the pipe.
pr Valve pS pr 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 In Class Exercise
• No. of unknowns and equations: • 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. current.
School of Mechanical Engineering Purdue University ME375 Hydraulic - 11 In Class Exercise
Electrical Analogy: School of Mechanical Engineering Purdue University ME375 Hydraulic - 12 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 the pump and reservoir, hydraulic actuation has the edge in power-to-weight ratio compared with power- toother cost effective actuation sources. Earth moving applications (wheel loaders, excavators, mining equipment, ...) 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. Can you analyze the cause and propose solutions?
M RV pS pr RV School of Mechanical Engineering Purdue University ME375 Hydraulic - 13 Motion Control of Hydraulic Cylinders
Let’s look at a simplified problem: Let’ 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. • Derive the input/output model and transfer function between qIN and V. • Draw the block diagram of the system. • Can this model explain the vibration when we suddenly close the valve?
V A C M
pr B qIN RV pS pSr pr pL School of Mechanical Engineering Purdue University 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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