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CHAPTER 6 MESB 374 System Modeling and Analysis Hydraulic (Fluid) Systems Hydraulic (Fluid) Systems • Basic Modeling Elements – Resistance – Capacitance – Inertance – Pressure and Flow Sources • Interconnection Relationships – Compatibility Law – Continuity Law • Derive Input/Output Models Variables • q : volumetric flow rate [m3/sec] ( current ) • V : volume [m3] ( charge ) • p : pressure [N/m2] ( voltage ) The analogy between the hydraulic system and the 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 q 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 gauge pressure, p*. Gauge pressure is the difference between the absolute pressure and the atmospheric pressure, i.e. p* p patmospheric Basic Modeling Elements • Fluid Resistance Describes any physical element with Ex: The flow that goes through an orifice or a the characteristic that the pressure valve and the turbulent flow that goes drop, Dp , across the element is through a pipe is related to the pressure proportional to the flow rate, q. drop by q k p12 p1 + Dp p2 + Dp p1 p2 Find the effective flow resistance of the q element at certain operating point ( q , p12). R R q q Dp p1 p2 p12 R q q 1 1 q Dp p12 R R p12 p12 1 dq k – Orifices, valves, nozzles and R d p12 q , p 2 p12 friction in pipes can be modeled 12 as fluid resistors. 2 p12 2q R 2 k k Basic Modeling Elements Ex: Consider an open tank with a constant • Fluid Capacitance cross-sectional area, A: Describes any physical element with the characteristic that the rate pr of change in pressure, p, in the element is proportional to the gh h difference between the input flow pC rate, qIN , and the output flow rate, qIN qOUT qOUT . pref + pCr pC gh pr pCr gh pC qOUT qIN d d qIN - qOUT C qIN qOUT Volum Ah Ah C dt dt d pCr gh C pC pref C pCr q IN qOUT & qIN qOUT Ah A 4 3 dt 1 24 C pCr pCr gh g – Hydraulic cylinder chambers, tanks, and accumulators are examples of fluid capacitors. Fluid 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 pr pC qIN M chamber volume C V h Recall the bulk modulus (b ) of a fluid is pC defined by: qIN qOUT dpCr dt b V dpCr V pC gh pr Mg pCr gh Mg A dV dV dt qIN qOUT d d A Volum Ah Ah dt dt pCr gh qIN qOUT Ah A V d C q p pCr gh g b dt Cr C Basic Modeling Elements • Fluid Inertance (Inductance) Ex: Consider a section of pipe with cross- Describes any physical element with the sectional area A and length L, filled characteristic that the pressure drop, Dp , across with fluid whose density is : the element is proportional to the rate of change (derivative) of the flow rate, q. F1 Ap1 p1 + Dp p2 F2 Ap2 F Ap 1 1 p1 + Dp p2 + Dp q p1 p2 A q L I q Start with force balance: F = ma I Dp p12 ( p1 p2 ) I d dt q I q & F F F 1 2 A p1 p2 Ap12 m LA dv d q Ap12 AL AL – Long pipes are examples of fluid dt dt A F m inertances. L dq a p12 A dt I L I A Basic Modeling Elements Voltage Source • Pressure Source (Pump) – An ideal pressure source of a pS + p1 p2 hydraulic system is capable of pS maintaining the desired pressure, q regardless of the flow required for what it is driving. p21 p2 p1 pS • Flow Source (Pump) Current Source – An ideal flow source is capable of p1 p2 delivering the desired flow rate, qS q regardless of the pressure required to drive the load. q qS Interconnection Laws • Compatibility Law • Continuity Law – The sum of the pressure drops – The algebraic sum of the flow rates around a loop must be zero. at any junction in the loop is zero. – Similar to the Kirchhoff’s voltage – This is the consequence of the law. conservation of mass. – Similar to the Kirchhoff’s current Dp j p ij 0 law. Closed Loop Closed Loop Any qj 0 Node p1 p2 B or q IN qOUT A C q1 q2 pr q1 q2 qo pr1 p12 p2 r 0 qo 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 In Class Exercise Derive the input/output model for the following fluid system. The pump supplies a constant 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. pr Valve ps C qc Ip q1 q2 Rp pS pr pr Ro pc p1 p2 R p3 R N v • Label the pressures at nodes and flow rates q1 Ip q2 p Rp p2 pc 1 • Write down element equations: p Ro i psc q1 Ro +s qc Rv Req p3 d qc C pcr _ C dt RN d pc1 I p q2 dt pr p1r q2 Req Equivalent electrical circuit In Class Exercise • No. of unknowns and equations: psc , q1 , qs , pcr , pc1 , q2 , p1r • Interconnection laws: we are interested in it Loop 1: psc pcr psr 0 R1q1 pcr psr 0 d Loop 2: pc1 p1r prc 0 I p q2 Req q2 pcr 0 dt dPcr Node 2: q1 qc q2 q1 C q2 dt • Eliminate intermediate variables and obtain I/O model: d pcr I p q2 Req q2 d 2 q2 R1CI p 2 R1CReq I p 2 R1 Req q2 psr dt dq d d dt dt R1 C pcr q2 I p q2 Req q2 psr dt dt q1 pcr 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. (Please refer to the previous slide) 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 M the pump and reservoir, hydraulic actuation has the edge in power-to-weight ratio compared with other 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 RV RV hydraulic cylinder connected to certain mechanical linkages (inertia load). The motion pS of the cylinder is regulated via a valve that is pr used to regulate the flow rate to the cylinder. It is well known that such system chatters during sudden stop and start. Can you analyze the cause and propose solutions? Motion Control of Hydraulic Cylinders v A C Let’s look at a simplified problem: M The input in the system to the right is the pr pL input flow rate qIN and the output is the B velocity of the mass, V. A: Cylinder bore area qIN C: Cylinder chamber capacitance RV B: Viscous friction coefficient between pS piston head and cylinder wall. pSr pr • Derive the input/output model and transfer function between qIN and V. • Draw the block diagram of the system. pL v • Can this model explain the vibration when we i fc qIN qc suddenly close the valve? q1 M C B pr Motion Control of Hydraulic Cylinders Element equations and interconnection equations: Hydraulic system Hydraulic-Mechanical Mechanical system d qc C pLr f c ApLr dt Mv f c Bv q1 Av qIN q1 qc Take Laplace transforms: Qc s CsPLr s Fc s APLr s MsV s Fc s BV s QIN s Q1 s Qc s Q1 s AV s Block diagram representation: QIN(s) 1 PLr s Fc s V s X s 1 1 - Cs A Ms B s Q1 s A V s Hydraulic System H-M Coupling Mechanical System Motion Control of Hydraulic Cylinders Transfer function between qIN and V: How would the velocity response look like V s A if we suddenly open the valve to reach GQV QIN s MCs BCs A2 2 constant input flow rate Q for some time T A and suddenly close the valve to stop the MC flow? 2 s2 B s A M MC 2n n 2 In reality, large M, small C Analyze the transfer function: Natural Frequency reasonable value of natural frequency 2 A n A MC very small damping ratio MC Damping Ratio B Oscillation cannot die out quickly 2A B C M B 2n M MC 2A M Chattering !! Steady State Gain A K MC 1 2 A A MC