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me375_sp11_hwk09_soln

me375_sp11_hwk09_soln - §mmmmra mwmmulucnam mwamonwmmnlm 3...

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Unformatted text preview: §mmmmra mwmmulucnam mwamonwmmnlm 3 ME 375 HOMEWORK #9 Out: Mar. 25'", 20]] Due: Apr. 1", 2011 Spring 2011 ’\_, Problem #1 (30%} Consider the following system diagram, which represents a pump driven water flow source for a fire truck. The accumulator with capacitance Cf is used to dampen the pressure and flow oscillations that occur when the end of the fire hose is quickly opened or closed. Flows Q1 and Q2 are volumetric flow rates. Determine the input output differential equation with Ps(t) as the input, and P3r = P3-P,, as the output. Known constants are R1, R2, I, P,, and Cf. PF] '4 ‘—P.;SP i Q : PU Q 5 . : QC. ‘- c ' = Ce 1 .3 = 3 70. (A Pr 9 I OH: I {a 4,. h " ' = ' o + “ . o + ‘ di .9; Q ' r= e a. “do. . : " I a. L ’ l = ' ' 4' 1 PP 4’ " + 7 0H:- R1. A477- 1. - C. _ F: I '. ' _‘ + ’ arr} Problem #2 (30%! Consider the following system diagram, representing the process of filling a remote tank in a batch process at a chemical plant. a) Assuming the shut-off valve is open Derive the input-output differential equation assuming that Ps(t) is the input, and P3X = P3-P,, is the output. Known constants are Rf, I, A., and Pr. Assume the density of the fluid in the tank is p. b) Assuming the shut-off valve is open. Derive the input-output differential equation assuming that Ps(t) is the input, and the flow into the tank, is the output. Known constants are Rf, I, A., and P,. Assume the density of the fluid in the tank is p. S'A ' M .U l_ A a l! a; _V e —— . _ 7 . Problem 3140%} I Consider the following system. In this system the voice coil actuator applies a force F to a spool valve which directs high supply pressure P5, or low return pressure P,, to one side or the other of actuator \I piston. However these pressures are not realized instantaneously in the volumes V1 and V2 due to the presence of fluid resistance, bulk modulus of the fluids in these volumes, and the inertia dynamics of the spool and actuator. A/D + C'l’U+D/A Pos ilinn \\\\\\\\\\\‘ . '- 33mm .7/ A Voice coil amplifier Hydraulic Supply With the force generated by the voice coil F considered as the input, derive the followin eou led e nations of motion: CM: M: .. . — :5p + — x,,xp + ZApxp = Jc,,(Ps — Pr) AP A? vav = F In so doing, assume that: R? _ P1 = R(xv)Q1 P2 — Pr = R(xv)Q2 I71+CP1=Q1 172 + 5P2 = ‘02 where: . ‘v' - R(x,,) is the flow resistance, that, as expected, is a function of the spool valve position. Specifically, assume that R(x,,) = 1/ xv (so that xv = 0 corresponds to an infinite flow resistance and therefore no flow, and that as xv increases the resistance decreases — which would be reasonable for xv values up to, but not exceeding those that correspond to the flow areas at full opening). - C is the fluid capacitance (assumed to be the same for both volumes V1 and V2). Also, as an additional hint, consider that, by inspection: Apip = V1 = 472 NOTE: you must show all your work in deriving the 2 coupled equations of motion. This is even more critical given the fact that we have given you the final result (the 2 coupled equations of motion) to check your work. —— __ A43 )SPJ {[1 ‘ \JJ. 3 —: " .3 ‘PLLAQ (I; - ab A ég/‘V-B/ ’- lag F V I- ’ 0., a in . r7 ~ __.s__ (Ag :9 .. : P - (gm) ‘ - P?) — 9: mp AWL f I I x I L- '- 'm“ I . I - I. A I L94. .0 J \l ‘ t ‘a ‘V AA 1 ...
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