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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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