Question

PROBLEM #2 (30%) A tank of cross section area Ar supplies fluid to a piston-cylinder shown below. Fluid is delivered to the top of the tank at a volumetric flow rate of qin. Let R represent the flow resistance in the pipe connecting the tank and the cylinder. The fluid is assumed to be incompressible. The piston area is Ap and its mass is m. You are to IGNORE the capacitance of the fluid in the cylinder. The fluicd capacitance for the tank is Cr. You are to find the transfer function relating the velocity of the piston v to the input volumetric flow rate qim using the steps below. (Hint: You may want to work in the Laplace domain.) in iston ylinder Im P2 Pa rarea of lank Ap -area of piston (a) Draw the free-body diagram of the piston and derive the relationship between the piston displacement r and the cylinder pressure p (b) Find the coupling relationship between the flow rate qout and the piston velocity u (c) Find the fluid elemental equation for the flow resistance R (d) Find the fluid elemental equation for the tank, relating (p1 -Pa) with input and output flow rates qm and gou. (e) Find the transfer function between the input flow rate qin and the piston velocity u-x Note the Laplace transform of is s .X(s)

Answer 1

om Sp XA Kx tcx.kxtCY一ㄧㄧㄧㄚ qi(S) Pi- Pa Po

ΔΡ dt dh dF (n-Au)

(S) -1S