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examI_s96 - EE 429 EXAM I 26 February 1996 Name ID DO NOT...

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EE 429 EXAM I 26 February 1996 Name: ID#: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO Problem Weight Score 1 35 2 35 3 30 Total 100 This test consists of three problems. Answer each problem on the exam itself; if you use additional paper, repeat the identifying information above, and staple it to the rest of your exam when you hand it in. The quality of your analysis and evaluation is as important as your answers. Your reasoning must be precise and clear; your complete English sentences should convey what you are doing. 1
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Problem 1: (35 points) Fluid flows play an important role in many physical systems. As an example, hydraulic fluid flows are used to move the control surfaces on airplanes. In this problem you will analyze the water tank system shown in Fig. 1. Water enters at the top of the tank with a mass flow rate of ω in ( t ) [Kg/sec] and exits at the bottom of the tank with flow rate ω out ( t ) [Kg/sec]. The fluid level h ( t ) satisfies ˙ h ( t ) = 1 A T ρ ( ω in ( t ) - ω out ( t ) ) where A T [m 2 ] is the cross-sectional area of the tank and ρ [Kg/m 3 ] is the density of water. This continuity relation is simply a statement of the conservation of matter. If the input and output flow rates are equal, then the fluid level remains constant. On the other hand, if ω in > ω out , then the fluid level h ( t ) increases. The fluid exits the tank through a discharge nozzle with cross-sectional area A N . The output flow rate is given by Toricelli’s equation ω out ( t ) = ρA N 2 g 1 - A 2 N A 2 T h ( t ) , where g = 9 . 8 m/sec 2 (use the symbol g in your analysis, do not substitute the numeric value for g ). . . h ( t ) A N A T ω in ω out Figure 1: Elementary system for studying fluid flow dynamics. 1. (5 points) Find a state-space representation using x ( t ) = h ( t ) as the state variable, u ( t ) = ω in ( t ) as the system input and y ( t ) = h ( t ) as the system output.
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