Exercises - Mechanical Engineering Department ME 304...

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Mechanical Engineering Department Middle East Technical University ME 304 CONTROL SYSTEMS EXERCISES Prof. Dr. Y. Samim Ünlüsoy I/ Introduction & Modeling PROBLEM I-1: Consider the swing axle type of independent suspension of a vehicle. The rigid half axle pivots about the fixed body point O. A strut, which consists of a spring and a damper coaxially placed, is pivoted on the half shaft at one end and on the body at the other. Neglect the masses of the strut (spring and damper) and half shaft. Wheel mass is m, the spring constant is k, and the damper damping coefficient is b. Assume the equilibrium position of the system to be the horizontal position of the half shaft. For small motions about the equilibrium position : a) Identify the elements and write down the elemental equations. Hint : Give a small rotation to the half shaft first. b) Write down the structural equation(s) and identify them as continuity or compatibility equation(s). c) Obtain the input output relation (input : force F on the mass, output : angle θ of the half shaft) Ans.: c) 22 bd kd mL F LL θ+ θ= && & m L d g θ O b k d L

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Mechanical Engineering Department Middle East Technical University ME 304 CONTROL SYSTEMS EXERCISES Prof. Dr. Y. Samim Ünlüsoy PROBLEM I-2: A fluid system consists of two tanks with cross-sectional areas of A 1 and A 2 . Valves of resistance coefficients R 1 and R 2 are fitted to the outlet of each tank. The fluid flows from the first tank into the second tank as illustrated in the figure. Fluid flows into the first tank and out of the second tank at rates of Q 1 and Q 2 . The density of fluid is denoted by ρ . Follow the procedure outlined in the first problem and obtain the input output relation (input : Q i , output : Q o ). Ans.: 2 oo 1212 11 22 io 2 dQ AA RR AR A R QQ gd t t ⎛⎞ + ⎜⎟ =+ + ρ ⎝⎠ ρ PROBLEM I-3: In order to harden a steel shaft, it is heated to a temperature of T os and then it is quenched in a small can containing m b kg of water at an initial temperature of T ob . Thus while the shaft is cooling down, the bath temperature will be increased as a result of the quenching process. The convective heat transfer coefficient, and the surface area and mass of the steel shaft are denoted by h, A, and m s , respectively. The specific heats of the steel shaft and the water are denoted by c s and c b . Obtain the differential equations governing the variations of the shaft temperature T s and bath temperature T b as functions of time. Ans.: 2 bb ss s s bb ss 2 mcmc dT dT (m c m c ) 0 hA dt dt ++ = , 2 b b 2 m c ) 0 hA dt dt + += Q o g h 1 Q i R 1 R 2 h 2 Q m A 1 A 2
Mechanical Engineering Department Middle East Technical University ME 304 CONTROL SYSTEMS EXERCISES Prof. Dr. Y. Samim Ünlüsoy PROBLEM I-4: A hydraulic accumulator is illustrated in the figure. It is used to damp out pressure pulses by storing fluid during pressure peaks and releasing fluid during periods of low pressure.

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Exercises - Mechanical Engineering Department ME 304...

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