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Cork Institute of Technology
Bachelor of Engineering (Honours) in Mechanical Engineering- Stage 3
(NFQ Level 8)
Summer 2006
Control Systems Engineering
(Time: 3 Hours)
Answer any FOUR Questions Examiners: Prof. M. Gilchrist ALL questions carry equal marks. Mr. J. E. Hegarty Dr. M. J. O’Mahony
Q1. The water level, h ( t ), is controlled by an open-loop system, as shown in Fig.Q1. A dc motor controlled by an armature current ia turns a shaft, opening a valve. The inductance of the dc motor is negligible, that is, L (^) a = 0. Also, the rotational friction of the motor shaft and valve is negligible, that is, b = 0. The height of the water in the tank is
h t ( ) = ∫ (1.6 ( ) θ t − h t ( ) ) dt
the motor constant is Km = 10, and the inertia of the motor shaft and valve is J = 6 x 10 -^3 kg-m 2. You may assume that K (^) m=K (^) b =K (^) T, i.e the back emf constant Kb is equal to the motor torque constant K (^) T.
Determine
(a) the differential equation for h ( t ) and v( t ) and (b) the transfer function H(s)/V(s) (25 marks)
Q2. An electro-hydraulic servo controlled system is shown in Fig. Q2. Assume that the rate of oil flow to the piston is given by: q = K (^) x x – K (^) p p. where x = Spool valve displacement p = Piston differential pressure Oil compressibility, leakage and damping are assumed to be negligible. The input displacement xi is applied to a potentiometer, while the output displacement y is detected by a potentiometer. Both potentiometers have sensitivity K 1. Construct a block diagram for the system and hence or otherwise obtain the transfer function Y(s)/X (^) i (s) for the system and comment. Leakage and compressibility effects can be assumed to be negligible.
The system parameters are as follows;
Load Mass M = 1000 kg Control valve constants K (^) x = 0.02 m^3 /s per m of valve opening K (^) p = 0.5 x 10-8^ m^3 /s per N/m^2 Potentiometer Sensitivity K 1 = 100 V/m Power amplifier Gain K 2 = 0.5 A per V Linear Transducer constant K 3 = 10 m per A Piston Area A = 0.01 m^2 (25 marks)
Q3. (i) State Mason’s signal-flow gain Formula. (10 marks)
(ii) Find the transfer function (()) 2
1 R s
C s for the multivariable system shown in Fig Q3.
(15 marks)
Q6. Sketch the root locus of the control system with the following open loop transfer function: G s H s ( ) ( ) = (^) s s ( + 5)(^ K s +7) (15 marks)
Determine the the value of K which will result in a marginally stable system. What is the natural frequency of oscillation for this value of K.? (5 x 2 marks)
Fig Q1.
Fig Q
Amplifier
K (^) a=
va ia
v (^) V (^) b
h(t)
θ(t) ω(t)
