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Cork Institute of Technology
Bachelor of Engineering (Honours) in Mechanical Engineering – Stage 3
(Bachelor of Engineering in Mechanical Engineering – Stage 3)
(NFQ – Level 8)
Summer 2005
Control Systems Engineering
(Time: 3 Hours)
Answer any FOUR Questions Examiners: Prof. J. Monaghan ALL questions carry equal marks. Mr. J. E. Hegarty Dr. M. J. O’Mahony
Q1. A winding oscillator consists of two steel spheres on each end of a long slender rod, as shown in Fig. Q1. The rod is hung on a thin wire that can be twisted many revolutions without breaking. The device will be wound up to 4000 degrees. Assume that the thin wire has a rotational spring constant of 2 x 10 -4^ Nm/rad and that the viscous friction coefficient of the sphere in air is 2 x 10 -4^ Nm/rad/s. The sphere has a mass of 1 kg. Obtain: (a) the differential equation of motion of the system (5 Marks) (b) the Laplace transform of the above; Θ(s) (10 Marks) (Assume that the initial conditions are θ (0) = θ 0 and^ ddt^ θ(0) = 0 )
(c) Given that 0 2 1 2 2 ( ) sin( 1 ), where tan^1 1 θ t θ e^ ζω^ n^ t ω (^) n ζ t φ φ ζ ζ ζ = −^ − + = −^ − − , determine how
long it will take until the motion decays to a swing of only 10 degrees. (10 Marks)
Q2. (i) Discuss the relative advantages and disadvantages of open loop and closed loop control systems. (10 Marks)
(ii) An automobile speed control system is shown in Fig. Q2. ∆D(s) is the load disturbance due to a percentage grade. The engine gain K (^) e varies within the range 10 – 1000 for various models of cars. The engine time constant τe, is 20 seconds.
(a) Determine the sensitivity of the system to changes in the engine gain K (^) e. (5 Marks) (b) Determine the effect of the load torque on the speed. (5 Marks) (c) Determine the constant percentage grade ∆D(s)=∆d/s for which the vehicle stalls (velocity V(s)=0) in terms of the gain factors. You may assume that the grade is constant and thus the steady state solution is sufficient. Assume that R(s) =30/s (km/hr) and that Ke K 1 >>1. (5 Marks)
Q3. (a) Using the Final Value Theorem derive an expression for the steady state error signal ess of a negative feedback control system in terms of forward loop transfer function G(s), the feedback loop transfer function H(s) and the input transfer function R(s). (8 marks)
(b) Explain what is meant by the Type No. of a control system and discuss its effect on the steady state response of a system to step, ramp and parabolic inputs. (7 marks)
(c) The system shown in Fig. Q3 controls the heading direction of a modern ship. The impact of a constant wind force is represented by the disturbance D(s). (i) Determine the steady-state error due to a disturbance D(s)=1/s (let R(s)=0) for K= and K=30. (5 marks) (ii) Show that the rudder can then be used to bring the ship deviation back to zero. (5 marks)
Q6. A control system is found to have the following open loop transfer function:
( ) ( ) (^) ( 2 2 5) G s H s K = (^) s s + s + (a) Find the asymptotes and draw them in the s-plane. (5 marks) (b) Find the angle of departure from the complex poles. (5 marks) (c) Determine the gain when the two roots lie on the imaginary axis. (5 marks) (d) Plot the root locus as K varies from 0 to ∞. (10 marks)
Fig Q1:Winding oscillator
Fig Q2: Automobile speed control
Percentage overshoot versus Damping ratio for a second order system Fig Q5(b)
The Optimum Coefficients of T(s) based on the ITAE criteria (Zero Steady-State Step Error Systems) Step Input
2 2 3 2 2 3 4 3 2 2 3 4 5 4 2 3 3 2 4 5 6 5 2 4 3 3 4 2 5 6
1.75 2. 2.1 3.4 2. 2.8 5.0 5.5 3. 3.25 6.60 8.60 7.45 3.
n n n n n n n n n n n n n n n n n n n n n
s s s s s s s s s s s s s s s s s s s s s
ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω
