UNIVERSITY COLLEGE LONDON
EXAMINATION FOR INTERNAL STUDENTS
MODULE CODE :
ASSESSMENT :
PATTERN
MODULE NAME :
DATE :
TIME :
MECH302P
MECH302PA
Dynamics and Contrc
Tuesday 22 May 2018
10:00
TIME ALLOWED : 3 hrs
This paper is suitable for candidates who attended classes for this
module in the following academic year(s):
Year
2016/17.2017/18
EXAMINATION PAPER CANNOT BE REMOVED FROM THE EXAM HALL. PLACE EXAM
PAPER AND ALL COMPLETED SCRIPTS INSIDE THE EXAMINATION ENVELOPE
2016/17-MECH302PA-001-EXAM-Mechanical Engineering 10
© 2016 University College London
TURN OVER
MECH302P 2017/18 2
Answer all questions.
Note: A table of equations is included at the end of the exam paper.
Section A: Dynamics
1. Answer the following questions.
a) What is resonance? What is the response of a system with no damping at its
resonance condition?
13 marks]
b) How many degrees of freedom has the system shown in Figure 1? The pulley
has a mass moment of inertia of/. You should justify your answer.
[4 marks]
J
s s / s s s s s s
1
m
Figure 1. dynamic system for question 1 (b).
c) In numerical calculations, describe briefly "round-off error" and "truncation error".
[3 marks]
d) In a passenger vehicle, briefly explain the mechanism by which low-frequency
road noise reaches the cabin.
[2 marks]
TURN OVER
MECH302P 2017/18
2. Solve the problems stated below for single degree of freedom (SDOF) systems.
a) The nondimensional Frequency Response Function (FRF) for a single degree of
freedom system is given in Figure 2. What is the damping ratio of the system?
[3 marks]
0 0.5 1
Frequency ratio, r
Figure 2. FRF of a SDOF system for question 2(a).
b) Calculate the natural frequency and damping ratio for the system in Figure 3
given the values m = 10 kg, c = 100 kg/s, kx = 4000 N/m, kz = 200 N/m, and
fc3 = 1000 N/m. Assume that no friction acts on the rollers. Is the system
overdamped, critically damped, or underdamped?
[8 marks]
Figure 3. The SDOF system for question 2(b).
c) The equation of motion for an underdamped SDOF system can be written as:
mx + 2^ma)nx + kx = F cos u)t
Find the response of the system for a case of m = 1.2 kg, ( = 0.01, a)n = 2.5
rad/s, F = 6 N, and u> = 10 rad/s with initial conditions x0 = 1 m/s and xo = lm.
[10 marks]
CONTINUED
MECH302P 2017/18
3.
a) Consider the system shown in Figure 4 consisting of two pendula connected by a
spring. Assume small angle oscillations.
i) Obtain the equations of motion for this system using Lagrange's method.
ii) Linearise the equations of motion and write them in matrix form.
[10 marks]
Figure 4. Two pendula of the same length connected by a spring for question 3(a).
b) The equations of motion of a two degree of freedom system are given as:
[2
W 1.4 -35 40
where the mass matrix has units of kg and the stiffness matrix is in N/m. The
natural frequencies and mass normalised mode shape of this system are:
o)x = 2.41 rad/s, o)2 = 6.91 rad/s
_ (0.52) £(2) _ ( 0.48 1
~ IO.57J' "1-0.62J
What is the response of the system to an initial condition of:
*° = (oilm and lo = (o)m/s-
[7 marks]
END OF SECTION A
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5 MECH302P 2017/18
Section B: Control
4.
a) The horizontal position of a gripper at the end of a robot arm is given by
y(t)= 2.6 + 7.1sin(0(t))
Where 9 is the angular position of the arm joint in radians.
i) Explain the principle of superposition and hence state why the relationship
above is nonlinear.
[3 marks]
ii) Linearise the relationship about the normal operating point where 6 = 0.3
radians.
[4 marks]
b) A multi-input, multi-output model has the following equations of motion where xu
X2 and X3 are the state variables and ui and uz are the two inputs.
x2 = Sx2 - O.lUi + 0.3u2
Write these equations in the conventional state-space matrix format: x = Ax + Bu
[4 marks]
CONTINUED
MECH302P 2017/18
5. During a gear-change, the engine control unit of a high-performance car controls
the engine speed (in RPM) to match the gearbox's input shaft speed. The transfer
function of the engine is given by:
W) 0.44
T(s) (0.16s +1)
Where ft(s) is the angular speed of the engine in RPM and r(s) is the throttle value.
a) Calculate the closed-loop transfer function if a unity-feedback control scheme is
used with a proportional gain k= 18.
[3 marks]
b) State the time constant and gain of this closed loop system.
[2 marks]
c) Starting with the engine running at 2,000 RPM, sketch a graph of the engine
speed in response to a step demand to increase speed by 2,000 RPM.
[4 marks]
d) Sketch a Bode plot of the Open Loop system with the proportional gain of 18
included.
[5 marks]
e) Using your plot from part (d), estimate the Phase margin and Gain margin.
[3 marks]
f) The control algorithms take 0.5 ms to calculate. Given the crossover frequency is
49.1 rad/s, is this delay significant to the performance?
[2 marks]
g) The following statement contains three factual errors. For each incorrect phrase,
state what the correct phrase should be.
"A Bode plot shows the frequency response of a system as the loop gain K is
increased from 0 to infinity. This can be obtained from any transfer function by
letting s -* jw, or can be measured experimentally. The upper plot shows the
magnitude ratio of the output relative to the input in dB which is a linear scale in
order to show large changes clearly. The lower plot shows the damping ratio of
the output relative to the input, usually expressed in degrees instead of radians.
Bode plots can be used to identify a phase margin or gain margin to predict the
stability of a closed loop system."
[4 marks]
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MECH302P2017/18
6. The vertical position response of a magnetic-levitation bearing can be
approximated by the following open-loop transfer function:
64°/(s) " (0.020s +
Where Y(s) is the levitation gap and I(s) is the control current to the
electro-magnetic coils.
a) Find the positions of the open loop poles for GP(s) and briefly comment on the
stability of the system without feedback.
[3 marks]
b) It is proposed to add a lead controller to GP to be used in closed-loop control.
Two alternatives are considered: Gd = (s - 9) or G& = (s + 20). Sketch a root
locus plot for each case and describe the likely effectiveness of each approach,
commenting specifically on stability.
[9 marks]
c) Choose any inherently unstable system which has not previously been mentioned
in this paper and describe how a controller acts to stabilise it. Specify:
i) Input or set-point
ii) Sensor
iii) Disturbance (any possible example)
[4 marks]
CONTINUED
MECH302P 2017/18
USEFUL FORMULAS FOR DYNAMICS:
In the following formulas, the symbols take their usual meanings. All the equations
are for damped systems and undamped system can be obtained from them.
Damped free response:
x(t) = i4c-<"«t sin(o)dt + $)
A = 7T^*
. -x( X0&d \0 = tan~1 . ,_B—\x0 + S(onx0J
Forced Harmonic excitation:
Unit impulse response (SDOF):
git) sina)dt
ma>d
Convolution integral:
xit) = I F(r)o(t - r)dTJo
Lagrange's equation:
TURN OVER
9 MECH302P 2017/18
END OF PAPER
END OF PAPER