Engineering Maths 2
UFQEFK-10-2
Mathematics for Mechanical Engineering
UFQEFB-20-2
Module leader: Dr Robert Laister, Room 2P34, email: Robert.Laister@uwe.ac.uk
Pre-requisites:
Students are expected to be familiar with the material covered in Engineering Maths 1 or
equivalent. In particular, you are expected to know how to: differentiate functions using the
product, quotient and chain rules; integrate products of functions by parts; manipulate and
conceptualise vectors, compute scalar (dot) and vector (cross) products; multiply matrices
and calculate determinants; manipulate algebraic fractions and use partial fractions.
Teaching and learning:
Each student should attend a two hour lecture and a one hour tutorial per week. The lecture
will present the underlying theory, accompanied by worked examples. During the tutorial
you may continue to work through the exercises set the previous week. Your tutor may also
take the opportunity to go through some examples with the whole class. You should use
this time to ask your tutor about any problems you are having in completing the exercises
or with concepts relating to the material covered in the lectures. It is essential that you
prepare for and attend tutorials.
Assessment:
UFQEFK-10-2 Engineering Mathematics 2:
20% coursework and 80% by examination (January 2006).
The coursework will consist of computer-based QuestionMark tests on Laplace Transforms
and on Vector Geometry. You are allowed to attempt each test twice. Your final coursework
mark is the average of your highest marks attained on each test. For example, if you score
34% and 78% on the Laplace Transform test, 48% and 62% on the Vector Geometry test,
then your final C/W mark will be (62+78)/2 = 70%. Further details on how to access and
complete the tests will be given during the first semester. The 2 hour written examination
will be held during the two assessment weeks in January 2006.
UFQEFB-20-2 Mathematics for Mechanical Engineering:
60% coursework and 40% by examination (January 2006).
The Jan 2006 exam and the computer-based tests will be the same as for UFQEFK-10-2
above. The coursework consists of two elements: (i) (semester 1) computer-based tests (1/5),
(ii) (semester 2) case study on Fourier series and numerical methods for partial differential
equations (4/5). (The fractions in brackets denote the relative weightings towards the overall
coursework mark).
Module web site: http://www.cems.uwe.ac.uk/∼rlaister/uqm008h2.html
Other learning resources: Maths and Stats Learning Centre 2P39... and the library of
course!
1
Weeks 1-3 Laplace Transforms
Introduction: Laplace transforms are a useful mathematical tool for solving differential
equations and for describing the input-output relationship of a linear system. The Laplace
transform can convert ordinary differential equations into (simpler) algebraic equations.
Engineers commonly use Laplace tranforms in the design and analysis of control systems.
Week 1: Introduction to Laplace Transforms
Learning outcomes. You should be able to:
(i) give the definition of the Laplace transform,
(ii) find the Laplace transforms of some common functions,
(iii) state and correctly use the linearity property,
(iv) state and correctly use the first shift theorem.
Reading:
Lecture notes plus handout on Laplace Transforms,
CDH pp 506–514 (taken from ‘Engineering Mathematics’
(1992) by Croft, Davison and Hargreaves).
Exercises:
Handbook exercise sheet:
Week1: Laplace transforms
2
Exercise Sheet: Week 1: Laplace Transforms
(1) Use tables to find the Laplace transforms of the following functions:
(a) sin 6t (b) cos (4t/3) (c) t4
(d) t2 t3
3t
4t
−t
(e) e
(f ) 1/e
(g) e sin 3t (h) (cos 7t)/e5t
(2) Using linearity find the Laplace transforms of the following functions:
(a) 3t2 − 4 (b) 3e2t + 4 sin t (c) 3t4 e5t + t
(3) Write down L(t), L(t2 ) and L(t3 ), where L denotes the Laplace transform operator.
Deduce that L(f (t)g(t)) 6= L(f (t))L(g(t)) in general.
(4) Suppose that the function f (t) is unknown but that its Laplace transform, F (s), is
known and is given by
F (s) =
3s2 − 1
.
s2 + s + 1
Using the First Shift Theorem find the Laplace transforms of the following functions:
(a) e−t f (t) (b) e3t f (t) (c) e−t/2 f (t)
(5) Use the Laplace transform to solve the following first order, linear differential equations
with the given initial conditions:
+ 4y = 1,
x(0) = 1 (b) 2 dy
dt
(a) x0 + x = 3,
y(0) = 4
Answers
(1)
(a)
(e)
(b) 9s29s+16 (c)
1
(f ) s+4
(g)
6
s2 +36
1
s−3
(2) (a)
6
s3
(4) (a)
3s2 +6s+2
s2 +3s+3
−
4
s
(b)
3
s−2
(b)
+
24
s5
3
s2 +2s+10
4
s2 +1
3s2 −18s+26
s2 −5s+7
(c)
(c)
(d)
(h)
72
(s−5)5
+
120
s6
s+5
s2 +10s+74
1
s2
12s2 +12s−1
4s2 +8s+7
(5) (a) x = 3 − 2e−t (b) 41 (1 + 15e−2t )
3
Week 2: Inverse Laplace transforms
Learning outcomes. You should be able to:
(i) Write down the inverse Laplace transforms of some standard functions.
(ii) Compute the inverse L.T. using partial fractions.
(iii) Compute the inverse L.T. by completing the square.
Reading:
Lecture notes plus handout on Laplace Transforms,
CDH pages 516–520.
Exercises:
Handbook exercise sheet:
Week2: Inverse Laplace tranforms
4
Exercise Sheet: Week 2: Inverse Laplace Transforms
(1) Use tables to
1
(a)
s
5
(e)
s−3
2
(i)
(s + 1)2
find the inverse Laplace transforms of:
2
8
6
(b) 3
(c)
(d)
s
s
s+4
2
s
20
(f ) 2
(g) 2
(h) 2
s +4
s +9
s + 16
9
(j)
(s + 3)3
(2) Using linearity (and some ingenuity!) find the inverse Laplace transforms of:
3
4
1
30
1
(a)
(b) − 3
(c) 2
(d)
2s
s s
s
3(s + 2)
3s − 7
s+4
5
6s + 17
(e) 2
(f )
(g)
(h)
s +9
(s + 4)2 + 1
(s + 4)2 + 1
(s + 4)2 + 1
s+5
6s + 9
7s + 3
(i) 2
(j) 2
(k) 2
s + 8s + 20
s + 2s + 10
s + 4s + 8
(3) Use partial fractions to find the inverse Laplace transforms of:
5s + 2
4s + 1
3s + 5
(a)
(b)
(c)
(s + 1)(s + 2)
s(s + 2)(s + 3)
(s + 1)(s2 + 3s + 2)
2
2s − 8
s + 4s + 4
1−s
(d)
(e) 3
(f )
2
2
(s + 2)(s + 7s + 6)
s + 2s + 5s
(s + 1)(s2 + 2s + 2)
2
s+4
3s − s + 8
(g) 2
(h)
s + 4s + 4
(s + 2)(s2 − 2s + 3)
Answers
(a) 1
(b) t2
(c) 8
(d) 6e−4t
3t
(f ) sin 2t
(g) cos 3t (h) 5 sin 4t
(1) (e) 5e
(i) 2te−t (j) (9/2)t2 e−3t
(2)
(a) 3/2
(b) 4 − t2 /2 (c) 30t
(d) e−2t /3
(e) 3 cos 3t − (7/3) sin 3t (f ) e−4t cos t (g) 5e−4t sin t
(h) 6e−4t cos t − 7e−4t sin t (i) e−4t cos 2t + (1/2)e−4t sin 2t
(j) 6e−t cos 3t + e−t sin 3t (k) 7e−2t cos 2t − (11/2)e−2t sin 2t
(a) 8e−2t − 3e−t
(c) e−t + 2te−t − e−2t
(3)
(e) 1/5(4 + e−t cos 2t + (11/2)e−t sin 2t)
(g) e−2t (1 + 2t)
5
(b) 1/6 + (7/2)e−2t − (11/3)e−3t
(d) 3e−2t − e−6t − 2e−t
(f ) e−t (2 −√
2 cos t √
− sin t)√
−t
(h) e (cos 2t + 2 sin 2t) + 2e−2t
Week 3: Laplace Transforms and Ordinary Differential Equations
Learning outcomes. You should be able to:
(i) compute the Laplace transform of first and second derivatives,
(ii) solve linear, second order differential equations via Laplace transforms.
Reading:
Lecture notes plus Laplace transforms handout
CDH pages 515, 526–533
Exercises:
Handbook exercise sheet:
Week 3: Laplace transforms and differential equations
6
Exercise Sheet: Week 3: Laplace transforms and differential equations
1 Solve the following linear second order differential equations, subject to the given
initial conditions:
(i) y 00 + y 0 + y = 1,
(ii) x00 + x = 2t,
y(0) = 1, y 0 (0) = 3;
x(0) = 0, x0 (0) = 5;
(iii) 2ẍ + ẋ − x = 27 cos 2t + 6 sin 2t,
(iv) ẍ + ẋ − 2x = 1 − 2t,
x(0) = −1, ẋ(0) = −2;
x(0) = 6, ẋ(0) = −11;
2
dz
− 4z = 4(cos 2t − 1),
dt2
d2 y
dy
(vi)
− 5 − 6y = 14e−t ,
2
dt
dt
dz
(0) = 0;
dt
dy
y(0) = 3,
(0) = 8;
dt
(v)
z(0) = 1,
2∗ Consider the following linear second order differential equation corresponding to a
harmonic oscillator with a sinusoidal input:
m
d2 x
+ kx = sin(ωt),
dt2
(1)
where m, k, and ω are given constants with m, k, > 0. Suppose the system
is initially perturbed from equilibrium with unit velocity. The initial condition is
therefore given by
dx
x(0) = 0,
(0) = 1.
(2)
dt
Solve the equation (??) subject to (??) for each of the following cases:
(i) = 0 (simple harmonic motion);
(ii) 6= 0 and ω 2 6= k/m (periodic and quasi-periodic motion);
(iii) 6= 0 and ω 2 = k/m (resonance).
(For part (iii) you’ll need to use the tables on p.508 of the Laplace transform handout
notes).
Now fix m = k = 0.01 and = 0.1. Plot the graph of your solution (x against t)
for 0 ≤ t ≤ 130 with the following values of the input frequency
(feel free to use a
√
graphical calculator or a computer): (a) ω = 1.142, (b) ω = 2, (c) ω = 1. What do
you observe?
Remark: (A) In a mechanical system the differential equation (??) arises in an undamped mass-spring system where the spring is attached to point which is oscillating
periodically. Here the mass is m, x is the displacement of the mass from its equilibrium position, k is the ‘stiffness’ of the spring and the forcing is sin ωt. (B) In an
electrical system (??) arises in a series LC circuit in which x is the current, L = m
is the inductance, C = 1/k is the capacitance and sin ωt is an alternating current
source.
7
Answers
1
√
√
(i) y = 1 + 2 3e−t/2 sin ( 3t/2);
(ii) x = 3 sin t + 2t;
(iii) x = −3 cos 2t + 2e−t ;
(iv) x = t + 6e−2t ;
(v) z = (1/4)e−2t + (1/4)e2t − (1/2) cos 2t + 1;
(vi) y = (8/7)e−t + (13/7)e6t − 2te−t ;
2
(i) x = (1/ω0 ) sin ω0 t, where ω02 = k/m;
ω 2 − ω 2 − (/m)ω
/m
(ii) x = 0
sin ω0 t + 2
sin ωt;
2
2
ω0 (ω0 − ω )
ω0 − ω 2
#
"
"
#
#
"
"
#
2ω0 + (/m)
(/m)
(iii) x =
sin ω0 t −
t cos ω0 t.
2
2ω0
2ω0
8
Weeks 4-6: Vector geometry of lines, planes and curves
Introduction: Certain physical quantities are fully described by a single number or scalar,
e.g. mass, speed. Other quantities are not fully decribed unless a direction is specified in
addition to a scalar. e.g. electric field, pressure, velocity. Such quantities are called vectors.
Vectors are a very useful way to describe positions and orientations in space. In particular,
we will use them to describe the geometry of lines, planes and curves in three-dimensional
space and to investigate some simple problems in particle motion.
Week 4: Vector geometry of planes
Learning outcomes. You should be able to:
(i) Define and compute the vector equation of a plane.
(ii) Define and compute the Cartesian equation of a plane.
(iii) Calculate the anlge between two planes.
(iv) Calculate the distance of a point from a plane.
Reading:
Lecture notes
Exercises:
Handbook exercise sheet:
Week 4: Vector geometry of planes.
9
Exercise Sheet: Week 4: Vector geometry of planes
1 The following planes have been expressed in Cartesian form. Rewrite them in vector
form:
(i) 2x − 3y + 4z = −1
(ii) y = 0
2 The following planes have been expressed in vector form. Rewrite them in Cartesian
form:
(i) r · (2i + j − 3k) = 3
(ii) r · (−j + 2k) = 0
3 Find the Cartesian equation of the plane passing through the three points P =
(−4, −1, −1), Q = (−2, 0, 1) and R = (−1, −2, −3).
4 Determine whether the given point A lies on the given plane P :
(i) P : x + z = 7; A = (12, −4, −5)
(ii) P : −x − y + 4z = 1; A = (1, 0, 1)
5 Determine whether the four points P = (1, 1, 1), Q = (3, 1, −1), R = (0, 1, −7) and
S = (2, 1, 3) lie in the same plane (i.e. coplanar).
6 Determine whether the following planes are parallel, perpendicular or neither:
(i) P1 : x − y + 2z = 0;
P2 : −3x + 3y − 6z = 7
(ii) P1 : −x − 11y + 2z = 1;
P2 : 2x + z = −3
(iii) P1 : 7x + 3y − 4z = −1;
P2 : 2x + 2y − z = 0
7 Find the acute angle of intersection between the two planes P1 : 2y − 2z = 1 and
P2 : 3x + y − z = 1.
8 Find the distance of the point (3, 1, −2) from the plane x + 2y − 2z = 4.
10
Answers
1
(i) r · (2i − 3j + 4k) = −1
(ii) r · j = 0
2
(i) 2x + y − 3z = 3
(ii) −y + 2z = 0
3 2y − z = −1
4
(i) Yes;
(ii) No
5 Coplanar
6
(i) Parallel;
(ii) Perpendicular;
(iii) Neither
7 64.76 degrees
8 5/3
11
Week 5: Vector geometry of lines
Learning outcomes. You should be able to:
(i) Define and compute the vector equation of a line.
(ii) Define and compute the parametric equation of a line.
(iii) Calculate the angle between two lines.
(iv) Find the intersection point of two lines.
(v) Find the intersection point of a line and a plane.
(vi) Calculate the angle between a line and a plane.
Reading:
Lecture notes
Exercises:
Handbook exercise sheet:
Week 5: Vector geometry of lines.
12
Exercise Sheet: Week 5: Vector geometry of lines
1 Find the vector equation of the line passing through the two points A and B for each
of the following:
(i) A = (3, −1, 5) and B = (0, 1, 1)
(ii) A = (0, 1, −4) and B = (−2, −3, 0)
2 Write down the parametric form of the lines in Q1 and find the acute angle between
them.
3 Determine whether the following lines are parallel:
(i) L1 : r = 2i + j − 3k + t(i − j) and L2 : r = j − 3k + s(i + j)
(ii) L1 : x = 1 − 3t, y = 4 + t, z = 2t and L2 : x = 7 + 6t, y = −2t, z = −3 − 4t
4 Determine whether the following lines intersect and find the point of intersection when
they do:
(i) L1 : r = 2i + j − 3k + t(i − j) and L2 : r = j − 4k + s(i + j)
(ii) L1 : x = 3 + 4t, y = 4 + t, z = 1 and L2 : x = 12t − 1, y = 7 + 6t, z = 5 + 3t
5 Determine whether the three points P , Q and R lie on the same line in each of the
following cases:
(i) P = (1, 1, 1) Q = (3, 1, −1) R = (9, 1, −7)
(ii) P = (1, 2, 1) Q = (0, 1, −3) R = (2, 2, 3)
Remark: points which lie on the same straight line are said to be colinear.
6 Let M denote the mid-point of the line segment joining the points P = (4, −4, 0) and
Q = (1, 3, 1). Find the parametric equation of the line passing through the points M
and R = (3, −6, 1).
7 Let T denote the triangle joining the points P = (2, −3, 1) and Q = (0, 3, 0) and
R = (2, 1, 1). Find the vector equation of the line passing through the corner R which
is perpendicular to T . (Hint: you may find the vector cross product useful).
8 Find the parametric equation of the line of intersection of the two planes P1 : 2y−2z =
1 and P2 : 3x + y + z = 1.
9 Find the point of intersection of the plane P : 2x − 3y + 4z + 7 = 0 and the line
L : r = 9i − j + 3k + t(−5i − j + k).
13
Answers
1
(i) r = 3i − j + 5k + t(3i − 2j + 4k)
(ii) r = j − 4k + t(i + 2j − 2k)
2
(i) x = 3 − 3t, y = 2t − 1, z = 5 − 4t
(ii) x = t, y = 2t + 1, z = −4 − 2t
Angle is 56.1 degrees.
3
(i) Not parallel
(ii) Parallel
4
(i) Don’t intersect
(ii) Intersect at (−17, −1, 1)
5
(i) Colinear
(ii) Not colinear
6 x = 5/2 + t, y = −1/2 − 11t, z = 1/2 + t
7 r = 2i + j + k + t(i − 2k)
8 x = t, y = 3(1 − 2t)/4, z = (1 − 6t)/4
9 (− 173
, − 43
, 49 )
3
3 3
14
Week 6: Vector Geometry of Curves
Learning outcomes. You should be able to :
(i) Define the parametric and vector equation of a curve.
(ii) Sketch simple curves (e.g. parabola, circle) given in parametric or vector form.
(iii) Define and compute the first and second derivative of a vector.
(iv) Compute the derivative of a scalar or vector product.
Reading:
Lecture notes
Exercises:
Handbook exercise sheet:
Week 6: Vector Geometry of Curves
15
Exercise sheet: Week 6: Vector Geometry of Curves
1) Find the velocity and acceleration for each of the following at the time indicated:
(i) r = i + (2t3 − t)j − tk;
−t
(ii) r = e i + cos tk;
t = 2.
t = 0.
(iii) r = t cos ti + t sin tj + tk;
t = 0.
2) If r = sin ti + cos tj + tk is the position vector of a particle at time t, show that the
speed of the particle is constant in time.
3) Let γ be a constant scalar and b and c constant vectors. If r = cos (γt)b + sin (γt)c
show that
d2 r
+ γ 2 r = 0 (centripetal motion)
dt2
dr
= γb × c (constant angular momentum)
(ii) r ×
dt
(i)
4) Sketch the curves in the x-y plane given by the following vector equations:
(i) r = 3ti − 3tj
(ii) r = ti + t2 j
(iii) r = et i − e−t j
In each case sketch the tangent (velocity) vector at time t = 1.
5) The path of a particle at time t is given by r = t2 i + 3j − 4tk. Describe this path
geometrically and find the time t0 at which the particle hits the plane with Cartesian
equation x + y + z = −1. Show that the particle strikes the plane tangentially at time
t0 .
Answers
1) (i) v = √
23j − k, a = 24j; (ii) v = −i, a = i − k; (iii) v = i + k, a = 2j
2) speed= 2
4) (i) straight line; (ii) parabola; (iii) hyperbola.
5) t0 = 2.
16
Weeks 7-9 Eigenvalues and Eigenvectors
Introduction: Many engineering problems can be formulated as systems of linear differential equations (continuous time) or linear difference equations (discrete time). These can be
written in matrix form. Associated to any square matrix there are special numbers (eigenvalues) and vectors (eigenvectors). The signs or magnitudes of the eigenvalues essentially
determine whether the solutions of a given system oscillate, decay to zero or grow unboundedly as time increases. We will concentrate on the mathematical techniques required to
compute eigenvalues and eigenvectors. Applications to difference and differential equations
will be given in weeks 8 and 9.
Week 7: Introduction to eigenvalues and eigenvectors
Learning outcomes. You should be able to:
(i) Define what is meant by an eigenvalue and an eigenvector of a square matrix.
(ii) Compute the eigenvalues and eigenvectors of 2 × 2 and 3 × 3 matrices.
Reading:
Lecture notes plus handout on ‘Eigenvalues and eigenvectors’
pages 189-195 (taken from Mathematical Techniques by Jordan and Smith).
Exercises:
Handout, Page 211-213, Problems Qns. 11.1 (a),(b), 11.2, 11.3,
11.4 (a),(b),(c), 11.22
17
Week 8: Discrete linear systems
Learning outcomes. You should be able to:
(i) Determine the eigenvalues and eigenvectors of the matrix An .
(ii) Solve systems of linear difference equations.
(iii) Determine whether the solutions of such a system decay to zero, oscillate or grow
unboundedly over time using only the eigenvalues.
Reading:
Lecture notes
Exercises:
Handbook exercise sheet:
Week 8: Discrete Linear Systems .
18
Exercise sheet: Week 8: Discrete Linear Systems
(1) Use eigenvalues and eigenvectors to find the solution of the system of discrete linear
equations xn+1 = Axn for each of the following:
"
(i) A =
2 −1
−1 2
#
"
, x0 =


2
0
#


1 −1 −1
1


3 
(ii) A = 
 0 1
 , x0 =  1 
0 3
1
−1
(2) Write the following systems of discrete linear equations in the form xn+1 = Axn and
hence find the solution:
(i)
xn+1 = xn + yn , x0 = −6,
yn+1 = 2xn + 2yn , y0 = 6.
(ii)
xn+1 = 2xn , x0 = 2,
yn+1 = 2yn + 2zn , y0 = 0,
zn+1 = 2yn − zn , z0 = 5.
(3) Using only the eigenvalues of the following systems, describe the behaviour of the
general solution of xn+1 = Axn for large n (i.e. as n → ∞):
"
(i) A =
2 3
4 6


1 1 2

(ii) A =  1 2 1 
.
2 1 1
#
(Recall exercises 11.1 and 11.4 from Week 7).
(4) (Just for fun!) The Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, . . . is generated by adding
together the previous two numbers in the sequence. Hence, if Fn denotes the nth
Fibonacci number, we have
Fn+2 = Fn+1 + Fn ,
with F0 = F1 = 1 (the first two Fibonacci numbers). By setting Gn = Fn+1 , show
that
"
Fn+1
Gn+1
#
"
=
0 1
1 1
#"
Fn
Gn
#
"
,
F0
G0
#
"
=
1
1
#
.
Hence obtain a formula for Fn in terms of n. Does this formula surprise you? Check
that your formula works by putting in n = 2, 3, 4, 5, . . ..
19
Answers
"
(1)
(i) xn =
1
1
#
"
n
+3


#
1
.
−1


0
1



n
(ii) x =  0  + (−2)  1 .
0
−1
"
(2)
(i) xn =
0
0
#
.






0
1
0



n
n
−1
0
(ii) x = 2(−2)n 
+
2(2)
+
3




 2 .
2
0
1
(3)
(i) λ = 0, 8, so unstable and |xn | → ∞ as n → ∞.
(ii) λ = 1, 4, −1, so unstable and |xn | → ∞ as n → ∞.
(4)
1
Fn = √
5
√ !n+1
1+ 5
1
−√
2
5
20
√ !n+1
1− 5
.
2
Week 9 Continuous linear systems
We will apply some of the techniques you have learned on eigenvalues, eigenvectors and
Laplace transforms to analyse the stability of systems of linear differential equations. These
ideas are central to the qualitiative understanding of dynamical and control systems engineering and design.
Learning outcomes. You should be able to :
(i) Define what is meant by ‘stability’ for differential systems.
(ii) Determine when a system of linear differential equations is stable.
Reading:
Lecture notes
Exercises:
Handbook exercise sheet:
Week 9: Continuous linear systems
21
Exercise sheet: Week 9: Continuous linear systems
(1) Use eigenvalues and eigenvectors to find the solution of the system of differential
equations ẋ = Ax, x(0) = x0 for each of the following:
"
(i) A =
3 −2
2 −2
#
"
, x0 =


−1
1
#
"
(ii) A =

−3 2
2 −3
#
"
, x0 =
3
1
#

1 −1 −1
−1



3 
(iii) A =  0 1
,
x
=

 0 
0
0 3
1
6
(2) Using only the eigenvalues decide whether the system of differential equations ẋ = Ax
is stable for each of the following:
"
(i) A =
"
(iii) A =
−2 1
1 −3
#
4
2
−13 −6
"
(ii) A =
2 −1
3 −2
#
#
(3) Recall the system of differential equations describing a series LCR circuit:
Q̇ = I,
1
R
I˙ = −
Q − I.
LC
L
Using only the eigenvalues determine the stability of this system in the two cases (i)
R2 > 4L/C and (ii) R2 < 4L/C. In each case describe the qualitative behaviour of
the solutions for large time i.e. as t → ∞. A sketch will do!
(4) An unforced, damped oscillator may be described via Hooke’s Law and Newton’s
Second Law by the differential equation
mẍ + cẋ + kx = 0.
Here mẍ is “mass × acceleration”, cẋ is a frictional force and kx is the tension in
the spring. By letting y = ẋ show that this equation can be written as the system of
differential equations
ẋ = y,
ẏ = −
k
c
x − y.
m
m
Determine the stability of this system in the two cases (i) c2 > 4mk and (ii) c2 < 4mk.
In each case describe the qualitative behaviour of the solutions as t → ∞.
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Answers
"
(1)
(i) x =
(ii) x = 2

1
2
#
"
1
1
"
−t
e

−
#
2
1
"
−t
e
+

#
e2t .
1
−1

#
e−5t .


1
−2
0

 t 
 4t


(iii) x =  0  e +  3  e + 3  −1  e−2t .
0
3
1
√
−5 ± 5
(2) (i) λ =
, stable.
2
(ii) λ = ±1, unstable.
(ii) λ = −1 ± j, stable.
(3)
(i) Eigenvalues real and negative. Hence stable. Solutions decay monotonically to
zero.
(ii) Eigenvalues complex with negative real part σ. Hence stable. Solutions oscillate
and decay to zero.
(4)
(i) Eigenvalues real and negative. Hence stable. Solutions decay monotonically to
zero.
(ii) Eigenvalues complex with negative real part σ. Hence stable. Solutions oscillate
and decay to zero.
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Weeks 10-12 Fourier Series
Introduction: The ability to analyse waveforms of various types is an important engineering tool. Fourier analysis provides a mathematical tool which enables the engineer to break
down a wave or signal into its various frequency components. This is particularly useful
in communications. Fourier series are also a useful mathematical tool for solving partial
differential equations describing heat flow or electrostatics problems.
Week 10: Introduction to Fourier Series
Learning outcomes. You should be able to:
(i) Determine the period of any given periodic function.
(ii) Sketch a periodic function (continuous and discontinuous).
(iii) Understand what a Fourier series is and how it approximates a given periodic function.
(iv) Compute integrals using integration by parts.
Reading:
Lecture notes.
Exercises:
Handbook exercise sheet:
Week 10: Integration and Periodic Functions.
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Week 11: More on Fourier Series
Learning outcomes. You should be able to:
(i) Calculate the coefficients an and bn of the Fourier series for any periodic function.
(ii) Sketch the Fourier series for any periodic function.
(iii) Determine the value of a Fourier series for any given argument.
Reading:
Lecture notes.
Exercises:
Handbook exercise sheet:
Week 11: Fourier Series
Week 12 Fourier series/Revision
We will use the final week to complete any outstanding work on Fourier series and, time
permitting, go through some exam-type questions.
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