MTH215e
Examination – July Semester 2017
Further Mathematical Methods and
Mechanics
Monday, 13 November 2017
4:00 pm – 6:00 pm
____________________________________________________________________________________
Time allowed: 2 hours
____________________________________________________________________________________
INSTRUCTIONS TO STUDENTS:
1. This examination contains SIX (6) questions and comprises FIVE (5) printed
pages (including cover page).
2. You must answer ALL questions.
3. This is an Open Book examination.
4. All answers must be written in the answer book. Marks will only be awarded if
FULL working is shown.
At the end of the examination
Please ensure that you have written your examination number on each answer book
used.
Failure to do so will mean that your work cannot be identified.
If you have used more than one answer book, please tie them together with the string
provided.
THE UNIVERSITY RESERVES THE RIGHT NOT TO MARK YOUR
SCRIPT IF YOU FAIL TO FOLLOW THESE INSTRUCTIONS.
MTH215e Copyright © 2017 Singapore University of Social Sciences (SUSS)
Examination – July Semester 2017
Page 1 of 6
Answer all questions. (Total 100 marks)
Question 1
A set of simultaneous equations, in three unknowns x, y and z, is given by
x – 2y + 2z = 1
–4x – y + 3z = –8
14x – y – 5z = 26.
(a)
Identify these equations in augmented matrix form, labelling the rows.
(1 mark)
(b)
By using Gauss elimination method, reduce the augmented matrix to upper
triangular form.
(6 marks)
(c)
Identify the solutions for x, y and z and state whether the set of equations has
•
•
•
a unique solution,
no solution, or
an infinite number of solutions.
If it has an infinite number of solutions, find its general solution.
(5 marks)
Question 2
(a)
Solve the eigenvalues and its corresponding eigenvectors of a 2×2 matrix given
by
23
− 14
3
− 73
.
− 53
(9 marks)
(b)
Hence, explain the general solution in matrix form, of the system of differential
equations
3x1 = 2x1 – 7x2
3x2 = –14x1 – 5x2.
(3 marks)
MTH215e Copyright © 2017 Singapore University of Social Sciences (SUSS)
Examination – July Semester 2017
Page 2 of 6
Question 3
A particle of mass 5 kg is constrained to move horizontally on a smooth surface, and
has two model dampers and a model spring attached to it as shown in Figure Q3. The
spring has a stiffness of 5 Nm-1 and a natural length of 2 m. The left damper has a
damping constant of 3 Nsm-1 and the right damper 6 Nsm-1. The distance between the
two fixed supports is 4 m with the position vectors x(t) and y(t) are indicated as shown
along the unit vector i.
Figure Q3
(a)
Draw a force diagram for the particle, indicating all the forces that cause the
oscillations.
(2 marks)
(b)
Express each of these forces in vector form.
(3 marks)
(c)
Apply Newton’s second law, express the equation of motion of the particle.
(4 marks)
(d)
What is the equilibrium position of the particle, measured from O if point A is
now stationary?
(2 marks)
(e)
If the particle is initially projected from its equilibrium position toward the
right, described briefly the subsequent motion of the particle.
(2 marks)
Question 4
A ball of mass m is moving with velocity 2i – 2j with respect to a Cartesian coordinate
system. It collides with a second ball of identical size and mass, which is initially at
rest. After the collision, the two balls have velocities u1i + v1j and u2i + v2j.
(a)
Apply the conservation of momentum, write down the momentum equation of
the system of two balls. Simplify this equation.
(4 marks)
(b)
Find the energy of the system before and after the collision.
(3 marks)
MTH215e Copyright © 2017 Singapore University of Social Sciences (SUSS)
Examination – July Semester 2017
Page 3 of 6
(c)
Show that if,
(u1i + v1j) • (u2i + v2j) = 0
then the collision is elastic.
(6 marks)
Question 5
A system of non-linear differential equations is given by
x = y 2 + x − 1
.
2
y
1
y
2
x
x
=
+
−
+
(a)
Calculate all the equilibrium points for the system of differential equations.
(8 marks)
(b)
For each of the equilibrium points (X0, Y0), define the excesses u and v by
x = u + X0 and y = v + Y0 and use these to explain in matrix form, a linear
approximation u = Mu for the system of equations near the equilibrium point
where u = (u,v)T.
(8 marks)
(c)
Hence in each case, find the eigenvalues and explain these equilibrium points.
Sketch the vector fields in the neighbourhood of each point.
(9 marks)
Question 6
Consider the longitudinal vibrations of two particles B and C, of mass 4 kg and 1 kg
respectively. Particle B is attached by a model spring of stiffness 6 Nm-1 to a fixed
point A. The two particles are connected in between by a model spring of stiffness 2
Nm-1. The displacements of the two particles from their equilibrium positions O1 and
O2, are denoted by x and y respectively, as shown in Figure Q6. You may ignore air
resistance and any other frictional forces.
Figure Q6
MTH215e Copyright © 2017 Singapore University of Social Sciences (SUSS)
Examination – July Semester 2017
Page 4 of 6
(a)
Draw force diagrams showing the forces acting on each particle.
(2 marks)
(b)
Describe in vector form, the changes in the spring forces acting on the particles,
when the particles are displaced from their equilibrium positions by distances x
and y respectively in the direction as shown, for the case y > x > 0.
(3 marks)
(c)
Describe the equation of motion of the two particles and show that it can be
written as
x
y = A
x
y ,
and hence write down the dynamic matrix A of the system in numerical form.
(5 marks)
(d)
Relate the eigenvalues of matrix A and hence write down the normal mode
angular frequencies of the system.
(4 marks)
(e)
Relate the eigenvectors of matrix A and hence write down the normal mode
displacement ratios of the system. State whether each normal mode is in-phase
or phase-opposed.
(6 marks)
(f)
Write down the displacement of each particle from its equilibrium position as a
function of time in matrix form. (Note: There should be four arbitrary
constants).
(2 marks)
(g)
If the particles are released from rest with the initial displacements
x − 0.2
y = 0.2
relate qualitatively the subsequent motion.
(3 marks)
----- END OF PAPER -----
MTH215e Copyright © 2017 Singapore University of Social Sciences (SUSS)
Examination – July Semester 2017
Page 5 of 6
Answers
1 −2 2 1
Q1(b) 0 −9 11 −4
0 0
0 0
1
Q1(c) x = ( 4 k + 17 )
9
1
y = ( 4 k + 11) z =k
9
1
Q2(a) Eigenvalues are λ = 3 with Eigenvector and
−1
1
λ = -4 with Eigenvector
2
x1
1 3t
1 −4 t
Q2(b)=
+
C
e
C
x
−1
2 e
2
y ) i R2 =
−3 ( x−6 x i H =
5(2 − x) i
Q3(b) R1 =
Q3(c) 4
x + 9 x + 5 x = 3 y + 10
Q3(d) xeq = 2 m
Q3(e) Damping Ratio = 0.9 This is a weak damping, so there is some initial
oscillation
Q4(a)
−2j
2i =
( u1+u2 ) i + ( v1+ v2 ) j
Q4(b) Before collision E1 = 4 m , E=
2
1
1
m ( u12 + v12 ) + m ( u22 + v22 )
2
2
Q4(c) E1 = E2 Collision is elastic
Q5(a) (1,0) and (0, -1) are equilibrium points
u 1 0 u
u 1 −2 u
Q5(b) at (1,0) =
.
at
(0,
-1)
v = −2 1 v
v 0 1 v
Q5(c) (1,0) is unstable source. (0,-1) is an unstable saddle point
−6 x i , ∆H 2 =
2 ( y- x ) i , ∆H 3 =
−∆H 2 =
−2 ( y- x ) i
Q6(b) ∆H1 =
1
y, =
y 2 x− 2 y
2
Q6(d) Normal mode angular frequencies are w1 = 1 , w2 = 3
Q6(e) When λ = −1 Displacement Ratio =2, normal mode is in-phase
When λ = −3 Displacement Ratio =-2, normal mode is phase-opposed.
x
1
1
Q6(f)=
y C 2 cos ( t + φ1 ) + D −2 cos 3 t + φ2
Q6(c)
x=
−2 x +
(
)
x −0.2
Q6(g) The displacement =
does not corresponds to any two of the normal
y 0.2
modes, so the oscillation is non-sinusoidal
MTH215e Copyright © 2017 Singapore University of Social Sciences (SUSS)
Examination – July Semester 2017
Page 6 of 6
MTH215 – 2016 Exam Final Answers
x1 = 10 , x2 = 5 and x3 = 5
There is virtually no change in the solutions between essential row interchanges
and partial pivoting which is usually stable. So, there is no evidence of induced
instability.
However, a change of 0.001 in one of the coefficient results in a maximum
change of approximately 0.002905 in the value of x1, which gives a scale factor of
2.905, so there is no evidence that the equations are ill-conditioned.
153
14
21
1.
(a)
(b)
2.
(a)
(b)
Eigenvalues are 4 and 10.
Eigenvectors are [1 –1]T and [1 1]T respectively.
(c)
trace B =
3.
(a)
(b)
(c)
H = – 10(x – 1)i and R = 3( y – x )i (use vector notations)
x 3x 10 x 3 y 10
xeq = 1m
4.
(a)
W1i = –2mg (sin 6 i – cos 6 j),
(b)
1
2
5.
, det B =
9700
6
W2i = 3mg i
(|T| – mg)
For particle 1,
ma =
For particle 2,
1
ma = mg 3 |T|
2
5
a=
(a)
Eigenvalues are 1 and 2 with eigenvectors are [1 2]T and [1 1]T respectively.
x 3 1 x 0 t
y 2 0 y 6e
(c)
(d)
(a)
(b)
(c)
g,
|T| =
9
5
(c)
(b)
6.
697
6
mg
xC
1 t
1 2t
y A2 e B 1e
C
x 1 t
1 2t 1 t
y 32e 21e 4e
x x( A x y )
y y ( B x y ) = V(x,y)
(0,0), (0,B) and (A, 0)
A 0
Near (0,0),
M =
0 B
A B 0
Near (0,B),
M =
B B
(d)
A A
M =
0 B A
(0,0) with eigenvalues 2 and 1, is an unstable source,
(0,1) with eigenvalues 1 and –1, is an unstable saddle point,
(2,0) with eigenvalues –2 and –1, is a stable sink.
(a)
(a)
Near (A,0),
7.
(b)
(c)
(d)
ΔH1 = –kx1 i, ΔH2 = 4k(x2 – x1)i,
ΔH4 = –kx2 i
x1 k 5 4 x1
x m 4 5 x
2
2
ΔH3 = –ΔH2 = –4k(x2 – x1)i
k
k
and 2 3
m
m
with displacement ratios 1 and –1 respectively.
x1
1 k
1 k
A
cos
t
B
cos
3
t
x
1 m
1 m
2
Normal modes angular frequencies are 1
MTH215e
Examination – July Semester 2016
Further Mathematical Methods and
Mechanics
Monday, 14 November 2016
4:00 pm – 6:00 pm
____________________________________________________________________________________
Time allowed: 2 hours
____________________________________________________________________________________
INSTRUCTIONS TO STUDENTS:
1. This examination contains SEVEN (7) questions and comprises SIX (6) printed
pages (including cover page).
2. You must answer ALL questions in section A and any TWO (2) questions in
section B.
3. This is an Open Book examination.
4. All answers must be written in the answer book. Marks will only be awarded if
FULL working is shown.
At the end of the examination
Please ensure that you have written your examination number on each answer book
used.
Failure to do so will mean that your work cannot be identified.
If you have used more than one answer book, please tie them together with the string
provided.
THE UNIVERSITY RESERVES THE RIGHT NOT TO MARK YOUR
SCRIPT IF YOU FAIL TO FOLLOW THESE INSTRUCTIONS.
MTH215e Copyright © 2016 SIM University
Examination – July Semester 2016
Page 1 of 6
SECTION A (Total 50 marks)
Answer all questions.
Question 1
(a)
Identify a set of simultaneous equations, in three unknowns x1, x2 and x3, by
Gauss elimination with partial pivoting, the stage below was reached.
2 3 6 − 3
0 2 − 3 7
0 3 − 2 0
Complete the process of solution by the same method, and obtain values for x1,
x2 and x3.
(7 marks)
(b)
A certain set of three simultaneous linear algebraic equations in three unknowns
x1, x2 and x3, by Gauss elimination in various ways. In the first place only
essential row interchanges are made. Secondly, the equations are solved by
Gauss elimination with partial pivoting. Thirdly, one coefficient is changed,
from 15 to 15.001, and the equations solved again using essential row
interchanges only. In each case the calculations are carried out to eight figure
accuracy, and the answers rounded. The results are summarised in Table Q1(b).
Method
Essential row interchanges
Partial pivoting
Changed coefficient
x1
6.789232
6.789233
6.786328
Solutions
x2
3.676731
3.676731
3.674532
x3
–0.234767
–0.234768
–0.237341
Table Q1(b)
Decide, on the basis of these results, whether or not the equations are likely to be
ill-conditioned or whether there might have been induced instability in any of the
methods. Give reasons for your answer.
(5 marks)
Question 2
A 2 × 2 matrix is given by
(a)
7 3
A=
.
3 7
Solve the eigenvalues of the matrix.
(4 marks)
(b)
Solve and explain the corresponding eigenvectors of the matrix.
(4 marks)
MTH215e Copyright © 2016 SIM University
Examination – July Semester 2016
Page 2 of 6
(c)
Let B be a matrix defined by B = (10I – A)–1 + A2. Solve the trace and the
determinant of B? Explain how you obtained your answers.
(4 marks)
Question 3
A particle A, of mass 1 kg, moves along a frictionless horizontal track. The particle is
attached to a fixed point O by a model spring, and on the right by a model damper as
shown in Figure Q3. The damping constant is 3 Ns m-1. The spring has stiffness 10 Nm-1
and natural length 1 metre. The displacement of the particle, x, is measured from O and
the forcing function y(t) is on the displacement at B.
Figure Q3
(a)
Draw a force diagram for the particle, indicating only the forces that cause the
oscillations, and define these forces.
(5 marks)
(b)
Express the equation of motion of the particle.
(5 marks)
(c)
What is the equilibrium position of the particle, measured from O if point B is
now stationary?
(3 marks)
Question 4
A particle of mass 2m lies on a frictionless plane which is inclined at an angle of 16 π to
the horizontal as shown in Figure Q4. The particle is connected by a light inextensible
string, passing over a small light frictionless pulley at the top of the inclined plane, to a
second particle of mass 3m, which is hanging freely. When release from rest, the
hanging particle moves downwards.
Figure Q4
MTH215e Copyright © 2016 SIM University
Examination – July Semester 2016
Page 3 of 6
(a)
Draw two force diagrams showing all the forces acting on each of the two
particles.
(4 marks)
(b)
Apply Newton’s second law and write down the equations of motion of each of
the two particles.
(5 marks)
(c)
Calculate the common acceleration of the two particles and the tension in the
string in terms of m and g, the magnitude of the acceleration due to gravity.
(4 marks)
MTH215e Copyright © 2016 SIM University
Examination – July Semester 2016
Page 4 of 6
SECTION B (Total 50 marks)
Answer 2 of the following 3 questions.
Question 5
(a)
Solve the eigenvalues and its corresponding eigenvectors of a 2×2 matrix given
by
3 − 1
2 0 .
(8 marks)
For the system of differential equations,
x = 3x – y
y = 2x + 6e–t.
(b)
Write and explain the system of differential equations in matrix form.
(2 marks)
(c)
Find the complementary function.
(3 marks)
(d)
Calculate the solutions of the system of differential equations which satisfy the
initial conditions, x(0) = 0 and y(0) = 0.
(12 marks)
Question 6
The question concerns a flock of sheep and a herd of cows sharing a field and
competing for the same food source, i.e. the grass. let x be the number of sheep and y be
the number of cows. The evolution of the two populations can be modelled by the
system of differential equations,
x = x( A − x − y )
.
y = y ( B − x − y )
where A and B are non-equal, positive constants representing the total food value of the
field of grass to the sheep and cows respectively.
(a)
Explain the system of differential equations in matrix form [ x , y ]T = V(x, y)
where V(x, y) is a vector field.
(2 marks)
(b)
Calculate all the equilibrium points for the system of differential equations.
(6 marks)
MTH215e Copyright © 2016 SIM University
Examination – July Semester 2016
Page 5 of 6
(c)
For each of the equilibrium points (X0, Y0) define the excesses u and v by
x = u + X0 and y = v + Y0 and use these to find a linear approximation u = M u
for the system of equations near the equilibrium point. Specify this matrix M for
each equilibrium point in terms of A and B.
(8 marks)
(d)
In each case, A = 2 and B = 1, deduce the eigenvalues for the matrix M. Hence
classify each equilibrium point and state if it is stable.
(9 marks)
Question 7
Three model springs AB, BC and CD, each of natural length l0, have stiffness k, 4k and k
respectively. A particle of mass m is attached to the springs at B and another particle of
mass m is attached at C as shown in Figure Q7. The end A and D of the springs are fixed
to two points which are a horizontal distance 3l0 apart and the system is free to oscillate
along a horizontal line AD.
Figure Q7
You may assume that the only forces acting on the particles in a horizontal direction are
those due to the springs.
(a)
Describe in vector form, the changes in the spring forces acting on the particles,
when the particles at B and C are displaced from their equilibrium positions by
distances x1 and x2 respectively in the direction away from the fixed point A, for
the case x2 > x1 > 0.
(3 marks)
(b)
Relate the equations of motion of the two particles.
(6 marks)
(c)
Relate the normal mode angular frequencies of the system and the corresponding
normal mode displacement ratios.
(10 marks)
(d)
Write down the general solution of the motion.
(2 marks)
(e)
Assuming the phase angle φ is zero, and with initial conditions, x1(0) = 0.1 and
x2(0) = 0.2, find the required solution of the motion.
(4 marks)
----- END OF PAPER ----MTH215e Copyright © 2016 SIM University
Examination – July Semester 2016
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