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Question/ 4
Consider the non-linear system of differential equations
i = f + x + I
i=12+?:+y+I
(a) Find the two equilibrium points of the system.
(8 marks)
(b) For each of the equilibrium points (Xo, Y
o)
, define the excesses zr and v by
x: tt * Xo and y: v + Io and use these to find a linear approximation
u = M u for the system of equations near the equilibrium point where
u: (2, v)r.
(9 marks)
(c) In each case find the eigenvalues for the matrix M. Hence classi$ each
equilibrium point.
(8 marks)
Quxtionf y’
The question concerns the system of differential equations
I *= y-3
[y=r’-ry+2
(a) Write the system in the form
L*,rf : vft’.v)
where Y(x,y) is a vector field.
(2 marks)
O) Calculate all the equilibrium points forthe system ofdifferential equations.
(6 marks)
(c) For each of the equilibrium points ({0, Io) define the excesses z and v by
x= u* Xo and y=v + Is aoduse theseto fmda linear approximation
t = M u for the system of equations near the equilibrium point.
(8 marks)
(d)
In each case calculate the eigenvalues for the matrix M. Hence classiff each
equilibrium point and sketch the vector fields in the neighbourhood of that point.
(9 marks)
Question$
Three model springs AB, BC and CD, each of natural length /0, have stifftress 2h,3k and
fr respectively. A particle of mass 2m is attached to the springs at B and another particle
of mass m, is atlached at C as shown in Figure Q7. The end, A and D of the springs are
fixed to two points which are aharizontal distance 3ls apaft and the system is free to
oscillate along a horizontalline AD.
Fieure O7
You may assume that the only forces acting on the particles in a horizontal direction are
those due to the springs.
(a)
(b)
(c)
Write down and 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 .r1 snd.r2 respectively in the direction away
from the fixed poirrt l, for the c&So.r2 > rr > 0.
(3 marks)
Write down the equations of motion of the two particles;
(6 marks)
Find and relate the normal mode angular frequencies of the system to the
corresponding normal mode displacement ratios.
(12 marks)
The initial positions of the particles are such that B is f /6 to the left of its
equilibrium position while C is { /o to the right of its equilibrium position. The
particles are released from rest from these positions. Describe the resultant
motion is one of the normal mode, and write down the angular frequency of the
motion’
(4 marks)
(d)
Question f
The question concens the system of differentiat equdions
[ *=ly-xy
\i=z*-*v’
(a) Write the system in the form
l*,ilr = v(ay)
where V(r, y) is a vector field,
(2 marks)
(b) Find all the equilibrium points for the system of differential equations.
(6 mafts)
(c) For each of the equilibrium points (Xo, Yo) define the exsesses u and v
by r = u * Xo and y = v + Io and use these to find a linear
approximation u = M u for the system of equations nearthe equilibrium point.
(7 marks)
(d) In each case find the eigenvalues for the matrix M. Hence classi$ each
equilibrium point and sketch the vector fields in the neigfrbourhood of that
point.
(10 marks)
Question $
Two model springs AA and AB have stiffiress 4k and 3& respectively, and equal natural
lengths /e. A particle of mass 2m is allashed to the springs at A and another, of mass 3za
at B. The end O of the first spring is fixed and the system is free to oscillate along a
horizontal line as shown in the figure.
OAB
tr’isure 06
You may assume that the only forces acting on the particles in a horizontal direction are
those due to the springs.
(a) Let xr and xz be the displacements of the particles A and.B from their respective
equilibrium positions, away from the fixed point O. Show that the equations of
motion of the particles are
?mir–:|fut+3fu2
fr*z= fut- luz.
(8 marks)
Find the normal mode angular frequencies of the system and the corresponding
normal mode displacement ratios.
(12 marks)
The initial positions of the particles are such that the springs OA wtd AB each
have length * /0. The particles are released from rest from these positions. Show
that the resultant motion is one of the normal modes, and write down
expressions for the displacements of the two particles at time r for this motion.
(5 marks)
(c)
4k
o)
Question 6
Three model springs AB, BC and CD each have stiffiress & and natural length /o are
arftrnge horizontally as shown in Figure Q6. Two particles of equal rnass nt are attached
to the springs at B and C, and the ends A and D arc fixed to two points a horizontal
distance 3ls apart. The system is free to move in the horizontal \ne AD.
4ieure O6
You may assume that the only forces acting on the particies in a horizontal direction are
those due to the springs.
(a) Let xr and xz be the displacements of the particles I and C from their
equilibrium positions. Show that the equations of motion of the particles are
frir= -2b1 + b2
friz: br – 2fu2 .
(8 marks)
(b) Find the normal mode angular frequencies of the system and the corresponding
normal mode displacement ratios.
(l l marks)
(c) The initial positions of the particles are such that B is { /o from ,4 and C is } /o
from D. The particles are released from rest from these positions. Show that the
resultant rnoiio, is one of the normal mode, and write down the angular
frequency of the motion.
(6 marks)
