Normal Modes Goals: Additional Resources: Problems:

Normal Modes
D. Jaksch 1
Goals:
• Learn how to describe coupled oscillating systems by coupled ODEs
• Give physical interpretations of the ODE solutions
• Understand the importance of normal modes as basic building blocks for describing linear coupled oscillations
Additional Resources:
Several concepts required for this problem sheet are explained in FR and also in RHB. Further problems are
contained in the lecturers’ problem sheets.
Problems:
1. Two simple pendula with positions x and y are of equal length l, but their bobs have different masses
m 1 and m 2 . They are coupled via a spring of spring constant k. Show that their equations of motion
are:
ẍ = − g l x − k m 1
(x − y) and ÿ = − g l y − k m 2
(y − x)
(a) By taking suitable linear combinations of the two equations of motion, obtain two uncoupled differential
equations for linear combinations of x and y. Hence find the normal mode frequencies and
the relative amplitudes. [Hint: One of these linear combinations is fairly obvious. For the other, it
may be helpful to consider the centre of mass of the two bobs.]
(b) Use the standard matrix method to find the frequencies and the relative amplitudes of the bobs
for the normal modes of the system. Solution: ω 2 1 = g/l, ω 2 2 = g/l + k/m 1 + k/m 2 , v 1 = (1, 1),
v 2 = (1, −m 1 /m 2 ).
(c) Write down the most general solutions for x and y as a function of time t. Solution: (x, y) =
v 1 (A sin(ω 1 t) + B cos(ω 1 t)) + v 2 (C sin(ω 2 t) + D cos(ω 2 t))
(d) At t = 0, both pendula are at rest, with x = 0 and y = Y 0 . They are then released. Determine the
subsequent motion of the system. Solution: (x, y) = Y 0 (v 1 cos(ω 1 t) − v 2 cos(ω 2 t))/(1 + m 1 /m 2 )
(e) At t = 0, both bobs are at their equilibrium positions: the first is stationary but the second is given
an initial velocity v 0 . Determine the subsequent motion of the system.
(f) Give two sets of initial conditions such that the subsequent motion of the pendula corresponds to
each of the normal modes.
We now specialize to the case m 1 = m 2 = m and k/m = αg/l with α ≪ 1.
(g) For the original initial conditions of part (d), show that
and determine D, ∆ and ¯ω.
y(t) = D cos(∆t) cos(¯ωt) and x(t) = D sin(∆t) sin(¯ωt)
(h) Sketch x(t) and y(t) for α = 0.105, and note that the oscillations are transferred from the first
pendulum to the second and back. Approximately how many oscillations does the second pendulum
have before the first pendulum is oscillating again with its initial amplitude?
1 These problems are based on problem sets by Profs GG Ross and previous Oxford prelims exam questions

2. The displacements of two coupled oscillators q 1 and q 2 with mass m are described by the coupled equations
m¨q 1 = k(q 2 − 2q 1 ) and m¨q 2 = k(q 1 − 2q 2 ),
where k is a constant. The normal modes are given by Q 1 = (q 1 + q 2 )/ √ 2 and Q 2 = (q 1 − q 2 )/ √ 2 which
evolve according to
Q 1 = A cos ω 1 t + B sin ω 1 t and Q 2 = C cos ω 2 t + D sin ω 2 t,
where ω 1 = √ k/m and ω 2 = √ 3k/m are the normal mode frequencies.
(a) Find expressions for q 1 (t) and q 2 (t).
(b) The forces appearing in the equations for q 1 and q 2 may be interpreted in terms of a potential energy
function V (q 1 , q 2 ), as follows. We write the equations as
Two equal masses m are connected as shown with two identical massless
springs, of spring constant k. Considering
m¨q 1 = − dV only motion
, and m¨q 2 = − dV in the vertical direction,
,
obtain the differential equations dq for 1 the displacements dq 2 of the two masses from
their equilibrium positions. Show that the angular frequencies of the normal
modes are given by
generalising “mẍ = dV/dx”. Show that V may be taken to be V = k(q 2 1 + q 2 2 − q 1 q 2 ).
(c) Rewrite V in terms of the normal mode coordinates. Solution: V = mω 2 1Q 2 1/2 + mω 2 2Q 2 2/2.
2
! = ±
( )
3 5 k / 2m
(d) Calculate the total kinetic energy K of the two masses. Solution: K = m( ˙Q 2 1 + ˙Q 2 2)/2.
(e) Write
Find
down
the
the
ratio
total
of
energy
the amplitudes
V + K in terms
of the
oftwo coordinates
masses
Qin 1
each
and Qseparate 2 and also
mode.
in terms of q 1 and
q 2 . Discuss similarities and differences of these two expressions.
Why does the acceleration due to gravity not appear in these answers?
3. Two masses m are fixed to springs at B and C as shown in Fig. 1. They are displaced by small distances
x 1 5. and x 2 *AB, from their BC, equilibrium and CD are positions identical alongsprings the line with of thenegligible springs, andmass, execute and small stiffness oscillations.
The springs have negligible masses.
constant k:
A B C D
Figure 1: Masses connected by springs.
(a) AB, BC, and CD are identical springs with stiffness constant k. Show that the angular frequencies
of the normal modes are
The masses m, fixed to the springs √ at B and C, are √ displaced by small distances
k 3k
x 1 and x 2 from their equilibrium ω 1 = positions along the line of the springs, and
m and ω 3 =
m
execute small oscillations. Show that the angular frequencies of the normal
modes are !
1
= k / m and !
2
= 3 k / m . Sketch how the two masses move in
each mode. Find x 1 and x 2 at times t > 0 if at t = 0 the system is at rest with
x 1 = a, x 2 = 0.
Sketch how the two masses move in each mode. Find x 1 and x 2 at times t > 0 if at t = 0 the system
is at rest with x 1 = a, x 2 = 0.
(b) Now consider the case where the springs AB and CD have stiffness constant k 0 , while BC has
stiffness constant k 1 . If C is clamped, B vibrates with frequency ν 0 = 1.81Hz. The frequency of
the lower frequency normal mode is ν 1 = 1.14Hz. Calculate the frequency of the higher frequency
normal mode, and the ratio k 1 /k 0 .
6. *The setup is as for question 5, except that in this case the springs AB and CD
have stiffness constant k 0 , while BC has stiffness constant k 1 . If C is clamped, B
vibrates with frequency ν 0 = 1.81 Hz. The frequency of the lower frequency
L d2 I 1
normal mode dt 2 is + I 1
ν 1 C= − 1.14 M d2 I 2
dt Hz. 2 = Calculate 0, and I 2
the Ld2 dtfrequency 2 + I 2
C − M d2 I 1
of dt the 2 = higher 0, frequency
normal mode, and the ratio k 1 /k 0 . (From French 5-7).
4. The currents I 1 and I 2 in two coupled LC circuits satisfy the equations
where 0 < M < L. Find formulae for the two possible frequencies at which the coupled system may
oscillate sinusoidally.
7. Two particles 1 and 2, each of mass m, are connected by a light spring of
stiffness k, and are free to slide along a smooth horizontal track. What are the
normal 2 d2 y
frequencies of this system? Describe the motion in the mode of zero
dx
frequency. 2 − 3 dy
dx + 2 dz
dx + 3y + z = d 2 y
e2x , and
dx
Why does a zero-frequency mode appear 2 − 3 dy
dx + dz + 2y − z = 0,
dx
in this problem, but not in
question 5, for example?
5. Solve the differential equations
Is it possible to have a solution to these equations for which y = z = 0 when x = 0?
Particle 1 is now subject to a harmonic driving force F cos!t . In the steady

V + K = mQ! 1
+ m! 1Q1 + mQ!
$ % $ 2
+ m!
2Q2 % = E1 + E2
& 2 2 ' & 2 2 '
where E 1 is the total energy of ‘oscillator’ Q 1 with frequency ω 1 , and similarly
for E 2 .
What is the expression for the total energy when written in terms of q
1
, q
2
, q!
1
,
and q!
2
? Discuss the similarities and differences.
6. Two particles 1 and 2, each of mass m, are connected by a light spring of stiffness k, and are free to
slide along a smooth horizontal track. What are the normal frequencies of this system? Describe the
(d) Find the equations of motion for Q 1 and Q 2 from Newton’s law in the form
motion in the mode of zero frequency. Why does a zero-frequency mode appear in this problem, but not
V
in problem 3, for example? mQ !! !
1
= "
Particle 1 is now subject to aQ
, V
mQ!! !
= " ,
!
2
harmonic 1 driving force F ! cos Q2
ωt . In the steady state, the amplitudes of
vibration and hence ofre-derive 1 and 2 are solution A and(2).
B respectively. Find A and B, and discuss qualitatively the behaviour of
the system as ω 2 is slowly increased from values near zero to values greater than 2k/m.
7. 4.* We consider the setup in Fig. 2.
Class problems
k
m
k
m
Figure 2: Masses connected by springs.
Two equal masses m are connected as shown with two identical massless springs, of spring constant k.
Considering only motion in the vertical direction, obtain the differential equationsW2Q: for the 3 displacements
of the two masses from their equilibrium positions. Show that the angular frequencies of the normal
modes are given by
ω1,2 2 = (3 ± √ 5)k
2m .
Find the ratio of the amplitudes of the two masses in each separate mode. Why does the acceleration
due to gravity not appear in these answers?
8. We extend problem 3 to n masses connected by identical springs. A Mathematica program to solve this
problem is given in Fig. 3. Discuss the solutions for frequencies and amplitude ratios produced by this
program.
In[1]:=
In[2]:=
In[3]:=
An_ :⩵ DiagonalMatrixTable2, n
DiagonalMatrixTable1, n 1, 1 DiagonalMatrixTable1, n 1, 1
Solve for n masses connected by springs
n ⩵ 100; vv ⩵ EigensystemNAn;
omegas ⩵ ReI Sqrtvv1; amplitudes ⩵ vv2;
Consider the l-th normal mode
l ⩵ 100; Printl, "th mode frequency is: ", omegasl
ListPlotamplitudesl, AxesLabel "mass position", "amplitude ratios"
Figure 3: Mathematica program for masses connected by springs.