USSC3002 Oscillations and Waves
Lecture 2 Normal Modes
Wayne M. Lawton
Department of Mathematics
National University of Singapore
2 Science Drive 2
Singapore 117543
Email matwml@nus.edu.sg
http://www.math.nus/~matwml
Tel (65) 6874-2749
1
SYSTEMS WITH TWO DEGREES OF FREEDOM
d
k T m
k T m
k T
1

3
2


u1
u2
x0
k  spring constant of each spring
m  mass of each particle
d  length of system
  natural or equilibrium length (of each spring)
0  extent or diameter of each ‘particle’
x1 , x2  particle positions
u1 ,u2  particle displacements
T1 , T2 , T3  spring tension forces (inward directed)
F1 , F2  particle net forces
Questions What are constants, variables, relations ? 2
CONSTANTS AND VARIABLES
d
k T m
k T m
k T
1
x0

3
2
u1

u2

k  spring constant (of each spring)
m  mass (of each particle)
CONSTANTS
d  length of system
  natural or equilibrium length (of each spring)
0  extent or diameter of each ‘particle’
x1 , x2 
u1 ,u2 
T1 , T2 , T3
F1 , F2 
particle positions
VARIABLES
particle displacements
 spring tension forces (inward directed)
particle net forces
3
GEOMETRIC RELATIONS
d
k T m
k T m
k T
1

x0
3
2
u1

u2

0  extent or diameter of each ‘particle’
x1 , x2  particle positions
u1 ,u2  particle displacements (from equilibrium)
d  3
since diameter of particles equal zero
x1    u1
x2  2  u2
4
HOOKS LAW RELATIONS
d
k T m
k T m
k T
1

3
2

u1
u2
x0
k  spring constant of each spring

u1 ,u2  particle displacements
T1 , T2 , T3  spring tension forces
T1  k u1
T2  k (u2  u1 )
T3  k u2
5
NET FORCE RELATIONS
d
k T m
k T m
k T
1
x0

3
2
u1

u2

T1 , T2 , T3  spring tension forces (inward directed)
F1 , F2  particle net forces
F1  T1  T2
F2  T2  T3
6
NEWTONIAN RELATIONS
d
k T m
k T m
k T
1
x0

3
2
u1

u2

F1 , F2  particle net forces
m  mass of each particle
x1 , x2  particle positions
x1 , x2  particle velocities
x1 , x2  particle accelerations
F1  mx1 F2  mx2 Newton’s 2nd Law
7
THE EQUATIONS OF MOTION
for the displacements of each particle obtained from
F1  mx1
F2  mx2
and
x1 
d2
dt 2
(  u1 )  u1
x2 
d2
dt 2
(2  u2 )  u2
are
m u1  F1  T2  T1  k (u2  2u1 )
m u2  F2  T3  T2  k (u1  2u2 )
8
MATRIX FORM OF THE EQUATIONS
m u1  F1  k (T2  T1 )  k (u2  2u1 )
m u2  F2  k (T3  T2 )  k (u1  2u2 )
u1 
u1 
u1 



u
 u 
 u 
u2 
u2 
u2 
is
u  M u
where
2

1


k
M m

 1 2 
9
NORMAL MODES
are solutions of
u  M u
u (t )  a cos ( t   ) v
v is a vector and a,  ,  are parameters
that have the form
where
as in the Simple Harmonic Motion of a single particle
on a spring or on a pendulum (with small swings).
Question Show that u is a normal mode if and only if
M v   v
2
10
EIGENVALUES AND EIGENVECTORS
From the preceding page, we learn that the normal
modes (if they exist) are related to eigenvectors with
negative eigenvalues of the Vibration Matrix M
Question What are the eigenvalues & eigenvectors of
 2  1
 1 2 


and how are they related to those of M ?
11
EIGENVALUES AND EIGENVECTORS
Answer The eigenvalues & eigenvectors of M are
 2  1 1 1
1
1
k
 1 2  1  1 1  M 1   m 1

 


 2  1  1 
1
1
1
k
 1 2   1  3  1  M  1  3 m  1

 
 
 
 
The corresponding normal angular frequencies are

k
m

3k
m
12
TUTORIAL 2 (Due Tuesday 23 August)
1. Set up the equations of motion for a system with
three particles with identical masses and four springs
with same equilibrium lengths and spring constants.
2. Compute the normal modes for this system and
compute their angular frequencies and periods.
3. Read p. 371-380 in the Halliday, Resnick, Walker
handouts, do problems 2E, 3E, 6P, 13E, 15P, p. 394
13