
Systems that require two independent coordinates
to describe their motion are called Two Degree of
Freedom Systems.

Systems that require two independent coordinates
to describe their motion are called Two Degree of
Freedom Systems.

The general rule for the computation of the
number of freedom
Number of
Degrees of
freedom
of the system
=
Number of masses in the system
x
number of possible types
of motion of each mass
There are two equations of motion for a two
degree of freedom system, one for each
mass/DOF.
 They are in the form of coupled differential
equation- that is, each equation involves all
coordinates
 If harmonic solution is assumed for each
coordinate, the equation lead to a frequency
equation that gives two natural frequencies




Given a suitable initial excitation, the system
vibrates at one of these natural frequencies
During free vibration at one of the natural
frequencies, the amplitude of the TDOF are
related in a specific manner and the
configuration is called a normal mode, principal
mode, or natural mode
If the system vibrates under the action of an
external harmonic force , the resulting forced
harmonic vibration takes place at the frequency
of the applied force
Resonance occurs when the forcing
frequency is equal to one of the natural
frequencies of the system
 The configuration of a system can be
specified by a set of independent coordinates
such as length, angle, or other physical
parameter. Generalized coordinates
 A set of coordinates which leads to an
uncoupled equations of motions are called
principal coordinates

m1x1  c1  c2 x1  c2 x2  k1  k2 x1  k2 x2  F1
m2 x2  c2 x1  c2  c3 x2  k2 x1  k2  k3 x2  F2




mxt   cx t   k xt   F t 
m1 0 
m  

 0 m2 
c1  c2
c  
  c2
 c2 

c2  c3 




mxt   cx t   k xt   F t 
k1  k 2
k   
  k2
 k2 

k 2  k3 
 x1 t 

x t   

 x2 t 

 F1 t 
F t   



 F2 t 




mxt   cx t   k xt   F t 
The solution involves four constant of integration.
From the initial conditions;
x1 t  0  x1 0, x1 t  0  x1 0
x2 t  0  x2 0, x2 t  0  x2 0
F1 t   F2 t   0
c1  c2  c3  0
m1x1 t   k1  k2 x1 t   k2 x2 t   0
m2 x2 t   k2 x1 t   k2  k3 x2 t   0
x1 t   X1 cost   
x2 t   X 2 cost   
X1 and X2 are constants the maximum amplitude
of x1 (t) and x2(t), φ is the phase angle.
 m   k  k X  k X cost     0
 k X  m   k  k X cost     0
2
1
1
2
1
2
2
2
2
1
2
2
3
2
Equation 5.7
 m   k  k X  k X
 k X  m   k  k X
2
0
2
0
2
1
1
2
1
2
2
2
nontrivial solution
det
1
2
 m1 2  k1  k2 
 k2
2
3
 k2
m2  k2  k3 
2
m1m2   k1  k2 m2  k1  k3 m1
2
 k1  k2 k2  k3   k2   0
4
Frequency or Characteristic equation
0
2
Natural frequencies of the system,
1  k1  k 2 m2  k 2  k3 m1 
 ,  

2
m1m2

2
1
2
2
 k  k m  k  k m 2
2
3
1
  1 2 2

m1m2


 k1  k 2 k 2  k3   k 
 4

m1m2


2
2
1/ 2
Frequencies ratios,
X 21  m112  k1  k2 
k2
r1  1 

X1
k2
 m212  k2  k3 
X 22   m122  k1  k2 
k2
r2  2  

X1
k2
 m222  k2  k3 
The normal modes of vibration (modal vectors),
 1  X 11   X 11 
X   1    1 
 X 2  r1 X 1 
 2   X 12    X 12  
X   2    
2  
 X 2  r2 X 1 
The free vibration solution or the motion in time,
 x11 t   X 11 cos1t  1  
 1
x t    1    1
  first mode
 x2 t  r1 X 1 cos1t  1 
 x12  t   X 12  cos2t  2  
 2 
x t    2    
  second mode
2 
 x2 t  r2 X 1 cos2t  2 
The unknown constant can be determine from
the initial conditions, x1 t  0  x1 0, x1 t  0  x1 0
x2 t  0  x2 0, x2 t  0  x2 0

 
X 1 t   X 1 cos 1  X 1 sin 1
1
1
2
1

2 1/ 2

1 
 r2 x1 0  x2 0 
2

r2 x1 0  x2 0 

r2  r1 
12

2
1


1 0  x2 0 
X
1
1   r2 x
1 sin 1
1  tan  1
  tan 

 X 1 cos 1 
1 r2 x1 0  x2 0
1/ 2
from the initial conditions,
2 

2 
 
2 
X 1  X 1 cos 2  X 1 sin 2
2

2 1/ 2

1 
r1 x1 0  x2 0 
2

 r1 x1 0  x2 0 

r2  r1 
22

2
2 


X
r1 x1 0  x2 0 
1
1 
1 sin 2
2  tan  2 
  tan 

 X 1 cos 2 
2  r1 x1 0  x2 0
1/ 2
Consider a torsional system consisting of two
discs mounted on a shaft as shown below.
 Parameters; k, J0 and Mt

J11  kt11  kt 2  2  1   M t1
J   k      k   M
2 2
t2
2
1
t3 2
t2
0


J11  kt1  kt 2 1  kt 2 2  M t1
0
J 22  kt 21  kt 2  kt 3  2  M t 2
Similar to the translational equations , but
substituting θ → x, J → m, kt → k
Free
Vibrations