Free Vibration Response of SDOF
Systems
Response: Free or forced
In this chapter: Free response, i.e. no
external forces are applied.
k
c
m
mx  cx  kx  0
x  2n x   n 2 x  0
k
natural frequency
m
c
 
damping ratio
2 km
n 
1
We know from theory of DE that x(t )  Ce st
Characteristic equation: s 2  2n s   n2  0
3 cases:
1. >1 (overdamped) two real solutions
2. =1 (critically damped) double real
solution
3. <1 (underdamped) two complex
solutions (most practical systems)
2
Case 1: Overdamped (1)
x(t )  c1e
(    2 1)nt
x(t)
x(0)
 c2 e
(    2 1)nt
Slope = x (0) here
 5
  1.1
c1 
c2 
t
x (0)  (   2  1 )   n x (0)
2 n  2  1
 x (0)  (    2  1 )   n x(0)
2 n  2  1
Observations: Free vibration response is an
exponentially decaying function. The
higher the damping factor the slower the
decay is.
3
Case 2: Critically damped (=1)
x(t )  c1e nt  c2 te nt
c1  x(0),
x(t)
c2   n x(0)  x (0)
Slope = x (0) here
x(0)
t
Observations: Free vibration response is an
exponentially decaying function, like the
response of overdamped systems. The
decay is the smaller than that of overdamped
systems.
4
Case 3: Underdamped systems (<1)
Consider special case where there is no
damping (i.e. system is undamped) first (i.e.
=0):
k
m
mx  kx  0 
x   n2 x  0
x(t )  x(0) cos( n t ) 
x (0)
n
sin( n t )  A cos( n t   )
where:
n 
k
x (0) 2 1/ 2
x (0)
, A  [ x (0) 2  {
} ] ,   tan -1 (
)
m
n
 n x (0)
5
Example of response of undamped system
.
1
0.75
0.5
0.25
x ( t)
0
0.25
0.5
0.75
1
0
3.14
6.28
9.42
12.56
t
T
2
n
6
.
Displacement, velocity and acceleration
4
acceleration
3
 n2 A
2
x ( t)
1
xd ( t)
0
xdd ( t)
1
velocity
n A
displacement
A
2
3
4
0
3.14
6.28
9.42
12.56
t
Observations:
 Response is a harmonic wave.
 The angular frequency is  n 
k
m
rad/sec. This frequency depends only
on the system- not on the initial
conditions. The higher the spring rate,
the higher the frequency of oscillation
is. The lower the mass, the higher the
frequency of oscillation is.
7
 Number of cycles per second: f 
n
2
(Hz).
 Velocity is also a harmonic wave.
Leads the displacement by a quarter
period or 90 degrees.
 Acceleration is also a harmonic wave.
Leads the displacement by a half period
or 180 degrees.
8
Harmonic motion: three representations
1. x(t )  c1e jnt  c2 e  jnt
2. x(t )  A1 cos(n t )  A2 sin(n t )
x (t )  A cos( n t   )
3.
1
where A  ( A 2  A22 ) 2
1
A
and   tan -1 ( 2 )
A1
9
Underdamped system (<1):
x (t )  e n t  A1 cos( d t )  A2 sin( d t )
where
A1  x (0)
 x (0)
x (0) 
A2  


2

d 
 1  

Therefore:
x(t )  en t A cos(d t   )
1
A  ( A12  A22 ) 2
A
  tan 1 ( 2 )
A1
 d  damped natural frequency   n 1   2
10
Example of underdamped system response
1
1
Ae
x( t)
 0.731
d = 6.25
nt
= 0.1
n= 6.28
0
Td 
1
0
1
0
t
2
d
2
3
3
Observations:
Underdamped system:
 Free vibration response is an oscillating
function. The amplitude decays with
time.
 The higher the damping ratio, the faster
the decay is.
11
 The frequency of oscillation is
 d   n 1   2 rad/sec, which is smaller
than the undamped natural frequency
 n . But for small values of the damping
ratio (say   0.1) the two frequencies are
practically equal.
Estimating damping from records of free
vibration response
 

4 2   2
where
 = logarithmic decrement= ln[
x(t )
]
x(t  Td )
If measurements are separated by n periods:
 = 1 ln[
n
x(t )
]
x(t  nTd )
12