1.7.4 Damping Influence on Free Vibration Decay
With free vibrations the level and character of the damping forces determine how
fast, and with which time envelope, vibrations decay. So if this is of concern,
careful consideration to damping is necessary. With linear viscous damping the
displacement (and thus also velocity and acceleration) amplitude will decay
exponentially with time, as illustrated by Fig. 1.2 in Sect. 1.2.2, and quantified
further below. With dry friction the decay envelope is different, e.g. pure Coulomb
friction/damping gives an acceleration amplitude that decays linearly with time
(Lorenz 1924; Lian 2005).
For linear viscous damping with damping ratio f one can show, using (1.5), that
the number n 1/r of free oscillations taken to reduce the initial maximum displacement amplitude by a factor of r is:
n 1=r ¼
ln r
2p
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
1
f
2
À 1
s
!
ln r
2pf
for f ! 0:
ð1:107Þ
Table 1.3 quantifies examples of using this relation to calculate n 1/2 and n 1/10 ,
i.e. the number of oscillation cycles required to, respectively, halve and decimate
the initial displacement amplitude. As a simple memo it takes about 10 oscillations
to half the amplitude at 1% damping; then just scale up and down in proportion:
about 100 oscillations to half the amplitude at 0.1% damping, etc.
1.7.5 Damping Influence on Resonance Buildup
With forced vibrations the level of damping also determines how quickly the
response settles into stationary vibrations, i.e. how fast the effect if initial conditions
(i.e. the free, unforced part of the vibrations) disappears. Or in mathematical terms:
The total response consists of the homogeneous solution plus the particular solution, and damping determines how fast the homogeneous part decays towards zero.
With very low damping it may take a very long time for stationary vibrations to
settle. This holds with real systems, and also with numerical simulation. So, if
damping is ignored in numerical simulation, the homogeneous part of the solution
Table 1.3 Freely damped oscillation cycles to reach half amplitude (n 1/2 ) or to decimate
amplitude (n 1/10 ) in dependency of damping ratio f
f (%)
n 1/2
n 1/10
10
1
3
1
1 1
3 7
0.1
110
365
0.01
1103
3664
34
1 Vibration Basics
With free vibrations the level and character of the damping forces determine how
fast, and with which time envelope, vibrations decay. So if this is of concern,
careful consideration to damping is necessary. With linear viscous damping the
displacement (and thus also velocity and acceleration) amplitude will decay
exponentially with time, as illustrated by Fig. 1.2 in Sect. 1.2.2, and quantified
further below. With dry friction the decay envelope is different, e.g. pure Coulomb
friction/damping gives an acceleration amplitude that decays linearly with time
(Lorenz 1924; Lian 2005).
For linear viscous damping with damping ratio f one can show, using (1.5), that
the number n 1/r of free oscillations taken to reduce the initial maximum displacement amplitude by a factor of r is:
n 1=r ¼
ln r
2p
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
1
f
2
À 1
s
!
ln r
2pf
for f ! 0:
ð1:107Þ
Table 1.3 quantifies examples of using this relation to calculate n 1/2 and n 1/10 ,
i.e. the number of oscillation cycles required to, respectively, halve and decimate
the initial displacement amplitude. As a simple memo it takes about 10 oscillations
to half the amplitude at 1% damping; then just scale up and down in proportion:
about 100 oscillations to half the amplitude at 0.1% damping, etc.
1.7.5 Damping Influence on Resonance Buildup
With forced vibrations the level of damping also determines how quickly the
response settles into stationary vibrations, i.e. how fast the effect if initial conditions
(i.e. the free, unforced part of the vibrations) disappears. Or in mathematical terms:
The total response consists of the homogeneous solution plus the particular solution, and damping determines how fast the homogeneous part decays towards zero.
With very low damping it may take a very long time for stationary vibrations to
settle. This holds with real systems, and also with numerical simulation. So, if
damping is ignored in numerical simulation, the homogeneous part of the solution
Table 1.3 Freely damped oscillation cycles to reach half amplitude (n 1/2 ) or to decimate
amplitude (n 1/10 ) in dependency of damping ratio f
f (%)
n 1/2
n 1/10
10
1
3
1
1 1
3 7
0.1
110
365
0.01
1103
3664
34
1 Vibration Basics