calculated by equating the work done by damping forces through one full vibration
cycle for, respectively, linear viscous damping the nonlinear damping law in
question. Table 1.1 lists the resulting equivalent linear viscous damping coefficients
c eq ; as appears these are not really system constants, but depend on vibration
frequency x and amplitude X.
1.7.3 Damping Measures and Their Relations
Table 1.2 summarizes some key damping parameters and their interrelations. Most
of these follows directly from the definitions and relations in Sects. 1.2-4, others
can be found in, e.g., Inman (2014), Crandall (1970), Lazan (1968), Adhikari
(2013).
Table 1.1 Damping force models, in terms of instantaneous displacement x = x(t) and velocity _
x,
and equivalent linear viscous damping constants c eq at vibration frequency x and amplitude X
Model name
Damping force
c eq
Source
Linear viscous damping
c_ x
c
Slow fluid flow
Air or quadratic damping
c 1 sgnð_ xÞ_ x
2 ¼ c 1 _
xj_ xj
8c1xX
3p
Fast fluid flow
Coulomb damping/friction
lsgnð_ xÞ ¼ l_ x=j_ xj
4l
pxX
Sliding friction
Displacement-squared damping
c 2 sgnð_ xÞx
2
4c2X
3px
Material damping
Solid or structural damping
c 3 sgnð_ xÞjxj
2c3
px
Internal damping
Table 1.2 Damping measures and interrelations for a linear SDOF-system with displacement
coordinate x(t), concentrated mass m, linear stiffness k, undamped natural frequency x 0 =
ffiffiffiffiffiffiffiffi ffi
k=m
p
;
oscillation period T = 2p/x 0 , (displacement/force) frequency response function H(x), energy loss
during a single oscillation cycle DE, maximum potential energy U max , and number of oscillation
cycles n
Symbol
Quantity
Interrelations
c
Viscous damping
coefficient
c ¼ fc cr ¼ 2f
ffiffiffiffiffiffi
km
p ¼ 2fmx 0
c cr
Critical viscous
damping coeff.
c cr ¼
c
f ¼ 2
ffiffiffiffiffiffi
km
p ¼ 2mx 0
f
Damping ratio
f ¼
c
ccr ¼
Dx3db
2x0 ¼
c
2
ffiffiffiffi
km
p
¼
c
2mx0 ¼
d
ffiffiffiffiffiffiffiffiffiffiffiffi
4p 2 þ d
2
p
%
d
2p
À
Á
d
Logarithmic
decrement
d ¼ ln
xðtÞ
xðt þ TÞ
¼
1
n ln
xðtÞ
xðt þ nTÞ ¼ fx 0 T % 2pf
Dx 3dB
3 dB bandwidth
D x3dB = Width of resonance peak in H(x) where the peak
response H(x 0 ) has dropped a factor
ffiffi ffi
2
p
(=3 dB)
η
Loss factor/
coefficient
g ¼
DE
2pUmax (=2f at resonance for linear visc. damp.)
Q
Quality factor
Q ¼
1
g
1.7 Damping: Types, Measures, Parameter Relations
33
cycle for, respectively, linear viscous damping the nonlinear damping law in
question. Table 1.1 lists the resulting equivalent linear viscous damping coefficients
c eq ; as appears these are not really system constants, but depend on vibration
frequency x and amplitude X.
1.7.3 Damping Measures and Their Relations
Table 1.2 summarizes some key damping parameters and their interrelations. Most
of these follows directly from the definitions and relations in Sects. 1.2-4, others
can be found in, e.g., Inman (2014), Crandall (1970), Lazan (1968), Adhikari
(2013).
Table 1.1 Damping force models, in terms of instantaneous displacement x = x(t) and velocity _
x,
and equivalent linear viscous damping constants c eq at vibration frequency x and amplitude X
Model name
Damping force
c eq
Source
Linear viscous damping
c_ x
c
Slow fluid flow
Air or quadratic damping
c 1 sgnð_ xÞ_ x
2 ¼ c 1 _
xj_ xj
8c1xX
3p
Fast fluid flow
Coulomb damping/friction
lsgnð_ xÞ ¼ l_ x=j_ xj
4l
pxX
Sliding friction
Displacement-squared damping
c 2 sgnð_ xÞx
2
4c2X
3px
Material damping
Solid or structural damping
c 3 sgnð_ xÞjxj
2c3
px
Internal damping
Table 1.2 Damping measures and interrelations for a linear SDOF-system with displacement
coordinate x(t), concentrated mass m, linear stiffness k, undamped natural frequency x 0 =
ffiffiffiffiffiffiffiffi ffi
k=m
p
;
oscillation period T = 2p/x 0 , (displacement/force) frequency response function H(x), energy loss
during a single oscillation cycle DE, maximum potential energy U max , and number of oscillation
cycles n
Symbol
Quantity
Interrelations
c
Viscous damping
coefficient
c ¼ fc cr ¼ 2f
ffiffiffiffiffiffi
km
p ¼ 2fmx 0
c cr
Critical viscous
damping coeff.
c cr ¼
c
f ¼ 2
ffiffiffiffiffiffi
km
p ¼ 2mx 0
f
Damping ratio
f ¼
c
ccr ¼
Dx3db
2x0 ¼
c
2
ffiffiffiffi
km
p
¼
c
2mx0 ¼
d
ffiffiffiffiffiffiffiffiffiffiffiffi
4p 2 þ d
2
p
%
d
2p
À
Á
d
Logarithmic
decrement
d ¼ ln
xðtÞ
xðt þ TÞ
¼
1
n ln
xðtÞ
xðt þ nTÞ ¼ fx 0 T % 2pf
Dx 3dB
3 dB bandwidth
D x3dB = Width of resonance peak in H(x) where the peak
response H(x 0 ) has dropped a factor
ffiffi ffi
2
p
(=3 dB)
η
Loss factor/
coefficient
g ¼
DE
2pUmax (=2f at resonance for linear visc. damp.)
Q
Quality factor
Q ¼
1
g
1.7 Damping: Types, Measures, Parameter Relations
33