numerically to determine (or wait before experimentally measuring) a stationary
response after changing initial conditions, or how fast vibration based sensors can
react to changing load.
1.7.6 Estimating Mass/Stiffness-Proportional
Damping Constants
The two constants a and b in the stiffness/mass-proportional model of linear viscous
damping (cf. Sect. 1.3.6) can be estimated by fitting the model-predicted damping
ratios (cf. (1.34)) to experimentally measured quantities. Let (x j , f j ), j = 1, n, n ! 2
be a set of n measured natural frequencies and damping ratios, and define:
r 1 ¼
1
n
X n
j¼1
1
x 2
j
; r 2 ¼
1
n
X n
j¼1
x
2
j ;
p 1 ¼
1
n
X n
j¼1
f j
x j
; p 2 ¼
1
n
X n
j¼1
f j x j :
ð1:111Þ
One can then show that the estimate of a and I minimizing the mean square error
between measured and model-predicted (by (1.34)) modal damping ratios is:
a ¼
2ðr 2 p 1 À p 2 Þ
r 1 r 2 À 1
; b ¼
2ðr 1 p 2 À p 1 Þ
r 1 r 2 À 1
:
ð1:112Þ
1.7.7 Mass/Stiffness Damping Proportionality
Constants for Beams
Consider the equation of motion for transverse vibrations of a beam with both
internal (cf. Sect. 1.4.6) and external linear viscous damping:
qA€ u þ EIu
0000
þ c 1 _
u þ c 2 _
u
0000
þ qðx; tÞ ¼ 0;
ð1:113Þ
where c 1 is the coefficients of external damping (proportional to velocity,
expressing e.g. external air resistance), while c 2 is the coefficient of internal
damping (proportional to rate of change of curvature or bending moment,
expressing energy loss associated with beam material deformation). Performing a
mode shape expansion as in Sect. 1.5.3, one arrives at a system in the form (1.29),
with a damping matrix of the form (1.30), where the mass- and
stiffness-proportionality constants are:
36
1 Vibration Basics