A continuous damped vibration absorber to reduce broad-band ...

β /k a
10 0
10
10
10
10
ARTICLE IN PRESS
D.J. Thompson / Journal of Sound and Vibration 311 (2008) 824–842 837
0.3 0.5 1
ω/ω
a
2 3
Fig. 8. Decay rate of beam with tuned absorbers, Za ¼ 0.4. —, single absorber with m ¼ 0.2 tuned to oa; – – –, two absorbers each with
mi ¼ 0.1 tuned to 0.72oa and 1.45oa;– – , three absorbers each with mi ¼ 0.067, tuned to 0.57oa, oa and 1.75oa; , ten absorbers
each with m i ¼ 0.02, tuned to frequencies between 0.38o a and 2.7o a.
Including damping in each of the springs and the beam itself, the imaginary part of the wavenumber is given
by a modified form of Eq. (15),
b ¼ Im kbð1 þ iZbÞ 1=4 1 o20 mð1 þ iZ
ð1 þ iZÞ
aÞ
o2 ðo2 =o2 aÞ ð1þ iZ ( )
1=4
. (36)
aÞ
Around and above oa the final term, related to the absorber, dominates and the decay rate can be estimated
using the relations in Section 4. For oboa,
Zb b kb
4 þ Zo20 4o2 þ Zamo2 a
4o2 . (37)
The first two terms represent the damping effect of the beam and foundation layer, respectively, given by
Eq. (6) and the third term is that of absorber above its tuning frequency, Eq. (16). This shows that the
damping effects can be simply combined by adding the separate decay rates.
For frequencies well below oa, Eq. (36) may be expressed as
( )
b kbð1 þ mÞ 1=4 Im ð1 þ iZbÞ 1=4 1 o20 o2 ð1 þ iZÞ
, (38)
ð1þmÞ which is equivalent to the result in Eq. (6) for a beam of mass mb 0 (1+m) on the elastic foundation. Thus the
absorber actually reduces the attenuation in this region, due to the increase in mass and the corresponding
shift in cut-off frequency o0 0 .
Fig. 9 shows the normalised decay rate in the form of b/ka for the beam on elastic foundation with absorber,
according to Eq. (36). In Fig. 9(a) the initial cut-off frequency for the beam on elastic foundation o0 is set to
0.3oa and m ¼ 0.2. For simplicity the beam damping loss factor Z is set to zero. In addition, the separate
results are shown for the beam on elastic foundation and the unsupported beam with absorber. As noted
above, the effective cut-off frequency due to the support stiffness is reduced, here by about 9%, and
1=4