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

Damping can be added as before by making s1 and s2 complex with loss factors Z1 and Z2, respectively. If the
beam is constrained, the mass ms 0 can vibrate freely on the combined stiffness of the two layers at the
frequency oa, given by
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s1 þ s2
oa ¼ . (40)
This corresponds to an anti-resonance of the support as seen at the beam and is the tuning frequency of the
two-layer support at which it acts as a neutraliser to the beam.
0 2
There are two frequencies at which mr o s(o) ¼ 0, corresponding to the condition of cut-off seen
in Section 3. These are the natural frequencies of the corresponding two-degree-of-freedom system shown in
Fig. 10(b). They can be found as
o 2 c ¼ ðo21 þ o2a Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðo
2
2 1 þ o2a Þ2
o
4
2 1o2 s
2,
(41a)
where o1 and o2 are given by
ARTICLE IN PRESS
o1 ¼
m 0 s
rffiffiffiffiffiffi
rffiffiffiffiffiffi
s1
s2
; o2 ¼ . (41b,c)
m 0 r
The two cut-off frequencies o c are plotted in Fig. 11 for different values of m ¼ m 0 s=m 0 b and k ¼ s 1/s 2. These
results have been normalised by the absorber tuning frequency o a from Eq. (40). The absorber bandwidth
[oa, oc2] can be seen to be smaller than the corresponding result from Eq. (12), shown by the circles, which
forms the limit as s2-0.
Examples of the wavenumbers are shown in Fig. 12, normalised to the free beam wavenumber at oa. These
results are given for m ¼ 4, which is typical of a railway track with concrete sleepers [30], although much larger
than considered in earlier sections.
As for the case of a single stiffness support, below the first cut-off frequency oc1, a low frequency ‘blocked’
region occurs where no waves propagate, the wavenumbers having equal real and imaginary parts. Free wave
ω 0 /ω a
10 0
D.J. Thompson / Journal of Sound and Vibration 311 (2008) 824–842 839
10 10 0
10
Fig. 11. Bounding frequencies of propagating wave behaviour shown relative to ‘absorber tuning frequency’ of two-layer foundation. —,
m ¼ 0.5; – – –, m ¼ 1; , m ¼ 2; – – –, m ¼ 4; o, limit for s2 ¼ 0.
s 1 /s 2
m 0 s
10 1