Metal Foams: A Design Guide

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Vibrating beams, tubes and disks
2R
2R
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2R
C1
3.52
9.87
22.4
9.87
2.68
C 2
1.44
2.94
Design formulae for simple structures 77
f = Natural frequency (s −1 )
m o = ρA = Mass/Length (kg/m)
ρ = Density (kg/m 3 )
A = Section area (m 2 )
I = See Figure 6.1
With A = 2πR
With A =
Disks
Figure 6.7 Vibrating beams, tubes and disks
I = πR 3 t
f 1 = C 2
2π
Beams,tubes
f 1 = C 1
2π
t 3
12
EI
m o 4
Et 3
m 1R 4 (1 − ν 2 )
m 1 = ρ t = Mass/Area (kg/m 2 )
t = Thickness (m)
R = Radius (m)
ν = Poisson's ratio
the sample are fixed. The moduli of foams scale as ⊲ / s⊳ 2 ,andthemassas
⊲ / s⊳. Thus the natural vibration frequencies of a sample of fixed dimensions
scale as f/fs D ⊲ / s⊳ 1/2 – the lower the density of the foam, the lower its
natural vibration frequencies. By contrast, the natural vibration frequencies
of panels of the same stiffness (but of different thickness) scale as f/fs D
⊲ / s⊳ 1/6 – the lower the density, the higher the frequency. And for panels
of equal mass (but of different thickness) the frequencies scale as f/fs D
⊲ / s⊳ 1/2 – the lower the density, the higher the frequency.