Metal Foams: A Design Guide

178 Metal Foams: A Design Guide
and the response is minimized by maximizing the material index:
Md D ⊲12.15⊳
More generally, the input x is described by a mean square (power) spectral
density:
� � k
ω
Sx ⊲ω⊳ D S0
⊲12.16⊳
ω0
where S0, ω0 and k are constants, and k typically has a value greater than
2. It can be shown (Cebon and Ashby, 1994) that the material index to be
maximized in order to minimise the response to x is
M 0 d
D ωk 1
1
D Mk 1
u
⊲12.17⊳
The selection to maximize M0 d can be performed by plotting a materials selection
chart with log⊲ ⊳ on the x-axis and log⊲Mu⊳ on the y-axis, as shown in
Figure 12.4. The selection lines have slope 1/⊲1 k⊳. The materials which lie
farthest above a selection line are the best choice.
Performance index for undamped vibration
log ω 1
−1/5
−1/3
−1
Loss coefficient log η
k = −∞
k = −6
k = −4
k = −2
Input
Lower
frequency
content
White
accn
White
velocity
Figure 12.4 Schematic diagram of a materials selection chart for
minimizing the RMS displacement of a component subject to an input with
spectral density S0 ⊲ω/ω0 ⊳ k
If k D 0, the spectrum of input displacement is flat, corresponding to a
‘white noise’ input, but this is unrealistic because it implies infinite power input
to the system. If k D 2, the spectrum of the input velocity is flat (or white),
which just gives finite power; for this case the selection line on Figure 12.4 has
a slope of 1. For larger values of k, the input becomes more concentrated at