Metal Foams: A Design Guide

176 Metal Foams: A Design Guide
Single low-frequency undamped input
For small values of ω/ω1 and low damping
jYj D ⊲ω/ω1⊳ 2 jXj ⊲12.7⊳
meaning that the response Y is minimized by making its lowest natural
frequency ω1 as large as possible. Real vibrating systems, of course, have
many modes of vibration, but the requirement of maximum ω1 is unaffected
by this. Further, the same conclusion holds when the input is an oscillating
force applied to the mass, rather than a displacement applied to the base. Thus
the material index Mu
Mu D ω1
⊲12.8⊳
should be maximized to minimize response to a single low-frequency
undamped input.
Consider, as an example, the task of maximizing ω1 for a circular plate. We
suppose that the plate has a radius R and a mass m1 per unit area, and that
these are fixed. Its lowest natural frequency of flexural vibration is
�
ω1 D C2
2
Et 3
m1R 4 ⊲1
2 ⊳
� 1/2
⊲12.9⊳
where E is Young’s modulus, is Poisson’s ratio and C2 is a constant (see
Section 4.8). If, at constant mass, the plate is converted to a foam, its thickness,
t, increases as ⊲ / s⊳ 1 and its modulus E decreases as ⊲ / s⊳ 2 (Section 4.2)
giving the scaling law
ω1
�
D
� 1/2
⊲12.10⊳
ω1,s
s
– the lower then density, the higher the natural vibration frequency.
Using the foam as the core of a sandwich panel is even more effective
because the flexural stiffness, at constant mass, rises even faster as the density
of the core is reduced.
Material damping
All materials dissipate some energy during cyclic deformation, through
intrinsic material damping and hysteresis. Damping becomes important when
a component is subject to input excitation at or near its resonant frequencies.
There are several ways to characterize material damping. Here we use the
loss coefficient which is a dimensionless number, defined in terms of energy
dissipation as follows. If a material is loaded elastically to a stress max (see