Metal Foams: A Design Guide

76 Metal Foams: A Design Guide
If the foam is not of low density, the indentation hardness is better
approximated by
� � ��
H ³ c 1 C 2
s
This means that foams are more vulnerable to contact loads than dense solids,
and that care must be taken in when clamping metal foams or joining them to
other structural members: a clamping pressure exceeding c will cause damage.
6.8 Vibrating beams, tubes and disks
(a) Isotropic solids
Anything vibrating at its natural frequency without significant damping can be
reduced to the simple problem of a mass, m, attached to a spring of stiffness,
K. The lowest natural frequency of such a system is
f D 1
2
� K
m
Specific cases require specific values for m and K. They can often be estimated
with sufficient accuracy to be useful in approximate modeling. Higher natural
vibration frequencies are simple multiples of the lowest.
The first box in Figure 6.7 gives the lowest natural frequencies of the flexural
modes of uniform beams with various end-constraints. As an example,
the first can be estimated by assuming that the effective mass of the vibrating
beam is one quarter of its real mass, so that
m D m0ℓ
4
where m0 is the mass per unit length of the beam (i.e. m is half the total mass
of the beam) and K is the bending stiffness (given by F/υ from Section 6.3);
the estimate differs from the exact value by 2%. Vibrations of a tube have a
similar form, using I and m0 for the tube. Circumferential vibrations can be
found approximately by ‘unwrapping’ the tube and treating it as a vibrating
plate, simply supported at two of its four edges.
The second box gives the lowest natural frequencies for flat circular disks
with simply supported and clamped edges. Disks with curved faces are stiffer
and have higher natural frequencies.
(b) Metal foams: scaling laws for frequency
Both longitudinal and flexural vibration frequencies are proportional to p E/ ,
where E is Young’s modulus and is the density, provided the dimensions of