76 Lecture 4 
escape from the dilemma represented by Eq. (10), but represents a means of 
obtaining a good practical compromise between the conflicting requirements 
with available materials. 
Snowdon has also shown [8, 9, 10] that substantial damping in a mount at 
high frequencies is useful in suppressing any peaks occurring in this region due 
to foundation resonances (as in Fig. 4.5). This comes about since resonant vi- 
bration of the foundation with anantinode belowthe mount will involve substantial 
deformation in the mount, the mounted mass remaining almost stationary. Those 
mounts having high damping at high frequencies may therefore be useful here. 
A similar effect is of course achieved by adding damping material to the founda- 
tion structure [11], and may in particular be more effective for the higher-order 
modes of the foundation. Moderate damping of the foundation has relatively little 
effect except near the foundation resonance frequencies. 
4.4. ADDITIONAL MASS IN VIBRATING STRUCTURES 
As might be expected, the transmission of vibration through a system may 
be reduced by making certain parts of it more massive, particularly in such a 
way as to introduce or emphasize differences in mechanical impedance between 
adjoining parts of a structure. Taking the idealized case of a mass m supported 
by an isolation spring at the center of a beam (Fig. 4.8), an additional mass can 
usefully be added either to mM itself, to the beam either distributed along its 
length or concentrated at the base of the spring, to the spring at some point 
along its length, or attached to M via an additional spring. These possibilities 
will be considered in turn. 
Increasing the mass of the mounted item merely reduces the resonance 
frequency 9 as M~“, with a corresponding gain in high-frequency isolation (for 
a rigid foundation) proportionaltoM. Alternatively, a stiffer spring, giving greater 
static stability, may be used for the same resonance frequency. The extra mass 
also serves to reduce the motion of the machine or equipment itself at high 
frequencies, and may be used to provide a more rigid base for the machine. 
The effect of adding a mass m to the base of the spring has been investigated 
by Snowdon [9, 10] from whose work Figs. 4.9 and 4.10 are taken. Figure 4,9 
shows the response ratio for a system in which the mass ™, of the supporting 
beam foundation is ha of the supported mass M, and in which masses m equaling 
0, 0.1, 0.2, and 1.0 times M are attached at the beam center. The natural fre- 
Fig. 4.8. The idealized simple mount and resonant 
foundation. 
M¢ 
