J.H. Janssen 331 
of series circuits." The natural frequencies of the beam correspond to the 
"resonant frequencies" of the series circuits. A hydraulic analog for a point- 
force excitation is shown in Fig. 17.7. The picture of "parallel connection of 
series circuits" for a beam seems to have a general validity for other mechan- 
ical structures as well. It agrees very well with experimentally observed reso- 
nance curves of mechanical structures which closely resemble those of the 
simple classic RLC series circuit. 
The behavior of these circuits is well known, and the noise impedance Z,,, 
if measured with half-octave band filters, is given by 
(25) 
where the symbols of Eqs. (7), (22), and (23) are used. Of course, Eq. (25) can 
be only approximately valid for a mechanical structure and only for one reso- 
nant frequency within the frequency band of the exciting force noise. Assuming 
the various natural frequencies within a given frequency band to be sufficiently 
separated, it is possible to derive an expression for the noise impedance of a 
plate or a beam as soon asthetotal number of the "peaks" in that band is known. 
For simply supported beams or plates, the noise impedances averaged over 
the beam or plate turn out to be given by 
Ss Mcph 
Zieh = On ONT ae (26) 
for point excitation of a rectangular plate of thickness h, width 5, and area 
S = 1b; and, 
al 2 
GB gf (27) 
for line excitation along the width b of a plate; and, 
al M2 Vc 1 
GE ce Og (28) 
for point excitation of a beam (length / and radius of gyration i,). In Fig. 17.8 
the reasonable agreement between a measured and the corresponding computed 
impedance from Eq. (26) is shown. Especially interesting is the deviation from 
the theoretical curve for frequencies f > f.,. Presumably it is due to the increased 
load caused by the efficient radiation of sound into the surrounding air; the order 
of magnitude for this effect is correct. 
A closer examination of the results given above raises at least two questions 
respecting their applicability to underwater noise control. Up to now, theory 
has introduced only structural data, but no coupling of the structure with a fluid 
medium. The first question, therefore, concerns the extra load due to water 
when in direct contact with a vibrating plate. The second has to do with the 
excitation. It was assumed that the exciting force acted in a direction parallel 
to the resulting velocity of the points on the beam. In principle, it must be pos- 
sible to handle situations like that shown in Fig. 17.5 with the aid of the method 
given. However, it is too simple a supposition that a diesel engine vibrates in 
a "normal" direction only. It is equally probable that its contact planes with 
springs move "parallel" to the beam, or otherwise, of course. 
