326 Lecture 17 
dB dBre 
exciter oakar 1kg/s 
acceleration(a) 
70 120 
60 
110 
130 kg 
(steel) 
sO 
oe 
Pee c 
Np WwW 
e) eo) 
(01) (to) 
fe) e) 
impedance level (L 7) 
+ 
(e) 
ss eee ae 60 
= 20 log|2mfF/a] _ afew \ciozeatins soleil 
sare ma inesset eer a ree 
eee oes 4 
-20 30 
1 2 4 8 i s2 6s) 125 280 Seo) 1 2 4 16 32 63 
————_ f Hz kHz 
Fig. 17.4. The impedance level L,, of a spring was measured. (Metalastik double-U-shear mount 
31/149 B/7; 130-kg "immobile" steel base, force gauge between mount and base, and "normal" exci- 
tation.) The impedance level L., of a foundation was also measured (3.2-m I-NP-16-beam on rubber 
blocks). Moreover, the "normal" isolation L,,; — Ly; Wasmeasuredfor the spring mounted on the beam 
(same point as for impedance measurement). The isolation is approximately equal toL,,;—L,,! 
———= Isolation (Lyj -Ly¢) 
(e) 
1 
(e) 
17.4. BENDING WAVES IN STRUCTURES 
A simple model of what possibly happens when a mechanical structure is 
excited by alternating forces can be based uponthe pure bending wave phenomenon 
in plates or in beams. This type of wave seems to be very important, moreover, 
for the radiation of sound by vibrating plates in direct contact with a fluid acous- 
tic medium. A plane longitudinal wave in some such medium, e.g., water, can be 
described, as is well known, by a differential equation showing the balancing of 
inertia and elasticity forces; e.g., 
po PE - Ky, 4 = 0 (11) 
where p, is the fluid density, u is the particle displacement in the x direction, 
and K,, is the fluid compression modulus. 
A similar though largely different equation may be derived for the plane 
bending waves in plates or in beams [9]: 
Ow O+w 
/\ LAE. jp Lue = 0 (12) 
EEE =” ee 
where p is the density and E the modulus of elasticity of the plate or beam ma- 
terial; A is the area and / is the axial moment of inertia typical of the cross 
section; and w is the particle displacement in the z direction, i.e., normal to 
the plate or beam. 
