330 
Lecture 17 
2) 
a pote eat 
avtA)= (ay @3(A) 8) 
thus allowing a quick determination of q,(A). 
The factor 9,(A) in the denominator of Eq. (17) may be called a force dis- 
tribution factor. It is equal to 1 for a point force applied in A. For a pressure 
distribution, it is given by 
a, (A)F 
A\ a ae 
Pv(A) 7p, as 
(20) 
where F is the amplitude of the total force [cf. Eq. (16)]. For the ratio @,/,(A) 
under the integral sign, ¢,/¢,(A) may be substituted (cf. [11]). 
The last term inthe denominator of (17) is the most illustrative. It is given by 
2 
M 
Zy=joM +R,+ = R,(1 + jv,Q,) (21) 
the well-known expression for the complex impedance of a series circuit, where 
Ry =2n at (22) 
Ov=—t (23) 
ee 
ary (24) 
Aline fa 
and 7 is the loss factor, f, is the vth natural frequency, f is the excitation fre- 
quency (@/27), and M is the total mass of the beam. Eventually, Eq. (17) turns 
out to be a complicated expression for what may be called a "parallel connection 
v(A) | F(A) 
AS 
\S “YS \ XX ~ 
SQ YOO YS AL WN 
ni SSAA OO SOY SYS 
. SX WV’ >» WS 
esa= 
Fig. 17.7. Hydraulic analog of a mechanical structure (e.g., beam or plate) 
excited by a sinusoidal point force F(A). The velocity v(A) at A depends upon 
the response of infinitely many "resonators" (mass on spring and mechanical 
resistance) numbered 1,2,3,...%. An electrical analog would be a parallel 
connection of series circuits. The values of the circuit components are given 
in Eqs. (17) through (24). 
