14 Lecture ] 
Also, A, =A. and Eq. (13) simplifies to 
@Ul ty, ONG AD 5 (14) 
Z2A2 ZA Zi 
For the tail section the condition for resonance is 
7; 
tats 
4 
1.6.3, Electromechanical Coupling Coefficient 
An expression for the coupling factor of a compound bar can be developed 
by considering the complete equivalent circuit and using Eq. (7). For the resona- 
tor in Fig. 1.4 in which /; =1,, the expression [16] is 
2 Ke 
se Tass , lad 53, Tess 1 
DNS,” MZ Gy DlSs 
where k is the coupling coefficient of the active material alone and s, is the 
compliance of the ith ply. The significance of the various material parameters 
can be more easily appreciated by simplifying the compound vibrator so that 
the front and back masses are of the same material and are of the same diam- 
eter as the active component, Equation (15) reduces to 
2 A: lates eee NY 2th ees 
SS Tae Cafolln = noe (16) 
where /; is the total length of the element. 
Equation (16) shows that for a high coupling coefficient, the compliance of the 
loading mass should be small compared with the compliance of the active ma- 
terial. The theoretical and experimental variations of coupling factor with the 
ratio of active length to total length of the vibrator are shown in Fig. 1.5. The 
agreement between the experimental points (whichare fora brass-loaded barium 
titanate sandwich) and theory is good. 
, -50 | 
z 
ua 
oO 
z -40 =| 
uw 
Ww 
° 
© .30 4 
(o) 
= 
& .20 4 me aye 
a BaT + BRASS Fig. 1.5. Variation of coupling co- 
8 ——— Rata efficient with the ratio of active to 
"10 inactive length for a compound-bar 
= element. 
te} 2 4 -6 8 1.0 
e ACTIVE LENGTH 
! TOTAL LENGTH 
