596 



SEISMIC METHODS 



[Chap. 9 



d(c — SM.^/4ira) = 0, from which it is seen that the second term repre- 

 sents a negative spring constant, also called negative stiffness: 



^**^ (9-936) 



Cm = 



ixa^ 



The presence of the magnetic field reduces the elastic spring constant and, 

 therefore, the natural frequency is 



/. 



«.= ./«- SM' 



m irma^ 



(9-93c) 



Uf 



10 



m¥ 



2 4 6 8 mm 



Fig. 9-104. Voltage output of experi- 

 mental reluctance detector as a function 

 of gap length. 



When the armature or magnet 

 moves, the flux <E> = SH changes, 

 since the gap width and hence the 

 reluctance increases or decreases, 

 respectively. In a balanced de- 

 tector the two armatures are so 

 wound that opposite flux changes 

 above and below produce an e.m.f . 

 in the same direction. The fliur 

 through the upper two gaps (disre- 

 garding the deflection d) is SM/a. 

 The same flux passes through the 

 lower gaps^ so that $ = 2SM/a. 

 If N is the total number of turns 

 on the two armatures, the induced 



, „ rf$ d^da ,, , 

 e.m.f. i; oc -- oc — - -— so that 

 dt da dt 



E = - 



2NSM da 

 a2 di 



(9-94a) 



By substitution of the time derivative of the steady state term in eq. 

 (9-89rf), 



E = - 



«' V(co^-a,Y + 4e 



2 2 

 CO 



sin 



(-^ + 2 + ^)- 



(9-946) 



The voltage output increases, therefore, in inverse proportion to the 

 square of the gap width which is verified by the experiment represented 

 in Fig. 9-104. Analysis of this curve gives 94-amm" millivolts for the 

 e.m.f. in this case. The deviation from the exponent —2 is probably 

 caused by the fact that in eq. (9-946) the variation of coo with gap width 



