790 



EXPLORATION GEOPHYSICS 



It is sometimes convenient to measure the square of the frequency from the point of 

 resonance. Introduce a new term (//) such that 



then 



//=■ = zer" - w/ = «/' — »'' + 2/1 V 



A = 



Az= 



V (2/iV -H-y+ 4hV (H" + »" - 2AV) 

 a 



\/jF+4F»^(T^=^F) 



(147) 



Now plot the values of the amplitude A against the square of the frequency, to 

 obtain a curve which is symmetrical about the ordinate for which the frequency cor- 

 responds to the frequency of resonance (Figure 478). 



. , , FREE RESONANCE FREQUENCY 



u/j. n i h . (WITHOUT DAMPING) 



FOR NATURAL FREQUENCY 

 (WITH DAMPING) 



I. FREQUENCY FOR MAXIMUM 

 u/a • n (l-zh ) AMPLITUDE 



Fig. 478. — Plot of amplitude versus square of frequency, showing 

 resonance conditions. 



From a consideration of Equation 143 it may be seen that for any impressed frequency 

 (w) the amplitude of the forced vibrations is less, the greater nh becomes. The ampli- 

 tude at resonance (H = 0) is (from Equation 147) 



^maa — 



2n'h Vl - h' 



(148) 



It is to be noted that amplitude becomes greater for small values of (nh). Mathemat- 

 ically the amplitude becomes infinite when the coefficient of the damping term equals 

 zero. 



Another useful expression in the study of a seismometer is the ratio of Amax to A, 

 often referred to as the sharpness of tuning. 



J 7 Jp Y 

 ^ ^'^\2n'hyi-h') 



(149) 



where A represents the amplitude for a particular value of //^ It is to be noted that the 

 sharpness of tuning is larger for small values of the damping coefficient (nh). 



The expression for the average kinetic energy due to a forced vibration may be 

 written as follows : 



£ = J4 w^W = % 



ma^zs/' 



(n' - zt/')' + Aur'h'n' 



(ISO) 



