786 



EXPLORATION GEOPHYSICS 



In comparing this expression with that for the free resonant frequency 



Wo = 2TTlT-y/FIM 



(131) 



we note that the natural frequency is less than it is when there is no damping. (Note 

 that w = Wo when /i = 0.) If we plot the displacement (3;) against time, it is clear that 

 the curve will lie between the two curves corresponding to the maximum and minimum 

 values of the cosine for any value of t : 



y = + Ae-"""* and 3^ = — Ae-"""* 



Fig. 476. — Showing damped oscillations of ji = Ae-''"* cos (wt + ft). 



Now consider the case when /i>l. Then the so-called angular frequency w becomes 

 imaginary from Equation 135. Therefore the motion will not be oscillatory, and the 

 solution involving oscillatory trigonometric functions is not suitable as a solution of 

 the equation of motion in the case where /i>l. For this case we may try y^iAe'"'. 

 Differentiating and substituting in Equation 133, 



A (a- — 2ahti -f n") e-"' = 

 Equation 138 is satisfied for all values of t if 



(138) 



a=: hn±n y/h^ — 1 

 Thus the complete solution of Equation 133 for the case when /i>l is 



A and B are the two arbitrary constants which may be used for the complete solution 

 of Equation 133. In this solution, when A>1, the exponent is always real and negative. 

 This means that the motion is aperiodic. If the inertia mass is pulled away from the 

 equilibrium position and released, the damping is so great that the inertia mass never 

 passes to the other side of the rest position, but its movement decreases exponentially 

 to zero with time. In this case, then, when /j>l, the system is over-damped; when 

 h ■= 1, the motion is said to be critically damped; when /j<l, the motion is under- 

 damped; and when h = 0, the motion is undamped. 



Frequency Response 



Figure 477 illustrates the response of a seismometer measured in 

 relative units plotted against the applied frequency in cycles per second. 



