SEISMIC METHODS 783 



The dots represent differentiation with respect to time as illustrated by the following : 



d^y J ' dy 



y=-^ and y = ^ 



Equation 125 does not contain the damping term associated with the dash-pot arrange- 

 ment shown by Figure 475. If this dash-pot is filled with some type of viscous oil, there 

 will be a damping resistance to the motion of the inertia mass proportional to the veloc- 

 ity. The equation of motion then becomes 



^'+1^ + 1^ = -^' ^'^^^ 



We shall solve this equation later, but first let us discuss some definitions and make 

 some of the more obvious observations with reference to the oscillatory system shown 

 schematically by Figure 475. Perhaps the easiest measurement to make on this loaded 

 spring arrangement is the period of free oscillation with no viscous liquid in the dash- 

 pot. The period of free oscillation may be defined as the time interval between succes- 

 sive passages in the same direction through the rest position. For this purpose, assume 

 that the motion of the vibrating mass is of the simple harmonic variety. This assump- 

 tion, when translated into the language of the physicist, states that if there is a restor- 

 ing acceleration proportional to the negative displacement, there is simple harmonic 

 motion. The displacement (y) may then be stated by the well known formula 



y = ym sin (nt + c) (127) 



This expression comes from a solution of the differential equation 



y = -n^y (128) 



To integrate, multiply both sides of this equation by 2 dy which gives 



2dyy=z2yydt=. diy)" =z — 2n^ydy 



and integrating y = — n^y^ -\- ci. 



To evaluate the constant of integration, consider the condition for a maximum (ym) 

 displacement which requires that the derivative vanish. Then 



— n^ym" -1- ci = and ci = n^y^m 



so that the result of the first integration is 



y = n y/ym" — 3'" 



Separating the variables and integrating the second time : 



dy V 



. z = ndt sm"^ -^^ =znt -{- c 



V Tm — y- ym 



Thus, 



y = ym sin (lit -\- c) (127) 



The definition of the period (T) of the oscillatory system corresponds to a difference 

 of 27r or 360° in the angle of Equation 127. Placing T as the symbol representing the 

 period, then from Equation 127 : 



»r = 27r or n=^ (129) 



