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EXPLORATION GEOPHYSICS 



Fundamental Theory of Seismometer 

 Based on the Mechanics of a Vibrating Particle 



The logical basis for the design of a seismometer to respond to the vertical com- 

 ponent of the earth's motion is a loaded vertical spring arrangement as shown 

 schematically by Figure 475. In actual practice the moving pendulum or inertia mass 

 is connected to some type of transducer which develops an oscillatory voltage corre- 



FiG. 475.— Vertical component seismometer pendu- 

 lum (schematic). M — mass suspended; S — suspension 

 spring; D — dash; P — dash pot containing viscous liquid. 



sponding to the motion of the inertia mass relative to the frame. However, Figure 475 

 is used to illustrate only the behavior and theory of the oscillatory system, and not 

 mechanical design. 



Let M = the mass of the pendulum 

 Lo = length of spring unloaded 

 L •=■ length of spring loaded and at rest 

 (L — y) = length of spring loaded and displaced 



P z=. force exerted by the spring per unit of extension 



In this theoretical discussion the weight of the spring has been neglected ; the condition 

 of the loaded spring may be expressed by the following relationship : 



Mg = FiL-Lo) 



(123) 



Consider the mass constrained to move along the Y axis. Assume the ordinate of the 



center of mass to be measured upward from the position of equilibrium. Z is the up- 

 ward acceleration of the frame caused by the motion of the earth. The upward accelera- 

 tion of the mass will be given by the differential equation 



My = -Mg + F (L-y - Lo) - MZ 



(124) 



Substituting F (L — Lo) for Mg and rearranging, the differential equation of motion 

 becomes 



y+ i- y + Z = 



(125) 



