582 SEISMIC METHODS [Chap. 9 



where c is a spring constant and is equal to the ratio of force and spring 

 elongation d (see page 591). 



2. Undamped free oscillations. In an undamped seismograph, as in a 

 mathematical pendulum (see page 97), the condition of equilibrium of 

 restoring and inertia forces is expressed by mg sin ^ + ml id ipjdi ) = 

 (see Fig. 9-96), so that in abbreviated differential notation after dividing 

 by ml, and letting sin tp = <p, 



vp + f V = 0. (9-84a) 



V 



Substituting a = Jv? (Fig. 9-96, where a is the recorded amplitude) and 

 utihzing eq. (9-82d), we have 



a + woO = 0, (9-846) 



whose solution is 



a = Ao sin(woi -|- ^) = Ao sin ajo(^ + ^o), (9-84c) 



which is identical in form with eq. 9-19 given on page 449. 



3. Free oscillations with friction. Actually the last equation does not 

 express completely the motion of the seismograph. Its amplitude de- 

 creases with time, since the kinetic energy is consumed and converted into 

 heat. This transformation acts as a brake on the amplitude. The manner 

 in which the amplitude decreases is dependent on the type of braking 

 resistance, of which the following are the more important types: (1) 

 Coulomb's friction, constant during the entire motion, (2) velocity damp- 

 ing, which is proportional to the velocity of motion, and (3) velocitj'- 

 square damping, proportional to the square of the velocity of motion. 

 Only the first and second will be discussed here. Details on the theory 

 of velocity-square damping may be obtained from the literature. 



In early seismograph construction, attempts were made to reduce the 

 natural movement by excessive friction. However, this makes the seis- 

 mograph inoperative for accelerations equal to or less than the frictional 

 force. It does not return to its zero position, while with velocity damping 

 it always returns. 



Although friction is undesirable in a seismograph, it is never possible to 

 avoid it altogether, since there is a certam amount of friction in the sus- 

 pension springs, electrical wires, and the like. The t5rpe of record obtained 

 with a seismograph under the influence of friction only is shown in Fig. 

 9-97. A line connecting the peaks is straight, while for a damped seis- 



»i L. S. Jacobsen, Seis. Soc. Amer. Bull., 20(3), 160-195 (Sept., 1930); B. Hague, 

 Alternating Current Bridge Methods (section on oscillographs), London (Pitman & 

 Sons), 1930. 



