584 SEISMIC METHODS [Chap. 9 



mographs a vane may be attached to the end of the magnification cone to 

 be free to move in an oil cup; or a series of vanes may be fastened to the 

 mass directly, arranged between partitions in an oil container. In elec- 

 trical detectors the receiver cases are generally filled to a level which gives 

 the desired amount of damping. It is preferable to use mineral oils of 

 low temperature coefiicient of viscosity (Nujol or automobile oils of vari- 

 ous S.A.E. ratings). 



Electromagnetic damping is most satisfactory from the point of view of 

 design, clean operation, and independence cf temperature. It is most 

 effective in detectors of low natural frequency and light mass. In a 

 Galitzin type seismometer a copper plate at the end of the coil-carrying 

 lever moves between two permanent . magnets whose spacing can be ad- 

 justed to obtain the desired damping rate. In the Wood-Anderson 

 torsion seismometer the mass moves between the poles of a permanent 

 magnet. In the Wenner seismometer pickups and galvanometer coils are 

 of comparable mass and damp one another; the damping rate may be 

 regulated by a shunt across the line. With an amplifier between pickup 

 and galvanometer the primary of the input transformer produces damping. 

 Additional damping may be produced by short-circuited turns or coil 

 frames of conductive material. 



If p is defined as damping resistance, or damping force on mass of unit 

 velocity, the damping acceleration equals pd/m, and 



d -I- ? d -f- cooa = 0. (9-85a) 



m 



Substituting a damping constant e = ^p/w (which has the dimension of a 

 frequency), 



d 4- 2ed + coo a = 0, (9-855) 



whose solution a = Bo-e"'' + Co-e"^', or 



a = Bo-e-t'-Vc^^^^] + Co.e-['+^(^'=^r'l (9-85c) 



If € > ojo , the exponent is negative, the motion overaperiodic, the damping 

 overcritical, and the mass creeps back to the zero position. If e = coo , 

 the mass returns to zero without overswing, and the motion is aperiodic 

 (critical damping). If e < coo , the mass oscillates about the zero position 

 with declining amplitudes. This is referred to as damped free oscillation, 

 comprising damping rates between zero and critical. In this case, the 

 term y/e^ — ul becomes j\/o}l — e^, so that by the use of Moivre's theorem 



a = Bo . e~" . sin V^^ - e^ i -h Co • e~" • cos \/wo - e' ^ (9-86a) 



Comparing this with eq. (9-18a), we see that a/w^ — e^ represents the 

 natural frequency reduced by damping. Hence, 



