SEISMIC METHODS 799 



turns = 0.4 n n i) that the magnetic energy Tc2 associated with the current i is given 

 by the equation 



^^^^0421211^. 10- (155) 



OTT 



where 4n, the flux created by the current /, is equal to the magnetomotive force around 

 the magnetic circuit (Figure 487) divided by the reluctance of the circuit. 



The magnetic energy Tea associated with the reaction of the current i on the per- 

 manent magnetic field is given by the equation : 



Te3 = ^«10-« (156) 



If each of the equations 154, 155 and 156 is expanded in a Maclaurin series and if 

 all terms involving powers of s higher than the first are dropped, the total kinetic 

 energy of the mechanical and electrical parts of the seismometer may be expressed 

 by the relation 



T = H mr- + ( Tei) .=0 + (^) s + y2<iLr) .=0 r + J^ (dU\ fs + (^) si 



^ ' »=o \ ds / s=0 «=0 



where (157) 



L. = ^ . 10- and X ^ ^ . 10- 



The potential energy of the system is due entirely to the mechanical part, be- 

 cause the electrical circuit is assumed to have no capacity; that is, 



F = F„ = ^ (158) 



where Cm is the compliance of the spring supporting the horseshoe magnets. 

 The total dissipation function is 



D = y2rmP + y2ri' (159) 



where rm is the mechanical resistance factor (determined by the side slip between 

 the moving horseshoes and the air) and r is the resistance of the armature coils. 



The general equations of motion of the combined electrical and mechanical systems 

 which constitute a seismometer may be obtained by using Lagrange's equations employ- 

 ing generalized coordinates.! In particular, most seismometers have two degrees of 

 freedom, one corresponding to the displacement of the moving mass and the other to 

 the flow of electrical charge or current, and the Lagrangian equations for a seismom- 

 eter are : 



d dT d(T-V) ,dD_. ..,„. 



and 



d^_ ^(T-V) ^^D^^ (161) 



dt -dq dq -Qq 



The symbols T, V, D, s and s have the meanings previously indicated ; d/dt denotes 

 as usual a time derivative ; q is the electrical charge ; q is the time rate of change 

 of electrical charge; / is the externally applied force; and e is the E.M.F. in the 

 electrical circuit. 



The motion of the electromechanical system constituting the seismometer is 

 specified when the quantities representing the kinetic energy (mechanical plus mag- 

 netic), the potential energy (mechanical plus electrical) and the dissipation function 

 (friction plus electrical heating) are substituted into the two Lagrangian equations 



t A discussion of Lagrange's equations is given in many texts on theoretical mechanics. See, 

 for example, A. Zivet and P. Field, Introduction to Analytical Mechanics, pp. 352-359 (Macmillan 

 Co., New York), 1936. A more generalized discussion is given by J. H. Jeans, Mathematical 

 Theory of Electricity and Magnetism, 5th Edition, pp. 489-498 (Cambr. tJniv. Press, 1937). 



