SEISMIC METHODS 785 



From a mathematical point of view it may be stated that the design is such that, 

 in addition to the acceleration of the mass proportional to the negative displacement, 

 there is an opposing acceleration to be added which is proportional to the velocity. 

 As applied to a seismometer of the loaded spring type, there exists, besides the simple 

 harmonic force of restitution, a damping resistance proportional to the velocity. The 

 equation of motion when the frame is at rest is given by 



■^ + f+¥-' = ° ("^' 



This equation represents a particle constrained to move along the Y axis as a damped 

 linear oscillator. In order to simplify the result of obtaining solutions of this Equatior 

 133, we shall choose coefficients so that 



2 hn = - — 

 M 



and n*= — 



M 



Rewriting Equation 133 using new coefficients 



3; + 2 hny + iv'y = 



At once a solution for this differential equation may be written because of its standard 

 form which has been discussed elsewheref in the many treatises on the mechanics of 

 vibrating particles and text books of differential equations. 



y = Ae-" cos (wt + /3) (134) 



Differentiating with respect to time and substituting in Equation 133 we have 



{2aiv — 2hnw} sin {wt -{- p) + {n^ + a^ — 2hna — mr'} cos (ivt -]- p) =0 



(135) 



This equation is satisfied for all values of (t) if the coefficients of the sine and cosine 

 functions vanish separately. This requires that 



a =: hn and w^ = m^ (1 — A^) 

 Thus, 



y = Ac-""* cos (zvt + /3) (136) 



is a solution of Equation 133 when /i<l. (For /^>1, Wo is the square root of minus a 

 positive number, and is therefore imaginary.) Since this solution contains two arbitrary 

 constants, A and /3, it is therefore the complete solution. It represents the damped vibra- 

 tions of the oscillator when not under forced vibration. The exponential factor indicates 

 that the amplitude decreases with time, falling to 1/e of its original value in the time, 



t = -—. The frequency with damping, called the natural frequency (not under forced 



hn 



vibration) is given by 



«, = „Vl-/r= V^-S <"'> 



t William V. Houston, Principles of Mathematical Physics, McGraw-Hill, 1934. 

 Leigh Page, Introduction to Theoretical Physics, Van Nostrand, 1928. 



