Creep-Enhanced Low-Frequency Sensitivity of Seismometers Creep-Enhanced Low-Frequency Sensitivity of Seismometers

Randall D. Peters
Physics Department
Mercer University
Macon, Georgia 31207

(Copyright August 2005)

Abstract

The frequency response of a seismometer is typically assumed to be the textbook case of a viscous damped, simple harmonic oscillator. Real mechanical oscillators are not ideal, and the damping at low frequencies, due to internal friction, is presently too poorly understood to describe from first principles. Even if the low-level motions were smooth (which they are not), the mean position of a seismic mass changes because of creep and creep recovery. This article shows that secondary creep can actually serve to increase the sensitivity of a seismometer at low frequencies.

1  Background

The equation of motion of a driven, viscous damped simple harmonic oscillator is the following linear expression:
..
x
 
 +  w0
Q
.
x
 
 +  w02 x  =  ad(t)
(1)
where w0 is the natural frequency of the oscillator, Q is its quality factor (2p times the energy divided by the energy loss per cycle), and ad(t) is the drive acceleration.

Here we assume a harmonic drive ad(t)  =  -w2 Ad ejwt where w is the frequency of the drive having amplitude Ad and j  =  (-1)1/2. Using the Steinmetz phasor method, one obtains the steady state solution to this well known equation. The result is frequently provided in terms of the Bode plot shown in fig. 1.

bode1.gif
Figure 1. Graph of the ratio of steady state oscillator amplitude to drive amplitude as a function of drive frequency-ratio r, where r  =  w/w0. The graph was generated with Q = 10.

The equation used to generate Fig. 1 is

| A
Ad
|        =                 w2
  


(w02-w2)2+ w02
Q2
w2
 
(2)

2  Creep Influence on the Equation of Motion

Creep and creep recovery cause the length of the spring to vary; consequently the zero mean position (implied) in Eq. (1) is changed to x0, yielding the new equation of motion
..
x
 
 +  w0
Q
.
x
 
 +  w02 (x - x0)  =  ad(t)
(3)
Once a system has `stabilized', primary creep (exponential) is no longer important except in response to major stress changes. Secondary creep, however, never disappears; since it is well known by material scientists that there is no level below which creep vanishes.

The secondary creep/creep recovery rate is assumed to be proportional to the acceleration; i.e.,

.
x0
 
  =  c ad(t)  =  -c w2 Ad ejwt
(4)
which yields for the steady state (variable) mean position
x0  =  j c w Ad ejwt
(5)

Substituting Eq.(5) into Eq.(3) and solving for the steady state solution with creep, yields an expression involving the sum of two terms. The first term of the new solution yields a Bode plot identical to Fig. 1, and the second term of the new solution yields the Bode plot shown in Fig. 2.

bode2.gif
Figure 2. Normalized Bode plot for the creep component of the new solution. The complete response is obtained by adding c w0 times Fig. 2 to Fig. 1.

The equation used to generate Fig. 2 is

| A
Ad
|creep      =        w0 r
  ____________
(1-r2)2+r2/Q2
 
       ,        r  =   w
w0
(6)

3  Consequence of Creep

The difference in shape of figures 1 and 2 gives rise to practical consequences. When driving at frequencies above, or in the vicinity of resonance (r = 1), creep is inconsequential. For frequencies below resonance, creep can yield an increased amplitude response. Observe that in this regime, the fall-off of Fig. 1 is 40 dB/decade, whereas that of Fig. 2 is only 20 dB/decade. If the constant c is not vanishingly small, then the creep could allow signals to be detected at low frequencies, that would otherwise be below noise.

4  Practical matters

This work is still too preliminary to quantify the practical importance of what has been observed. The following figure shows that creep should not be summarily disregarded.

tors-creep.gif
Figure 3. Illustration of creep in an unconventional torsion pendulum.

The instrument in this case was a torsion-gravity pendulum acting as a tiltmeter (remotely similar to a `garden gate' horizontal seismometer). Because the tungsten torsion wire of the pendulum is tied to both the top and bottom of the case, the instrument is tilt sensitive. For that reason the mean position could be easily adjusted using a piezo-translator, since the restoration depends on the Earth's gravitational field as well as the shear modulus of the wire.

It is seen that changes of only a few hundred micro-radians are accompanied by a significant creep response-for the ratio of drive frequency to natural frequency in the vicinity of 0.04. Clearly the creep coefficient of Eq.(5) is for this case too large to be ignored for its influence on the instrument's response at low frequencies.


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On 12 Aug 2005, 15:58.