Tilt sensitivity of swinging-gate horizontal sensor.

For a vertical pendulum, the period for small displacements is 2 PI sqrt(L / g), where g is the acceleration of gravity, and L is the length of the pendulum.  In a swinging gate sensor, the hinge tilts forward by an angle theta.  The mass is constrained to move in a plain with a tilt of theta.  The "down-hill" gravitational force is g sin(theta), so the period becomes:

2 PI sqrt(L / (g sin(theta))).

If the horizontal sensor is tilted to the side by an angle beta, then this changes the direction that the "hinge" is leaning, and thus the center position of the boom.  The angular rotation of the boom due to a tilt, beta, is:

 arc tan(sin(beta)/tan(theta)). 

 The displacement of the mass is thus:

L sin(beta)/tan(theta).

Acceleration of gravity, g = 980 cm/s**2

Effect of boom length on hinge tilt and sideways tilt sensitivity
Period fixed at 10 s    
SidewaysTilt,
beta = .0231 deg
   
   
 

Hinge tilt

L sin(beta)/tan(theta)
Boom Length L (cm) theta (deg) Disp from sidetilt (cm)
1 0.0231 1.0008
5 0.1154 1.0008
10 0.2308 1.0008
15 0.3462 1.0008
20 0.4616 1.0008
25 0.5770 1.0008
30 0.6924 1.0007
35 0.8079 1.0007
40 0.9233 1.0007
45 1.0387 1.0007
50 1.1541 1.0006
55 1.2696 1.0006
60 1.3850 1.0005
65 1.5004 1.0005
70 1.6159 1.0004
75 1.7313 1.0004
For a given seismometer period, sideways-tilt sensitivity does not vary with boom length.
Variable Period for Three Different Boom Lengths
Sideways Tilt, beta = 0.0231 deg  
  2 PI sqrt(L / g) 2 PI sqrt(L / (g sin(theta)))     L sin(beta)/tan(theta)
Boom Length L (cm) Vertical Period (s) Period (s) sin theta theta (deg) Disp from tilt (cm)
1 0.2007 2 0.0101 0.5770 0.0400
1 0.2007 5 0.0016 0.0923 0.2502
1 0.2007 10 0.0004 0.0231 1.0008
1 0.2007 15 0.0002 0.0103 2.2518
1 0.2007 20 0.0001 0.0058 4.0033
1 0.2007 25 0.0001 0.0037 6.2551
1 0.2007 30 0.0000 0.0026 9.0074
10 0.6347 2 0.1007 5.7801 0.0398
10 0.6347 5 0.0161 0.9233 0.2502
10 0.6347 10 0.0040 0.2308 1.0008
10 0.6347 15 0.0018 0.1026 2.2518
10 0.6347 20 0.0010 0.0577 4.0033
10 0.6347 25 0.0006 0.0369 6.2551
10 0.6347 30 0.0004 0.0256 9.0074
75 1.7382 2 0.7553 49.0539 0.0262
75 1.7382 5 0.1209 6.9413 0.2484
75 1.7382 10 0.0302 1.7313 1.0004
75 1.7382 15 0.0134 0.7694 2.2516
75 1.7382 20 0.0076 0.4328 4.0032
75 1.7382 25 0.0048 0.2770 6.2550
75 1.7382 30 0.0034 0.1923 9.0073
   
For a given boom length, the hinge tilt decreases and the sideways tilt sensitivity increases as the period is increased.

Back