## 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