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 |
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|
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. | |||||||||||