From: S-T Morrissey sean@...........

Date: Tue, 15 Feb 2000 13:59:41 -0600 (CST)

Dewayne, You ask about the proper angle of the boom of a horizontal seismometer. This has to do with the restoring force that makes it swing back to center. This force is a portion of gravity, namely g*sine(i), where i is the angle with respect to the horizontal, measured in radians. In the case of a SG, the boom is vertical, but horizontal in the Lehman design. There it is generally assumed that the boom support is adjusted so the boom is parallel to the base, so the base angle adjusts the period. IF the boom is exactly horizontal, it will experience NO gravitational restoring force, so it will swing back and forth aimlessly. If the mass end is lowered slightly, it will swing in a shallow curve in the gravitational field, with the minimum being at the bottom or center of the swing. Here is a repeat of the calculation of the actual numbers you can expect: SO ... Lets consider some formulae of interest for the horizontal pendulum: (assuming that the restoring force by the hinges and/or pivot are minimal): The natural period: Tn = 2*pi*sqrt(L/(g*sine i)) where L is the boom length in cm, g=980cm/sec^2, i is the angle that the boom makes wrt the horizontal, (if i is measured in radians, (360 degrees = 2*pi radian, or 1 radian = 57.3 degrees), and i is small, sine i = i). For example, a 40 cm boom hanging vertically (mass at the bottom) as a simple tick-tock pendulum ( an SG design at an angle of 90 degrees) has a period of 1.3 seconds. (a one second clock pendulum is 24.8 cm). But the pendulum supports or hinges can be arranged in a "garden gate" configuration as is the Lehman and most long-period horizontals. When tilted horizontally to about 4 degrees, the period is 5 seconds. At about a 1 degree ((2*pi/360) radian) angle, it is 10 seconds, and at about 0.23 deg. it is 20 seconds. However, if we increase the boom length by times 4 to 160cm, (an impractical 60 inches), we also get a period of 20 seconds. So the period is changing with the square root of the boom length as well as the inverse of the square root of the angle the boom makes with the horizontal. In general, a practical boom length is 10" to 15", with a baseplate of 15" to 24" long and about half as wide at the leveling end (for a horizontal; leveling for a vertical, as shown above, is nowhere as critical, and 4" to 8" widths are workable). It is important to note that the size of the mass determines nothing of the period or sensitivity to tilting. Any reasonable size will work; larger is better for overcomming any torque of the hinges or flexures, to the point where the mass/boom structure begins to distort any part of the suspension. (THe size of the mass IS a factor in a VBB fedback system). The total mass of the boom should be less than 10% of the main mass, which includes the sensor coils. The tilt sensitivity of a seismometer is therefore a function of the square of the operating period Tn. For a HORIZONTAL: (where z is the displacement, and phi is the tilt) z = (g * Tn^2 / 4 * pi^2) * phi For a VERTICAL: z = (g * Tn^2 / 8 * pi^2) * phi^2) (vertical) Note the vertical sensor responds to the SQUARE of the tilt. BUT .. Since the angle is always small and less than 1, the square of a small angle (measured in radians) is smaller than the original number. So conversely the horizontal is MORE sensitive to tilt of the base (at a right angle to the boom) by the square of the tilt angle. SO what does this mean in comparing the tilt noise of a vertical compared with a horizontal of the same period. Suppose the seis is in a corner of the garage or basement. Then suppose that when you walk up to the site you deflect the floor by 1 micron (10^-6 meter) when you are 1 meter away (Or your neighbor parks his Humvee 100 meters away and deflects the neighborhood by 0.1 millimeter.) In both cases the tilt is delta(L)/L or 10^-6 radian. So if your horizontal seis has a period of 10 seconds, the mass will offset 24.8 microns. This is a large number; the 6-second microseisms run about 2 to 4 microns. HOWever: if you have a vertical seis, the displacement from a 10^-6 tilt is 10^-6th of the horizontal. Conversely, it takes a floor deflection if 1mm at 1 meter distance to get the same 24.8 microns movement on a vertical. So when you push the operating period from 10 to 20 seconds, the tilt sensitivity increases by 4. Even a modest period increase demands a good site for the instrument. The WWNSS (worldwide network of standard seismographs) originally tried to operate the long period sensors at 30 seconds, but so many were always at the stops that they backed off to 15 seconds as the standard. You can test the tilt sensitivity using your leveling screw, which I guess is something like 40 threads/inch. In the above formula, "phi" is the angle in radians, so if you turn the screw one turn and if the base support width is 10 inches, the tilt is 1/40" divided by 10", or 0.0025 radians. Using the formula above for a horizontal sensor: The displacement then is 0.062cm times the square of the period. If Tn is 10 seconds, it is 6.2cm; if Tn is 100 seconds, one turn of th 40 tpi screw will try to move the boom 620 cm. Even if 1/100 turn can be used, the displacement is still quite large. THis is why VBB instruments have a feature that allows a shorter period to be switched in for setup at installation, and a motor-driven lever of 100:1 to level the sensor in operation. A typical tilt noise level for the 360-second STS-1 is equivalent to about 6 nanoradians. For practical operation of a home-made VBB, an operating period of 20 to 40 seconds would be preferred. Regards, Sean-Thomas _____________________________________________________________________ Public Seismic Network Mailing List (PSN-L)

Larry Cochrane <cochrane@..............>