http://people.csail.mit.edu/jaffer/Seismo

# Buoyancy Seismometer

The essential elements of a vertical seismometer are a weight and a spring. Vertical seismometers are more difficult to build than horizontal ones because the spring must support the mass. The lower spring constants which give longer time constants require longer springs to support a given weight.

Springs run the gamut from coils of steel to magnetic levitation. The crucial property of the spring is that the force it exerts is proportional to the amount it is stretched.

Buoyancy seems to have been overlooked as a vertical spring in this application. Parallelepiped objects with uniform density exert force proportional to their elevation away from equilibrium.

yL
The depth of liquid in the tank.
yM
is the submerged depth of the block.
HM
is the total height of the block.
AM
is the cross-sectional area of the block.
AL
is the area of the tank.
DM
is the density (weight per volume) of the block.
DL
is the density of the liquid.
The volume of liquid being constant constrains the variables thus:
 yL · (AL - AM) = AM · yM

In equilibrium the weight of the displaced liquid must equal the total weight of the block:
 yL · DL · (AL - AM) = HM · DM · AM
 yM · DL = HM · DM

The force acting on the block by the liquid when the block is offset d vertically from equilibrium is the weight of that liquid being displaced:
 Fd = d · AM · DL · g

The spring constant k is then this force per the displacement d:
 k = AM · DL · g

From an initial displacement d the acceleration of the block will be:
 d · DL · g HM · DM

We want the block to have high inertia, to be accelerated little; it is the motion of the seismometer housing that is measured. So the block's density should be slightly less than the density of the liquid; but more importantly, the block should be as tall as possible.

A uniform density block which is taller than it is wide tends to tip toward horizontal. To prevent this I ballasted the bottom of the block with a weight. Although the density of the combination is not uniform, it would only affect the equations were the block to be displaced by its full height.

If outfitted with position transducers and angle encoders this arrangement is capable of 6-axis seismic sensing. The vertical and lateral distances of the block from points on the housing encode movement along those three axes. The angles between the block and the housing sense rotation around those axes. The ballast is responsible for the restoring torques around the lateral axes.

The photograph shows a laser-projection vertical seismometer constructed using these principles. A ballasted block of pressure-treated wood floats in a tub of water snugly fit into a wooden box.

Connected to that block and the seismometer housing are stiff wire yokes encircling the laser housing near its center of gravity. These yokes transmit vertical movement into angular motion of the laser. A wire clip and cardboard baffle keep the yokes from wandering out of position.

Flexible coils of thin wire connect the battery holder terminals to the laser. The white sheets above and below the view meet to form a cardboard wind-screen.

With the wooden box clamped to a steel post supporting a (foot) bridge between two buildings, the laser spot projected on a wall was seen to deflect a couple degrees when people walked through the bridge.