Download spring mass, M inertial framework ground motion, y(t) mass position

GS 388 Lab 1
B. L. Isacks
1
Seismographs
SEISMOGRAPHS
The simplest vertical seismograph is shown below:
x=0
x=xo
equilibrium
position:
Mg = Kxo
ground
ground motion, y(t)
seismograph frame
inertial framework
mass, M
mass position, x(t)
spring
The mass M hangs from the frame on a spring with spring constant K. The force applied
to the mass by the spring is proportional to the displacement of the mass, x, measured as shown
relative to the frame. The constant of proportionality is K: Force = K•x. At equilibrium when the
ground is not moving, the mass is at position xo such that the force on the mass due to gravity,
M•g, is balanced by K•xo. When the ground and attached frame move, the mass moves relative
to the frame, and the force due to displacement, x (relative to x=xo) are balanced by the mass
times the acceleration relative to an inertial framework. The motion of the ground relative to the
inertial framework is measured by y, as shown, so the motion of the mass relative to the inertial
GS 388
B.L. Isacks
2
Lab 1 Vertical Seismograph
framework is given by (y - x + constant). Since we are looking at motions relative to equilibrium
values, we can ignore the constant. Thus, substituting in Newton's law,
force = mass • acceleration
we have
d2(y –x)
K (x– xo) = M
=M (y –x)
dt 2
or
x + K (x– xo) = y
M
This is the equation for a simple harmonic oscillator, or a simple linear system with input ground
motion, y, and output mass motion, x. The natural frequency of the oscillator, fo, is
1
fo = 2π
K
M
If the motion of the mass relative to the frame is impeded slightly in proportion to its
velocity, as is characteristic of any real physical system, we have to add a force on the left side of
the first equation equal to a constant times the velocity of the mass relative to the frame. This
will give the damped harmonic oscillator equation.
Imagine sine wave input, y = A sin(2πf), where f is varied. The response, x, will have the
same frequency but will have an amplitude that depends on the input frequency, f. For f>>fo, the
mass will remain nearly stationary and the output x will have nearly the same amplitude as the
input. For f<<fo, the motion is too slow and the mass will tend to move with the frame, so the
output amplitude will be very small. Thus the seismograph is most sensitive to ground motions
with frequencies greater than the natural frequency, fo.
A practical seismograph must measure the motion of the mass relative to the frame. the
early seismographs used mechanical levels to amplify the motion and move a pen writing on
paper (the seismograph record). Optical-mechanical systems (light reflecting from a mirror
moved by the mass and recorded photographically) greatly improved response and sensitivity,
but the major breakthrough was use of electrical techniques to sense the motion of the mass.
Typically a magnet is attached to the mass, and a wire coil attached to the frame, so that motion
of the magnetic’s magnetic field in the coil (or sometimes vice-versa) generates an electrical
signal in the coil, which can then drive a sensitive recording galvanometer. For the past several
decades the electrical signals have been electronically amplified, digitized, and recorded in
digital form, ready for extensive signal processing analyses by modern digital computers.