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## The mechanical pendulum

The simplest physical model for the mechanical part of an inertial seismometer is a mass-and-spring system (a spring pendulum) with viscous damping (Fig. 5).

We assume that the seismic mass is constrained to move along a straight line, without rotation (i.e. it performs a pure translation). The mechanical elements are a mass of M kilograms, a spring with a stiffness S (measured in Newtons per meter), and a damping element with a constant of viscous friction R (in Newtons per meter per second). Let the time-dependent ground motion be x(t), the absolute motion of the mass y(t), and its motion relative to the ground z(t) = y(t)-x(t). An acceleration of the mass results from any external force f(t) acting on the mass, and from the forces transmitted by the spring and the damper:

 (24)

Since we are interested in the relationship between z(t) and x(t), we rearrange this into

 (25)

We observe that an acceleration of the ground has the same effect as an external force of magnitude acting on the mass in the absence of ground acceleration. We may thus simulate a ground motion x(t) by applying a force to the mass while the ground is at rest. The force is normally generated by sending a current through an electromagnetic transducer, but it may also be applied mechanically.

Next: Transfer functions of mechanical Up: Basic Theory Previous: The transfer function of
Erhard Wielandt
2002-11-08