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).

  

Figure 5: Damped harmonic oscillator

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 $\ddot y(t)$ 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:

$$M \ddot y(t) = f(t) - S z(t) -R \dot z(t)$$ (24)

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

$$M \ddot z(t) + R \dot z(t) + S z(t) = f(t) - M \ddot x(t)$$ (25)

We observe that an acceleration $\ddot x(t)$ of the ground has the same effect as an external force of magnitude $f(t) = -M \ddot x(t)$ acting on the mass in the absence of ground acceleration. We may thus simulate a ground motion x(t) by applying a force $-M \ddot x(t)$ 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.