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: