A simple model of a seismometer is shown in Figure 3.60. The top view shows how a pen attached to the seismometer mass traces out a pattern on a paper grid that is on a roller. The side view provides more details of the model. The absolute motion of

the ground is defined by \(y(t)\), the relative motion of the mass recorded by the seismometer is \(x(t)\), and parameter values are \(m=1 \mathrm{~kg}, c=3 \mathrm{~N}\)-s \(/ \mathrm{m}\), and \(k=15 \mathrm{~N} / \mathrm{m}\). The seismometer is initially stationary until ground motion initiates motion, governed by the differential equation,

\[ \ddot{x}+2 \zeta \omega_{n} \dot{x}+\omega_{n}^{2} x=F(t) / m \]

where

\[ F(t)=-\ddot{y}=a \omega_{b}^{2} \sin \omega_{b} t \]

with \(a=15 \mathrm{~mm}\) and \(\omega_{b}=3 \mathrm{rad} / \mathrm{s}\). Derive the response in general, and then for the specific parameter values. Discuss the possibilities of resonance.

\section*{Problems for Section 3.8 - Periodic but Not Harmonic Excitation}

Transcribed Image Text:

6000 m x Top View Side View Figure 3.60: A schematic of a seismograph. Ground shaking oscillates the base, which then shakes the body, the motion of which is recorded on the drum.