Vibrations and Waves · Problem 4-6

Coupled oscillators and normal modes — A.P. French

Physics — Waves · seismograph-forced-oscillations

Problem

Imagine a simple seismograph consisting of a mass

M

hung from a spring on a rigid framework attached to the earth, as shown. The spring force and the damping force depend on the displacement and velocity relative to the earth's surface, but the dynamically significant acceleration is the acceleration of

M

relative to the fixed stars. (a) Using

y

to denote the displacement of

M

relative to the earth and

\eta

to denote the displacement of the earth's surface itself, show that the equation of motion is

\frac{d^2y}{dt^2} + \gamma \frac{dy}{dt} + \omega_0^2 y = -\frac{d^2\eta}{dt^2}

(b) Solve for

y

(steady-state vibration) if

\eta = C \cos \omega t

. (c) Sketch a graph of the amplitude

A

of the displacement

y

as a function of

\omega

(supposing

C

the same for all

\omega

). (d) A typical long-period seismometer has a period of about 30 sec and a

Q

of about 2. As the result of a violent earthquake the earth's surface may oscillate with a period of about 20 min and with an amplitude such that the maximum acceleration is about

10^{-9}

m/sec

^2

. How small a value of

A

must be observable if this is to be detected?

Related theory

Forced Oscillations

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