Vibrations and Waves · Problem 4-15

Coupled oscillators and normal modes — A.P. French

Physics — Waves · resonance-quality-factor

Problem

The free oscillations of a mechanical system are observed to have a certain angular frequency

\omega_1

. The same system, when driven by a force

F_0 \cos \omega t

(where

F_0 =

const. and

\omega

is variable), has a power resonance curve whose angular frequency width, at half-maximum power, is

\omega_1/5

. (a) At what angular frequency does the maximum power input occur? (b) What is the

Q

of the system? (c) The system consists of a mass

m

on a spring of spring constant

k

. In terms of

m

and

k

, what is the value of the constant

b

in the resistive term

-bv

? (d) Sketch the amplitude response curve, marking a few characteristic points on the curve.