Plots of the amplitude and phase of the steady-state response of a damped oscillator to a sinusoidal driving force as functions of driving frequency.

k=9,m=1

d=b/(2*m)

a=sqrt(W^2-d^2)

W=sqrt(k/m)

A,B – amplitude; k – spring constant; m – mass, b – damping constant; d – damping parameter; a – angular frequency; W – natural frequency

F_0=slider([0,5,10])

b=slider([0,5,40])

function(X,x,d)=[F_0/m]/sqrt([W^2-x^2]^2+4*d^2*x^2)

y=function(X,x,g),in(g,set(0.4,0.6,1))

y=function(X,x,d)

P=atan([2*d*f/(W^2-f^2)])

function(P,x)=branch(if(atan([2*d*x/(W^2-x^2)]),x<W),if(atan([2*d*x/(W^2-x^2)])+pi,x>W))

prime(y)=function(P,prime(x))

prime(y)=a*pi/2,in(a,set(1,2))

R=sqrt(W^2-(2*d^2))

prime(x)=W

x=W

x=R

F0 – magnitude of driving force; f – frequency of driving force; X – amplitude of driven oscillation; P – phase of driven oscillation; R – resonant frequency


Author: David A. Craig <http://web.lemoyne.edu/~craigda/>

Graph of the formula

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