Plot of response of a damped oscillator to a sinusoidal driving force as a function of time.

k=9,m=1

b=slider([0,10,100])

d=b/(2*m)

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

W=sqrt(k/m)

A=3

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

Underdamped:

A*e^(-(d*x))*cos([a*x])

Envelope:

y^2=A^2*e^(-(2*d*x))

Undamped:

A*cos([W*x])

Critically damped:

B=0

[A+B*x]*e^(-(d*x))

A*e^(-(W*x))

Overdamped:

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

[B*e^(-(c*x))+A*e^(c*x)]*e^(-(d*x))

Full solution:

function(H,x)=branch(if(A*e^(-(d*x))*cos([a*x]),d<W),if([A+B*x]*e^(-(d*x)),d=W),if([B*e^(-(c*x))+A*e^(c*x)]*e^(-(d*x)),d>W))

Driving:

F_0=slider([0,10,20])

f=slider([0,10,50])

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

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

T=slider([0,1])

p=2*pi*T

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


Steady state:

function(H,x)+X*cos([f*x+P-p])

Envelope top & bottom:

e^(-(d*x))

-e^(-(d*x))

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

Graph of the formula

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