Phase space trace of (under-)damped oscillator; third dimension is time.

k=9,m=1

b=slider([0,6,100])

d=b/(2*m)

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

W=sqrt(k/m)

T=2*pi/W

A – amplitude; k – spring constant; m – mass, b – damping constant; d – damping parameter; a – angular frequency; W – natural frequency; f – position; g – velocity

A=1

function(f,x)=A*e^(-(d*x))*cos([a*x])

function(g,x)=-(a*A*e^(-(d*x))*sin([a*x]))-(d*function(f,x))

function(k,x)=X*cos([f*x-P])

function(l,x)=-(f*X*sin([f*x-P]))

vector(x,y)=vector(function(f,n*t)+function(k,n*t),function(g,n*t)+function(l,n*t))

vector(x,y)=vector(function(f,n)+function(k,n),function(g,n)+function(l,n))

vector(prime(x),prime(y),prime(z))=vector(function(f,n*t)+function(k,n*t),function(g,n*t)+function(l,n*t),n*t)


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

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)])

Poincare Section:

prime(z)=m,in(m,set(0,T,2*T,3*T,4*T,5*T,6*T,7*T,8*T,9*T,10*T))


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

Graph of the formula

This file was created by Graphing Calculator 3.5.
Visit Pacific Tech to download the helper application to view and edit these equations live.