GraphingCalculator 4;
Window 47 6 862 1440;
PaneDivider 325;
FontSizes 14;
Slider 0 100;
SliderSteps 500;
SliderControlValue 0;
2D.Scale 0.5 0.25 2 1;
2D.BottomLeft -3.265625 -0.875;
2D.GraphPaper 0;
Text "Plot of response of a damped oscillator to a sinusoidal driving force as a function of time.
Version 0.92, 4-24-14


With k=9 and m=1, the natural frequency W=3, so critical damping (W=d) occurs when b=6.
Change range of b to [0,1] to see very light damping.

To do:
(i) Add choice of initial conditions that match in all three cases (x(0)=A v_x(0)=0).";
Color 4;
Expr k=9,m=1;
Color 5;
MathPaneSlider 1;
Expr b=slider([0,10,100]);
Color 8;
Expr d=b/(2*m);
Color 7;
Expr a=sqrt(W^2-d^2);
Color 7;
Expr W=sqrt(k/m);
Color 2;
Expr A_0=3;
Text "A0 – undamped amplitude; k – spring constant; m – mass, b – damping constant; d – damping parameter; a – angular frequency; W – natural frequency; c – overdamped ""frequency""; p = phase constant

Three sets of initial conditions such that all three solutions match at t=0.   

The first set has x(0)=A and phase constant p=0 but does not try to match initial velocities: A=A0, B=0,C=A,D=A,G=0,p=0
The second set has x(0)=A and phase constant p=0 while matching initial velocities, v_x(0)=-dA: A=A0, B=A/2,C=A/2,D=A,G=0,p=0
The third set matches the initial position and velocity of the undamped case, x(0)=A0 and v_x(0)=0: A=A0/cosp, B=(c-d)/2c*A0, C=(c+d)/2c*A0,D=A0,G=d*A0,p=-atan(d/a)";
Expr A=A_0,B=0,C=A_0,D=A_0,G=0,p=0;
Text "Underdamped:";
Color 17;
Expr A*e^(-(d*x))*cos([a*x+p]),x>0;
Text "Envelope:";
Color 17;
Expr y^2=A^2*e^(-(2*d*x)),x>0,a>0;
Text "Undamped:";
Color 17;
Expr A_0*cos(W*x),x>0;
Text "Critically damped (d=W; occurs at b=6 with k=9 and m=1)(putting exponent = -dx in full solution changes critical solution with changing b):";
Color 17;
Expr [D+G*x]*e^(-(W*x)),x>0;
Color 17;
Expr A*e^(-(W*x)),x>0;
Text "Overdamped:";
Color 3;
Expr c=sqrt(d^2-W^2);
Color 17;
Expr [B*e^(-(c*x))+C*e^(c*x)]*e^(-(d*x)),x>0;
Text "Full solution:";
Expr function(H,x)=branch(if(A*e^(-(d*x))*cos([a*x+p]),d<W),if([D+G*x]*e^(-(d*x)),d=W),if([B*e^(-(c*x))+C*e^(c*x)]*e^(-(d*x)),d>W));
Text "Driving:";
Color 2;
Expr F_0=slider([0,10,20]);
Color 4;
MathPaneSlider 10;
Expr f=slider([0,10,100]);
Color 3;
Expr function(F,x)=F_0*cos([f*x-p]),x>0;
Color 6;
Expr X=[F_0/m]/sqrt([W^2-f^2]^2+4*d^2*f^2);
Color 5;
Expr P=atan([2*d*f/(W^2-f^2)]);
Color 3;
Expr T=slider([0,1]);
Color 8;
Expr Q=2*pi*T;
Text "F0 – magnitude of driving force; f – angular frequency of driving force; Q – phase of driving force; X – amplitude of driven oscillation; P – phase of driven oscillation";
Text "
Steady state, transient piece, and driving force:";
Color 17;
Expr function(H,x)+X*cos([f*x-P-Q]),x>0;
Color 17;
Expr function(H,x),x>0;
Color 17;
Expr function(F,x),x>0;
Text "Animated:";
Grain 1;
Expr vector(n*t,function(H,n*t)+X*cos([f*n*t-P-Q]));
Color 3;
Expr vector(n,function(H,n)+X*cos([f*n-P-Q]));
Text "Envelope top & bottom:";
Color 17;
Expr A*e^(-(d*x)),x>0,a>0;
Color 17;
Expr -(A*e^(-(d*x))),x>0,a>0;
Text "

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