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MatLab Spring Damping and Energy

  1. Dec 4, 2006 #1
    This is the entire problem. I am looking for help with part 3.

    For the mass-spring system with damping, we obtain the following DE
    my′′ + gammay′ + ky = 0, or y′′ + cy′ + omega^2y = 0, where c = gamma/m and omega^2 = k/m. Given m = 1,k = 4, gamma= 1, and initial conditions y(0) = 0.1, y′(0) = 0.
    (1) Plot solutions y and v = y′ = dy/dt as functions of time.
    (2) Plot v vs y (phase plot). Comment on the behavior of the curve.
    (3) Plot the quantity E = 1/2mv^2 + 1/2ky^2 as a function of time. What do you observe?
    (4) Show that dE/dt < 0 for gamma > 0 while dE/dt > 0 for gamma < 0 (analytically or by graphic).


    For part 1
    function Q2part1
    m = 1; % mass [kg]
    k = 4; % spring constant [N/m]
    gamma = 1; % friction coefficient [Ns/m]
    omega = sqrt(k/m); c = gamma/m;
    y0 = 0.1; v0 = 0; % initial conditions
    [t,Y] = ode45(@f,[0,10],[y0,v0],[],omega,c); % solve for 0<t<10
    y = Y(:,1); v = Y(:,2); % retrieve y, v from Y
    figure(1); plot(t,y,'b+-',t,v,'ro-'); % time series for y and v
    %-------------------------------------------
    function dYdt = f(t,Y,omega,c)
    y = Y(1); v = Y(2);
    dYdt = [ v ;-c*(v)-omega^2*y];

    for part 2

    function Q2part2
    m = 1;
    k = 4;
    gamma = 1;
    omega = sqrt(k/m); c = gamma/m;
    y0 = 0.1; v0 = 0;
    [t,Y] = ode45(@f,[0,10],[y0,v0],[],omega,c);
    y = Y(:,1); v = Y(:,2);
    figure(1); plot(v,y);
    function dYdt = f(t,Y,omega,c)
    y = Y(1); v = Y(2);
    dYdt = [ v ;-c*(v)-omega^2*y];


    I know that i need to plug in formulas for y (and v as well?) so that I have E(t), but is it as simple as solving for and plugging in y? I am in need of direction here I suppose...a nice big fat hint to put me in the correct direction.:biggrin:
     
  2. jcsd
  3. Dec 4, 2006 #2
    I don't know which part you need help with, but if you are on part three you should look into a contour plot.

    For example, here is a way to do an energy contour of a undamped oscillator with a mass of 4.5kg and a spring constant of 2N/m:

    [x y]=meshgrid(-1:0.01:1,-1:0.01:1);

    m=4.5;
    k=2;

    contour(x,y,((1/2)*m*y.^2)+((1/2)*k*x.^2),4)
    title('Energy Contour Plot for Undamped Oscillations')
    ylabel('Velocity')
    xlabel('Position')
     
  4. Dec 4, 2006 #3
    Oh whoops, they asked for it as a function of time... which I don't quite understand. Maybe you could overlay the potential energy vs. time and the kinetic energy vs. time onto one plot.
     
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