MATLAB tutorial

Closed Form Solution

The following is the Matlab code for closed form solution for a brake pedal problem.
x is theta and xdot is theta_dot.

clear all;

%
%   necessary input constant 
%

N = 5000;
m = 5;
k0 = 1000;
c0 = 0.7;
k = 500;
l = 0.3;
a = 0.1;
I = m/12*(2*a)^2;           

wn = sqrt((k0+k*a^2)/I)
zeta = c0/(2*wn*I)
wd = wn*sqrt(1-zeta^2);

%  wd is the damped natural frequency

t_final = 0.5;              %  calculation time
n = 5000;                   %  number of data points

At the beginning of the program, we need to write some constant inputs. n is the number of the data points.

x_0 = Need to fill this in;   %  initial displacement for homogenous 
                              %  solution
xdot_0 = Need to fill this in;                    
                              %  initial velocity for homogeenous 
                              %  solution

Then, initial conditions are followed.

X_0 = sqrt(x_0^2+((xdot_0+zeta*wn*x_0)/wd)^2);
phi = acos(x_0/X_0);

%  X_0 is the amplitude of response and
%  phi is the phase angle of the response

%  Remember that this is about complimentary (homogeneous) solution
%  therefore, only thing you have to include is initial conditions

Calculate the amplitude and the phase angle with given initial conditions.

t = 0:t_final/n:t_final;
x = X_0*exp(-zeta*wn*t).*cos(wd*t+phi);  %  displacement x(t)
xdot = X_0*exp(-zeta*wn*t).*(-zeta*wn*cos(wd*t+phi)-wd*sin(wd*t+phi));
                                         %  velocity xdot(t)

And we can determine the displacement x(t) and velocity xdot(t).

Click here for the entire Matlab source code for closed form solution.