MATLAB tutorial

Closed Form Solution

The following is the MATLAB code for closed form solution of mass-spring-damper system.


clear all;

m = 2;                      %  mass of the system in meter
k = 5000;                   %  stiffness of the spring (N/m)
c = You need to set this value;  %  damping coefficient (N*s/m)

wn = sqrt(k/m);             %  natural frequency (rad/sec)
zeta = c/(2*wn*m);          %  viscous damping factor
wd = YOU ADD OMEGA_DAMPED HERE !!;

%  wd is the damped natural frequency

t_final = 3;                 %  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 = 1;                    %  initial displacement for homogeenous 
                            %  solution
xdot_0 = 0;                 %  initial velocity for homogeenous solution

Then, initial conditions are followed.

X_0 = CHANGE this line for amplitude ;
phi = CHANGE this line for phase angle ;

%  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.