Matlab tutorial

Mass-Spring system

Here is a mass-spring system example. The procedure for deriving equation of motion for 1 degree of freedom mass-spring system will be described. The purpose of this example is to let you be familiar with setting up the equation of motion and solve for the motion of the system using Matlab.

Problem Description

The below is the drawing of the mass-spring system with FreeBody Diagram. We will solve for the homogeneous (transient) part of the response of the system, which means the external excitation force is zero. We will define the natural frequency of the system. After that, we will set up equation of motion (with FBD and force balance equation), get the closed form of the solution and try to solve it using Matlab.

Freebody Diagram

Deriving the Equation of Motion and Closed Form Solution

We can write down the equation of motion for this 1 degree of freedom system with FBD above using force balance. Click here for details.

Matlab Codes

Here is Matlab code for the closed form solution.

m = 2;                      %  mass of the system in meter
k = 5000;                   %  stiffness of the spring (N/m)
wn = sqrt(k/m);             %  natural frequency (rad/sec)

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.

xh_0 = 1;                    %  initial displacement for homogeneous 
solution
xh_dot_0 = 0;                %  initial velocity for homogeneous 
solution

Then, initial conditions are followed.

X_0 = sqrt(xh_0^2+(xh_dot_0/wn)^2);
phi = acos(xh_0/X_0);

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

t = 0:t_final/n:t_final;
x = X_0*cos(wn*t+phi);           %  displacement x(t)
x_dot = -wn*xh_0*sin(wn*t+phi);  %  velocity x_dot(t)

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

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

Results

The displacement x(t) and the velocity v(t) are shown.