Numerical Iintegration In MATLAB And Working  Model 2D Assessment

The System 

Figure 1: Spring-mass-damper system

Recall the equation for the system in Figure 1. This will be used as a basis for the MATLAB program.

Also recall the representation of this in terms of damping coefficient: 

Use m = 1 kg, k = 40 N/m. Set the initial value for y,y₀ = 0.1 m. Evaluate b for the critically damped  system. For the Working Model version, make the spring at least 0.2 m long and ensure the mass  is clamped securely to it in the model. 

Some useful information 

Frequency can be expressed in radians/sec (w), or Hz (f), where: 

and the period of oscillation is given by: 

With no damping, b = 0, the system oscillates at the natural frequency given by w_n=(k/m)^1/2When damped, provided it is underdamped, it oscillates at the damped natural radiancy

,

or for frequency: 

where f_n is the damped natural frequency and

The assignment 

Your task is to ascertain how well Euler integration can show this and how this compares to a more  complicated numerical integration method as used in Solidworks motion. Use the Pendulum  example on myLearning as an example of how to generate the report for the comparison. 

In the Solidworks motion settings, set the animation frames/s to match that of the time step used in  MATLAB so that you can make a fair comparison. Also look at the integrator values. 

From your simulations, make a table of results so that you can compare the results for over-damped  response. Model values of damping ratio (b) from 0 to 12 kg/s and determine the period of oscillation (T) for each in both the MATLAB Euler integration and Solidworks (SW). From this you can evaluate  damped frequency and calculate the analytical (f_da) from equation  (6). Plot these (frequency versus damping ratio) on a graph. 

Evaluate the differences between 1) the analytical and MATLAB and 2) the analytical and SW. Using  statistical analysis determine which is more accurate and whether the differences are statistically  significant (you will need to use Minitab for this). 

You might want to create a spreadsheet of the results in order to plot f_d against b. It might look  something like this: