Modeling a system
Visual Cue  Narration 

Show first slide 
In this tutorial, we will see how to model a physical system using the ODE representation of the system and then convert it to a more familiar transfer function representation in Scilab. 
Show slide with MassSpringDamper 
The system that we will attempt to model will be a massspringdamper based system. We assume that the values of the mass, the spring constant and the damping coefficient are known to us. 
We know from basic Systems theory that the system can be modeled using the ODE:  
Change slide to show this equation: <math> m\ddot{x}(t) + b\dot{x}(t) + k x(t) = F(t) </math> 
The first element in the expression gives the force due to the mass from Newton's second law. The second element gives the force that is exerted in order to move the damper. The third element gives the force that is exerted by a spring that is displaced from its mean position. 
By taking the Laplace transform of this equation, we get the following equation:  
<math> m s^2 X(x) + b s X(s) + k X(s) = F(s) </math> 

which can be simplified by rearranging the terms around the equality to:  
<math> \frac{X(s)}{F(s)} = \frac{1}{m s^2 + b s + k} </math> 

We use a scaling factor of k to obtain our transfer function:  
<math> G(s) = \frac{k}{m s^2 + b s + k} </math> 

Switch to Scilab and enter the following values: m = 1 b = 10 k = 20 s = %s 

Our system is represented in the transfer function form as:  
System = k/(m*s^2 + b*s + k) Switch to presentation 

We compare the system with the standard representation of a second order model. Using this comparison, we can obtain the values of the natural frequency omega_n and damping factor zeta. We compute these in Scilab.  
Switch to Scilab: wn = sqrt(k/m) zeta = (b/m)/(2*wn) 
We note that this system is overdamped since zeta is greater than one. 
We have successfully modeled this system in Scilab. Now, we must analyze this sytem to understand its characteristics and to decide how we must proceed in order to control the system and obtain the response we desire. 