Few weeks back SolidThinking Activate (equivalent to Matlab Simulink) is released under its math and system modelling portfolio. Being a Simulink user thought of exploring it with the classical spring and damper system, a single input single output (SISO) system.

**The system**

Image source: wiki

Where,

m = 1 kg;

k = 1 N/m;

c = 0.2 N / (m/sec)

Governing equations by using Newton’s Third law:

∑F = m. x^{’’}(t)

=> **F(t) – k.x(t) – c.x’(t) = m.x’’(t)** …………………..*eq.1.1*

**Direct modelling of the differential equations**:

At this point, as you know, you can directly model the **eq.1.1 **in ACTIVATE (or in Simulink):

Upon simulation, I got the response (displacement) for unit step functions as input force:

**Modelling of transfer functions**

You can, of course, model the system using the transfer function. And to develop a transfer function of the system, you must take Laplace transform of ** eq.1.1**:

** F(t) – k.x(t) – c.x’(t) = m.x’’(t)**

- F(s) – k.X(s) – c.s.X(s) = m.s
^{2}.X(s) *X(s) / F(s) = 1 / [m.s*^{2}+ c.s + k]…………………eq.1.2

So, we got the transfer function. I used the transfer function in ACTIVATE:

And the simulation response by this method:

Please observe that in both the method, the max displacement value is 1.6 mm. I simulate changing the mass from 1kg to 0.5 kg, max. displacement increased to 1.8mm:

Further I tried decreasing the damping coefficient from 0.2 to 0.1 and found:

If you are an existing matlab simuink user, you will found SoildThinking ACTIVATE effortless. It has nice clean GUI and a help section with some tutorials.