If you have just started learning Simulink, one of the easiest tasks is solving a simple ordinary differential equation. In fact, most of the beginning guides you will find through web searches will probably be similar to the example I am going to provide. Let’s start by assuming you have the following common spring-damper system. For reference, *m* is mass, *c* is the damper coefficient, *k* is the spring coefficient, *x* is the position, *x-prime* is the velocity, *x-double-prime* is the acceleration, and *f(t)* is a step-input function with a magnitude of 3.

We begin first by solving for the second derivative of *x*. In this case, it ends up solving to:

Now, it is time to place this into Simulink using the following blocks:

- 2 integrator blocks
- 3 gain blocks
- 1 sum block
- 1 step input block
- 1 scope output block

Step 1) Connect two integrator blocks together to simulate a double integration as seen below:

Step 2) Add the appropriate gain blocks to simulate *c*x’* and *kx*.

Step 3) Add the sum block to simulate *f(t) – cx’ – kx*.

Step 4) Add the gain block after the summation to simulate the multiplication of *(1 / m)* and the step input function as the third input to the sum block added in Step 3. Make sure the step input function has the properties of **Step Time = 0, Initial Value = 0, Final Value = 3**.

Step 5) Add the scope block for output after the second integration to view the plotted contents of the numerical solution.… Continue reading