Laplace transform interactive demo

Use this demo to play around with a common problem addressed through Laplace transforms - a damped, oscillating system.

Pick a physical scenario, adjust the parameters, and compare the external forcing with the response of the system

$$m x'' + c x' + k x = F(t).$$

Focus on what changes when you alter the damping, stiffness, timing, and forcing type. The aim is to help you see why switched, delayed, and pulsed inputs are much easier to manage with Laplace transforms compared to solving the problem in the time domain.
Quick preset

Choose the system
0.2 5
0 8
0.5 25
-3 3
-5 5
3 20

Choose the input
Forcing type
-10 10
0 10
0.1 10
0.1 3

Try these experiments

  1. Keep the same forcing and change only the damping. Watch the oscillations shrink or disappear.
  2. Keep the same system and switch between a step, a delayed step, and a pulse.
  3. Move the sine-burst frequency closer to the system's natural timescale and compare the size of the response.
  4. Turn the forcing off and see what initial conditions alone can do.

Main lesson

You can treat all of these cases in one organised framework. Laplace transforms turn differentiation into algebra, keep the initial conditions visible, and handle delayed inputs with factors like $e^{-cs}$.