Free oscillations

To understand the behavior of a system at resonance we have to know how this system behaves when it is oscillating in a free way, that is to say, without any external force acting on it.

Let us have a look to the most simple system: The harmonic oscillator composed of a mass linked to a spring in an horizontal plane. If we denote by x the position of the mass relative to that corresponding to the equilibrium one, the equation describing the evolution of the system is:

In this differential equation k is the spring constant, m the mass connected to the spring and g represents the friction (losses) that the environment exerts on the vibrating mass. The equation states that the acceleration of the mass is (1/m) times the force from the spring, proportional to the displacement of the mass relative to its equilibrium position, plus a resistive force proportional to the velocity of the mass.

Click the START button to start an animation corresponding to an undamped oscillation (zero friction).

Look at the time diagram r[x] to have a look to the time evolution of the position, x, of the particle: a sinusoidal curve.
Play with the simulation, change the mass of the particle and observe the result.