Forced oscillations 

This situation corresponds to the system being driven by an external force, different to those acting on what we have called the free oscillating system. For instance if we have a pendulum (mass acted by the tension of a string, gravitational force and friction), if the mass has a charge and, superimposed, there is an horizontal time varying electric field, the pendulum is forced by the electric field and a resonance effect could be induced. In this case, the system is the mass-rope-gravity-friction set and the external force is the one associated to the Electric Field.

Let us have a look to a mass spring system:
In the simulator screen there it is also shown the force (arrow) acting on the mass. The frequency has been selected close to the natural frequency of the system in such a way that a maximum transfer of energy from the external force to the system is achieved. The kinetic, potential and total energy of the system are shown in a separate window (yellow, blue and black lines, respectively). These energies increase without limit because in this ideal experiment no provision for energy losses has been introduced.
The behavior of these systems is represented again by a differential equation, coincident with the one previously shown plus a new term associated to the external force. In case this external force is harmonic at a frequency w, the equation is:
where a 0 =F 0 /m is the amplitude of the external force per unit mass; we have used b (=g/2m) to represent the friction.