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.