As you have probably noticed in our various demos
most mechanical systems do not continue oscillating forever
and this is because friction gradually transforms the
mechanical energy into internal energy such as heat.
The result is a steady decay in the amplitude of oscillation.
In general it can be complicated to cope with dissipation in
oscillating systems. We shall consider the simple case of
a dissipative force which is proportional to but opposed to the
velocity.
Where b is a constant with units Ns/m which controls how much dissipation.
Note that because of the minus sign this force always dissipates
energy since it is always opposed to the direction of motion.
In fact we have that the power dissipated is
Including the dissipative force, 
, in Newton's second law we obtain a differential equation of the form
To solve this differential equation it is most convenient
to use complex numbers. If you have not yet learned about
them don't despair. I show you this to make you aware
of how elegantly math and physics work together in the theory
``linear'' systems.
The most elegant way to solve this differential equation
is to basically guess a solution of the form
where
Here 
and z are constants to be determined and 
signifies taking the real part of the complex
number 
since I am allowing for z to be
a complex number. To establish conditions on and z I insert
rather than x(t) in Eq. 16. This
simplifies the calculations significantly and still ensures that
x(t) is a solution. I first
calculate derivatives of
These results are then inserted in Eq. 16 to give
Thus I have converted the differential equation into a simple second
order polynomial whose solutions are
Where I introduced
The corresponding functional form for x(t) depends on whether
z is complex or real. Specifically
The corresponding expressions for x(t) are
where
is the modified resonance frequency of the damped system. The first solution is simply a damped harmonic oscillation. We still observe
oscillations albeit with an envelope amplitude function which
decreases with time:
Sometimes we introduce a life-time, 
(not to be confused
with torque!) through
in terms of which
Note that he mechanical energy decreases with a life-time,
The other solution is an over-damped solution which shows no discernible oscillations before the mechanical energy is completely dissipated .