A harmonic oscillator obeys
the simple 1D version of Hooke’s law:
to displace the system away from its equilibrium,
the needed force scales linearly with the displacement :
Where is a system-specific proportionality constant,
called the spring constant,
since a spring is a good example of a harmonic oscillator,
at least for small displacements.
Hooke’s law is also often stated for
the restoring force instead:
Let a mass be attached to the end of the spring.
After displacing it, we let it go ,
so Newton’s second law for the restoring force demands that:
leading to the following equation for :
Where is the natural frequency of the system.
This differential equation has the following general solution:
Where and are constants determined by the initial conditions.
For example, for and , the solution becomes:
When using Lagrangian
or Hamiltonian mechanics,
we need to know the potential energy
added to the system by a displacement to .
This equals the work done by the displacement,
and is therefore given by:
If there is a friction force affecting the system,
then the oscillation amplitude will decrease,
or it might not oscillate at all.
We define using a viscous damping coefficient :
Both and are acting on the system,
so Newton’s second law states that:
This can be rewritten in the following conventional form
by defining the damping coefficient ,
which determines the expected behaviour of the system:
The general solution is found from the roots of the auxiliary quadratic equation:
tells us that the behaviour changes substantially
depending on the damping coefficient ,
with three possibilities: or or .
If , there is underdamping:
the system oscillates with exponentially decaying
amplitude and reduced frequency .
The general solution is:
If , there is critical damping:
the system returns to its equilibrium point in minimum time.
The general solution is given by:
If , there is overdamping:
the system returns to equilibrium slowly.
The general solution is as follows,
In the differential equations given above,
the right-hand side has always been zero,
meaning that the oscillator is not affected by any external forces.
What if we put a function there?
Obviously, there exist infinitely many to choose from,
and each needs a separate analysis.
However, there is one type of that deserves special mention,
Where is a constant force, is an arbitrary phase,
and the frequency is not necessarily .
We solve this case for in detail.
Consider the complex version of the equation:
Inserting the ansatz ,
for some constant :
Where has already been divided out.
We isolate this equation for :
We would like to rewrite this in polar form ,
which turns out to be as follows:
For brevity, let us define the impedance
and the phase shift
in the following way:
Returning to the original ansatz ,
we take its real part to find :
Two things are noteworthy here.
Firstly, and are out of phase by ; there is some lag.
This is caused by damping, because if , it disappears .
Secondly, the amplitude of depends on and .
This brings us to resonance,
where the amplitude can become extremely large.
Actually, resonance has two subtly different definitions,
depending on which one of and is a free parameter,
and which one is fixed.
If the natural is fixed and the driving is variable,
we find for which resonance occurs by minimizing the amplitude denominator .
We thus find:
Meaning the resonant is lower than ,
and resonance can only occur if .
However, if the driving is fixed and the natural is is variable,
the problem is bit more subtle:
the damping coefficient
depends on .
This leads us to:
Surprisingly, the damping does not affect , if is given.
However, in both cases, the damping does matter for the eventual amplitude:
leads to ,
and resonance disappears or becomes negligible for .
- M.L. Boas,
Mathematical methods in the physical sciences, 2nd edition,