A harmonic oscillator is any physical system that varies above and below its mean value with a characteristic frequency, f. Common examples of harmonic oscillators include pendulums, masses on springs, and RLC circuits.
Most harmonic oscillators, at least approximately, solve the differential equation:
where t is time, b is the damping constant, ωo is the characteristic angular frequency, and Aocos(ωt) represents something driving the system with amplitude Ao and angular frequency ω. x is the measurement that is oscillating; it can be position, current, or nearly anything else. The angular frequency is related to the frequency, f, by:
A simple harmonic oscillator is simply an oscillator that is neither damped nor driven. So the equation to describe one is:
The above never actually exists, since there will always be friction or some other resistance, but two approximate examples are a mass on a spring and an LC circuit.
In the case of a mass hanging on a spring, Newton's Laws, combined with Hooke's law for the behavior of a spring, states that:
where k is the spring constant, m is the mass, y is the position of the mass, and a is its acceleration. Rewriting the equation, we obtain:
The easiest way to solve the above equation is to recognize that when d2z/dt2 ∝ -z, z is some form of sine. So we try the solution:
where A is the amplitude, δ is the phase shift, and ω is the angular frequency. Substituting, we have:
and thus (dividing both sides by -A cos(ωt + δ)):
The above formula reveals that the angular frequency of the solution is only dependent upon the physical characteristics of the system, and not the initial conditions (those are represented by A and δ). That means that what was labelled ω is in fact ωo. This will become important later.
AC LC circuit.
a few notes about what the response of the circuit to different AC frequencies.
weighted spring underwater
Note well: underdamped, critically damped
Notes for above apply, transient vs steady state response, and quality factor.
For a more complete description of how to solve the above equation, see the article on Differential equations.