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Harmonic oscillator
Introduction
A harmonic oscillator is a special type of physical system that varies above and below its mean value with a characteristic frequency, f . The system must satisfy that the returning force is proportionate to the displacement, i.e.
Lagrangian is in the form of
Hamiltonian is
periodically is a harmonic oscillator. One important property of a harmonic oscillator is that the period of the oscillation is independent of the oscillation amplitude .
Common examples of harmonic oscillators include pendulums (in small angles), masses on springss , and RLC circuits .
The following article discusses the harmonic oscillator in terms of classical mechanics . See the article quantum harmonic oscillator for a discussion of the harmonic oscillator in quantum mechanics .
Full mathematical definition
Most harmonic oscillators, at least approximately, solve the differential equation:
t is time, b is the damping constant, ωo is the characteristic angular frequency , and A o cos(ωt ) represents something driving the system with amplitude A o 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:
Important terms
Amplitude : maximal displacement from the equilibrium .
Period : the time it takes the system to complete an oscillation cycle.
Frequency : the number of cycles the system performs per unit time (usually measured in Hertz = 1/s).
Angular frequency :
Phase : how much of a cycle the system completed (system that begins is in phase zero, system which completed half a cycle is in phase ).
Initial conditions: the state of the system at t = 0, the beginning of oscillations.
Simple harmonic oscillator
A simple harmonic oscillator is simply an oscillator that is neither damped nor driven. So the equation to describe one is:
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:
k is the spring constant, m is the mass, y is the position of the mass, and a is its acceleration. Noting that acceleration is the second derivative of position, we can rewrite the equation as follows:
d 2 z /dt 2 ∝ -z , z is some form of sine. So we try the solution:
A is the amplitude , δ is the phase shift, and ω is the angular frequency . Substituting, we have:
A cos(ωt + δ)):
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.
Driven harmonic oscillator
Satisfies equation:
AC LC circuit.
a few notes about what the response of the circuit to different AC frequencies.
Damped harmonic oscillator
Satisfies equation:
weighted spring underwater
Note well:
underdamped, critically damped
Damped, driven harmonic oscillator
equation:
transient (the solution for damped undriven harmonic oscillator, homogenous ODE) and the steady state (particular solution of the unhomogenous ODE).
The steady state solution is
impedance
phase of the oscillation.
One might see that for certain frequency the amplitude (relative to a given ) is maximal. This occurs for the frequency
resonance of displacement
example:
RLC circuit
Notes for above apply, transient vs steady state response, and quality factor .
A final note on mathematics
For a more complete description of how to solve the above equation, see the article on differential equations .
See also: normal mode .
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