Harmonic oscillator

harmonic oscillatorsharmonic oscillationdamped harmonic oscillatorharmonicsimple harmonic oscillatorclassical harmonic oscillatorSpring-mass systemdriven harmonic oscillatorvibration dampingdamped
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x: :where k is a positive constant.wikipedia
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Oscillation

oscillatorvibrationoscillators
If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).
The systems where the restoring force on a body is directly proportional to its displacement, such as the dynamics of the spring-mass system, are described mathematically by the simple harmonic oscillator and the regular periodic motion is known as simple harmonic motion.

Clock

timepiecemechanical clockanalog clock
Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as clocks and radio circuits.
The timekeeping element in every modern clock is a harmonic oscillator, a physical object (resonator) that vibrates or oscillates at a particular frequency.

Pendulum

pendulumssimple pendulumpendula
Mechanical examples include pendulums (with small angles of displacement), masses connected to springs, and acoustical systems.
For small swings the pendulum approximates a harmonic oscillator, and its motion as a function of time, t, is approximately simple harmonic motion:

RLC circuit

LRC circuitRLCRLC circuits
Other analogous systems include electrical harmonic oscillators such as RLC circuits. This type of system appears in AC-driven RLC circuits (resistor–inductor–capacitor) and driven spring systems having internal mechanical resistance or external air resistance.
The circuit forms a harmonic oscillator for current, and resonates in a similar way as an LC circuit.

Simple harmonic motion

simple harmonic oscillatorharmonic motionharmonic
If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).
In the diagram, a simple harmonic oscillator, consisting of a weight attached to one end of a spring, is shown.

Relaxation (physics)

relaxation timerelaxationrelax
In physics, the adaptation is called relaxation, and τ is called the relaxation time.
:model damped unforced oscillations of a weight on a spring.

Q factor

quality factorQQ-factor
The Q factor of a damped oscillator is defined as :
Q factor is a parameter that describes the resonance behavior of an underdamped harmonic oscillator (resonator).

Damping ratio

dampingdampedoverdamped
The boundary solution between an underdamped oscillator and an overdamped oscillator occurs at a particular value of the friction coefficient and is called critically damped.
The damping ratio is a dimensionless measure describing how oscillations in a system decay after a disturbance.

Parametric oscillator

parametric resonanceelectric vibratorParametric generator
A parametric oscillator is a driven harmonic oscillator in which the drive energy is provided by varying the parameters of the oscillator, such as the damping or restoring force.
A parametric oscillator is a driven harmonic oscillator in which the oscillations are driven by varying some parameter of the system at some frequency, typically different from the natural frequency of the oscillator.

Transient (oscillation)

transienttransientselectrical transient
The general solution is a sum of a transient solution that depends on initial conditions, and a steady state that is independent of initial conditions and depends only on the driving amplitude F_0, driving frequency \omega, undamped angular frequency \omega_0, and the damping ratio \zeta.
Mathematically, it can be modeled as a damped harmonic oscillator.

Differential equation

differential equationsdifferentialsecond-order differential equation
Solving this differential equation, we find that the motion is described by the function
* Homogeneous second-order linear constant coefficient ordinary differential equation describing the harmonic oscillator:

Angular frequency

angular rateangular speedangular frequencies
If the spring is assumed to be ideal and massless with no damping, then the motion is simple and harmonic with an angular frequency given by

Capacitor

capacitorscapacitivecondenser
This type of system appears in AC-driven RLC circuits (resistor–inductor–capacitor) and driven spring systems having internal mechanical resistance or external air resistance.
The current and voltage reverse direction, forming a harmonic oscillator between the inductance and capacitance.

Linear response function

linear response theorylinear responsesusceptibility
is the absolute value of the impedance or linear response function, and
Consider a damped harmonic oscillator with input given by an external driving force h(t),

Nondimensionalization

characteristic unitcharacteristic unitsnondimensionalize
This is done through nondimensionalization.
The result of the definition is the universal oscillator equation.

Resonance

resonantresonant frequencyresonance frequency
For a particular driving frequency called the resonance, or resonant frequency, the amplitude (for a given F_0) is maximal.
Γ is a parameter dependent on the damping of the oscillator, and is known as the linewidth of the resonance.

Quantum harmonic oscillator

harmonic oscillatorharmonic oscillatorsquantum oscillator
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator.

Anharmonicity

anharmonicanharmonic oscillatoranharmonic oscillators
In classical mechanics, anharmonicity is the deviation of a system from being a harmonic oscillator.

Hooke's law

spring constantforce constantelasticity tensor
Hooke's law gives the relationship of the force exerted by the spring when the spring is compressed or stretched a certain length:
A mass m attached to the end of a spring is a classic example of a harmonic oscillator.

Elastic pendulum

Spring pendulum
In physics and mathematics, in the area of dynamical systems, an elastic pendulum (also called spring pendulum or swinging spring) is a physical system where a piece of mass is connected to a spring so that the resulting motion contains elements of both a simple pendulum and a one-dimensional spring-mass system.

Torsion spring

torsion balancetorsion pendulumtorsion coefficient
If the free balance is twisted and released, it will oscillate slowly clockwise and counterclockwise as a harmonic oscillator, at a frequency that depends on the moment of inertia of the beam and the elasticity of the fiber.

Classical mechanics

Newtonian mechanicsNewtonian physicsclassical
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x: