# 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),

### Exponential decay

**mean lifetimedecay constantexponentially**

### 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.

### The Feynman Lectures on Physics

**Feynman Lectures on PhysicsSix Easy PiecesFeynman lectures**

### 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: