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- This article is about the harmonic oscillator in classical mechanics. For its use in quantum mechanics, see quantum harmonic oscillator.

In classical mechanics, a harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force $F$ proportional to the displacement $x$ according to Hooke's law:

- $F\; =\; -k\; x\; ,$

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).

If a frictional force (damping) proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. In such situation, the frequency of the oscillations is smaller than in the non-damped case, and the amplitude of the oscillations decreases with time.

If an external time-dependent force is present, the harmonic oscillator is described as a driven oscillator.

Mechanical examples include pendula (with small angles of displacement), masses connected to springs, and acoustical systems. Other analogous systems include electrical harmonic oscillators such as RLC circuits (see Equivalent systems below). The analysis of the harmonic oscillator exhibits the paradox that there is almost no ideal, or perfect, harmonic oscillator in nature or by manufacture, yet it is a system of profound importance in mathematics, physics and applied science.

- $F\; =\; -k\; x$

Using Newton's Second Law of motion,

- $F\; =\; m\; a\; =\; -k\; x\; ,$

The acceleration, $a$ is equal to the second derivative of $x$.

- $m\; frac\{mathrm\{d\}^2x\}\{mathrm\{d\}t^2\}\; =\; -k\; x$

If we define $\{omega\_0\}^2\; =\; k/m$, then the equation can be written as follows,

- $frac\{mathrm\{d\}^2x\}\{mathrm\{d\}t^2\}\; +\; \{omega\_0\}^2\; x\; =\; 0$

Define $dot\; x\; =\; frac\{mathrm\{d\}x\}\{mathrm\{d\}t\}$.

We observe that:

- $frac\{mathrm\{d\}^2\; x\}\{mathrm\{d\}\; t^2\}\; =\; ddot\; x\; =\; frac\{mathrm\{d\}dot\; \{x\}\}\{mathrm\{d\}t\}frac\{mathrm\{d\}x\}\{mathrm\{d\}x\}=frac\{mathrm\{d\}dot\; \{x\}\}\{mathrm\{d\}x\}frac\{mathrm\{d\}x\}\{mathrm\{d\}t\}=frac\{mathrm\{d\}dot\{x\}\}\{mathrm\{d\}x\}dot\; \{x\}$

- $frac\{mathrm\{d\}\; dot\{x\}\}\{mathrm\{d\}x\}dot\; x\; +\; \{omega\_0\}^2\; x\; =\; 0$

- $mathrm\{d\}\; dot\{x\}cdot\; dot\; x\; +\; \{omega\_0\}^2\; x\; cdot\; mathrm\{d\}x\; =\; 0$

- $dot\{x\}^2\; +\; \{omega\_0\}^2\; x^2\; =\; K$

- $dot\{x\}^2\; =\; A^2\; \{omega\_0\}^2-\{omega\_0\}^2\; x^2$

- $dot\{x\}\; =\; pm\; \{omega\_0\}\; sqrt\{A^2\; -\; x^2\}$

- $frac\; \{mathrm\{d\}x\}\{pm\; sqrt\{A^2\; -\; x^2\}\}\; =\; \{omega\_0\}mathrm\{d\}t$

- $begin\{cases\}\; arcsin\{frac\; \{x\}\{A\}\}=\; omega\_0\; t\; +\; phi\; arccos\{frac\; \{x\}\{A\}\}=\; omega\_0\; t\; +\; phi\; end\{cases\}$

and has the general solution

- $x\; =\; A\; cos\; \{(omega\_0\; t\; +\; phi)\}\; ,$

where the amplitude $A\; ,$ and the phase $phi\; ,$ are determined by the initial conditions.

Alternatively, the general solution can be written as

- $x\; =\; A\; sin\; \{(omega\_0\; t\; +\; phi)\}\; ,$

where the value of $phi\; ,$ is shifted by $pi/2\; ,$ relative to the previous form;

or as

- $x\; =\; C\_1\; sin\{omega\_0\; t\}\; +\; C\_2\; cos\{omega\_0\; t\}\; ,$

where $C\_1\; ,$ and $C\_2\; ,$ are the constants which are determined by the initial conditions.

The frequency of the oscillations is given by

- $$

displaystyle

f

=

frac{omega_0} {2 pi}

=frac{1}{2 pi} sqrt{frac{k}{m}}

The kinetic energy is

- $K\; =\; frac\{1\}\{2\}\; m\; left(frac\{mathrm\{d\}x\}\{mathrm\{d\}t\}right)^2\; =\; frac\{1\}\{2\}\; k\; A^2\; sin^2(omega\_0\; t\; +\; phi)$.

and the potential energy is

- $U\; =\; frac\{1\}\{2\}\; k\; x^2\; =\; frac\{1\}\{2\}\; k\; A^2\; cos^2(omega\_0\; t\; +\; phi)$

so the total energy of the system has the constant value

- $E\; =\; frac\{1\}\{2\}\; k\; A^2.$

- $frac\{mathrm\{d\}^2x\}\{mathrm\{d\}t^2\}\; +\; \{omega\_0\}^2x\; =\; A\_0\; cos(omega\; t),$

where $A\_\{0\}$ is the driving amplitude and $omega$ is the driving frequency for a sinusoidal driving mechanism. This type of system appears in AC LC (inductor-capacitor) circuits and idealized spring systems lacking internal mechanical resistance or external air resistance.

- $frac\{mathrm\{d\}^2x\}\{mathrm\{d\}t^2\}\; +\; frac\{b\}\{m\}\; frac\{mathrm\{d\}x\}\{mathrm\{d\}t\}\; +\; \{omega\_0\}^2x\; =\; 0,$

where $b$ is an experimentally determined damping constant satisfying the relationship $F\; =\; -bv$. An example of a system obeying this equation would be a weighted spring underwater if the damping force exerted by the water is assumed to be linearly proportional to $v$.

The frequency of the damped harmonic oscillator is given by

- $omega\_1\; =\; sqrt\{omega\_0^2\; -\; R\_m^2\}$

- $R\_m=frac\{b\}\{2m\}.$

- $mfrac\{mathrm\{d\}^2x\}\{mathrm\{d\}t^2\}\; +\; r\; frac\{mathrm\{d\}x\}\{mathrm\{d\}t\}\; +\; kx=\; F\_0\; cos(omega\; t).$

The general solution is a sum of a transient (the solution for damped undriven harmonic oscillator, homogeneous ODE) that depends on initial conditions, and a steady state (particular solution of the nonhomogenous ODE) that is independent of initial conditions and depends only on driving frequency, driving force, restoring force, damping force,

The steady-state solution is

- $x(t)\; =\; frac\{F\_0\}\{Z\_m\; omega\}\; sin(omega\; t\; -\; phi)$

where

- $Z\_m\; =\; sqrt\{r^2\; +\; left(omega\; m\; -\; frac\{k\}\{omega\}right)^2\}$

is the absolute value of the impedance or linear response function

- $Z\; =\; r\; +\; ileft(omega\; m\; -\; frac\{k\}\{omega\}right)$

and

- $phi\; =\; arctanleft(frac\{omega\; m\; -\; frac\{k\}\{omega\}\}\{r\}right)$

is the phase of the oscillation relative to the driving force.

One might see that for a certain driving frequency, $omega$, the amplitude (relative to a given $F\_0$) is maximal. This occurs for the frequency

- $\{omega\}\_r\; =\; sqrt\{frac\{k\}\{m\}\; -\; 2left(frac\{r\}\{2\; m\}right)^2\}$

and is called resonance of displacement.

In summary: at a steady state the frequency of the oscillation is the same as that of the driving force, but the oscillation is phase-offset and scaled by amounts that depend on the frequency of the driving force in relation to the preferred (resonant) frequency of the oscillating system.

Example: RLC circuit.

- $frac\{mathrm\{d\}^2x\}\{mathrm\{d\}t^2\}\; +\; frac\{b\}\{m\}\; frac\{mathrm\{d\}x\}\{mathrm\{d\}t\}\; +\; \{omega\_0\}^2x\; =\; A\_0\; cos(omega\; t)$

where t is time, b is the damping constant, ω_{o} 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

- $f\; =\; frac\{omega\}\{2\; pi\}.$

- Amplitude: maximal displacement from the equilibrium.
- Period: the time it takes the system to complete an oscillation cycle. Inverse of frequency.
- Frequency: the number of cycles the system performs per unit time (usually measured in hertz = 1/s).
- Angular frequency: $omega\; =\; 2\; pi\; f$
- 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 $pi$).
- Initial conditions: the state of the system at t = 0, the beginning of oscillations.

- $frac\{mathrm\{d\}^2q\}\{mathrm\{d\}\; tau^2\}\; +\; 2\; zeta\; frac\{mathrm\{d\}q\}\{mathrm\{d\}tau\}\; +\; q\; =\; 0$

is known as the universal oscillator equation since all second order linear oscillatory systems can be reduced to this form. This is done through nondimensionalization.

If the forcing function is f(t) = cos(ωt) = cos(ωt_{c}τ) = cos(ωτ), where ω = ωt_{c}, the equation becomes

- $frac\{mathrm\{d\}^2q\}\{mathrm\{d\}\; tau^2\}\; +\; 2\; zeta\; frac\{mathrm\{d\}q\}\{mathrm\{d\}tau\}\; +\; q\; =\; cos(omega\; tau).$

The solution to this differential equation contains two parts, the "transient" and the "steady state".

$q\_t\; (tau)\; =\; begin\{cases\}\; e^\{-zetatau\}\; left(c\_1\; e^\{tau\; sqrt\{zeta^2\; -\; 1\}\}\; +\; c\_2\; e^\{-\; tau\; sqrt\{zeta^2\; -\; 1\}\}\; right)\; \&\; zeta\; >\; 1\; mbox\{(overdamping)\}\; e^\{-zetatau\}\; (c\_1+c\_2\; tau)\; =\; e^\{-tau\}(c\_1+c\_2\; tau)\; \&\; zeta\; =\; 1\; mbox\{(critical\; damping)\}\; e^\{-zeta\; tau\}\; left[c\_1\; cos\; left(sqrt\{1-zeta^2\}\; tauright)\; +c\_2\; sinleft(sqrt\{1-zeta^2\}\; tauright)\; right]\; \&\; zeta\; <\; 1\; mbox\{(underdamping)\}\; end\{cases\}$

The transient solution is independent of the forcing function. If the system is critically damped, the response is independent of the damping.

- $frac\{mathrm\{d\}^2\; q\}\{mathrm\{d\}tau^2\}\; +\; 2\; zeta\; frac\{mathrm\{d\}q\}\{mathrm\{d\}tau\}\; +\; q\; =\; cos(omega\; tau)\; +\; isin(omega\; tau)\; =\; e^\{\; i\; omega\; tau\}\; .$

Supposing the solution is of the form

- $,!\; q\_s(tau)\; =\; A\; e^\{i\; (omega\; tau\; +\; phi\; )\; \}\; .$

Its derivatives from zero to 2nd order are

- $q\_s\; =\; A\; e^\{i\; (omega\; tau\; +\; phi\; )\; \},\; frac\{mathrm\{d\}q\_s\}\{mathrm\{d\}\; tau\}\; =\; i\; omega\; A\; e^\{i\; (omega\; tau\; +\; phi\; )\; \},\; frac\{mathrm\{d\}^2\; q\_s\}\{mathrm\{d\}\; tau^2\}\; =\; -\; omega^2\; A\; e^\{i\; (omega\; tau\; +\; phi\; )\; \}\; .$

Substituting these quantities into the differential equation gives

- $,!\; -omega^2\; A\; e^\{i\; (omega\; tau\; +\; phi)\}\; +\; 2\; zeta\; i\; omega\; A\; e^\{i(omega\; tau\; +\; phi)\}\; +\; A\; e^\{i(omega\; tau\; +\; phi)\}\; =\; (-omega^2\; A\; ,\; +\; ,\; 2\; zeta\; i\; omega\; A\; ,\; +\; ,\; A)\; e^\{i\; (omega\; tau\; +\; phi)\}\; =\; e^\{i\; omega\; tau\}\; .$

Dividing by the exponential term on the left results in

- $,!\; -omega^2\; A\; +\; 2\; zeta\; i\; omega\; A\; +\; A\; =\; e^\{-i\; phi\}\; =\; cosphi\; -\; i\; sinphi\; .$

Equating the real and imaginary parts results in two independent equations

- $A\; (1-omega^2)=cosphi\; qquad\; 2\; zeta\; omega\; A\; =\; -\; sinphi.$

Squaring both equations and adding them together gives

- $left\; .\; begin\{matrix\}A^2\; (1-omega^2)^2\; =\; cos^2phi\; (2\; zeta\; omega\; A)^2\; =\; sin^2phi\; end\{matrix\}\; right\; \}\; Rightarrow\; A^2[(1-omega^2)^2\; +\; (2\; zeta\; omega)^2]\; =\; 1.$

By convention the positive root is taken since amplitude is usually considered a positive quantity. Therefore,

- $A\; =\; A(zeta,\; omega)\; =\; frac\{1\}\{sqrt\{(1-omega^2)^2\; +\; (2\; zeta\; omega)^2\}\}.$

Compare this result with the theory section on resonance, as well as the "magnitude part" of the RLC circuit. This amplitude function is particularly important in the analysis and understanding of the frequency response of second-order systems.

- $tanphi\; =\; -\; frac\{2\; zeta\; omega\}\{\; 1\; -\; omega^2\}\; =\; frac\{2\; zeta\; omega\}\{omega^2\; -\; 1\}\; Rightarrow\; phi\; equiv\; phi(zeta,\; omega)\; =\; arctan\; left(frac\{2\; zeta\; omega\}\{omega^2\; -\; 1\}\; right\; ).$

This phase function is particularly important in the analysis and understanding of the frequency response of second-order systems.

- $,!\; q\_s\; (tau)\; =\; A(zeta,omega)\; cos(omega\; tau\; +\; phi(zeta,omega))\; =\; Acos(omega\; tau\; +\; phi).$

The solution of original universal oscillator equation is a superposition (sum) of the transient and steady-state solutions

- $,!\; q(tau)\; =\; q\_t\; (tau)\; +\; q\_s\; (tau).$

For a more complete description of how to solve the above equation, see linear ODEs with constant coefficients.

Translational Mechanical | Torsional Mechanical | Series RLC Circuit | Parallel RLC Circuit |
---|---|---|---|

Position $x,$ | Angle $theta,$ | Charge $q,$ | Voltage $e,$ |

Velocity $frac\{dx\}\{dt\},$ | Angular velocity $frac\{dtheta\}\{dt\},$ | Current $frac\{dq\}\{dt\},$ | $frac\{de\}\{dt\},$ |

Mass $M,$ | Moment of inertia $I,$ | Inductance $L,$ | Capacitance $C,$ |

Spring constant $K,$ | Torsion constant $mu,$ | Elastance $1/C,$ | Susceptance $1/L,$ |

Friction $gamma,$ | Rotational friction $Gamma,$ | Resistance $R,$ | Conductance $1/R,$ |

Drive force $F(t),$ | Drive torque $tau(t),$ | $e,$ | $di/dt,$ |

Undamped resonant frequency $f\_n,$: | |||

$frac\{1\}\{2pi\}sqrt\{frac\{K\}\{M\}\},$ | $frac\{1\}\{2pi\}sqrt\{frac\{mu\}\{I\}\},$ | $frac\{1\}\{2pi\}sqrt\{frac\{1\}\{LC\}\},$ | $frac\{1\}\{2pi\}sqrt\{frac\{1\}\{LC\}\},$ |

Differential equation: | |||

$Mddot\; x\; +\; gammadot\; x\; +\; Kx\; =\; F,$ | $Iddot\; theta\; +\; Gammadot\; theta\; +\; mu\; theta\; =\; tau,$ | $Lddot\; q\; +\; Rdot\; q\; +\; q/C\; =\; e,$ | $Cddot\; e\; +\; dot\; e/R\; +\; e/L\; =\; dot\; i,$ |

A conservative force is one that has a potential energy function. The potential energy function of a harmonic oscillator is:

- $V(x)\; =\; frac\{1\}\{2\}\; k\; x^2$

Given an arbitrary potential energy function $V(x)$, one can do a Taylor expansion in terms of $x$ around an energy minimum ($x\; =\; x\_0$) to model the behavior of small perturbations from equilibrium.

- $V(x)\; =\; V(x\_0)\; +\; (x-x\_0)\; V\text{'}(x\_0)\; +\; frac\{1\}\{2\}\; (x-x\_0)^2\; V^\{(2)\}(x\_0)\; +\; O(x-x\_0)^3$

Because $V(x\_0)$ is a minimum, the first derivative evaluated at $x\_0$ must be zero, so the linear term drops out:

- $V(x)\; =\; V(x\_0)\; +\; frac\{1\}\{2\}\; (x-x\_0)^2\; V^\{(2)\}(x\_0)\; +\; O(x-x\_0)^3$

The constant term V(x_{0}) is arbitrary and thus may be dropped, and a coordinate transformation allows the form of the simple harmonic oscillator to be retrieved:

- $V(x)\; approx\; frac\{1\}\{2\}\; x^2\; V^\{(2)\}(0)\; =\; frac\{1\}\{2\}\; k\; x^2$

Thus, given an arbitrary potential energy function $V(x)$ with a non-vanishing second derivative, one can use the solution to the simple harmonic oscillator to provide an approximate solution for small perturbations around the equilibrium point.

Assuming no damping and small amplitudes, the differential equation governing a simple pendulum is

- $\{mathrm\{d\}^2thetaover\; mathrm\{d\}t^2\}+\{gover\; ell\}theta=0.$

The solution to this equation is given by:

- $theta(t)\; =\; theta\_0cosleft(sqrt\{gover\; ell\}tright)\; quadquadquadquad\; |theta\_0|\; ll\; 1$

where $theta\_0$ is the largest angle attained by the pendulum. The period, the time for one complete oscillation , is given by $2pi$ divided by whatever is multiplying the time in the argument of the cosine

- $T\_0\; =\; 2pisqrt\{ellover\; g\}quadquadquadquad\; |theta\_0|\; ll\; 1.$

The angular speed of the turntable is the pulsation of the pendulum.

In general, the pulsation-also known as angular frequency, of a straight-line simple harmonic motion is the angular speed of the corresponding circular motion.

Therefore, a motion with period T and frequency f=1/T has pulsation

- $omega=2pi\; f\; =\; frac\{2pi\}\{T\}.$

In general, pulsation and angular speed are not synonymous. For instance the pulsation of a pendulum is not the angular speed of the pendulum itself, but it is the angular speed of the corresponding circular motion.

When a spring is stretched or compressed by a mass, the spring develops a restoring force. Hooke's law gives the relationship of the force exerted by the spring when the spring is compressed or stretched a certain length:

- $F\; left(t\; right)\; =-kx\; left(t\; right)$

where F is the force, k is the spring constant, and x is the displacement of the mass with respect to the equilibrium position.

This relationship shows that the distance of the spring is always opposite to the force of the spring.

By using either force balance or an energy method, it can be readily shown that the motion of this system is given by the following differential equation:

- $F(t)\; =\; -kx(t)\; =\; m\; frac\; \{mathrm\{d\}^\{2\}\}\{mathrm\{d\}\{t\}^\{2\}\}\; x\; left(t\; right)\; =\; ma.$

...the latter evidently being _law_of_resultant_force.

If the initial displacement is A, and there is no initial velocity, the solution of this equation is given by:

- $x\; left(t\; right)\; =Acos\; left((sqrt\; \{k/m\})\; tright).$

In terms of energy, all systems have two types of energy, potential energy and kinetic energy. When a spring is stretched or compressed, it stores elastic potential energy, which then is transferred into kinetic energy. The potential energy within a spring is determined by the equation $U\; =\; 1/2,k\{x\}^\{2\}.$

When the spring is stretched or compressed, kinetic energy of the mass gets converted into potential energy of the spring. By conservation of energy, assuming the datum is defined at the equilibrium position, when the spring reaches its maximum potential energy, the kinetic energy of the mass is zero. When the spring is released, the spring will try to reach back to equilibrium, and all its potential energy is converted into kinetic energy of the mass.

- Serway, Raymond A.; Jewett, John W. (2003).
*Physics for Scientists and Engineers*. Brooks/Cole. ISBN 0-534-40842-7. - Tipler, Paul (1998).
*Physics for Scientists and Engineers: Vol. 1*. 4th ed., W. H. Freeman. ISBN 1-57259-492-6. - Wylie, C. R. (1975).
*Advanced Engineering Mathematics*. 4th ed., McGraw-Hill. ISBN 0-07-072180-7.

- Q factor
- Normal mode
- Quantum harmonic oscillator
- Anharmonic oscillator
- Parametric oscillator
- Critical speed
- Radial harmonic oscillator

- Harmonic Oscillator from The Chaos Hypertextbook
- Simple Harmonic oscillator on PlanetPhysics
- A-level Physics experiment on the subject of Damped Harmonic Motion with solution curve graphs

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Last updated on Wednesday October 08, 2008 at 06:53:50 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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