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

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

where $k$ is a positive constant.

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 (see RLC circuit).

Simple harmonic oscillator

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The simple harmonic oscillator has no driving force, and no friction (damping), so the net force is just

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

Using Newton's Second Law

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

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

$m\; \backslash frac\{d^2x\}\{dt^2\}\; =\; -k\; x$

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

$\backslash frac\{d^2x\}\{dt^2\}\; +\; \{\backslash omega\_0\}^2\; x\; =\; 0$

and has the general solution

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

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

Alternatively, the general solution can be written as

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

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

or as

$x\; =\; A\; \backslash sin\{\backslash omega\_0\; t\}\; +\; B\; \backslash cos\{\backslash omega\_0\; t\}\; \backslash ,$

where $A\; \backslash ,$ and $B\; \backslash ,$ are the constants which are determined by the initial conditions, instead of $A\; \backslash ,$ and $\backslash phi\; \backslash ,$ in the previous forms.

The frequency of the oscillations is given by

$f\; =\; \backslash frac\{\backslash omega\_0\}\{2\backslash pi\}$

The kinetic energy is

$T\; =\; \backslash frac\{1\}\{2\}\; m\; \backslash left(\backslash frac\{dx\}\{dt\}\backslash right)^2\; =\; \backslash frac\{1\}\{2\}\; k\; A^2\; \backslash sin^2(\backslash omega\_0\; t\; +\; \backslash phi)$.

and the potential energy is

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

so the total energy of the system has the constant value

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

Driven harmonic oscillator

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A driven harmonic oscillator satisfies the nonhomogeneous second order linear differential equation

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

where $A\_\{0\}$ is the driving amplitude and $\backslash omega$ 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.

Damped harmonic oscillator

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A damped harmonic oscillator satisfies the second order differential equation

$\backslash frac\{d^2x\}\{dt^2\}\; +\; \backslash frac\{b\}\{m\}\; \backslash frac\{dx\}\{dt\}\; +\; \{\backslash omega\_0\}^2x\; =\; 0,$

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 $b$.

Damped, driven harmonic oscillator

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This satisfies the equation

$m\backslash frac\{d^2x\}\{dt^2\}\; +\; r\; \backslash frac\{dx\}\{dt\}\; +\; kx=\; F\_0\; \backslash cos(\backslash omega\; t).$

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 unhomogenous ODE) that is independent of initial conditions and depends only on driving frequency, driving force, restoring force, damping force, and inertial moment of the oscillator (see also kernel and image).

The steady-state solution is

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

where

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

is the absolute value of the impedance

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

and

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

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

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

$\{\backslash omega\}\_r\; =\; \backslash sqrt\{\backslash frac\{k\}\{m\}\; -\; \backslash frac\{r^2\}\{2\; m^2\}\}$

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.

Full mathematical definition

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Most harmonic oscillators, at least approximately, solve the differential equation:

$\backslash frac\{d^2x\}\{dt^2\}\; +\; b/m\; \backslash frac\{dx\}\{dt\}\; +\; \{\backslash omega\_0\}^2x\; =\; A\_0\; \backslash cos(\backslash omega\; t)$

where t is time, b is the damping constant, ω_o is the characteristic angular frequency, and A_ocos(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

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

Important terms

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

Simple harmonic oscillator

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A simple harmonic oscillator is simply an oscillator that is neither damped nor driven. So the equation to describe one is:

$\backslash frac\{d^2x\}\{dt^2\}\; +\; \{\backslash omega\_0\}^2x\; =\; 0.$

Physically, 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 attached to a spring, Newton's Laws, combined with Hooke's law for the behavior of a spring, states that:

$-k\; x\; =\; ma\; \backslash ,$

where k is the spring constant

m is the mass

x is the position of the mass

a is its acceleration.

Because acceleration a is the second derivative of position x, we can rewrite the equation as follows:

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

The most simple solution to the above differential equation is

$x\; =\; A\; \backslash cos(\backslash omega\; t\; +\; \backslash delta)\; \backslash ,$

and the second derivative of that is

$\backslash frac\{d^2\; x\}\{dt^2\}\; =\; -A\; \backslash omega^2\; \backslash cos(\backslash omega\; t\; +\; \backslash delta)$

where A is the amplitude, δ is the phase shift, and ω is the angular frequency.

Plugging these back into the original differential equation, we have:

$-A\; k\; \backslash cos(\backslash omega\; t\; +\backslash delta)\; =\; -A\; m\; \backslash omega^2\; \backslash cos(\backslash omega\; t\; +\; \backslash delta).\; \backslash ,$

Then, after dividing both sides by $-A\; \backslash cos(\backslash omega\; t\; +\; \backslash delta)\; \backslash ,$ we get:

$k\; =\; m\; \backslash omega^2\; \backslash ,$

or, as it is more commonly written:

$\backslash omega\; =\; \backslash sqrt\{\backslash frac\{k\}\{m\}\}.$

The above formula reveals that the angular frequency ω 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). We will label this ω as ω_o from now on. This will become important later.

Universal oscillator equation

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The equation

$\backslash frac\{d^2q\}\{d\; \backslash tau^2\}\; +\; 2\; \backslash zeta\; \backslash frac\{dq\}\{d\backslash tau\}\; +\; q\; =\; 0$

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_cτ) = cos(ωτ), where ω = ωt_c, the equation becomes

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

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

Transient solution

The solution based on solving the ordinary differential equation is for arbitrary constants c₁ and c₂ is

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

Steady-state solution

Apply the "complex variables method" by solving the auxiliary equation below and then finding the real part of its solution:

$\backslash frac\{d^2\; q\}\{d\backslash tau^2\}\; +\; 2\; \backslash zeta\; \backslash frac\{dq\}\{d\backslash tau\}\; +\; q\; =\; \backslash cos(\backslash omega\; \backslash tau)\; +\; i\backslash sin(\backslash omega\; \backslash tau)\; =\; e^\{\; i\; \backslash omega\; \backslash tau\}\; .$

Supposing the solution is of the form

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

Its derivatives from zero to 2nd order are

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

Substituting these quantities into the differential equation gives

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

Dividing by the exponential term on the left results in

$\backslash ,\backslash !\; -\backslash omega^2\; A\; +\; 2\; \backslash zeta\; i\; \backslash omega\; A\; +\; A\; =\; e^\{-i\; \backslash phi\}\; =\; \backslash cos\backslash phi\; -\; i\; \backslash sin\backslash phi\; .$

Equating the real and imaginary parts results in two independent equations

$A\; (1-\backslash omega^2)=\backslash cos\backslash phi\; \backslash qquad\; 2\; \backslash zeta\; \backslash omega\; A\; =\; -\; \backslash sin\backslash phi.$

Amplitude part

Squaring both equations and adding them together gives

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

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

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

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.

Note that the variables in these equations ought to be identified before showing the equation.

Phase part

To solve for φ, divide both equations to get

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

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

Full solution

Combining the amplitude and phase portions results in the steady-state solution

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

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

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

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

Relationship to RLC circuit

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Comparing a mechanical harmonic oscillator with an RLC circuit, the following correspond:

• F (force) $\backslash Leftrightarrow$ V (electric potential)
• x (position) $\backslash Leftrightarrow$ Q (charge)
• k (spring constant) $\backslash Leftrightarrow\; \backslash frac\{1\}\{C\}$ (electrical elastance (reciprocal of capacitance)
• v (velocity) $\backslash Leftrightarrow$ I (electric current)
• b (damping factor) $\backslash Leftrightarrow$ R (electrical resistance)
• a (acceleration) $\backslash Leftrightarrow\; \backslash frac\{dI\}\{dt\}\; \backslash ,$ (rate of change of current)
• m (mass) $\backslash Leftrightarrow$ L (inductance)

Applications

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The problem of the simple harmonic oscillator occurs frequently in physics because of the form of its potential energy function:

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

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.

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

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

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

The constant term is arbitrary and thus may be dropped, and a coordinate transformation allows the form of the simple harmonic oscillator to be retrieved:

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

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.

Examples

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Simple Pendulum

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

$\{d^2\backslash theta\backslash over\; dt^2\}+\{g\backslash over\; \backslash ell\}\backslash theta=0$

Solution to this equation is given by:

$\backslash theta(t)\; =\; \backslash theta\_0\backslash cos\backslash left(\backslash sqrt\{g\backslash over\; \backslash ell\}t\backslash right)\; \backslash quad\backslash quad\backslash quad\backslash quad\; |\backslash theta\_0|\; \backslash ll\; 1$

where $\backslash theta\_0$ is the largest angle attained by the pendulum. Period, the time for one complete oscillation (time for the bob to return to its starting position), is given by $2\backslash pi$ divided by whatever is multiplying the time in the argument of the cosine

$T\_0\; =\; 2\backslash pi\backslash sqrt\{\backslash ell\backslash over\; g\}\backslash quad\backslash quad\backslash quad\backslash quad\; |\backslash theta\_0|\; \backslash ll\; 1$

Pendulum swinging over turntable

Simple harmonic motion can in some cases be considered to be the one-dimensional projection of two-dimensional circular motion. Consider a long pendulum swinging over the turntable of a record player. On the edge of the turntable there is an object. If the object is viewed from the same level as the turntable, a projection of the motion of the object seems to be moving backwards and forwards on a straight line. It is possible to change the frequency of rotation of the turntable in order to have a perfect synchronization with the motion of the pendulum.

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

$\backslash omega=2\backslash pi\backslash cdot\; f\; =\; \backslash frac\{2\backslash pi\}\{T\}$

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.

Spring-mass system

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

$Fs\; \backslash left(\; t\; \backslash right)\; =kx\; \backslash left(\; t\; \backslash right)$

where Fs is the force, k is the spring constant, and the 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:

$m\; \backslash frac\; \{d^\{2\}\}\{d\{t\}^\{2\}\}\; x\; \backslash left(\; t\; \backslash right)\; +kx(t)=0$

If the initial displacement is A, and there is no initial velocity, the solution of this equation is given by: $x\; \backslash left(\; t\; \backslash right)\; =A\backslash cos\; \backslash left(\; (\backslash sqrt\; \{k/m\})\; t\backslash right)$

Energy variation in the spring-damper system

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 $U\; =\; 1/2\backslash ,k\{x\}^\{2\}$

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.

References

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See also

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• Normal mode
• Quantum harmonic oscillator
• Anharmonic oscillator
• Q factor

Classical mechanics | Ordinary differential equations

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