NationMaster - Encyclopedia: 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: Image File history File links No higher resolution available. ... Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... For other uses, see Force (disambiguation). ... Hookes law accurately models the physical properties of common mechanical springs for small changes in length. ...

$F = -k x ,$

where $k$ is a positive constant. In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ...

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). Simple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... It has been suggested that pulse amplitude be merged into this article or section. ... For other uses, see Frequency (disambiguation). ... It has been suggested that pulse amplitude be merged into this article or section. ...

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. For other uses, see Force (disambiguation). ... Damping is any effect, either deliberately engendered or inherent to a system, that tends to reduce the amplitude of oscillations of an oscillatory system. ... This article is about velocity in physics. ... For other uses, see Frequency (disambiguation). ... It has been suggested that pulse amplitude be merged into this article or section. ...

If an external time-dependent force is present, the harmonic oscillator is described as a driven oscillator. For other uses, see Force (disambiguation). ...

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). For other uses, see Pendulum (disambiguation). ... For other uses, see Spring. ... Acoustics is the branch of physics concerned with the study of sound (mechanical waves in gases, liquids, and solids). ... An RLC circuit (also known as a resonant circuit or a tuned circuit) is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. ...

Simple harmonic oscillator

The simple harmonic oscillator has no driving force, and no friction (damping), so the net force is just: For other uses, see Friction (disambiguation). ... Damping is any effect, either deliberately engendered or inherent to a system, that tends to reduce the amplitude of oscillations of an oscillatory system. ...

$F = -k x$

Using Newton's Second Law of motion, Newtons laws of motion are the three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. ...

$F = m a = -k x ,$

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:

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

and substituting

$frac{mathrm{d} dot{x}}{mathrm{d}x}dot x + {omega_0}^2 x = 0$

$mathrm{d} dot{x}cdot dot x + {omega_0}^2 x cdot mathrm{d}x = 0$

integrating

$dot{x}^2 + {omega_0}^2 x^2 = K$

where K is the integration constant, set K = (A ω₀)² In calculus, the indefinite integral of a given function (i. ...

$dot{x}^2 = A^2 {omega_0}^2-{omega_0}^2 x^2$

$dot{x} = pm {omega_0} sqrt{A^2 - x^2}$

$frac {mathrm{d}x}{pm sqrt{A^2 - x^2}} = {omega_0}mathrm{d}t$

integrating, the results (including integration constant ϕ) are

$begin{cases} arcsin{frac {x}{A}}= omega_0 t + phi arccos{frac {x}{A}}= omega_0 t + phi end{cases}$

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. It has been suggested that pulse amplitude be merged into this article or section. ... This article is about a portion of a periodic process. ...

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 For other uses, see Frequency (disambiguation). ...

$displaystyle f = frac {omega_0} {2 pi} = frac{1}{2 pi} sqrt{frac{k}{m}}$

The kinetic energy is The cars of a roller coaster reach their maximum kinetic energy when at the bottom of their path. ...

$T = frac{1}{2} m left(frac{mathrm{d}x}{mathrm{d}t}right)^2 = frac{1}{2} k A^2 sin^2(omega_0 t + phi)$ .

and the potential energy is Potential energy can be thought of as energy stored within a physical system. ...

$U = frac{1}{2} k x^2 = frac{1}{2} k A^2 cos^2(omega_0 t + phi)$

so the total energy of the system has the constant value In classical physics, the total energy of an object is the sum of its potential energy and its kinetic energy. ...

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

Driven harmonic oscillator

A driven harmonic oscillator satisfies the nonhomogeneous second order linear differential equation In mathematics, a linear differential equation is a differential equation of the form Ly = f, where the differential operator L is a linear operator, y is the unknown function, and the right hand side f is a given function. ...

$frac{mathrm{d}^2x}{mathrm{d}t^2} + {omega_0}^2x = A_0 cos(omega t),$

where $A 0$ is the driving amplitude and $ω$ 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. For other uses, see Frequency (disambiguation). ... City lights viewed in a motion blurred exposure. ... An inductor is a passive electrical device employed in electrical circuits for its property of inductance. ... See Capacitor (component) for a discussion of specific types. ... For a solid object moving through a fluid or gas, drag is the sum of all the aerodynamic or hydrodynamic forces in the direction of the external fluid flow. ...

Damped harmonic oscillator

Main article: Damping

A damped harmonic oscillator satisfies the second order differential equation Damping is any effect, either deliberately engendered or inherent to a system, that tends to reduce the amplitude of oscillations of an oscillatory system. ...

$frac{mathrm{d}^2x}{mathrm{d}t^2} + frac{b}{m} frac{mathrm{d}x}{mathrm{d}t} + {omega_0}^2x = 0,$

where $b$ is an experimentally determined damping constant satisfying the relationship $F = - b v$ . 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

$omega_1 = sqrt{omega_0^2 - R_m^2}$

where

$R_m=frac{b}{2m}.$

Damped, driven harmonic oscillator

This satisfies the equation

$mfrac{mathrm{d}^2x}{mathrm{d}t^2} + r frac{mathrm{d}x}{mathrm{d}t} + kx= F_0 cos(omega t).$

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, Transient means passing with time. ... In mathematics, homogeneous may refer to: a homogeneous polynomial, in algebra a homogeneous function a homogeneous differential equation a homogeneous system of linear equations, in linear algebra homogeneous coordinates a homogeneous number a homogeneous space for a Lie group G, or more general transformation group a homogeneous ideal in a... HELLO EVERYONE!! Steady state is a more general situation than Dynamic equilibrium. ...

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 Mechanical impedance is a measure of how much a structure resists motion when subjected to a given force. ... A linear response function describes the input-output relationship of a signal transducer such as a radio turning electromagnetic waves into music or a neuron turning synaptic input into a response. ...

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

and

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

is the phase of the oscillation relative to the driving force. This article is about a portion of a periodic process. ...

One might see that for a certain driving frequency, $ω$ , 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. This article is about resonance in physics. ... In Newtonian mechanics, displacement is the vector that specifies the position of a point or a particle in reference to an origin or to a previous position. ...

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. An RLC circuit (also known as a resonant circuit or a tuned circuit) is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. ...

Full mathematical definition

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

$frac{mathrm{d}^2x}{mathrm{d}t^2} + frac{b}{m} frac{mathrm{d}x}{mathrm{d}t} + {omega_0}^2x = A_0 cos(omega t)$

where t is time, b is the damping constant, ω_o is the characteristic angular frequency, and A_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 It has been suggested that this article or section be merged into Angular velocity. ... It has been suggested that this article or section be merged into Angular velocity. ...

$f = frac{omega}{2 pi}.$

Important terms

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: $ω = 2π f$
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 $π$ ).
Initial conditions: the state of the system at t = 0, the beginning of oscillations.

It has been suggested that pulse amplitude be merged into this article or section. ... A standard definition of mechanical equilibrium is: A system is in mechanical equilibrium when the sum of the forces, and torque, on each particle of the system is zero. ... For other uses, see Frequency (disambiguation). ... For other uses, see Frequency (disambiguation). ... This article is about the SI unit of frequency. ... It has been suggested that this article or section be merged into Angular velocity. ... This article is about a portion of a periodic process. ... In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. ...

Simple harmonic oscillator

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

$frac{mathrm{d}^2x}{mathrm{d}t^2} + {omega_0}^2x = 0.$

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. For other uses, see Spring. ... An LC circuit consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C. When connected together, an electrical current can alternate between them at an angular frequency of where L is the inductance in henries, and C is the capacitance in farads. ...

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:

$-k x = ma ,$

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: In physics, Hookes law of elasticity states that if a force (F) is applied to an elastic spring or prismatic rod (with length L and cross section A), its extension is linearly proportional to its tensile stress Ïƒ and modulus of elasticity (E): or It is named after the 17th... For other uses, see Mass (disambiguation). ... Acceleration is the time rate of change of velocity and/or direction, and at any point on a velocity-time graph, it is given by the slope of the tangent to the curve at that point. ...

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

The most simple solution to the above differential equation is Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of partial differential equations. ...

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

and the second derivative of that is

$frac{mathrm{d}^2 x}{mathrm{d}t^2} = -A omega^2 cos(omega t + delta)$

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: It has been suggested that pulse amplitude be merged into this article or section. ... It has been suggested that this article or section be merged into Angular velocity. ...

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

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

$k = m omega^2 ,$

or, as it is more commonly written:

$omega = sqrt{frac{k}{m}}.$

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. It has been suggested that this article or section be merged into Angular velocity. ...

Universal oscillator equation

The equation

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

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. Nondimensionalization refers to the partial or full removal of units from a mathematical equation by a suitable substitution of variables. ...

If the forcing function is f(t) = cos(ωt) = cos(ωt_cτ) = 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".

Transient solution

The solution based on solving the ordinary differential equation is for arbitrary constants c₁ and c₂ is In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ...

$q_t (tau) = begin{cases} e^{-zetatau} left( c_1 e^{tau sqrt{zeta^2 - 1}} + c_2 e^{- tau sqrt{zeta^2 - 1}} right) & zeta > 1 mbox{(overdamping)} e^{-zetatau} (c_1+c_2 tau) = e^{-tau}(c_1+c_2 tau) & zeta = 1 mbox{(critical damping)} e^{-zeta tau} left[ c_1 cos left(sqrt{1-zeta^2} tauright) +c_2 sinleft(sqrt{1-zeta^2} tauright) right] & zeta < 1 mbox{(underdamping)} end{cases}$

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

Steady-state solution

Apply the "complex variables method" by solving the auxiliary equation below and then finding the real part of its solution: Plot of the function f(x)=(x2-1)(x-2-i)2/(x2+2+2i). ...

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

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

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

Amplitude part

Log-log plot of the frequency response of an ideal harmonic oscillator.

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. This article is about resonance in physics. ... An RLC circuit (also known as a resonant circuit or a tuned circuit) is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. ... Frequency response is the measure of any systems response to frequency, but is usually used in connection with electronic amplifiers and similar systems, particularly in relation to audio signals. ...

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

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

This phase function is particularly important in the analysis and understanding of the frequency response of second-order systems. Frequency response is the measure of any systems response to frequency, but is usually used in connection with electronic amplifiers and similar systems, particularly in relation to audio signals. ...

Full solution

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

$,! q_s (tau) = A(zeta,omega) cos(omega tau + phi(zeta,omega)) = Acos(omega tau + phi).$

The solution of original universal oscillator equation is a superposition (sum) of the transient and steady-state solutions The term superposition can have several meanings: Quantum superposition Law of superposition in geology and archaeology Superposition principle for vector fields Superposition Calculus is used for equational first-order reasoning This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

$,! q(tau) = q_t (tau) + q_s (tau).$

For a more complete description of how to solve the above equation, see linear ODEs with constant coefficients. In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ...

Equivalent systems

Harmonic oscillators occurring in a number of areas of engineering are equivalent in the sense that their mathematical models are identical (see universal oscillator equation above). Below is a table showing analogous quantities in four harmonic oscillator systems in mechanics and electronics. If analogous parameters on the same line in the table are given numerically equal values, the behavior of the oscillators will be the same.

Translational Mechanical	Torsional Mechanical	Series RLC Circuit	Parallel RLC Circuit
Position $x,$	Angle $theta,$	Charge $q,$	Voltage $e,$
Velocity $v,$	Angular velocity $omega,$	Current $frac{dq}{dt},$	$de/dt,$
Mass $M,$	Moment of inertia $I,$	Inductance $L,$	Capacitance $C,$
Spring constant $K,$	Torsion constant $mu,$	Elastance $1/C,$	Susceptance $1/L,$
Friction $C,$	Rotational friction $Gamma,$	Resistance $R,$	Conductance $1/R,$
Drive force $F(t),$	Drive torque $tau(t),$	$de/dt,$	$di/dt,$
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:
$Mddot x + Cdot x + Kx = F,$	$Iddot theta + Gammadot theta + mu theta = tau,$	$Lddot q + Rdot q + q/C = ddot e,$	$Cddot e + dot e/R + e/L = ddot i,$

Look up charge in Wiktionary, the free dictionary. ... International safety symbol Caution, risk of electric shock (ISO 3864), colloquially known as high voltage symbol. ... This article is about velocity in physics. ... Angular velocity describes the speed of rotation and the orientation of the instantaneous axis about which the rotation occurs. ... Look up current in Wiktionary, the free dictionary. ... For other uses, see Mass (disambiguation). ... Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI units kg m2, Former British units slug ft2), is the rotational analog of mass. ... An electric current i flowing around a circuit produces a magnetic field and hence a magnetic flux Î¦ through the circuit. ... Capacitance is a measure of the amount of electric charge stored (or separated) for a given electric potential. ... Hookes law accurately models the physical properties of common mechanical springs for small changes in length. ... This article or section does not adequately cite its references or sources. ... Capacitance is a measure of the amount of electric charge stored (or separated) for a given electric potential. ... In electrical engineering, the susceptance (B) is the imaginary part of the admittance. ... For other uses, see Friction (disambiguation). ... This article or section does not adequately cite its references or sources. ... Electrical resistance is a measure of the degree to which an electrical component opposes the passage of current. ... Conductance can refer to: Electrical conductance, the reciprocal of electrical resistance. ... For other uses, see Force (disambiguation). ... For other senses of this word, see torque (disambiguation). ... This article is about resonance in physics. ...

Applications

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

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

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. Series expansion redirects here. ...

$V(x) = V(x_0) + (x-x_0) V'(x_0) + frac{1}{2} (x-x_0)^2 V^{(2)}(x_0) + O(x-x_0)^3$

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(x₀) is arbitrary and thus may be dropped, and a coordinate transformation allows the form of the simple harmonic oscillator to be retrieved: In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ...

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

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.

Examples

Simple pendulum

A simple pendulum exhibits simple harmonic motion under the conditions of no damping and small amplitude.

Assuming no damping and small amplitudes, the differential equation governing a simple pendulum is Image File history File links Simple_pendulum_height. ... Image File history File links Simple_pendulum_height. ... A gravity pendulum is a weight on the end of a rigid rod, which, when given some initial lift from the vertical position, will swing back and forth under the influence of gravity over its central (lowest) point. ...

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

The solution to this equation is given by:

$theta(t) = theta_0cosleft(sqrt{gover ell}tright) quadquadquadquad |theta_0| ll 1$

where $θ 0$ is the largest angle attained by the pendulum. The period, the time for one complete oscillation , is given by $2π$ divided by whatever is multiplying the time in the argument of the cosine In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

$T_0 = 2pisqrt{ellover g}quadquadquadquad |theta_0| ll 1.$

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 word projection can mean more than one thing. ... In physics, circular motion is rotation along a circle: a circular path or a circular orbit. ... For other uses, see Pendulum (disambiguation). ... Edison cylinder phonograph from about 1899 The phonograph, or gramophone, was the most common device for playing recorded sound from the 1870s through the 1980s. ... Synchronization (or Sync) is a problem in timekeeping which requires the coordination of events to operate a system in unison. ...

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. Angular frequency is a measure of how fast an object is rotating In physics (specifically mechanics and electrical engineering), angular frequency ω (also called angular speed) is a scalar measure of rotation rate. ... Angular frequency is a measure of how fast an object is rotating In physics (specifically mechanics and electrical engineering), angular frequency ω (also called angular speed) is a scalar measure of rotation rate. ...

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.

Spring-mass system

Spring-mass system in equilibrium (A), compressed (B) and stretched (C) states.

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: Hookes law accurately models the physical properties of common mechanical springs for small changes in 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:

$m frac {mathrm{d}^{2}}{mathrm{d}{t}^{2}} x left( t right) +kx(t)=0.$

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

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

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 $U = 1/2,k{x}^{2}.$ Potential energy can be thought of as energy stored within a physical system. ... The cars of a roller coaster reach their maximum kinetic energy when at the bottom of their path. ...

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

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.

External links

Categories: Mechanical vibrations | Ordinary differential equations

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Harmonic oscillator - Wikipedia, the free encyclopedia (1810 words)
	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 such situation, the frequency of the oscillations is smaller than in the non-damped case, and the amplitude of the oscillations decreases with time.
	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.

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Encyclopedia > Harmonic oscillator

Contents

Simple harmonic oscillator

Driven harmonic oscillator

Damped harmonic oscillator

Damped, driven harmonic oscillator

Full mathematical definition

Important terms

Simple harmonic oscillator

Universal oscillator equation

Transient solution

Steady-state solution

Amplitude part

Phase part

Full solution

Equivalent systems

Applications

Examples

Simple pendulum

Pendulum swinging over turntable

Spring-mass system

References

See also

External links