Oscillation is the repetitive or
periodic variation, typically in
time, of some measure about a central value (often a point of
equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging
pendulum and
alternating current. Oscillations can be used in physics to approximate complex interactions, such as those between atoms.
Oscillations occur not only in mechanical systems but also in
dynamic systems in virtually every area of science: for example the beating of the
human heart (for circulation),
business cycles in
economics,
predator–prey
Predation is a biological interaction where one organism, the predator, kills and eats another organism, its prey. It is one of a family of common feeding behaviours that includes parasitism and micropredation (which usually do not kill th ...
population cycles in
ecology, geothermal
geyser
A geyser (, ) is a spring characterized by an intermittent discharge of water ejected turbulently and accompanied by steam. As a fairly rare phenomenon, the formation of geysers is due to particular hydrogeological conditions that exist only in ...
s in
geology, vibration of strings in
guitar and other
string instrument
String instruments, stringed instruments, or chordophones are musical instruments that produce sound from vibrating strings when a performer plays or sounds the strings in some manner.
Musicians play some string instruments by plucking the ...
s, periodic firing of
nerve cells in the brain, and the periodic swelling of
Cepheid variable stars in
astronomy. The term ''
vibration'' is precisely used to describe a mechanical oscillation.
Oscillation, especially rapid oscillation, may be an undesirable phenomenon in
process control
An industrial process control in continuous production processes is a discipline that uses industrial control systems to achieve a production level of consistency, economy and safety which could not be achieved purely by human manual control. I ...
and
control theory (e.g. in
sliding mode control), where the aim is
convergence to stable state. In these cases it is called chattering or flapping, as in
valve chatter, and
route flapping.
Simple harmonic
The simplest mechanical oscillating system is a
weight attached to a
linear spring subject to only
weight and
tension. Such a system may be approximated on an air table or ice surface. The system is in an
equilibrium state when the spring is static. If the system is displaced from the equilibrium, there is a net ''restoring force'' on the mass, tending to bring it back to equilibrium. However, in moving the mass back to the equilibrium position, it has acquired
momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
which keeps it moving beyond that position, establishing a new restoring force in the opposite sense. If a constant
force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
such as
gravity is added to the system, the point of equilibrium is shifted. The time taken for an oscillation to occur is often referred to as the oscillatory ''period''.
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. In the spring-mass system, oscillations occur because, at the
static equilibrium displacement, the mass has
kinetic energy which is converted into
potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of potential energy include the gravitational potentia ...
stored in the spring at the extremes of its path. The spring-mass system illustrates some common features of oscillation, namely the existence of an equilibrium and the presence of a restoring force which grows stronger the further the system deviates from equilibrium.
In the case of the spring-mass system,
Hooke's law states that the restoring force of a spring is:
By using
Newton's second law, the differential equation can be derived:
where
The solution to this differential equation produces a sinusoidal position function:
where is the frequency of the oscillation, is the amplitude, and is the
phase shift of the function. These are determined by the initial conditions of the system. Because cosine oscillates between 1 and −1 infinitely, our spring-mass system would oscillate between the positive and negative amplitude forever without friction.
Two-dimensional oscillators
In two or three dimensions, harmonic oscillators behave similarly to one dimension. The simplest example of this is an
isotropic
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
oscillator, where the restoring force is proportional to the displacement from equilibrium with the same restorative constant in all directions.
This produces a similar solution, but now there is a different equation for every direction.
Anisotropic oscillators
With
anisotropic
Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physic ...
oscillators, different directions have different constants of restoring forces. The solution is similar to isotropic oscillators, but there is a different frequency in each direction. Varying the frequencies relative to each other can produce interesting results. For example, if the frequency in one direction is twice that of another, a figure eight pattern is produced. If the ratio of frequencies is irrational, the motion is
quasiperiodic. This motion is periodic on each axis, but is not periodic with respect to r, and will never repeat.
Damped oscillations
All real-world oscillator systems are
thermodynamically irreversible. This means there are dissipative processes such as
friction or
electrical resistance which continually convert some of the energy stored in the oscillator into heat in the environment. This is called damping. Thus, oscillations tend to decay with time unless there is some net source of energy into the system. The simplest description of this decay process can be illustrated by oscillation decay of the harmonic oscillator.
Damped oscillators are created when a resistive force is introduced, which is dependent on the first derivative of the position, or in this case velocity. The differential equation created by Newton's second law adds in this resistive force with an arbitrary constant . This example assumes a linear dependence on velocity.
This equation can be rewritten as before:
where
.
This produces the general solution:
where
.
The exponential term outside of the parenthesis is the
decay function and is the damping coefficient. There are 3 categories of damped oscillators: under-damped, where ; over-damped, where ; and critically damped, where .
Driven oscillations
In addition, an oscillating system may be subject to some external force, as when an AC
circuit
Circuit may refer to:
Science and technology
Electrical engineering
* Electrical circuit, a complete electrical network with a closed-loop giving a return path for current
** Analog circuit, uses continuous signal levels
** Balanced circu ...
is connected to an outside power source. In this case the oscillation is said to be ''
driven''.
The simplest example of this is a spring-mass system with a
sinusoidal
A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in m ...
driving force.
where
This gives the solution:
where
and
The second term of is the transient solution to the differential equation. The transient solution can be found by using the initial conditions of the system.
Some systems can be excited by energy transfer from the environment. This transfer typically occurs where systems are embedded in some
fluid
In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
flow. For example, the phenomenon of
flutter
Flutter may refer to:
Technology
* Aeroelastic flutter, a rapid self-feeding motion, potentially destructive, that is excited by aerodynamic forces in aircraft and bridges
* Flutter (American company), a gesture recognition technology company acqu ...
in
aerodynamics occurs when an arbitrarily small displacement of an
aircraft wing (from its equilibrium) results in an increase in the
angle of attack
In fluid dynamics, angle of attack (AOA, α, or \alpha) is the angle between a reference line on a body (often the chord line of an airfoil) and the vector representing the relative motion between the body and the fluid through which it is m ...
of the wing on the air flow and a consequential increase in
lift coefficient, leading to a still greater displacement. At sufficiently large displacements, the
stiffness
Stiffness is the extent to which an object resists deformation in response to an applied force.
The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is.
Calculations
The stiffness, k, of a b ...
of the wing dominates to provide the restoring force that enables an oscillation.
Resonance
Resonance occurs in a damped driven oscillator when ω = ω
0, that is, when the driving frequency is equal to the natural frequency of the system. When this occurs, the denominator of the amplitude is minimized, which maximizes the amplitude of the oscillations.
Coupled oscillations
The harmonic oscillator and the systems it models have a single
degree of freedom. More complicated systems have more degrees of freedom, for example, two masses and three springs (each mass being attached to fixed points and to each other). In such cases, the behavior of each variable influences that of the others. This leads to a ''coupling'' of the oscillations of the individual degrees of freedom. For example, two pendulum clocks (of identical frequency) mounted on a common wall will tend to synchronise. This
phenomenon was first observed by
Christiaan Huygens
Christiaan Huygens, Lord of Zeelhem, ( , , ; also spelled Huyghens; la, Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor, who is regarded as one of the greatest scientists of ...
in 1665. The apparent motions of the compound oscillations typically appears very complicated but a more economic, computationally simpler and conceptually deeper description is given by resolving the motion into
normal mode
A normal mode of a dynamical system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixed frequencies. ...
s.
The simplest form of coupled oscillators is a 3 spring, 2 mass system, where masses and spring constants are the same. This problem begins with deriving Newton's second law for both masses.
The equations are then generalized into matrix form.
where
,
, and
The values of and can be substituted into the matrices.
These matrices can now be plugged into the general solution.
The determinant of this matrix yields a quadratic equation.
Depending on the starting point of the masses, this system has 2 possible frequencies (or a combination of the two). If the masses are started with their displacements in the same direction, the frequency is that of a single mass system, because the middle spring is never extended. If the two masses are started in opposite directions, the second, faster frequency is the frequency of the system.
More special cases are the coupled oscillators where energy alternates between two forms of oscillation. Well-known is the
Wilberforce pendulum, where the oscillation alternates between the elongation of a vertical spring and the rotation of an object at the end of that spring.
Coupled oscillators are a common description of two related, but different phenomena. One case is where both oscillations affect each other mutually, which usually leads to the occurrence of a single, entrained oscillation state, where both oscillate with a ''compromise frequency''. Another case is where one external oscillation affects an internal oscillation, but is not affected by this. In this case the regions of synchronization, known as
Arnold Tongues
In mathematics, particularly in dynamical systems, Arnold tongues (named after Vladimir Arnold) Section 12 in page 78 has a figure showing Arnold tongues. are a pictorial phenomenon that occur when visualizing how the rotation number of a dynami ...
, can lead to highly complex phenomena as for instance chaotic dynamics.
Small oscillation approximation
In physics, a system with a set of conservative forces and an equilibrium point can be approximated as a harmonic oscillator near equilibrium. An example of this is the
Lennard-Jones potential
The Lennard-Jones potential (also termed the LJ potential or 12-6 potential) is an intermolecular pair potential. Out of all the intermolecular potentials, the Lennard-Jones potential is probably the one that has been the most extensively studied ...
, where the potential is given by: