Vibrational Modes
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A normal mode of a
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
is a pattern of motion in which all parts of the system move
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 ...
ly with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixed frequencies. These fixed frequencies of the normal modes of a system are known as its
natural frequencies The fundamental frequency, often referred to simply as the ''fundamental'', is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. In ...
or
resonant frequencies Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillati ...
. A physical object, such as a building, bridge, or molecule, has a set of normal modes and their natural frequencies that depend on its structure, materials and boundary conditions. The most general motion of a system is a superposition of its normal modes. The modes are normal in the sense that they can move independently, that is to say that an excitation of one mode will never cause motion of a different mode. In mathematical terms, normal modes are
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
to each other.


General definitions


Mode

In the wave theory of physics and engineering, a mode in a
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
is a
standing wave In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect ...
state of excitation, in which all the components of the system will be affected sinusoidally at a fixed frequency associated with that mode. Because no real system can perfectly fit under the standing wave framework, the ''mode'' concept is taken as a general characterization of specific states of oscillation, thus treating the dynamic system in a ''linear'' fashion, in which linear superposition of states can be performed. Classical examples include * In a mechanical dynamical system, a vibrating rope is the most clear example of a mode, in which the rope is the medium, the stress on the rope is the excitation, and the displacement of the rope with respect to its static state is the modal variable. * In an acoustic dynamical system, a single sound pitch is a mode, in which the air is the medium, the sound pressure in the air is the excitation, and the displacement of the air molecules is the modal variable. * In a structural dynamical system, a high tall building oscillating under its most flexural axis is a mode, in which all the material of the building -under the proper numerical simplifications- is the medium, the seismic/wind/environmental solicitations are the excitations and the displacements are the modal variable. * In an electrical dynamical system, a resonant cavity made of thin metal walls, enclosing a hollow space, for a particle accelerator is a pure standing wave system, and thus an example of a mode, in which the hollow space of the cavity is the medium, the RF source (a Klystron or another RF source) is the excitation and the electromagnetic field is the modal variable. * When relating to
music Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact definitions of music vary considerably around the world, though it is an aspect ...
, normal modes of vibrating instruments (strings, air pipes, drums, etc.) are called "
overtones An overtone is any resonant frequency above the fundamental frequency of a sound. (An overtone may or may not be a harmonic) In other words, overtones are all pitches higher than the lowest pitch within an individual sound; the fundamental i ...
". The concept of normal modes also finds application in other dynamical systems, such as
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
,
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
,
atmospheric dynamics Meteorology is a branch of the atmospheric sciences (which include atmospheric chemistry and physics) with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did not ...
and
molecular dynamics Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of the ...
. Most dynamical systems can be excited in several modes, possibly simultaneously. Each mode is characterized by one or several frequencies, according to the modal variable field. For example, a vibrating rope in 2D space is defined by a single-frequency (1D axial displacement), but a vibrating rope in 3D space is defined by two frequencies (2D axial displacement). For a given amplitude on the modal variable, each mode will store a specific amount of energy because of the sinusoidal excitation. The ''normal'' or ''dominant'' mode of a system with multiple modes will be the mode storing the minimum amount of energy for a given amplitude of the modal variable, or, equivalently, for a given stored amount of energy, the dominant mode will be the mode imposing the maximum amplitude of the modal variable.


Mode numbers

A mode of vibration is characterized by a modal frequency and a mode shape. It is numbered according to the number of half waves in the vibration. For example, if a vibrating beam with both ends pinned displayed a mode shape of half of a sine wave (one peak on the vibrating beam) it would be vibrating in mode 1. If it had a full sine wave (one peak and one trough) it would be vibrating in mode 2. In a system with two or more dimensions, such as the pictured disk, each dimension is given a mode number. Using
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
, we have a radial coordinate and an angular coordinate. If one measured from the center outward along the radial coordinate one would encounter a full wave, so the mode number in the radial direction is 2. The other direction is trickier, because only half of the disk is considered due to the anti-symmetric (also called skew-symmetry) nature of a disk's vibration in the angular direction. Thus, measuring 180° along the angular direction you would encounter a half wave, so the mode number in the angular direction is 1. So the mode number of the system is 2–1 or 1–2, depending on which coordinate is considered the "first" and which is considered the "second" coordinate (so it is important to always indicate which mode number matches with each coordinate direction). In linear systems each mode is entirely independent of all other modes. In general all modes have different frequencies (with lower modes having lower frequencies) and different mode shapes.


Nodes

In a one-dimensional system at a given mode the vibration will have nodes, or places where the displacement is always zero. These nodes correspond to points in the mode shape where the mode shape is zero. Since the vibration of a system is given by the mode shape multiplied by a time function, the displacement of the node points remain zero at all times. When expanded to a two dimensional system, these nodes become lines where the displacement is always zero. If you watch the animation above you will see two circles (one about halfway between the edge and center, and the other on the edge itself) and a straight line bisecting the disk, where the displacement is close to zero. In an idealized system these lines equal zero exactly, as shown to the right.


In mechanical systems


Coupled oscillators

Consider two equal bodies (not affected by gravity), each of
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
''m'', attached to three springs, each with
spring constant In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring (device), spring by some distance () Proportionality (mathematics)#Direct_proportionality, scales linearly with respect to that ...
''k''. They are attached in the following manner, forming a system that is physically symmetric: : where the edge points are fixed and cannot move. We'll use ''x''1(''t'') to denote the horizontal
displacement Displacement may refer to: Physical sciences Mathematics and Physics *Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
of the left mass, and ''x''2(''t'') to denote the displacement of the right mass. If one denotes acceleration (the second
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
of ''x''(''t'') with respect to time) as \scriptstyle \ddot x, the
equations of motion In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Ver ...
are: :\begin m \ddot x_1 &= - k x_1 + k (x_2 - x_1) = - 2 k x_1 + k x_2 \\ m \ddot x_2 &= - k x_2 + k (x_1 - x_2) = - 2 k x_2 + k x_1 \end Since we expect oscillatory motion of a normal mode (where ω is the same for both masses), we try: :\begin x_1(t) &= A_1 e^ \\ x_2(t) &= A_2 e^ \end Substituting these into the equations of motion gives us: :\begin -\omega^2 m A_1 e^ &= - 2 k A_1 e^ + k A_2 e^ \\ -\omega^2 m A_2 e^ &= k A_1 e^ - 2 k A_2 e^ \end Since the exponential factor is common to all terms, we omit it and simplify: :\begin (\omega^2 m - 2 k) A_1 + k A_2 &= 0 \\ k A_1 + (\omega^2 m - 2 k) A_2 &= 0 \end And in
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
representation: :\begin \omega^2 m - 2 k & k \\ k & \omega^2 m - 2 k \end \begin A_1 \\ A_2 \end = 0 If the matrix on the left is invertible, the unique solution is the trivial solution (''A''1, ''A''2) = (''x''1, ''x''2) = (0,0). The non trivial solutions are to be found for those values of ω whereby the matrix on the left is
singular Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular homology * SINGULAR, an open source Computer Algebra System (CAS) * Singular or sounder, a group of boar, ...
i.e. is not invertible. It follows that the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of the matrix must be equal to 0, so: : (\omega^2 m - 2 k)^2 - k^2 = 0 Solving for \omega, we have two positive solutions: :\begin \omega_1 &= \sqrt \\ \omega_2 &= \sqrt \end If we substitute ω1 into the matrix and solve for (''A''1, ''A''2), we get (1, 1). If we substitute ω2, we get (1, −1). (These vectors are
eigenvector In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s, and the frequencies are
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s.) The first normal mode is: :\vec \eta_1 = \begin x^1_1(t) \\ x^1_2(t) \end = c_1 \begin 1 \\ 1 \end \cos Which corresponds to both masses moving in the same direction at the same time. This mode is called antisymmetric. The second normal mode is: :\vec \eta_2 = \begin x^2_1(t) \\ x^2_2(t) \end = c_2 \begin 1 \\ -1 \end \cos This corresponds to the masses moving in the opposite directions, while the center of mass remains stationary. This mode is called symmetric. The general solution is a superposition of the normal modes where ''c''1, ''c''2, φ1, and φ2, are determined by the
initial condition In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted ''t'' = 0). For ...
s of the problem. The process demonstrated here can be generalized and formulated using the formalism of
Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Lou ...
or
Hamiltonian mechanics Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
.


Standing waves

A
standing wave In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect ...
is a continuous form of normal mode. In a standing wave, all the space elements (i.e. (''x'', ''y'', ''z'') coordinates) are oscillating in the same
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
and in
phase Phase or phases may refer to: Science *State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter), a region of space throughout which all physical properties are essentially uniform * Phase space, a mathematic ...
(reaching the equilibrium point together), but each has a different amplitude. The general form of a standing wave is: : \Psi(t) = f(x,y,z) (A\cos(\omega t) + B\sin(\omega t)) where ''ƒ''(''x'', ''y'', ''z'') represents the dependence of amplitude on location and the cosine\sine are the oscillations in time. Physically, standing waves are formed by the
interference Interference is the act of interfering, invading, or poaching. Interference may also refer to: Communications * Interference (communication), anything which alters, modifies, or disrupts a message * Adjacent-channel interference, caused by extr ...
(superposition) of waves and their reflections (although one may also say the opposite; that a moving wave is a superposition of standing waves). The geometric shape of the medium determines what would be the interference pattern, thus determines the ''ƒ''(''x'', ''y'', ''z'') form of the standing wave. This space-dependence is called a normal mode. Usually, for problems with continuous dependence on (''x'', ''y'', ''z'') there is no single or finite number of normal modes, but there are infinitely many normal modes. If the problem is bounded (i.e. it is defined on a finite section of space) there are
countably many In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
normal modes (usually numbered ''n'' = 1, 2, 3, ...). If the problem is not bounded, there is a continuous spectrum of normal modes.


Elastic solids

In any solid at any temperature, the primary particles (e.g. atoms or molecules) are not stationary, but rather vibrate about mean positions. In insulators the capacity of the solid to store thermal energy is due almost entirely to these vibrations. Many physical properties of the solid (e.g. modulus of elasticity) can be predicted given knowledge of the frequencies with which the particles vibrate. The simplest assumption (by Einstein) is that all the particles oscillate about their mean positions with the same natural frequency ''ν''. This is equivalent to the assumption that all atoms vibrate independently with a frequency ''ν''. Einstein also assumed that the allowed energy states of these oscillations are harmonics, or integral multiples of ''hν''. The spectrum of waveforms can be described mathematically using a Fourier series of sinusoidal density fluctuations (or thermal
phonons In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phonon is an excited state in the quantum mechanic ...
). Debye subsequently recognized that each oscillator is intimately coupled to its neighboring oscillators at all times. Thus, by replacing Einstein's identical uncoupled oscillators with the same number of coupled oscillators, Debye correlated the elastic vibrations of a one-dimensional solid with the number of mathematically special modes of vibration of a stretched string (see figure). The pure tone of lowest pitch or frequency is referred to as the fundamental and the multiples of that frequency are called its harmonic overtones. He assigned to one of the oscillators the frequency of the fundamental vibration of the whole block of solid. He assigned to the remaining oscillators the frequencies of the harmonics of that fundamental, with the highest of all these frequencies being limited by the motion of the smallest primary unit. The normal modes of vibration of a crystal are in general superpositions of many overtones, each with an appropriate amplitude and phase. Longer wavelength (low frequency)
phonons In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phonon is an excited state in the quantum mechanic ...
are exactly those acoustical vibrations which are considered in the theory of sound. Both longitudinal and transverse waves can be propagated through a solid, while, in general, only longitudinal waves are supported by fluids. In the
longitudinal mode A longitudinal mode of a resonant cavity is a particular standing wave pattern formed by waves confined in the cavity. The longitudinal modes correspond to the wavelengths of the wave which are reinforced by constructive interference after many ...
, the displacement of particles from their positions of equilibrium coincides with the propagation direction of the wave. Mechanical longitudinal waves have been also referred to as ''compression waves''. For
transverse mode A transverse mode of electromagnetic radiation is a particular electromagnetic field pattern of the radiation in the plane perpendicular (i.e., transverse) to the radiation's propagation direction. Transverse modes occur in radio waves and microwav ...
s, individual particles move perpendicular to the propagation of the wave. According to quantum theory, the mean energy of a normal vibrational mode of a crystalline solid with characteristic frequency ''ν'' is: : E(\nu) = \frach\nu + \frac The term (1/2)''hν'' represents the "zero-point energy", or the energy which an oscillator will have at absolute zero. ''E''(''ν'') tends to the classic value ''kT'' at high temperatures : E(\nu) = kT\left + \frac\left(\frac\right)^2 + O\left(\frac\right)^4 + \cdots\right/math> By knowing the thermodynamic formula, : \left( \frac\right)_ = \frac the entropy per normal mode is: : \begin S\left(\nu\right) &= \int_0^T\fracE\left(\nu\right)\frac \\
0pt PT, Pt, or pt may refer to: Arts and entertainment * ''P.T.'' (video game), acronym for ''Playable Teaser'', a short video game released to promote the cancelled video game ''Silent Hills'' * Porcupine Tree, a British progressive rock group ...
&= \frac - k\log\left(1 - e^\right) \end The free energy is: : F(\nu) = E - TS=kT\log \left(1-e^\right) which, for ''kT'' >> ''hν'', tends to: :F(\nu) = kT\log \left(\frac\right) In order to calculate the internal energy and the specific heat, we must know the number of normal vibrational modes a frequency between the values ''ν'' and ''ν'' + ''dν''. Allow this number to be ''f''(''ν'')d''ν''. Since the total number of normal modes is 3''N'', the function ''f''(''ν'') is given by: : \int f(\nu)\,d\nu = 3N The integration is performed over all frequencies of the crystal. Then the internal energy ''U'' will be given by: : U = \int f(\nu)E(\nu)\,d\nu


In quantum mechanics

In
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, a state \ , \psi \rang of a system is described by a
wavefunction A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
\ \psi (x, t) which solves the
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
. The square of the absolute value of \ \psi , i.e. : \ P(x,t) = , \psi (x,t), ^2 is the
probability density In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
to measure the particle in
place Place may refer to: Geography * Place (United States Census Bureau), defined as any concentration of population ** Census-designated place, a populated area lacking its own Municipality, municipal government * "Place", a type of street or road ...
''x'' at
time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...
 ''t''. Usually, when involving some sort of
potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple re ...
, the wavefunction is decomposed into a superposition of energy eigenstates, each oscillating with frequency of \omega = E_n / \hbar . Thus, one may write : , \psi (t) \rang = \sum_n , n\rang \left\langle n , \psi ( t=0) \right\rangle e^ The eigenstates have a physical meaning further than an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
. When the energy of the system is measured, the wavefunction collapses into one of its eigenstates and so the particle wavefunction is described by the pure eigenstate corresponding to the measured
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
.


In seismology

Normal modes are generated in the Earth from long wavelength
seismic waves A seismic wave is a wave of acoustic energy that travels through the Earth. It can result from an earthquake, volcanic eruption, magma movement, a large landslide, and a large man-made explosion that produces low-frequency acoustic energy. ...
from large earthquakes interfering to form standing waves. For an elastic, isotropic, homogeneous sphere, spheroidal, toroidal and radial (or breathing) modes arise. Spheroidal modes only involve P and SV waves (like
Rayleigh waves Rayleigh waves are a type of surface acoustic wave that travel along the surface of solids. They can be produced in materials in many ways, such as by a localized impact or by piezo-electric transduction, and are frequently used in non-destructi ...
) and depend on overtone number ''n'' and angular order ''l'' but have degeneracy of azimuthal order ''m''. Increasing ''l'' concentrates fundamental branch closer to surface and at large ''l'' this tends to Rayleigh waves. Toroidal modes only involve SH waves (like
Love waves In elastodynamics, Love waves, named after Augustus Edward Hough Love, are horizontally polarized surface waves. The Love wave is a result of the interference of many shear waves (S-waves) guided by an elastic layer, which is ''welded'' to a ...
) and do not exist in fluid outer core. Radial modes are just a subset of spheroidal modes with ''l=0''. The degeneracy does not exist on Earth as it is broken by rotation, ellipticity and 3D heterogeneous velocity and density structure. It may be assumed that each mode can be isolated, the self-coupling approximation, or that many modes close in frequency
resonate Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillat ...
, the cross-coupling approximation. Self-coupling will solely change the phase velocity and not the number of waves around a great circle, resulting in a stretching or shrinking of standing wave pattern. Modal cross-coupling occurs due to the rotation of the Earth, from aspherical elastic structure, or due to Earth's ellipticity and leads to a mixing of fundamental spheroidal and toroidal modes.


See also

*
Antiresonance In the physics of coupled oscillators, antiresonance, by analogy with resonance, is a pronounced minimum in the amplitude of an oscillator at a particular frequency, accompanied by a large, abrupt shift in its oscillation phase. Such frequencies ...
*
Critical speed In solid mechanics, in the field of rotordynamics, the critical speed is the theoretical angular velocity that excites the natural frequency of a rotating object, such as a shaft, propeller, leadscrew, or gear. As the speed of rotation approache ...
*
Harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'': \v ...
*
Harmonic series (music) A harmonic series (also overtone series) is the sequence of harmonics, musical tones, or pure tones whose frequency is an integer multiple of a ''fundamental frequency''. Pitched musical instruments are often based on an acoustic resonator su ...
*
Infrared spectroscopy Infrared spectroscopy (IR spectroscopy or vibrational spectroscopy) is the measurement of the interaction of infrared radiation with matter by absorption, emission, or reflection. It is used to study and identify chemical substances or function ...
*
Leaky mode Leakey may refer to: *Leakey, Texas, U.S., a city *Leakey (crater), a lunar impact crater *Leakey (surname) *7958 Leakey, an asteroid *Leakey Independent School District, a public school district based in Leakey, Texas See also *Leak (disambiguati ...
*
Mechanical resonance Mechanical resonance is the tendency of a mechanical system to respond at greater amplitude when the frequency of its oscillations matches the system's natural frequency of vibration (its ''resonance frequency'' or ''resonant frequency'') close ...
*
Modal analysis Modal analysis is the study of the dynamic properties of systems in the frequency domain. Examples would include measuring the vibration of a car's body when it is attached to a shaker, or the noise pattern in a room when excited by a loudspeak ...
*
Mode (electromagnetism) The mode of electromagnetic radiation describes the field pattern of the propagating waves. Electromagnetic modes are analogous to the normal modes of vibration in other systems, such as mechanical systems. Some of the classifications of electroma ...
*
Quasinormal mode Quasinormal modes (QNM) are the modes of energy dissipation of a perturbed object or field, ''i.e.'' they describe perturbations of a field that decay in time. Example A familiar example is the perturbation (gentle tap) of a wine glass with a kni ...
*
Sturm–Liouville theory In mathematics and its applications, classical Sturm–Liouville theory is the theory of ''real'' second-order ''linear'' ordinary differential equations of the form: for given coefficient functions , , and , an unknown function ''y = y''(''x'') ...
*
Torsional vibration Torsional vibration is angular vibration of an object—commonly a shaft along its axis of rotation. Torsional vibration is often a concern in power transmission systems using rotating shafts or couplings where it can cause failures if not contr ...
*
Vibrations of a circular membrane A two-dimensional elastic membrane under tension can support transverse vibrations. The properties of an idealized drumhead can be modeled by the vibrations of a circular membrane of uniform thickness, attached to a rigid frame. Due to the ph ...


Sources

* * *


External links


Harvard lecture notes on normal modes
{{DEFAULTSORT:Normal Mode Ordinary differential equations Classical mechanics Quantum mechanics Spectroscopy Singular value decomposition Articles containing video clips