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A **normal mode** of an oscillating 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 the fixed frequencies. These fixed frequencies of the normal modes of a system are known as its natural frequencies or resonant frequencies. 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. When relating to music, normal modes of vibrating instruments (strings, air pipes, drums, etc.) are called "harmonics" or "overtones".

**Oscillation** is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. The term *vibration* is precisely used to describe mechanical oscillation. Familiar examples of oscillation include a swinging pendulum and alternating current.

**Natural frequency**, also known as **eigenfrequency**, is the frequency at which a system tends to oscillate in the absence of any driving or damping force.

In mechanical systems, **resonance** is a phenomenon that occurs when the frequency at which a force is periodically applied is equal or nearly equal to one of the natural frequencies of the system on which it acts. This causes the system to oscillate with larger amplitude than when the force is applied at other frequencies.

- General definitions
- Mode
- Mode numbers
- Nodes
- In mechanical systems
- Coupled oscillators
- Standing waves
- Elastic solids
- In quantum mechanics
- In seismology
- See also
- Sources
- External links

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 to each other.

The **superposition principle**, also known as **superposition property**, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input *A* produces response *X* and input *B* produces response *Y* then input produces response.

In mathematics, **orthogonality** is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms. Two elements *u* and *v* of a vector space with bilinear form *B* are **orthogonal** when *B*(*u*, *v*) = 0. Depending on the bilinear form, the vector space may contain nonzero self-orthogonal vectors. In the case of function spaces, families of orthogonal functions are used to form a basis.

In physics and engineering, for a dynamical system according to wave theory, a **mode** is a standing wave state of excitation, in which all the components of the system will be affected sinusoidally under a specified fixed frequency.

In mathematics, a **dynamical system** is a system in which a function describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a lake.

In physics, a **wave** is a disturbance that transfers energy through matter or space, with little or no associated mass transport. Waves consist of oscillations or vibrations of a physical medium or a field, around relatively fixed locations. From the perspective of mathematics, waves, as functions of time and space, are a class of signals.

In physics, a **standing wave**, also known as a **stationary wave**, is a wave which 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 time, and the oscillations at different points throughout the wave are in phase. The locations at which the amplitude is minimum are called nodes, and the locations where the amplitude is maximum are called antinodes.

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.

As classical examples, there are:

- 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, normal modes of vibrating instruments (strings, air pipes, drums, etc.) are called "harmonics" or "overtones".
- The concept of normal modes also finds application in optics, quantum mechanics, and molecular dynamics.

**Music** is an art form and cultural activity whose medium is sound organized in time. General definitions of music include common elements such as pitch, rhythm, dynamics, and the sonic qualities of timbre and texture. Different styles or types of music may emphasize, de-emphasize or omit some of these elements. Music is performed with a vast range of instruments and vocal techniques ranging from singing to rapping; there are solely instrumental pieces, solely vocal pieces and pieces that combine singing and instruments. The word derives from Greek μουσική . See glossary of musical terminology.

**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, ultraviolet, and infrared light. Because light is an electromagnetic wave, other forms of electromagnetic radiation such as X-rays, microwaves, and radio waves exhibit similar properties.

**Quantum mechanics**, including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.

Most dynamical system can be excited under several modes. Each mode is characterized by one or several frequencies, according to the modal variable field. For example, a vibrating rope in the 2D space is defined by a single-frequency (1D axial displacement), but a vibrating rope in the 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.

From all the modes of a dynamical system, the *normal* or *dominant* mode of a system, 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, will be the mode imposing the maximum amplitude of the modal variable.

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, 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 antisymmetric (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 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.

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.

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.

Consider two equal bodies (not affected by gravity), each of mass *m*, attached to three springs, each with spring constant *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 of the left mass, and *x*_{2}(*t*) to denote the displacement of the right mass.

If one denotes acceleration (the second derivative of *x*(*t*) with respect to time) as , the equations of motion are:

Since we expect oscillatory motion of a normal mode (where ω is the same for both masses), we try:

Substituting these into the equations of motion gives us:

Since the exponential factor is common to all terms, we omit it and simplify:

And in matrix representation:

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 i.e. is not invertible. It follows that the determinant of the matrix must be equal to 0, so:

Solving for , we have two positive solutions:

If we substitute ω_{1} into the matrix and solve for (*A*_{1}, *A*_{2}), we get (1, 1). If we substitute ω_{2}, we get (1, −1). (These vectors are eigenvectors, and the frequencies are eigenvalues.)

The first normal mode is:

Which corresponds to both masses moving in the same direction at the same time. This mode is called antisymmetric.

The second normal mode is:

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}, φ_{1}, and φ_{2}, are determined by the initial conditions of the problem.

The process demonstrated here can be generalized and formulated using the formalism of Lagrangian mechanics or Hamiltonian mechanics.

A standing wave 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 and in phase (reaching the equilibrium point together), but each has a different amplitude.

The general form of a standing wave is:

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 (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 normal modes (usually numbered *n* = 1, 2, 3, ...). If the problem is not bounded, there is a continuous spectrum of normal modes.

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).

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 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, 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 modes, 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:

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

By knowing the thermodynamic formula,

the entropy per normal mode is:

The free energy is:

which, for *kT* >> *hν*, tends to:

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:

The integration is performed over all frequencies of the crystal. Then the internal energy *U* will be given by:

In quantum mechanics, a state of a system is described by a wavefunction which solves the Schrödinger equation. The square of the absolute value of , i.e.

is the probability density to measure the particle in place *x* at time *t*.

Usually, when involving some sort of potential, the wavefunction is decomposed into a superposition of energy eigenstates, each oscillating with frequency of . Thus, one may write

The eigenstates have a physical meaning further than an orthonormal basis. 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.

Normal modes are generated in the earth from long wavelength seismic waves 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) 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) and do not exist in fluid outer core. Radial modes are just a subset of spheroidal modes with l=0. The degeneracy doesn’t exist on Earth as it is broken by rotation, ellipticity and 3D heterogeneous velocity and density structure.

We either assume that each mode can be isolated, the self-coupling approximation, or that many modes close in frequency resonant, the cross-coupling approximation. Self-coupling will change just the phase velocity and not the number of waves around a great circle resulting in a stretching or shrinking of standing wave pattern. Cross-coupling can be caused by rotation of Earth leading to mixing of fundamental spheroidal and toroidal modes, or by aspherical mantle structure or Earth’s ellipticity.

- Blevins, Robert D. (2001).
*Formulas for natural frequency and mode shape*(Reprint ed.). Malabar, Florida: Krieger Pub. ISBN 978-1575241845. - Tzou, H.S.; Bergman, L.A., eds. (2008).
*Dynamics and Control of Distributed Systems*. Cambridge [England]: Cambridge University Press. ISBN 978-0521033749. - Shearer, Peter M. (2009).
*Introduction to seismology*(2nd ed.). Cambridge: Cambridge University Press. pp. 231–237. ISBN 9780521882101.

**Frequency** is the number of occurrences of a repeating event per unit of time. It is also referred to as **temporal frequency**, which emphasizes the contrast to spatial frequency and angular frequency. The

In classical mechanics, a **harmonic oscillator** is a system that, when displaced from its equilibrium position, experiences a restoring force *F* proportional to the displacement *x*:

In mechanics and physics, **simple harmonic motion** is a special type of periodic motion or oscillation motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.

The **wave equation** is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves or light waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics.

The **quantum harmonic oscillator** is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.

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. Often designated a quasiparticle, it represents an excited state in the quantum mechanical quantization of the modes of vibrations of elastic structures of interacting particles.

In special relativity, a **four-vector** is an object with four components, which transform in a specific way under Lorentz transformation. Specifically, a four-vector is an element of a four-dimensional vector space considered as a representation space of the standard representation of the Lorentz group, the (½,½) representation. It differs from a Euclidean vector in how its magnitude is determined. The transformations that preserve this magnitude are the Lorentz transformations, which include spatial rotations and boosts.

In physics, a **wave vector** is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave, and its direction is ordinarily the direction of wave propagation.

In applied mechanics, **bending** characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element.

In theoretical physics, the (one-dimensional) **nonlinear Schrödinger equation** (**NLSE**) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides and to Bose-Einstein condensates confined to highly anisotropic cigar-shaped traps, in the mean-field regime. Additionally, the equation appears in the studies of small-amplitude gravity waves on the surface of deep inviscid (zero-viscosity) water; the Langmuir waves in hot plasmas; the propagation of plane-diffracted wave beams in the focusing regions of the ionosphere; the propagation of Davydov's alpha-helix solitons, which are responsible for energy transport along molecular chains; and many others. More generally, the NLSE appears as one of universal equations that describe the evolution of slowly varying packets of quasi-monochromatic waves in weakly nonlinear media that have dispersion. Unlike the linear Schrödinger equation, the NLSE never describes the time evolution of a quantum state. The 1D NLSE is an example of an integrable model.

In classical mechanics, **anharmonicity** is the deviation of a system from being a harmonic oscillator. An oscillator that is not oscillating in harmonic motion is known as an anharmonic oscillator where the system can be approximated to a harmonic oscillator and the anharmonicity can be calculated using perturbation theory. If the anharmonicity is large, then other numerical techniques have to be used.

The **vibrational partition function** traditionally refers to the component of the canonical partition function resulting from the vibrational degrees of freedom of a system. The vibrational partition function is only well-defined in model systems where the vibrational motion is relatively uncoupled with the system's other degrees of freedom.

Structural analysis is mainly concerned with finding out the behavior of a physical structure when subjected to force. This action can be in the form of load due to the weight of things such as people, furniture, wind, snow, etc. or some other kind of excitation such as an earthquake, shaking of the ground due to a blast nearby, etc. In essence all these loads are dynamic, including the self-weight of the structure because at some point in time these loads were not there. The distinction is made between the dynamic and the static analysis on the basis of whether the applied action has enough acceleration in comparison to the structure's natural frequency. If a load is applied sufficiently slowly, the inertia forces can be ignored and the analysis can be simplified as static analysis. **Structural dynamics**, therefore, is a type of structural analysis which covers the behavior of structures subjected to dynamic loading. Dynamic loads include people, wind, waves, traffic, earthquakes, and blasts. Any structure can be subjected to dynamic loading. Dynamic analysis can be used to find dynamic displacements, time history, and modal analysis.

**Vibration** is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin *vibrationem*. The oscillations may be periodic, such as the motion of a pendulum—or random, such as the movement of a tire on a gravel road.

**Precursors** are characteristic wave patterns caused by dispersion of an impulse's frequency components as it propagates through a medium. Classically, precursors precede the main signal, although in certain situations they may also follow it. Precursor phenomena exist for all types of waves, as their appearance is only predicated on the prominence of dispersion effects in a given mode of wave propagation. This non-specificity has been confirmed by the observation of precursor patterns in different types of electromagnetic radiation as well as in fluid surface waves and seismic waves.

The **vibration of plates** is a special case of the more general problem of mechanical vibrations. The equations governing the motion of plates are simpler than those for general three-dimensional objects because one of the dimensions of a plate is much smaller than the other two. This suggests that a two-dimensional plate theory will give an excellent approximation to the actual three-dimensional motion of a plate-like object, and indeed that is found to be true.

**Kapitza's pendulum** or **Kapitza pendulum** is a rigid pendulum in which the pivot point vibrates in a vertical direction, up and down. It is named after Russian Nobel laureate physicist Pyotr Kapitza, who in 1951 developed a theory which successfully explains some of its unusual properties. The unique feature of the Kapitza pendulum is that the vibrating suspension can cause it to balance stably in an inverted position, with the bob above the suspension point. In the usual pendulum with a fixed suspension, the only stable equilibrium position is with the bob hanging below the suspension point; the inverted position is a point of unstable equilibrium, and the smallest perturbation moves the pendulum out of equilibrium. In nonlinear control theory the Kapitza pendulum is used as an example of a parametric oscillator that demonstrates the concept of "dynamic stabilization".

**Cavity optomechanics** is a branch of physics which focuses on the interaction between light and mechanical objects on low-energy scales. It is a cross field of optics, quantum optics, solid-state physics and materials science. The motivation for research on cavity optomechanics comes from fundamental effects of quantum theory and gravity, as well as technological applications.

The Rayleigh's quotient represents a quick method to estimate the natural frequency of a multi-degree-of-freedom vibration system, in which the mass and the stiffness matrices are known.

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