NationMaster - Encyclopedia: Normal mode

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. The frequencies of the normal modes of a system are known as its natural frequencies or resonant frequencies. A physical object, such as a building or bridge, has a set of normal modes (and frequencies) that depend on its structure and composition. Look up mode in Wiktionary, the free dictionary. ... Image File history File links 1D_normal_modes_(280_kB). ... Image File history File links 1D_normal_modes_(280_kB). ... Oscillation is the variation, typically in time, of some measure as seen, for example, in a swinging pendulum. ...

It is common to use a spring-mass system to illustrate a deformable structure. When such a system is excited at one of these natural frequencies, all of the masses move at the same frequency. The phases of the masses are either exactly the same or exactly opposite. The practical significance of this can be illustrated by a mass-spring model of a building. If an earthquake excites the system near one of the natural frequencies, the displacement of one floor with respect to another will be maximum. Obviously, buildings can only withstand this displacement up to a certain point. Being able to model a building and find its normal modes is an easy way to check the safety of a building's design. The concept of normal modes also finds application in wave theory, optics and quantum mechanics. A wave is a disturbance that propagates through space or spacetime, transferring energy and momentum and sometimes angular momentum. ... For the book by Sir Isaac Newton, see Opticks. ... Fig. ...

1 Example - normal modes of coupled oscillators
2 Standing waves
3 Normal modes in quantum mechanics
4 See also
5 External links

Example - normal modes of coupled oscillators

Consider two bodies (not affected by gravity), each of mass M, attached to three springs with stiffness K. They are attached in the following manner: This article or section is in need of attention from an expert on the subject. ... Stiffness is the resistance of an elastic body to deflection or deformation by an applied force. ...

where the edge points are fixed and cannot move. We'll use x₁(t) to denote the displacement of the leftmost mass, and x₂(t) to denote the displacement of the rightmost. Image File history File links Two_masses. ... In Newtonian mechanics, displacement is one of two subtly different quantities measuring distance. ...

If we denote the second derivative of x(t) with respect to time as x″, the equations of motion are: For a non-technical overview of the subject, see Calculus. ...

$M ddot x_1 = - K x_1 + K (x_2 - x_1) ,$

$M ddot x_2 = - K x_2 + K (x_1 - x_2) ,$

Since we expect oscillatory motion, we try:

$x_1(t) = A_1 e^{i omega t} ,$

$x_2(t) = A_2 e^{i omega t} ,$

Substituting these into the equations of motion gives us:

$-omega^2 M A_1 e^{i omega t} = - 2 K A_1 e^{i omega t} + K A_2 e^{i omega t} ,$

$-omega^2 M A_2 e^{i omega t} = K A_1 e^{i omega t} - 2 K A_2 e^{i omega t} ,$

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

$(omega^2 M - 2 K) A_1 + K A_2 = 0 ,$

$K A_1 + (omega^2 M - 2 K) A_2 = 0 ,$

And in matrix representation:

$begin{bmatrix} omega^2 M - 2 K & K K & omega^2 M - 2 K end{bmatrix} begin{pmatrix} A_1 A_2 end{pmatrix} = 0$

For this equation to have a non-trivial solution, the matrix on the left must be singular, therefore the determinant of the matrix must be equal to 0, so: In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ... In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

$(omega^2 M - 2 K)^2 - K^2 = 0 ,$

Solving for $ω$ , we have two solutions:

$omega_1 = sqrt{frac{K}{M}}$ ,

$omega_2 = sqrt{frac{3 K}{M}}$ ,

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.) In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In linear algebra, a scalar λ is called an eigenvalue (in some older texts, a characteristic value) of a linear mapping A if there exists a nonzero vector x such that Ax=λx. ...

The first normal mode is:

$begin{pmatrix} x_1(t) x_2(t) end{pmatrix} = c_1 begin{pmatrix} 1 1 end{pmatrix} cos{(omega_1 t + phi_1)}$

and the second normal mode is:

$begin{pmatrix} x_1(t) x_2(t) end{pmatrix} = c_2 begin{pmatrix} 1 -1 end{pmatrix} cos{(omega_2 t + phi_2)}$

The general solution is a superposition of the normal modes where c₁, c₂, φ₁, and φ₂, are determined by the initial conditions of the problem. 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. ... 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. ...

The process demonstrated here can be generalized and formulated using the formalism of Lagrangian mechanics or Hamiltonian mechanics. Lagrangian mechanics is a re-formulation of classical mechanics introduced by Joseph Louis Lagrange in 1788. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ...

Standing waves

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. A standing wave, also known as a stationary wave, is a wave that remains in a constant position. ... FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. ... This article is about a portion of a periodic process. ... 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. ...

Image:Standing-wave05.png Image File history File links Standing-wave05. ...

The general form of a standing wave is:

Ψ(t) = f (x, y, z)(A cos(ω t) + B sin(ω t))

where f(x, y, z) represents the dependence of amplitude on location and the cosinesine 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 f(x, y, z) form of the standing wave. This space-dependence is called a normal mode. Interference of two circular waves - Wavelength (decreasing bottom to top) and Wave centers distance (increasing to the right). ... 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. ...

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 (a discrete infinity of ) normal modes (usually numbered n = 1,2,3,...). If the problem is not bounded, there is a continuous spectrum of normal modes. In mathematics the term countable set is used to describe the size of a set, e. ... The word spectrum (plural, spectra) has many uses: // Common nouns The Spectrum article explains why so many things are called by this name The spectrum of activity of a drug The political spectrum of opinion The economic spectrum The bipolar spectrum, in psychology The autistic spectrum, in psychology In the...

The allowed frequencies are dependent on the normal modes, as well as on physical constants of the problem (density, tension, pressure, etc.) which set the phase velocity of the wave. The range of all possible normal frequencies is called the frequency spectrum. Usually, each frequency is modulated by the amplitude at which it has arisen, creating a graph of the power spectrum of the oscillations. In physics, density is mass m per unit volume V. For the common case of a homogeneous substance, it is expressed as: where, in SI units: Ï (rho) is the density of the substance, measured in kgÂ·m-3 m is the mass of the substance, measured in kg V is... The use of water pressure - the Captain Cook Memorial Jet in Lake Burley Griffin in Canberra, Australia. ... The phase velocity of a wave is the rate at which the phase of the wave propagates in space. ... Familiar concepts associated with a frequency are colors, musical notes, radio/TV channels, and even the regular rotation of the earth. ... The power spectrum is a plot of the portion of a signals power (energy per unit time) falling within given frequency bins. ...

When relating to music, normal modes of vibrating instruments (strings, air pipes, drums, etc.) are called "harmonics". // Music is an art form consisting of sound and silence expressed through time. ... In acoustics and telecommunication, the harmonic of a wave is a component frequency of the signal that is an integral multiple of the fundamental frequency. ...

Normal modes in quantum mechanics

In quantum mechanics, a state $| psi rang$ of a system is described by a wavefunction $psi (x, t)$ which solves the Schrödinger equation. The square of the absolute value of $psi$ ,i.e. Fig. ... This article discusses the concept of a wavefunction as it relates to quantum mechanics. ... For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ...

$P(x,t) = |psi (x,t)|^2$

is the probability density to measure the particle in place x at time t. In quantum mechanics, a probability amplitude is a complex number-valued function which describes an uncertain or unknown quantity. ... Distance is a numerical description of how far apart objects are at any given moment in time. ... A pocket watch, a device used to tell time Look up time in Wiktionary, the free dictionary. ...

Usually, when involving some sort of potential, the wavefunction is decomposed into a superposition of energy eigenstates, each oscillating with frequency of $omega = E_n / hbar$ . Thus, we may write In physics, a potential may refer to the scalar potential or to the vector potential. ... 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. ... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...

$|psi (t) rang = sum_n |nrang leftlangle n | psi ( t=0) rightrangle e^{-iE_nt/hbar}$

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. In mathematics, an orthonormal basis of an inner product space V(i. ... The framework of quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications. ...

External links

Java simulation of coupled oscillators.
Java simulation of the normal modes of a string, drum, and bar.

Categories: Ordinary differential equations | Classical mechanics | Quantum mechanics | Spectroscopy | Singular value decomposition

Results from FactBites:

Normal mode - Wikipedia, the free encyclopedia (674 words)
	Normal modes in an oscillating system are special solutions where all the parts of the system are oscillating with the same frequency (called normal frequencies or allowed frequencies).
	The concept of normal modes is of vital importance in wave theory, optics and quantum mechanics.
	The allowed frequencies are dependent on the normal modes, as well on physical constants of the problem (density, tension, pressure, etc.) which sets the phase velocity of the wave.

Mode - Wikipedia, the free encyclopedia (145 words)
	Mode (computer interface), distinct method of operation within a computer program, in which the same user input can produce different results depending of the state of the computer
	Normal mode, patterns of vibration in acoustics, electromagnetic theory, etc, longitudinal mode or transverse mode
	Transport mode refers to the means of transport being used.

More results at FactBites »

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Contents

Standing waves

Normal modes in quantum mechanics

See also

External links