normal mode: Definition from Answers.com

For other types of mode, see mode.

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Various normal modes in a 1D-lattice.

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, bridge or molecule, has a set of normal modes (and corresponding frequencies) that depend on its structure and composition.

The normal modes of a mechanical system are single frequency solutions to the equations of motion; the most general motion of the 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 many systems this is equivalent to reducing a collection of coupled oscillators to a set of decoupled, effective oscillators.

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 the same, such that they all pass through both equilibrium and maximum amplitude simultaneously. 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 - depending on the mode - can be maximum. Obviously, buildings can only withstand this displacement up to a certain point. Modeling a building by finding its normal modes is an easy way to check the safety of the building's design. The concept of normal modes also finds application in wave theory, optics, quantum mechanics, and molecular dynamics.

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, each with spring constant K. They are attached in the following manner:

where the edge points are fixed and cannot move. We'll use x₁(t) to denote the horizontal displacement of the leftmost mass, and x₂(t) to denote the displacement of the rightmost. Further, we'll assume for simplicity that the masses are equal, i.e. m₁ = m₂ = M.

If we denote the second derivative of x(t) with respect to time as $\ddot x$ , the equations of motion are:

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

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

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:

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

Solving for $ω$ , we have two solutions:

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

$\omega_2 = \sqrt{\frac{3 K}{M}}.$

If we substitute ω₁ into the matrix and solve for (A₁, A₂), we get (1, 1). If we substitute ω₂, we get (1, −1). (These vectors are eigenvectors, and the frequencies are eigenvalues.)

The first normal mode is:

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

Which corresponds to both masses moving in the same direction at the same time. Hence, the frequency is the same as if the two masses were connected by a rigid rod.

The second normal mode is:

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

This corresponds to the masses moving in the opposite directions, while the center of mass remains stationary. 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 process demonstrated here can be generalized and formulated using the formalism of Lagrangian mechanics or Hamiltonian mechanics.

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.

The general form of a standing wave is:

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

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.

When relating to music, normal modes of vibrating instruments (strings, air pipes, drums, etc.) are called "harmonics" or "overtones".

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.

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

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 $\omega = E_n / \hbar$ . Thus, we may write

$|\psi (t) \rang = \sum_n |n\rang \left\langle n | \psi ( t=0) \right\rangle e^{-iE_nt/\hbar}$

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.

External links

Java simulation of coupled oscillators.
Java simulation of the normal modes of a string, drum, and bar.
Photograph of a cup of coffee vibrating at a normal mode frequency

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normal mode

Contents

Example — normal modes of coupled oscillators

Standing waves

Normal modes in quantum mechanics

See also

External links