Normal mode

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For other types of mode, see mode.

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.

[edit] 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.

[edit] 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".

[edit] Elastic solids

See: Einstein solid and Debye model

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).

The fundamental and the first six overtones of a vibrating string. The mathematics of wave propagation in crystalline solids consists of treating the harmonics as an ideal Fourier series of sinusoidal density fluctuations (or atomic displacement waves).

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 (or acoustic) 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 (or optical) 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 υ is:

$E(v)=\frac{1}{2}hv+\frac{hv}{e^{hv/kt}-1}$

The term (1/2)hυ 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

$E(v)=kT\left[1+\frac{1}{12}\frac{hv^2}{kT}+O\left(\frac{hv}{kT}\right)^4+\ldots\right]$

The entropy per normal mode is:

$\begin{align} S\left(v\right)&=\int_0^T\frac{d}{dT}E\left(v\right)\frac{dT}{T}\\ &=\frac{E\left(v\right)}{T}-k\log\left(1-e^{-\frac{hv}{kT}}\right)\\ \end{align}$

The free energy is:

$F(v)=E-TS=kT\log \left(1-e^{-\frac{hv}{kT}}\right)$

which, for kT >> hν, tends to:

$F(v)=kT\log \left(\frac{hv}{kT}\right)$

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)dν. Since the total number of normal modes is 3N, the function f (ν) is given by:

$\int f(v)dv=3N$

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

$U=\int f(v)E(v)dv$

[edit] 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.

[edit] See also

Specific types:
- Longitudinal mode
- Transverse mode
Physical applications:
Mathematical tools:

[edit] 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

Normal mode

From Wikipedia, the free encyclopedia

Contents

[edit] Coupled oscillators

[edit] Standing waves

[edit] Elastic solids

[edit] Quantum mechanics

[edit] See also

[edit] External links

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