Normal mode

A normal mode of a dynamical 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 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. The most general motion of a linear system is a superposition of its normal modes. The modes are "normal" in the sense that they move independently. An excitation of one mode will never cause excitation of a different mode. In mathematical terms, normal modes are orthogonal to each other. .

A normal mode of a dynamical 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 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. The most general motion of a linear system is a superposition of its normal modes. The modes are "normal" in the sense that they move independently. An excitation of one mode will never cause excitation of a different mode. In mathematical terms, normal modes are orthogonal to each other. . ===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, 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== In the analysis of conservative systems with small displacements from equilibrium, important in acoustics, molecular spectra, and electrical circuits, the system can be transformed to new coordinates called normal coordinates. Each normal coordinate corresponds to a single vibrational frequency of the system and the corresponding motion of the system is called the normal mode of vibration. \begin{align} 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{align} Since we expect oscillatory motion of a normal mode (where is the same for both masses), we try: \begin{align} x_1(t) &= A_1 e^{i \omega t} \\ x_2(t) &= A_2 e^{i \omega t} \end{align} Substituting these into the equations of motion gives us: \begin{align} -\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} \end{align} Omitting the exponential factor (because it is common to all terms) and simplifying yields: \begin{align} (\omega^2 m - 2 k) A_1 + k A_2 &= 0 \\ k A_1 + (\omega^2 m - 2 k) A_2 &= 0 \end{align} 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 If the matrix on the left is invertible, the unique solution is the trivial solution . The non trivial solutions are to be found for those values of 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: (\omega^2 m - 2 k)^2 - k^2 = 0 Solving for , the two positive solutions are: \begin{align} \omega_1 &= \sqrt{\frac{k}{m}} \\ \omega_2 &= \sqrt{\frac{3 k}{m}} \end{align} Substituting into the matrix and solving for , yields . Substituting results in . (These vectors are eigenvectors, and the frequencies are eigenvalues.) The first normal mode is: \vec \eta_1 = \begin{pmatrix} x^1_1(t) \\ x^1_2(t) \end{pmatrix} = c_1 \begin{pmatrix}1 \\ 1\end{pmatrix} \cos{(\omega_1 t + \varphi_1)} 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{pmatrix} x^2_1(t) \\ x^2_2(t) \end{pmatrix} = c_2 \begin{pmatrix} 1 \\ -1 \end{pmatrix} \cos{(\omega_2 t + \varphi_2)} 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 , , , and 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. \Psi(t) = f(x,y,z) \left[A\cos(\omega t) + B\sin(\omega t)\right] where 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 form of the standing wave. This space-dependence is called a normal mode. Usually, for problems with continuous dependence on 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 ). If the problem is not bounded, there is a continuous spectrum of normal modes. E(\nu) = \frac{1}{2}h\nu + \frac{h\nu}{e^{h\nu/k_\text{B}T} - 1} The term represents the "zero-point energy", or the energy which an oscillator will have at absolute zero. tends to the classic value at high temperatures E(\nu) = k_\text{B}T\left[1 + \frac{1}{12} \left(\frac{h\nu}{k_\text{B}T}\right)^2 + \mathcal{O}{\left(\left(\frac{h\nu}{k_\text{B}T}\right)^4\right)} \right] where is the Boltzmann constant. By knowing the thermodynamic formula, \left( \frac{\partial S}{\partial E}\right)_{N,V} = \frac{1}{T} the entropy per normal mode is: \begin{align} S(\nu) &= \int_0^T\frac{d}{dT}E\left(\nu\right)\frac{dT}{T} \\[10pt] &= \frac{E\left(\nu\right)}{T} - k_\text{B} \log\left(1 - e^{-\frac{h\nu}{k_\text{B}T}}\right) \end{align} The free energy is: F(\nu) = E - TS = k_\text{B}T\log \left(1-e^{-\frac{h\nu}{k_\text{B}T}}\right) which, for , tends to: F(\nu) = k_\text{B}T\log \left(\frac{h\nu}{k_\text{B}T}\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 and . Allow this number to be . Since the total number of normal modes is , the function is given by: \int f(\nu)\,d\nu = 3N The integration is performed over all frequencies of the crystal. Then the internal energy will be given by: U = \int f(\nu)E(\nu)\,d\nu ==In seismology== 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 and angular order but have degeneracy of azimuthal order . Increasing concentrates fundamental branch closer to surface and at large 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 . 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, 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.

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