Eigenmode

Eigenmodes or normal modes are special movements of a vibratory system. These are the movements that the system performs when it is left to its own devices. This includes uniform movement and all natural vibrations . The latter are free and undamped vibrations and are approximately regarded as harmonic . The overall movement of the system can be represented as a superposition of different eigenmodes. The number of eigenmodes of such a system is related to its degrees of freedom: The system can have as many natural frequencies as there are degrees of freedom and has as many eigenmodes as degrees of freedom.

In simple terms, a system oscillates at its natural frequency (engl. Natural frequency ) when neither suggestion nor acting damping. If, on the other hand, there is an excitation which results in an oscillation with a particularly large amplitude, one speaks of a resonance frequency . In weakly damped systems, resonance frequencies and natural frequencies are close to one another.

The natural frequencies of a system depend on its geometry, its structure and its material properties. The natural frequencies of the string of a musical instrument are determined, for example, by their length, their material and their tension. The same applies to all vibratory systems.

The word Eigenmode is derived from the English fashion or Latin mode , which in both cases means “manner”, and from eigenvalue , a term from algebra . From the point of view of theoretical physics, the eigenmodes form a discrete basis for all movements of the system. They result from the equations of motion of the system as eigenvectors of this system of equations. The frequencies of the eigenmodes are called the eigenfrequencies of the system; they are the eigenvalues ​​of the system of equations of motion. The uniform motion is represented as an eigenmode with zero frequency.

theory

The Lagrangian of a system with degrees of freedom is ${\ displaystyle f}$

${\ displaystyle L (q_ {1}, \ dots, q_ {f}, {\ dot {q}} _ {1}, \ dots, {\ dot {q}} _ {f}) = {\ frac { 1} {2}} \ sum _ {i, j = 1} ^ {f} m_ {ij} (q_ {1}, \ dots, q_ {f}) {\ dot {q}} _ {i} { \ dot {q}} _ {j} -U (q_ {1}, \ dots, q_ {f})}$

where is the mass matrix and the potential. When approximating the Lagrangian up to the second order around the equilibrium coordinates and neglecting the constant term, this becomes ${\ displaystyle m_ {ij}}$${\ displaystyle U}$${\ displaystyle q ^ {0}}$

${\ displaystyle L = {\ frac {1} {2}} \ sum _ {i, j = 1} ^ {f} m_ {ij} (q_ {1} ^ {0}, \ dots, q_ {f} ^ {0}) {\ dot {q}} _ {i} {\ dot {q}} _ {j} - {\ frac {1} {2}} \ sum _ {i, j = 1} ^ { f} {\ frac {\ partial ^ {2} U} {\ partial q_ {i} \ partial q_ {j}}} {\ bigg |} _ {q_ {i, j} = q_ {i, j} ^ {0}} (q_ {i} -q_ {i} ^ {0}) (q_ {j} -q_ {j} ^ {0})}$

respectively with the coordinate transformation and the abbreviations as well as briefly ${\ displaystyle x = qq ^ {0}}$${\ displaystyle T_ {ij} = T_ {ji} = m_ {ij} (q_ {1} ^ {0}, \ dots, q_ {f} ^ {0})}$${\ displaystyle V_ {ij} = V_ {ji} = \ partial ^ {2} U / \ partial q_ {i} \ partial q_ {j} | _ {q_ {i, j} = q_ {i, j} ^ {0}}}$

${\ displaystyle L = {\ frac {1} {2}} \ sum _ {i, j = 1} ^ {f} \ left (T_ {ij} {\ dot {x}} _ {i} {\ dot {x}} _ {j} -V_ {ij} x_ {i} x_ {j} \ right)}$

The equations of motion of the system result from the Lagrangian equations

${\ displaystyle \ sum _ {j_ {1}} ^ {f} T_ {ij} {\ ddot {x}} _ {j} = - \ sum _ {j = 1} ^ {f} V_ {ij} x_ {j} \ Leftrightarrow T {\ ddot {x}} = - Vx}$

where both and is matrices and a -dimensional vector. Since the kinetic energy is always greater than zero, positive is definite . In order for the system to be in a stable or indifferent equilibrium , it must be positive semidefinite. In particular, all eigenvalues ​​of and are nonnegative. ${\ displaystyle T}$${\ displaystyle V}$ ${\ displaystyle f \ times f}$${\ displaystyle x}$${\ displaystyle f}$${\ displaystyle T}$ ${\ displaystyle V}$${\ displaystyle T}$${\ displaystyle V}$

The approach to solving the equation is:

${\ displaystyle x (t) = A \ exp (- \ mathrm {i} \ omega t)}$

This leads to the eigenvalue problem

${\ displaystyle (V- \ omega ^ {2} T) A = 0}$.

In order to solve this non-trivial, the determinant must disappear. This is a polynomial of degree in and therefore has complex zeros. However, the nonnegativity of the eigenvalues ​​of and ensures that they are all real and nonnegative. Physically this can be interpreted as follows: Assuming there were a zero in the negative or complex, then it would have an imaginary part and the solution would diverge. This is in contradiction to the stable equilibrium assumption. ${\ displaystyle \ det (V- \ omega ^ {2} T)}$${\ displaystyle f}$${\ displaystyle \ omega ^ {2}}$${\ displaystyle f}$${\ displaystyle T}$${\ displaystyle V}$${\ displaystyle \ omega}$

The (positive) roots of the roots of the polynomial

${\ displaystyle P ^ {(f)} (\ omega ^ {2}) = \ det (V- \ omega ^ {2} T)}$

are the natural frequencies of the system described by and . A system with degrees of freedom therefore has at most natural frequencies. ${\ displaystyle \ omega _ {k}}$${\ displaystyle T}$${\ displaystyle V}$${\ displaystyle f}$${\ displaystyle f}$

The natural vibrations of the system are the eigenvectors of the eigenvalue problem that make up the equation ${\ displaystyle f}$${\ displaystyle f}$

${\ displaystyle (V- \ omega _ {k} ^ {2} T) A ^ {(k)} = 0}$

fulfill. In particular, every multiple of an eigenvector is also an eigenvector. This means that they can be normalized and multiplied by a complex constant . ${\ displaystyle c_ {k}}$

If several natural frequencies coincide, then the equation does not have full rank and some components of the associated can be chosen freely. If the matrix has an eigenvalue of zero, there is an indifferent equilibrium. Then a natural frequency of the system is also zero. In this case the eigenvalue equation is such that the solution is uniform motion of the system. ${\ displaystyle A ^ {(k)}}$${\ displaystyle V}$${\ displaystyle {\ ddot {x}} = 0}$

The general solution of the equation system for the oscillation of the system is therefore a superposition of its natural oscillations and possibly a uniform movement

${\ displaystyle x (t) = \ sum _ {k = 1 \ atop \ omega _ {k} \ neq 0} ^ {f} \ operatorname {Re} \ left (c_ {k} A ^ {(k)} \ exp (- \ mathrm {i} \ omega _ {k} t) \ right) + \ sum _ {k = 1 \ atop \ omega _ {k} = 0} ^ {f} A ^ {(k)} \ left (x_ {k} ^ {0} + C_ {k} t \ right)}$

For each degree of freedom there are either 2 real or 1 complex free parameters. This results in constants that have to be determined by initial conditions . ${\ displaystyle 2f}$

Normal coordinates

The normal coordinates of the system are defined as ${\ displaystyle Q}$

${\ displaystyle Q = a ^ {- 1} x}$

in which

${\ displaystyle a = \ left ({A_ {i}} ^ {(k)} \ right)}$

is the matrix of the eigenvectors. This matrix of eigenvectors diagonalizes both and , because it follows from the symmetry of${\ displaystyle V}$${\ displaystyle T}$${\ displaystyle V}$

${\ displaystyle \ left (\ omega _ {k} ^ {2} - \ omega _ {l} ^ {2} \ right) \ sum _ {i, j = 1} ^ {f} T_ {ij} a_ { il} a_ {jk} = 0}$

so that for all eigenvalues ​​that are not degenerate, all non-diagonal elements of must vanish. A corresponding normalization of the eigenvectors leads to the orthonormality relation${\ displaystyle a ^ {\ mathrm {T}} Ta}$

${\ displaystyle a ^ {\ mathrm {T}} Ta = 1}$

For degenerate eigenvalues, the eigenvectors can also be chosen so that this matrix becomes diagonal. It can also be shown that also diagonalized. The equation of motion can be written as ${\ displaystyle a}$${\ displaystyle V}$${\ displaystyle \ lambda = \ left (\ omega _ {k} ^ {2} \ delta _ {kl} \ right)}$

${\ displaystyle Va = Ta \ lambda}$

written in such a way that the assertion follows directly by multiplying by from the left. ${\ displaystyle a ^ {\ mathrm {T}}}$

Thus, a coordinate transformation decouples the deflections from the equilibrium position into the normal coordinates by means of the system of equations, because: ${\ displaystyle x}$${\ displaystyle Q}$${\ displaystyle x = aQ}$

${\ displaystyle L = {\ frac {1} {2}} \ left ({\ dot {x}} ^ {\ mathrm {T}} T {\ dot {x}} - x ^ {\ mathrm {T} } Vx \ right) = {\ frac {1} {2}} \ left ({\ dot {Q}} ^ {\ mathrm {T}} {\ dot {Q}} - Q ^ {\ mathrm {T} } \ lambda Q \ right) = {\ frac {1} {2}} \ sum _ {k} \ left ({\ dot {Q}} _ {k} ^ {2} - \ omega _ {k} ^ {2} Q_ {k} ^ {2} \ right)}$

In particular is

${\ displaystyle Q_ {k} = \ operatorname {Re} \ left (c_ {k} \ exp (- \ mathrm {i} \ omega _ {k} t) \ right)}$

Examples

Spring pendulum

A spring pendulum is a system on which a mass is suspended from a spring and which can only move in one dimension. It has only one degree of freedom, the deflection from the rest position . For the spring pendulum applies and , where is the spring constant and the mass. Therefore, the matrix equation simplifies to a scalar equation ${\ displaystyle V = D}$${\ displaystyle T = m}$${\ displaystyle D}$${\ displaystyle m}$

${\ displaystyle (D- \ omega ^ {2} m) A = 0}$

with a first degree polynomial in ${\ displaystyle \ omega ^ {2}}$

${\ displaystyle P (\ omega ^ {2}) = \ det \ left (D- \ omega ^ {2} m \ right) = D- \ omega ^ {2} m = 0 \ Leftrightarrow \ omega ^ {2} = {\ frac {D} {m}}}$

and an eigenvector

${\ displaystyle A = 1}$.

So the solution is

${\ displaystyle x (t) = \ operatorname {Re} \ left (c \ exp \ left (- \ mathrm {i} {\ sqrt {\ frac {D} {m}}} t \ right) \ right)}$

CO 2 molecule

As a first approximation, a carbon dioxide molecule can be viewed as three masses, of which the outer two identical masses are connected to the middle mass by springs. Since the bonds are both alike, the spring constants are both . The indices are chosen in such a way that the atoms are numbered consecutively from left to right and it is also assumed that the molecule can only move along the molecular axis, i.e. only valence vibrations but not deformation vibrations are taken into account. Therefore there are three degrees of freedom of the system: The distances of the three masses from their equilibrium position. Then applies with ${\ displaystyle m_ {O}}$${\ displaystyle m_ {C}}$${\ displaystyle D}$

${\ displaystyle T = {\ begin {pmatrix} m_ {O} && \\ & m_ {C} & \\ && m_ {O} \ end {pmatrix}}}$
${\ displaystyle V = D {\ begin {pmatrix} 1 & -1 & \\ - 1 & 2 & -1 \\ & - 1 & 1 \ end {pmatrix}}}$

for the determinant of the system

${\ displaystyle P ^ {(k)} (\ omega ^ {2}) = \ omega ^ {2} (D- \ omega ^ {2} m_ {O}) (\ omega ^ {2} m_ {C} m_ {O} -k (2m_ {O} + m_ {C}))}$

Its three zeros are included

${\ displaystyle \ omega _ {1} ^ {2} = 0, \ qquad \ omega _ {2} ^ {2} = {\ frac {D} {m_ {O}}}, \ qquad \ omega _ {3 } ^ {2} = {\ frac {D} {m_ {O}}} \ left (1 + 2 {\ frac {m_ {O}} {m_ {C}}} \ right)}$

and the eigenvectors are

${\ displaystyle A ^ {(1)} = {\ begin {pmatrix} 1 \\ 1 \\ 1 \ end {pmatrix}}, \ qquad A ^ {(2)} = {\ begin {pmatrix} 1 \\ 0 \\ - 1 \ end {pmatrix}}, \ qquad A ^ {(3)} = {\ begin {pmatrix} 1 \\ - 2m_ {O} / m_ {C} \\ 1 \ end {pmatrix}} }$.

This gives the general solution too

${\ displaystyle x (t) = {\ begin {pmatrix} 1 \\ 1 \\ 1 \ end {pmatrix}} \ left (x_ {1} ^ {0} + C_ {1} t \ right) + \ operatorname {Re} \ left ({\ begin {pmatrix} 1 \\ 0 \\ - 1 \ end {pmatrix}} c_ {2} \ exp \ left (- \ mathrm {i} {\ sqrt {\ frac {D} {m_ {O}}}} t \ right) + {\ begin {pmatrix} 1 \\ - 2m_ {O} / m_ {C} \\ 1 \ end {pmatrix}} c_ {3} \ exp \ left ( - \ mathrm {i} {\ sqrt {{\ frac {D} {m_ {O}}} \ left (1 + 2 {\ frac {m_ {O}} {m_ {C}}} \ right)}} t \ right) \ right)}$.

The first natural oscillation is the translation of the entire molecule, the second describes the opposite oscillation of the two outer oxygen atoms, while the carbon atom remains at rest, and the third describes the uniform oscillation of the two outer atoms, with the middle atom oscillating in opposite directions.

Vibrating string

A vibrating string has an infinite number of degrees of freedom and, accordingly, an infinite number of natural frequencies. However, these must meet the boundary conditions of the problem. The wave equation is

${\ displaystyle {\ frac {\ partial ^ {2} u} {\ partial x ^ {2}}} - {\ frac {1} {c_ {k} ^ {2}}} {\ frac {\ partial ^ {2} u} {\ partial t ^ {2}}} = 0}$

where is the deflection of the string and the phase velocity of the wave. The solution to the wave equation for a solid is ${\ displaystyle u (x, t)}$${\ displaystyle c_ {k}}$${\ displaystyle k}$

${\ displaystyle u_ {k} (x, t) = \ operatorname {Re} \ left (c_ {k} \ exp (- \ mathrm {i} (\ omega _ {k} t-kx)) \ right)}$

with . The relationship between and is called the dispersion relation of the system. For a string is a constant that depends on the tension and the linear mass density of the string. ${\ displaystyle \ textstyle c_ {k} = {\ frac {\ omega _ {k}} {k}}}$${\ displaystyle \ omega _ {k}}$${\ displaystyle k}$${\ displaystyle \ textstyle c_ {k} = {\ sqrt {S / \ rho}}}$ ${\ displaystyle S}$${\ displaystyle \ rho}$

The boundary conditions on the vibrating string is that the ends are firmly clamped and therefore a string of length for all${\ displaystyle L}$${\ displaystyle t}$

${\ displaystyle u (0, t) = u (L, t) = 0}$

have to be. This leads to the boundary condition

${\ displaystyle k = {\ frac {\ pi n} {L}}}$

with any and thus countable infinitely many different and correspondingly many . The natural frequencies of the string are therefore ${\ displaystyle n \ in \ mathbb {N}}$${\ displaystyle k}$${\ displaystyle \ omega _ {k}}$

${\ displaystyle \ omega _ {n} = {\ sqrt {\ frac {S} {\ rho}}} {\ frac {\ pi n} {L}}}$

and the general solution of the wave equation is a superposition over all natural vibrations:

${\ displaystyle u (x, t) = \ operatorname {Re} \ left (\ sum _ {n} c_ {n} \ exp \ left (- \ mathrm {i} {\ frac {\ pi n} {L} } \ left ({\ sqrt {\ frac {S} {\ rho}}} tx \ right) \ right) \ right)}$

Normal vibrations of molecules

An atomic molecule has degrees of freedom. Of these, 3 are translational degrees of freedom and in the case of a linear molecule 2 or in the case of an angled molecule 3 are rotational degrees of freedom. This leaves degrees of freedom or vibration degrees of freedom that correspond to natural frequencies not equal to zero. The symmetries of these molecular vibrations can be described by the group-theoretic character tables. The normal vibrations of a degenerate natural frequency different from zero represent a basis for an irreducible representation of the point group of the vibrating molecule. ${\ displaystyle N}$${\ displaystyle 3N}$${\ displaystyle 3N-5}$${\ displaystyle 3N-6}$

In the above example, the other two normal vibrations are the neglected transverse vibrations of the atoms in the two other spatial directions that are not in the line of the atoms.

Quantum mechanics

In quantum mechanics , the state of a system is represented by a state vector , which is a solution to the Schrödinger equation${\ displaystyle | \ psi (t) \ rangle}$

${\ displaystyle H | \ psi (t) \ rangle = \ mathrm {i} \ hbar {\ frac {\ partial} {\ partial t}} | \ psi (t) \ rangle}$

is. If the Hamilton operator is not time dependent, then there is a formal solution to the Schrödinger equation

${\ displaystyle | \ psi (t) \ rangle = \ exp \ left (- {\ tfrac {\ mathrm {i}} {\ hbar}} Ht \ right) | \ psi (0) \ rangle}$

Since the Hamilton operator has a complete system of eigen-states, the energy eigen-states , it can be developed in these. With follows ${\ displaystyle H | n \ rangle = E_ {n} | n \ rangle}$

${\ displaystyle | \ psi (t) \ rangle = \ sum _ {n} \ exp \ left (- {\ tfrac {\ mathrm {i}} {\ hbar}} E_ {n} t \ right) | n \ rangle \ langle n | \ psi (0) \ rangle}$

The quantum mechanical natural frequencies do not describe an oscillation in the spatial space, but a rotation in the Hilbert space on which the state vector is defined. ${\ displaystyle \ omega _ {n} = E_ {n} / \ hbar}$

Technical examples

Resonance of a loudspeaker
• A bell that is struck then vibrates with the natural frequencies. The vibration decays over time due to damping . Higher frequencies are attenuated faster than lower ones.
• A tuning fork is constructed in such a way that apart from the lowest natural frequency, hardly any other natural vibrations are excited.
• Natural frequencies can be excited in buildings. If the neighbors are playing music, it can happen that the frequency of a bass tone matches the natural frequency of the building wall. The vibrations of the wall stimulated by the music are then sometimes audible even if the music itself would not be perceptible.
• Like most musical instruments, drums have several natural frequencies.
• In loudspeakers , the partial vibrations of the membranes worsen the reproduction quality.