Normal mode
A normal mode in an oscillating system is the frequency at which a deformable structure will oscillate when disturbed. Normal modes are also known as natural frequencies or resonant frequencies. There is a set of these frequencies that are unique to each structure.
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 practial significance of this can be illustrated by a mass-spring model of a building.
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A
normal mode in an oscillating system is the frequency at which a deformable structure will oscillate when disturbed. Normal modes are also known as natural frequencies or resonant frequencies. There is a set of these frequencies that are unique to each structure.
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 practial 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.
Example - normal modes of coupled oscillators
Consider two bodies , each of
mass M, attached to three springs with stiffness
K. They are attached in the following manner:
where the edge points are fixed and cannot move. We'll use
x1 to denote the displacement of the leftmost mass, and
x2 to denote the displacement of the rightmost.
If we denote the second
derivative of
x with respect to time as
x″, the equations of motion are:
Since we expect oscillatory motion, we try:
Substituting these into the equations of motion gives us:
Since the exponential factor is common to all terms, we omit it and simplify:
And in matrix representation:
For this equation to have a non-trivial solution, the determinant of the matrix on the left must be equal to 0, so:
Solving for , we have:
If we substitute into the matrix and solve for , we get . If we substitute , we get
.
The first normal mode is:
and the second normal mode is:
The general solution is a superposition of the
normal modes where
c1,
c2, φ
1, and φ
2, 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 are oscillating in the same
frequency and in phase , but each has a different amplitude.
The general form of a standing wave is:
where
f represents the dependence of amplitude on location and the cosine\sine are the oscillations in time.
Physically, standing waves are formed by the
interference of waves and their reflections . The geometric shape of the medium determines what would be the interference pattern, thus determines the
f 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 mode, but there are infinitely many normal modes. If the problem is bounded there are countably many normal modes . 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 on physical constants of the problem which sets 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 a vibrating instruments are called "harmonics".
Normal modes in quantum mechanics
In
quantum mechanics, a state of a system is described by a wavefunction which solves the Schrödinger equation. The square of the absolute value of ,i.e.
is the probability 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 . Thus, we may write
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.
See also
- Physical applications:
- Waves
- Optics
- harmonic oscillator
- vibrational spectroscopy
- quantum theory
- Schrödinger equation
- Wavefunction
- Measurement in quantum mechanics
- harmonic series
- Mathematical tools:
External links
- .
- Java simulation of the normal modes of a , , and .
