Normal mode Totally Explained

Everything about Normal Mode totally explained

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

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:
ť

where the edge points are fixed and can't move. We'll use x₁(t) to denote the displacement of the leftmost mass, and x₂(t) to denote the displacement of the rightmost.
If we denote the second derivative of x(t) with respect to time as

ddot x

, the equations of motion are:
ť

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:
ť

x_1(t) = A_1 e^

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.

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