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 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. In music, normal modes of vibrating instruments (strings, air pipes, drums, etc.) are called "harmonics" or "overtones".
The most general motion of a 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 mathematical terms, normal modes are orthogonal to each other.
I am trying to analyse the dynamics of a cluster of 79 atoms.
The system can be described with:
##\omega^2 \vec x = \tilde D\vec x##
Where ##\omega^2## (the eigenvalues) are the squares of the vibration frequencies for each mode of motion, ##\tilde D## is the "dynamical matrix" which is a...
Homework Statement
Let's say that I have a potential ##U(x) = \beta (x^2-\alpha ^2)^2## with minima at ##x=\pm \alpha##. I need to find the normal modes and vibrational frequencies. How do I do this?
Homework Equations
##U(x) = \beta (x^2-\alpha ^2)^2##
##F=-kx=-m\omega ^2 x##
##\omega =...
Homework Statement
In the book "Quantum Optics" written by Scully and Zubairy, there is an equation (1.1.5). The equation is presented directly and not explained how to be deduced. The content is as follows.
Homework Equations
The Attempt at a Solution
I know the solution should have the form...
Homework Statement
I have to determine the frequencies of the normal modes of oscillation for the system I've uploaded.
Homework Equations
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I determined the following differential equations for the coupled system:
\ddot{x_A}+2(\omega_0^2+\tilde{\omega_0}^2)x_A-\omega_0^2x_B = 0...
If I have a system where the following is found to describe the motion of three particles:
The normal modes of the system are given by the following eigenvectors: $$(1,0,-1), (1,1,1), (1,-2,1)$$
How can I find the corresponding eigenfrequencies? It should be simple... What am I missing?
Homework Statement
Given the system in the image below, I need to find the equation of motions for the coupled system. The surface where the block moves is frictionless. The red line is position where the block is at equilibrium. At equilbrium x1 and x2 = 0. After finding the equation of motion...