Linked Questions

1
vote
1answer
895 views

Normal modes of oscillation: how to find them? [duplicate]

Are normal modes the eigenvectors of the matrix $(\omega ^2 T- V)$ where $T$ is the matrix of kinetic energy and $V$ is the matrix of potential energy? Is it the only way to express them? How can I ...
8
votes
2answers
362 views

Interpreting physical meaning of normal modes

What really is a normal mode? Maybe it's because of my teachers but I find it really abstract. I know that "numerically" corresponds to the eigenvectors of the equation $\ddot{X}= -M^{-1}KX$ ...
3
votes
3answers
660 views

How to calculate the potential energy of coupled oscillators?

The equations of motion that describe the above situation is given by: $$ m \ddot{x_1} = -2kx_1 + kx_2 $$ $$ m \ddot{x_2} = -2kx_2 + kx_1 $$ Now I want to work out the potential energy of this system. ...
1
vote
2answers
796 views

Why do two masses connected to each other by a spring have the same frequency of oscillation?

Why do two masses, connected to each other by a spring, and each connected to a wall by a spring, have the same frequencies of oscillation when perturbed? In solving for the motion of the masses, ...
3
votes
0answers
281 views

Why is it that a coupled mass-spring system will always produce a diagonalizable matrix?

If you take a system like the one in the image, and you do the $y=x'$ trick to turn it into a first-order system of equations ($x_{1}$ or $x_{2}$ being the displacement of the mass $m_{1}$ or $m_{2}$ ...
1
vote
3answers
143 views

Plucked string eigenvalues/harmonic frequencies: integer multiples (or not)

I'm trying to derive a model of a plucked string from Newton's second law. My derivation results in $$ω_n = C\cdot\sqrt{n},\, n=1,2,3\dots\text{integer}$$ I think it should be $$ω_n = C\cdot n,\, n=1,...
1
vote
3answers
136 views

Confused about behaviour of spring mass system

I am writing some code that will plot the behaviour of a system consisting of 4 springs and 3 masses. They are arranged in the configuration (s:spring, m:mass) ...
1
vote
2answers
214 views

Congruence transformations of matrices

From the book Analytical Mechanics by Fowles and Cassiday I am studying classical coupled harmonic oscillators. These are systems that are governed by a system of linear second order differential ...
2
votes
1answer
108 views

Intuition for normal modes of a beaded string

These questions are inspired by the following the paper http://www.soton.ac.uk/~stefano/courses/PHYS2006/chapter7.pdf on 'Normal Modes of a Beaded String'. Problem Statement Given a recurrence ...
1
vote
1answer
80 views

Equations of motion of two bodies attached to three springs

I've been tasked with describing the equations of motion of two bodies attached via three springs, as visualized below. Let $x_1(t)$ and $x_2(t)$ denote the $x$-displacements of boxes $m_1$ and $m_2$ ...
0
votes
1answer
107 views

Find the frequency of oscillation of two masses connected by a spring [duplicate]

Question: Two masses $m_1$ and $m_2$, constrained to the $x$-axis, are coupled by a light spring of stiffness $s$ and natural length $l$. If $x$ is the extension of the spring and $x_1$ and $x_2$ ...
0
votes
0answers
81 views

Lagrangian for a series of $N$ one-dimensional coupled oscillators

The Problem I have been given the following Lagrangian of a series of $N$ one-dimensional coupled oscillators, with distance a. I have also been given the boundary conditions: $y_0=0=y_{N+1},$ but ...
1
vote
0answers
57 views

What are normal modes?

In normal modes analysis the differential equations of the system are Fourier transformed and the Fourier monochromatics are found. I think these monochromatics are usually called normal modes of the ...
0
votes
1answer
32 views

Normal modes of coupled oscillators

For two pendulums of mass $m_1$ and $m_2$, coupled by a spring of constant k, both suspended by strings of length $l$, the following matrix equality results from their equations of motion: $$ \omega^2 ...