Normal modes Definition and 81 Threads

Consider a system of ##n## coupled oscillators. ##n## particles are constrained to move in the ##x## direction. ##x_j## measures the displacement of the ##j##th particle from equilibrium. From Newton's 2nd law $$m_j\ddot{x_j}=F_j\tag{1}$$ "Because the system is linear, we expect that we can...

This exercise comes from Kleppner and Kolenkow, 2nd ed., problem 6-3. I'm using a solution key as a study reference, but the solution key is coming to a pretty different conclusion. Mostly the issue is in the equations of motion for this system. I'm not sure if there's something I'm...

The first part is trivial not sure where to go on the second part.

I was doing the exercise as follows: I am not sure if you agree with me, but i disagree with the solution given. I was expecting that the kinect energy of the mass ##m## (##T_2##) should be $$T_2 = \frac{m((\dot q+lcos(\theta)\dot \theta)^2 + (lsin(\theta) \dot \theta)^2)}{2}$$ I could be...

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...

## \frac {\partial^2 \psi} {\partial t^2} = v^2 \frac {\partial^2 \psi} {\partial x^2} ## has solution ## \psi (x, t) = \sum_{m=0}^\infty A_m \sin(k_mx + \alpha_m)sin(\omegat + \beta_m) ## The boundary conditions I can discern $$ \psi (0, t) = 0 $$ $$ \frac {\partial \psi} {\partial x} (L, t)...

Hi, I know there's are 2 normal modes because the system has 2 mass. I did the Newton's law for both mass. ##m\ddot x_1 = -\frac{mgx_1}{l} -k(x_1 - x_2)## (1) ##m\ddot x_2 = -\frac{mgx_2}{l} +k(x_1 - x_2)## (2) In the pendulum mode ##x_1 = x_2## and in the breathing mode ##x_1 = -x_2## I get...

We need to find the normal modes of this system: Well, this system is a little easy to deal when we put it in a system and solve the system... That's not what i want to do, i want to try my direct matrix methods. We have springs with stiffness k1,k2,k3,k4 respectively, and block mass m1, m2...

A mass ##m## is restricted to move in the parabola ##y=ax^2##, with ##a>0##. Another mass ##M## is hanging from this first mass using a spring with constant ##k## and natural lenghth ##l_0##. The spring is restricted to be in vertical position always. The coordinates for the system are ##x##...

I am not sure if i get this part of a book i am using: Why are the mwo²xa acting too? Is not 2kxa enough?

This is a question from an exercise I don't have the answers to. I have been trying to figure this out for a long time and don't know what to do after writing mx''¨(t)=−kx(t)+mg I figure that the frequency ω=√(k/m) since the mg term is constant and the kx term is the only term that changes. I...

My understanding is that you can describe the complicated motion of atoms in a crystal as a sum of standing waves (normal modes). A phonon is an excitation of a normal mode in the sense that it increases the vibration amplitude of that normal mode and the energy of that mode by a quantized...

I found the equations of motion as ##m\frac{\mathrm{d}^2x_1 }{\mathrm{d} t^2} = -\frac{mg}{l}x_1 + k(x_2-x_1)## and ##m\frac{\mathrm{d}^2x_2 }{\mathrm{d} t^2} = -\frac{mg}{l}x_2 + k(x_1-x_2)## I think the k matrix might be ##\begin{bmatrix} mg/l + k & -k \\ -k & mg/l + k \end{bmatrix}##...

Let's try inputting a solution of the following form into the two-dimensional wave equation: $$ \psi(x, y, t) = X(x)Y(y)T(t) $$ Solving using the method of separation of variables yields $$ \frac {v^2} {X(x)} \frac {\partial^2 X(x)} {\partial x^2} + \frac {v^2} {Y(y)} \frac {\partial^2 Y(y)}...

I'm looking at what should be just a simple spring system where four identical springs are holding up a square, load-bearing pallet plate in a warehouse. Now, someone says the equation of motion for the vertical normal mode of vibration is simply d2z/dt2 = -4(k/m)z. Right away however, I see no...

Why would normal modes occur in the coupled oscillator system I.e. why the parts of system would oscillate with constant angular frequency and constant phase difference ?

Hello! I am reading some representation theory (the book is Lie Algebra in Particle Physics, by Georgi, part 1.17) and the author solves a problem of 3 bodies connected by springs forming a triangle, aiming to find the normal modes. He builds a 6 dimensional vector formed of the 3 particles and...

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...

I'm just looking for any tips one might have for finding normal modes quickly? The GRE always seems to have a question or two on them and I have no idea how they expect someone to do a problem like that in the time given. I know that there is normally, in the problems given, a symmetric and an...

So, my question is what does the "normal" part mean when one talks about normal frequencies and normal modes in coupled oscillations. Does it have to do with the normal coordinates that one uses when solving some problems, or with normal in the sense of orthogonal. Thanks for your help.

I think I miss something about energy of a mechanical wave. In absence of dissipation the mechanical energy transported by an harmonic wave is constant. $$E=\frac{1}{2} A^2 \omega^2 m$$ But, while studying normal modes on a rope, I find that the mechanical energy of a normal mode (still...

Homework Statement I have to determine the frequencies of the normal modes of oscillation for the system I've uploaded.Homework Equations [/B] 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 Here's the problem. I was able to find the a_n and b_n values, my question is mainly on part (c), how do I find which modes are missing? The function is odd, so even modes should disappear, but cos(n*pi) doesn't disappear, it's either +1 or -1. I'd greatly appreciate any...

Not a textbook/homework problem so I'm not using the format (hopefully that's ok). Can someone offer an explanation of normal modes and how to calculate the degrees of freedom in a system of coupled oscillators? From what I've seen the degrees of freedom seems to be equal to the number of...

Homework Statement So, a string with length L and a mass of M is given tension T. Find the frequencies of the smallest three modes of transverse motion. Then compare with a massless string with the same tension and length, but there are 3 masses of M/3 equally spaced. So this is problem #1...

Homework Statement The system is conformed by two blocks with masses m (on the left) and M (on the right), and two springs on the left/right has the spring constant of k. The middle spring has a spring constant of 4k. Friction and air resistance can be ignored. All springs are massless. Find...

Homework Statement A room has two opposing walls which are tiled. The remaining walls, floors, and ceiling are lined with sound-absorbent material. The lowest frequency for which the room is acoustically resonant is 50Hz. (a) Complex noise occurs in the room which excites only the lowest two...

I'd like to ask a couple of questions. As a solid object gets bigger, the molecular orbital (combinations of all single atom orbitals) has greater size too? For a one inch square object (of closely packed molecules like crystals), what is its molecular orbital size compared to a one foot square...

Why are they important? I've been learning about them quite a bit and have no idea why they are significant. What is the motivation for their discovery, their use, or even their mention in physics for that matter. All I really know is that when you look/have solutions with the same angular...

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...

Homework Statement http://imagizer.imageshack.us/v2/275x215q90/661/kIVMcC.png Mathematical pendulum is the part of the oscillating circuit. The system is in a constant uniform magnetic field. Oscillations is small. Find the normal modes of oscilations. Homework Equations ## \begin{cases}...

Hey guys, I'm trying to understand the properties of normal modes of the electromagnetic field inside an arbitrary cavity, but I'm having some trouble. By definition, for a normal mode we have \mathbf{E}(\mathbf{x},t) = \mathbf{E}_0 (\mathbf{x}) e^{i \omega_1 t} and \mathbf{B}(\mathbf{x},t) =...

I don't really understand boundary conditions and I've been trying to research it for ages now but to no real avail. I understand what boundary conditions are, I think. You need them along with the initial conditions of a wire/string in order to describe the shape of motion of the string. I...

Homework Statement The displacement, y(x; t), of a tight string of length, L, satisfies the conditions y(0, t) = \frac{\delta y}{\delta x}(L,t) = 0 The wave velocity in the string is v. a) Explain what is meant by a normal mode. Give the form of the displacement, y(x; t), for the normal...

Homework Statement So I'm given two horizontal masses coupled by two springs; on the left there is a wall, then a spring with k_{1}, then a mass, then a spring with k_{2}, and finally another mass, not attached to anything on the right. The masses are equal and move to the right with x_{1}...

Homework Statement A molecule consists of three identical atoms located at the vertices of a 45 degree right triangle. Each pair of atoms interacts by an effective spring potential, with all spring constants equal to k. Consider only planar motion of this molecule. What are 6 normal modes and...

What does one mean by normal mode of an object? Why is it 3n-5 for linear particles, 3n-6 for non linear particle where n is the number of particle.

Hi. I'd like to find the normal modes / harmonics, displayed in a dB - Hz graph, of a given 3D object, namely a wind instrument shaped like a hyperbolic con with holes. I'm trying to perform simulations on Comsol Multyphysics, but I don't know how to do it. Is Comsol the best suited software...

Hey! So If I have a stretched string of length L fastened at one end, and I am moving the other end sinusoidally, will a standing wave appear ONLY if I move the other end with one of the normal-mode frequencies of the string? If not, will the resulting wave be a moving wave which is a...

Homework Statement I have a system of two masses m1 and m2 coupled by two springs with constants k1 and k2. If m1 and m2 are equal what would be the normal modes for this system? Homework Equations Equations of motion for the system: \begin{align*} m_1\ddot{x}_1 &=...

Homework Statement A bar with mass m and length l is attached at one end to a string (also length l) and is swinging back and forth. Find the normal modes of oscillation. Homework Equations L=T-U, and the Lagrange-euler equation, I=(1/12)ml^2The Attempt at a Solution So my idea is this. Use...

Homework Statement Homework Equations The Attempt at a Solution When i do the matrix multiplication of the 2x2 and 2x1 matrix, I get 2 conflicting solutions that don't match at all! So which one do i take to find ratio of X and Y?

Homework Statement A thin uniform rod of length 2b is suspended by two vertical light strings, both of fixed length l, fasted to the ceiling. Assuming only small displacements from equilibrium, find the Lagrangian of the system and the normal frequencies. Find and describe the normal...

Hi, I'm studying Small Oscillations and I'm having a problem with normal modes. In some texts, there is written that normal modes are the eigenvectors of the matrix $V- \omega^2 V$ where V is the matrix of potential energy and T is the matrix of kinetic energy. Some of them normalize the...

The title, basically. If we're at a temperature below the critical temperature (let's just say for a Type 1 superconductor) and an applied magnetic field less than the critical magnetic field, it will be in the superconducting state. But if we increase the field beyond the critical point, it...

Homework Statement Two horizontal frictionless rails make an angle θ with each other. Each rail has a bead of mass m on it and the beads are connected by a spring with spring constant k and relaxed length=zero.Assume that one of the rails is positioned a tiny distance above the other so that...

Homework Statement Two equal masses (m) are constrained to move without friction, one on the positive x-axis and one on the positive y axis. They are attached to two identical springs (force constant k) whose other ends are attached to the origin. In addition, the two masses are connected to...

Hi, why does the energy of the system equals the sum of the energy of the modes? The book I'm reading only states it, it doesn't prove it.