Normal modes Definition and 90 Threads

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.

I am trying to find a succinct definition of normal coordinates and normal modes - relating to numerous coupled harmonic oscillators. What exactly do the normal coordinates describe about these coupled harmonic oscillators? I know that for normal modes, all parts of the system move with...

What is the difference between normal modes and propagation modes? I am confused. I know that normal modes are natural modes of vibration of a system. And, propagating modes are ... ? Thanks.

Homework Statement Hi everyone! I'm trying to solve a problem involving 3 equal masses (m) arranged in a 45-45-90 right triangle, each mass connected to the other two by a spring, and all the springs have the same spring constant (k). So it should look like this: Where I have made a...

Homework Statement ***This is problem 11.29 in Taylor's Classical Mechanics*** A thin rod of length 2b and mass m is suspended by its two ends with two identical vertical springs (force constant k) that are attached to the horizontal ceiling. Assuming that the whole system is constrained to...

Hello! For a coupled two-body oscillator we write the general solution as: x1(t)=C1-Cos[ω-t+ψ1-]+C1+Cos[ω+t+ψ1+] x2(t)=C2-Cos[ω-t+ψ2-]+C2+Cos[ω+t+ψ2+] Where we determine C1-/C2- and C1+/C2+ from the normal mode condition. We call ψ1-=ψ2-=ψ- and ψ1+= ψ2+=ψ+, and we end up with 4...

Homework Statement I'm stuck at understanding how to find the kinetic and potential energy matrices such that the determinant |V- \omega ^2 T|=0 when solved for \omega, gives the normal modes (characteristic frequencies?) of the considered system. For example in Goldstein's book for a molecule...

Homework Statement 1. A double pendulum, consisting of a pair, each of mass m and length l, is released from rest with the pendulums displaced but in a straight line. Find the displacements of the pendulums as functions of time. === So... this is a problem from Kibble's Classical Mechanics...

Homework Statement The first part of the problem is just finding the Lagrangian for a system with 2 d.o.f. and using small angle approximations to get the Lagrangian in canonical/quadratic form, not a problem. I am given numerical values for mass, spring constants, etc. and am told to find the...

Homework Statement Two equal masses are held on a frictionless track by 3 equal springs, attached to two rigid posts. If either of the masses is clamped, the period (t=2pi/w) of one oscillation is three seconds. If both masses are free, what is the periods of oscillation of both normal...

So I have a spring problem where I have two masses and three springs hanging down from a ceiling. Each spring has a different force constant and each mass has a different weight. |.....| |Spring 1...|Spring 3 Mass 1...|...

I am having problems understanding the "systematic method" for determining the frequencies and amplitude ratios of normal modes when a system has more than one degree of freedom. I think I initially have problems setting up the differential equation that describes the motion. Here is the...

Homework Statement Please see question attached Homework Equations The Attempt at a Solution Ok so I've been able to do the first few parts and have derived that Wm,n = c pi / L (m^2 + n^2) I've thus been able to show that the second lowest freq is a factor of root(5/2)...

Dear Friends, I am new to this forum and I am not sure whether I am writing my Query in the correct section or not...If not Kindly guide me where to post this... I am confused about what are normal modes of propagation in isotropic or anisotropic media?? or How one can define Normal modes...

Have a project to do on an acoustic resonance chamber, a loudspeaker attached to a perspex box with a copper pipe, there was a microppohne inside the chamber connected to oscillloscope. We investigating a few variables in most detail was route [1] [1]damping by chainging materials in the...

Can someone provide me a link that explains and provides a proof for the following principles: 1. Fermat's Principle that light always takes the path that minimizes the time taken 2. Solution to a Fourier Series and why all periodic motion can be represented as an infinite sum of sines and...

Homework Statement In textbooks, I often see the sum of the first two normal modes given in the equation attached (on the right). I'm wondering how they arrive at that equation based on the general formula (on the left). I tried subbing in n= 1 and 2 in the general formula, but I'm not sure...

Homework Statement The first circuit has a capacitor with capacitance c and an inductor with inductance L. In series with this is another capacitor which is connected to the next loop in the circuit. It look something like http://imgur.com/YJDaD.png" Sorry for the crude drawing...

there are three molecules joined through three springs of equal length 'l' forming an equilateral triangle , if all three molecules are displaced by equal lengths, they start oscillating linearly. write the simulation for both, the original problem and the linear one.

Homework Statement Consider the CO2 molecule as a system made of a central mass m_2 connected by equal springs of spring constant k to two masses m_1 and m_3 a) set up and solve the equations for the two normal modes in which the masses oscillate along the line joining their centers (the...

1. Homework Statement Someone studying a dynamical system in another field of science tells you that when they attempt to model the experiment they’ve been examining they obtain the following set of coupled ordinary differential equations. \dot{x}= -Ax + By \dot{y}= -Cx In what follows you...

A string with one end fixed as U(x=0,t)=0. The other end is attached to a massless ring which moves frictionlessly along a rod at x=L a) Explain the boundary condition at x=L should be d/dx U(x,t) = 0. b) Find the normal modes for the wave equation d2/dt2 U(x,t) = c2 * d2/dx2 U(x,t) with the...

Homework Statement Consider a mass M whose motion is confined to a flat, smooth two-dimensional surface. Label the locations in this surface using the Cartesian coordinates (x, y). The mass is attached to two identical springs, each of length ℓ and spring constant k. One spring has one of its...

Homework Statement A block of mass M can move along a smooth horizontal track. Hanging from the block is a mass m on a light rod of length l that is free to move in a vertical plane that includes the line of motion of the block. Find the frequency and displacement patterns of the normal...

Homework Statement A uniform rod of length a hangs vertically on the end of an inelastic string of length a, the string being attached to the upper end of the rod. What are the frequencies of the normal modes of oscillation in a vertical plane? Answer: \omega^2 = (5 \pm \sqrt{19})g/a...

Homework Statement I'm reading Landau's Mechanics, in section 23, he discusses the oscillations with more than one degree of freedom, the Lagrangian is L = \frac{1}{2}\left(m_{ik}\dot{x}_i\dot{x}_k - k_{ik}x_ix_k\right) where m_{ik},k_{ik} are symmetric constants, and the summation over...

I have a coupled oscillator system that is three pendulums attached together by two springs. The first part of the problems asks to argue, using symmetry arguments, that there are two "obvious" normal modes: one with w^2=g/l and another with w^2=g/l + k/m. I understand that these two frequencies...

Hello, I have a question about standing sound waves in an open pipe caused by a speaker, I know that normal mode frequencies are given by f_n = n\frac{v}{4L}, \quad n=1,2,3,... if we put a microphone into the pipe, and if the frequency of the speaker is a normal mode frequency, we see...

Homework Statement A particle P of mass 3m is suspended from a fixed point O by a massless linear spring with strength alpha. A second particle Q of mass 2m is in turn suspended from P by a second spring of the same strength. The system moves in the vertical straight lie through O . Find...

I'm looking for an in depth and comprehensive treatment of the theory of normal modes; any suggestions?

Hi Can someone help me with the following questions please (see attachment)? I really need some help on the following: i). Drawing a force diagram for each particle (I really hate drawing these). As a guess for m1, am I right in thinking that H and N point up and W points down? But...

[SOLVED] Springs, Normal Modes, and Center of Mass coordinates 1.) Problem The problem of the linear triatomic molecule can be reduced to one of two degrees of freedom by introducing coordinates y_{1} = x_{2} - x_{1}, y_{2} = x_{3} - x_{2}, and eliminating x_{2} by requiring that the center...

Homework Statement hey, this is my first time using this forum and was wondering if i could have some help with this normal modes question. suspended from a ceiling is in order: a spring of spring constant (k_{2}), a mass (m), a spring with spring constant (k_{1}), and another mass (m)...

I Have a question that is bugging me because I can't get the answer out here's the question: A uniform rod of length a hangs vertically on the end of an inelastic string of length a, the string being attached to the upper end of the rod. What are the frequencies of the normal modes of...

I Have a question that is bugging me because I can't get the answer out here's the question: A uniform rod of length a hangs vertically on the end of an inelastic string of length a, the string being attached to the upper end of the rod. What are the frequencies of the normal modes of...

Homework Statement Discuss the motion of a continuous string when the initial conditions are q'(x,0) = 0 and q(x,0) = Asin(3πx/L). Resolve the solution into normal modes. Show that if the string is driven at an arbitrary point, none of the normal modes with nodes at the driving point will be...

Hi see the attached picture... 2 coupled masses, each suspended from spring in gravitational field... also entire construction can vibrate only vertically... I need to write lagrangian for this system in the following form...

I have just been solving some systems of ODEs to find the nomal mode solutions. Something about it has been bugging me though. In a simple case where we have a system of two linear ODEs representing a two-mass spring system, we assume that the solution is a normal mode and so find a general...

Hello, I'm preparing for my condensed matter exam and I'm trying to solve problem 3a) of chapter 22 in Ashcroft & Mermin. The problem is basically to prove that the dispersion relation of a diatomic linear chain will reduce to the monoatomic one when the coupling constants are equal...

I have a book here that says that the number of normal modes is equal to the number of degree of freedom of a system. I'm wondering if this applies to just about any system or only to systems in which every coordinates can oscillate harmonically. The question is not very well formulated, so I...