What is Normal modes: Definition and 89 Discussions

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.

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  1. dford

    Normal modes of four coupled oscillating masses (Kleppner and Kolenkow)

    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...
  2. J

    Normal modes of oscillation

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

    Normal modes: Spring and pendulum

    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...
  4. Mr_Allod

    Analysing the Normal Modes and Dynamics of a Cluster of Atoms

    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...
  5. B

    Solving the wave equation for standing wave normal modes

    ## \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)...
  6. H

    Coupled oscillators -- period of normal modes

    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...
  7. LCSphysicist

    Spring constant matrix and normal modes (4 springs and 3 masses)

    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...
  8. J

    I Normal modes of vibration from the total energy

    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##...
  9. LCSphysicist

    Why Do mω²xa and 2kxa Both Act in Normal Modes Oscillations?

    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?
  10. VapeL

    Equation of motion and normal modes of a coupled oscillator

    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...
  11. C

    I How can phonons "travel" if they are excitations of normal modes?

    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...
  12. P

    Two pendula connected by a spring - normal modes

    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}##...
  13. ContagiousKnowledge

    Normal modes of a rectangular elastic membrane

    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)}...
  14. T

    Normal modes of a system of springs

    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...
  15. R

    I Normal modes in a coupled system

    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 ?
  16. S

    I Normal modes using representation theory

    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...
  17. 1

    Help finding the vibrational frequencies and normal modes

    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 =...
  18. Tspirit

    How to deduce the solution of normal modes of a cavity?

    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...
  19. Crush1986

    Testing Mastering Normal Modes: Tips for Solving Physics GRE Questions Quickly

    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...
  20. AntonPannekoek

    I What is “normal” about normal frequencies and normal modes?

    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.
  21. C

    Mechanical energy in an harmonic wave and in normal modes

    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...
  22. RicardoMP

    Normal Modes and Normal Frequencies

    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...
  23. S

    I Normal Modes: Finding Eigenfrequencies

    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?
  24. S

    For a string with fixed ends, which normal modes are 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...
  25. N

    Normal modes and degrees of freedom in coupled oscillators

    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...
  26. ChickenTarm

    How do I find the normal modes of massless string w/ masses?

    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...
  27. S

    Finding the normal modes for a oscillating system

    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...
  28. H

    Normal modes of continuous systems

    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...
  29. L

    Molecular Orbital & Normal Modes

    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...
  30. PhysicsKid0123

    Why are normal modes important when analyzing waves or oscillations?

    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...
  31. J

    Help with coupled spring and pendulum system

    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...
  32. sergiokapone

    Harmonic oscillations of the electromechanical system (normal modes)

    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}...
  33. D

    Normal modes of electromagnetic field

    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) =...
  34. R

    Can somebody explain boundary conditions, for normal modes, on a wire?

    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...
  35. R

    Form of the displacement, y(x,t), for the normal modes of a string

    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...
  36. Y

    MATLAB Plotting normal modes in Matlab from data in Patran

    Hi, Could You help me with problem mentioned in the topic? I copied translational data (3 column vectors) from modal analysis in Patran (from f06 file). Now I have problem in reproducing the results in Matlab. I am trying to draw normal modes in Matlab, but I haven't got much luck so far...
  37. C

    Normal modes of oscillation

    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}...
  38. N

    Normal Modes of a Triangle Shaped Molecule

    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...
  39. ajayguhan

    Understanding Normal Modes of Objects - Linear vs Non-Linear Particles

    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.
  40. carllacan

    Finding normal modes of a 3D object

    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...
  41. N

    Normal modes of a string thought experiment

    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...
  42. S

    Normal modes of a two mass, two spring system?

    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 &=...
  43. B

    Help with finding normal modes of a bar swinging on a string

    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...
  44. U

    Normal Modes - 2 springs question

    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?
  45. brainpushups

    Normal Modes; Rod suspended from strings

    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...
  46. B

    Small oscillations: How to find normal modes?

    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...
  47. V

    How fast can you 'switch' between superconducting and normal modes?

    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...
  48. retro10x

    Finding Normal Modes (completely stumped)

    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...
  49. G

    Finding Normal Modes of Oscillation with matrix representations

    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...
  50. T

    Normal modes and system's energy

    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.
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