Questions tagged [coupled-oscillators]

Harmonic oscillators may have several degrees of freedom linked to each other so the behavior of each influences that of the others. For example, two pendulum clocks (of identical frequency) mounted on a common wall will tend to synchronize. The apparent motions of the compound oscillations typically appears very complicated, but a more economic, computationally simpler and conceptually deeper description follows resolving the motion into [normal-modes].

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Expression for total potential energy in coupled systems

I was reading through applications of Lagrangian mechanics and the case of coupled oscillators. The example provided is the famous two pendula length $l$ mass $m$ hanging from the ceiling connected by ...
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How to calculate properties of a coupled electromagnetic / mechanical oscillator?

I'd like to study a class of systems which are (essentially) coupled electromagnetic/acoustic oscillators which can act as antennas for an electromagnetic field, but also can vibrate mechanically with ...
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Stability around an equilibrium point in a normal mode

So, I was reading the chapter of small oscillations in Landau and Lifshitz's book of Mechanics. We assume solutions of the equations of motion that are in the form of $X_a=Ae^{iω_at}$ where $A$ is an ...
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Cohen-Tannoudji coupled harmonic oscillator

In the Cohen-Tannoudji QM book they say that the potential $$V(x)=\frac{1}{2}m\omega^2(x_1-a)^2 + \frac{1}{2}m\omega^2(x_2+a)^2 + \lambda m\omega^2(x_1-x_2)^2$$ describes two classical coupled ...
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Equipartition and coupled harmonic oscillator system

This question has to do with analyzing how equipartition sets in for a system such as a coupled harmonic oscillator system. Take, for example, a system as shown in the figure: $\xi$ denoting the ...
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Off-diagonal terms of a frequency response matrix

If I have coupled system of two harmonic oscillators. $$\ddot{x}_1+\Gamma\dot{x}_1+kx_1-kx_2=0$$ $$\ddot{x}_2+\Gamma\dot{x}_2+kx_2-kx_1=0.$$ Then I can fourier transform the equations of motion and ...
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Coupled quantum harmonic oscillators (exact $\neq$ perturbative)

Suppose we are given the Hamiltonian $$\hat H = \hat H_0 + \hat H_p(\varepsilon) = \frac 1 {2m}(\hat p_1^2 + \hat p_2^2) +\frac 1 2 m \omega^2(\hat x_1^2 + \hat x_2^2) + \varepsilon m\omega^2\hat x_1\...
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Spin-Spin Hamiltonian in coupled harmonic oscillator

I was reading about identical particles and i came across this example: Consider two electrons with spin 1/2. The Hamiltonian for this system is: $$Η=\frac{p_1^2}{2m}+\frac{p_2^2}{2m}+\frac{1}{2}m\...
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What are the normal modes of a velocity-dependent equation of motion?

I'm trying to find the normal models of a particle with charge $q$ and mass $m$ in a $3$-dimensional harmonic oscillator potential with an applied uniform magnetic field $B=B_0 \hat{z}$. The potential ...
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Recommendations for good books on mechanical vibrations

I'm looking for books that explain/model vibration concepts such as multi degree of freedom vibrations from a mechanical vibrations standpoint (not waves). I'd prefer if it has proofs for most ...
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Vibrational spectrum of a mass disordered chain

Consider a linear spring-mass disordered chain with a large number of masses (say $10^6$ masses). The spring constant $k_i$ of each spring is set to 1. The chain consists of atoms of mass 1 and mass 2 ...
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How to deal with coupled equations of motions in equilibrium analysis?

Consider a system including two generalized coordinates $q_1$ and $q_2$ whose dynamics is supposed to be obtained using first-kind Euler-Lagrange (E-L) formalism $$\frac{d}{dt}\frac{\partial L}{\...
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Vibrational Spectra of a mass disordered chain

Consider a linear spring-mass disordered chain with a large number of masses (say $10^6$ masses). The spring constant $k_i$ of each spring is set to 1. The chain consists of atoms of mass 1 and mass 2 ...
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Landau mechanics - Normal modes of oscillation

In Landau's Mechanics book there's a section in which he explains small oscillations in systems with $s \geq 1$ degrees of freedom. He writes the kinetic and potential energies as $$ T = \sum_{i, k} \...
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Doubt about finding normal modes in Molecular vibrations

In the book introduction to classical Mechanics by Kleppner and Kolenkow, while dealing with the analysis of molecular vibrations in a poliatomic molecule, they propose the following method, in order ...
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Coupled oscilators doubts and cases

I'm having some troubles with some questions of coupled oscillators, there are not difficult questions but i doubt in my reasoning and i I have not found anything about this doubts. First of all, i ...
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Instability of coupled non-linear oscillators

Consider a bunch of interacting oscillators (e.g., a chain of atoms), interacting due to anharmonicity in the potential energy. You can Taylor expand the force on each oscillator about equilibrium ...
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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 ...
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Dynamics of a linear chain of harmonic oscillators

Let's consider a linear chain of particles with harmonic nearest neighbor interaction: Assuming all particles have the same mass, Equations of motion are (with periodic boundary conditions): $$m\ddot{...
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Releasing items in the hold of a ship makes it easier to rock a ship?

There's a scene in the movie Pirates of the Caribbean III (At World's End) wherein the characters are trying to flip their pirate ship upside down as it floats in the water. To do this, they try to ...
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Why coupled oscillators tend to seek integer frequency ratios?

In this document, the author writes (page 225) Coupled oscillators have a tendency to seek frequency ratios which can be expressed as rational numbers with small numerators and denominators. For ...
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Coupled Oscillator's Stiffness and speed of light

In Schwabl book (Advanced Quantum Mechanics) page 258, in his triumph to show the relation between the coupled oscillators and Klein-Gordon equation he finds the following relation which is the ...
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Two-coupled oscillator, problem understanding general solution

I want to find the general solution of this system: First, let's assume that the system is symmetric, i.e the masses are equal. By using newtons second law for rotation on the points where the rope ...
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Energy states of particle in potential $V(x,y)= x^2 + y^2 + xy$

How can I find the energy of a particle in a 2D potential of form $V(x,y)= x^2 + y^2 + xy$? It looks to have a close relation with Quantum Harmonic Oscillators, is it related to it? What could we say ...
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Two-coupled oscillator: Doubt in finding normal modes and natural frequency

I want to find the natural frequency of a two coupled oscillator system like this- My book does it this way but I don't really get it. The equations of motion for the pendula are- $$I\frac{d^...
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Infinite Coupled Masses, symmetry, and the simultaneous diagonal theorem for infinite dimensional vector spaces

In The Physics of Waves by Georgi, in Chapter 4, we show that, in a coupled system of masses connected by springs, a transformation that preserves some symmetry $S$ commutes with $K^{-1}M$. From my ...
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Derivation of wave equation

I learned that the wave equation derivation is below. Suppose $q$ is the displacement on $y$ component, $T$ is string tension, $d$ is the interval of two particles in $x$ direction. Equation of ...
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Difference between resonance and the synchronization of coupled oscillator

I was reading about phase synchronization of coupled oscillator where the oscillators are synchronized by an applied field. Now the coupled oscillators are synchronized. So my question is that what is ...
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Coupled oscillators in Hamiltonian formalism - problem with diagonalization

I have a problem with simple coupled oscillator system. I tried to solve single oscillator with Hamiltonian, and then coupled system of two, but when I try to put coupling constant $k^\prime=0$ in my ...
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Energy Eigenvalue for SHO Classical and Quantum

Let's assume we are given a potential for coupled harmonic oscillator: $$U = \frac{k_1(x_1^2 +x_3^2)+k_2 x^2+k_3 (x_1x_2 + x_2x_3)}{2}$$ If I solve the normal modes of the oscillator I get the ...
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Solving a system of three masses and two springs

Let's say $m_1$ is attached to $m_3$ via a spring of constant $k_1$ and $m_3$ is attached to $m_2$ via a spring of constant $k_2$. Just to simplify the problem we can make $m_1=m_2=m_3$ and $k_1=k_2$. ...
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Can someone please explain the meaning of the circled paragraph?

why does the off diognal elements of the matrix mediate with the coupling differential equation?
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Why are all solutions to this system of pendulum differential equations a linear combination of the two given solutions?

I am currently trying to do a lab report for a coupled pendulums experiment in which we find the following linear system of second order differential equations (describing the position as a function ...
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What is the link between the rotating wave approximation and the algebraic representation of a dynamical system?

In analyzing a system of two coupled oscillators, I noticed a rather interesting correspondence between the so-called "rotating wave approximation" (RWA) for solving differential equations and the ...
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Recovering symmetry in coupled oscillators

Consider a pair of LC oscillators, one with capacitance $C_1$ and inductance $L_1$ and the other with capacitance $C_2$ and inductance $L_2$. Suppose they're connected through a capacitor $C_g$. We ...
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Coupled quantum oscillator: Field theory

Consider two masses $m$ connected by a spring with a spring constant $k$. Each mass is also connected to the wall using the same springs. The Hamiltonian is $$ H = \frac{p_1^2 + p_2^2}{2m} + \frac{k}{...
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Change of coordinates of Lagrangian

Consider the system above ($m_1$, $m_2$, and $m_3$ are connected by springs of stiffnesses $k_1$ and $k_2$, respectively. Also, $m_1 \neq m_2 \neq m_3$). The Lagrangian is $$L(x_{1},x_{2},x_{3},\dot ...
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Lagrangian corresponding to these equations of motion [closed]

I have the following equations of motion for a system with two degrees of freedom: $$\ddot{q_1}+q_1^2-q_2^2=0$$ and $$\ddot{q_2}+2q_1q_2=0.$$ I have tried to deduce the Lagragian corresponding ...
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Vibrational Question [closed]

A friend recently asked me this question, which I am not even sure how to comprehend... Three masses are arranged at the vertices of an equilateral triangle and are connected by springs along the ...
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Derivation of the dynamical matrix

I am following a derivation of the dynamical matrix given at http://physik.uni-graz.at/~pep/Lehre/PP/DynMat.pdf. Here T and W are kinetic and potential energies. $T=\sum_{n\alpha i}\frac{M_{\alpha}}{...
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Coupled pendulum, energy exchange period as a function of different lengths

The energy exchange period $T_{x}$ of a coupled pendulum with coupling strength $\mu$ in the symmetric case(where the two natural frequencies with equal masses and equal lengths are the same, i.e. $\...
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Solution of the coupled non-linear oscillators by using perturbation theory [closed]

The integration shown here, $$∫_{-\infty}^{+∞}x^r\mathrm{Exp}[−x^2]\mathrm{H_n}^2[x]\mathrm{d}x,$$ appears when we try to calculate the spectrum of the perturbed non-linear oscillators by using ...
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Several spring coupled: can such a movement happen or is it only theoretical?

We have 6 particles. We couple them 2 by 2 with a spring of strength $K$ (as in the picture below). We then have 3 harmonic oscillators. Then we couple each oscillator by a spring of strength $S\ll K$ ...
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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) ...
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Coupled pendulums at half height

Suppose we have the system described below (poor quality but it'll do the trick). We have two pendulums of mass $m$ coupled by a string of constant $k$ placed at a height $a$ from the top (as shown). ...
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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, ...
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Equation of coupled springs : where does this potential come from?

In this document, we try to derive the equation of two coupled springs as in this picture. At the bottom of the page 2, they say : it would be more efficient to introduce the potential energy ...
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Guitar strings struck out of phase

Do guitar strings struck out of phase with one another force each other to begin vibrating in phase with one another? I ask because wouldn’t chords sound more dissonant from time to time if this did ...
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Coordinates change for separating the Hamiltonian of a quantum system

Are there general methods, tips or tricks for choosing the correct change of coordinates so that the Hamiltonian of a quantum system becomes separable? Referring to Shankar's Principles of Quantum ...
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mechanical analogy to signal propagation on coaxial cable

EDIT: The analogy is wrong if we think of voltage propagation, I confused and in the eletrical signal, it actually happens contrary to what the analogy predicts. However, for current, it holds. I am ...