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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Potential energy of a three coupled pendulum [closed]

For a three coupled pendulum, coupled by springs of force constant $k$, (generalized coordinates used are angles $\theta_1$, $\theta_2$, $\theta_3$). In the potential energy equation, there is a term ...
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Chaos synchronization & Desynchronization- terms and concepts

I am new to the world of chaos theory & control. I am reading the paper, "Synchronization of chaotic systems " https://aip.scitation.org/doi/full/10.1063/1.4917383 I have some basic questions ...
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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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Orthonormalization of eigenamplitudes

Assuming $(-\omega^2 \hat m + \hat k)\vec{a}=0$ where $\vec a$ is the eigenamplitude of the eigenfrequency $\omega$ , $\hat m$ is the mass matrix and $\hat k$ is the matrix of the potential constants. ...
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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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Coupled Harmonic Oscillator (Forced Vibration)

I derived two equations for a 2DOF harmonic oscillator system, declared state variable equations, and placed them into matrix form: $Ax' + Bx = C$. I have a Matlab script to determine the constants ($...
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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 ...
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Numerically Modeling Coupled Oscillators Point Masses

I seek to model the motion of two coupled oscillating point masses as shown below: Note that x1(t) models the leftmost point mass and x2(t) is the motion of the rightmost point mass. I would like to ...
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Contradiction? Synchronized (steady) state and power flow

I am trying to investigate power flow in a simplified power system model with a few connected rotating motors. The dynamics is captured in the differential equations $$\frac{d^2 \phi_i}{dt^2} = ...
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Ott-Antonsen-Ansatz

im reading a paper https://doi.org/10.1063/1.2930766 about the Ott-Antonsen-Ansatz that is used to describe the dynamics of global coupled oscillator. There is a computational step from equation (4),(...
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Normal modes: how to get reduced masses from displacement vectors, atomic masses and vibrational frequencies

I'm calculating normal vibrational modes in a large molecular system. My goal is to obtain, for each normal mode, the vibrational frequency, the list of displacement vectors and the reduced mass. I'...
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Is there a relation between large-scale oscillations and small-scale oscillations?

From Neural oscillation - Wikipedia: Oscillatory activity in the brain is widely observed at different levels of organization and is thought to play a key role in processing neural information. In ...
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Coupled many body quantum harmonic oscillator in 3 Dimension

Can coupled many body quantum harmonic oscillator be solved exactly ? $$H=\sum_{i}\dfrac{\vec{p}_{i}^{2}}{2m}+\dfrac{1}{2}\sum_{k,j}K(\vec{R_{k}}-\vec{R}_{j})^{2}$$. Is there any reference to solve ...
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Energy Transfer Between Coupled Oscillators

This problem on energy transfer between coupled oscillators is from Introduction to Mechanics Kleppner and Kolenkow. Question Consider again the two-pendulum and spring system we have just discussed....
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Clarifying and simplifying what 'mode' means in terms of SHM

I am dealing with a string-coupled pendulum, where two pendulum are tied onto one string as seen in image 1. (Image attributed to Young-ki Cho available from pre-view at DeepDyve) The symmetrical ...
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Understanding the Terms in Coupled Springs Differential Equation

I am teaching differential equations and I got myself totally confused about the physics of a problem. Consider a coupled spring system in series: there is a mass $m_1$ on a horizontal track which is ...
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Coupled Oscillators with unknown spring constants

For a problem, I had the following coupled oscillator system for objects A, B and C of mass 0.13 kg and I am given two of the three proper vectors given as $$\left|1\right\rangle =\left(\begin{array}{...