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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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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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}{...
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Systematic way of decoupling a coupled oscillators Hamiltonian

I have been faced with the Hamiltonian $$H = P_1^2/2m_1 + P_2^2/2m_2 + (k/2) x_1^2 + (k/2)x_2^2 + (K/2)(x_1-x_2)^2$$ I'm trying to find a systematic way to decouple it other than guessing. So, I wrote ...
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Simple coupled quantum harmonic oscillator

I am having trouble finding the eigenvalues for the Hamiltonian $$ H = \frac{P_1^2}{2M} + \frac{P_2^2}{2m} + \frac{K}{2}x_1^2 + \frac{k}{2}(x_1 - x_2)^2$$ Even though I can find a basis where the $...
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Double Pendulum - equation of angle with respect to time [closed]

First of all I am in grade 12, last year of my IB diploma programme. I'm familiar with derivatives and integrals but nothing as complex as these Lagrangians, Hamiltonians or other university-level ...
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What is the generalised harmonic oscillator kernel?

So the kernel for the Lagrangian of the 1D harmonic oscillator: $$\mathcal{L} = \tfrac12 m \dot{x}^2 - \tfrac12 m \omega^2 x^2$$ is solvable (from the Wikipedia entry on Path Integral Formulation) ...
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Why is energy usually concentrated on low frequency modes in dynamics?

In all structural dynamics applications I have seen, the motion is mostly governed by low frequency modes. For example, a pretty accurate approximation of buildings dynamics can be obtained with the ...
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How do we know that the vibrational eigenmodes of a system are able to fully describe all possible motions of the system?

In the classical case of identical masses coupled with springs, in a 3D lattice like structure. The equations of motion with the harmonic approximation are given by: $$M\ddot{u}_{m}^{\alpha} = \sum_{...
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Normal modes of cylinder and two pendulums

I am trying to understand(falsify) interpretation of the answer to the following problem: Suppose we have cylinder of mass M which can oscillate along the horizontal line. There are two pendulums ...
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Why is it that a coupled mass-spring system will always produce a diagonalizable matrix?

If you take a system like the one in the image, and you do the $y=x'$ trick to turn it into a first-order system of equations ($x_{1}$ or $x_{2}$ being the displacement of the mass $m_{1}$ or $m_{2}$ ...
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Has the asymptotic theory of eigenvalues of infinite matrixes already been applied to vibrations analysis?

My question is reffering to the masses/springs model of a material, like the one presented in this article http://www.laserpablo.com/baseball/Kagan/UnderstandingCOR-v2.pdf. If one treates a long ...
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Resonance curves for coupled pendulums

I'm looking for the resonance curves of coupled pendulums (a) and coupled spring pendulums (b). The background: I want to find analogies for the oscillating curcuit as for inductive coupled ones I ...
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In a coupled pendulum, how is the equation of motion described

This is from Hobson, Riley, Bence Mathematical Methods problem 9.1 The question is: There is a hint given: If someone could just help me understand the physics part, I can try to do the ...
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In a four mass six spring vibration, how is the kinetic energy represented

This is from Hobson, Riley, Bence Mathematical Methods, p 322. A spring system is described as follows (they are floating in air like molecules): The equilibrium positions of four equal masses M of a ...
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How to determine the critical damping factors for a multi-DOF mass-spring system [closed]

How do I choose $ b_1 $, $ b_2 $ and $b_3$ to make the masses $m_1$, $m_2$ and $m_3$ have a critical damping behavior? If I have just one mass $m$ and one spring $k$ and one damper $b$, the damper ...
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Can there exist harmonic oscillator with asymmetric coupling?

In Classical Mechanics textbooks usually, for a coupled harmonic oscillator with two masses, coupling is taken to be same in both directions (i.e coupling constant w.r.t to m1 is same as that with ...
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Coefficient of coupling in coupled oscillators

My question is how and on which things this quantity p,the extent of coupling depends? Why the force exerted by one on other is proportional to its acceleration?
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Spring-Mass-Pendulum “via Newton's Laws”

Good Night everyone: I have one problem here that I KNOW how to solve using Lagragian Dynamics. But, I really want to know how to solve using Vector decomposition, Newton's Laws, first-year physics ...
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How do I write the Lagrangian for a system with 2 different locations of oscillation?

I have a system where there is a particle placed in each of the minima of the potential $$U(x)=\beta(x^2-\alpha^2)^2.$$ The particles are also connected by a massless spring where the equilibrium ...
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Coupled Mode Equation for Fiber Couplers

I have a very specific question about coupled mode equations in fiber couplers. I read from the book "Applications of Nonlinear Fiber Optics", Govind P. Agrawal. My question is in page 65 at the ...
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Coupled oscillators with imaginary frequency?

A typical problem in Lagrangian mechanics is a system of two coupled bodies under small oscillation. The typical way to do these problems is: Write the Lagrangian e.g. $L(\theta_1, \theta_2, \dot \...