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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Physical Interpretation of Normal Modes

For a simple case of frictionless coupled oscillators shown in the figure below: (Image: two pendula of equal length and equal masses suspended from a level ceiling and connected by a spring) (and ...
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Couple-Mode Theory: Coupling Coefficients of Two Resonators

I'm currently reading H. Haus book on "Wave and Fields in Optoeletronics". On the chapter of Coupling of Two Resonators Mode, the rate equations for the fields are introduced: $$ \frac{da_1}{...
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Can concepts like “critical damping” or “resonant frequency” be applied to more complex systems than just a spring and damper in parallel?

I am trying to do some modeling analysis by representing materials with parallel systems of springs and dampers. In the simplest case with just one spring parallel to a damper, we have the traditional ...
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Why does resonance occur at only standing wave frequencies in a fixed string?

I could not understand why only certain discrete frequencies are allowed in a fixed-ended string. How does the string behave if we excite it with frequencies other than resonant frequencies?
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What does resonance conceptually mean in coupled Oscillators?

In Forced oscillations, resonant frequency is the frequency at which amplitude is maximum. So what do resonant frequencies mean in coupled oscillators (since they are not corresponding to maximum ...
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References for the meaning and conditions for resonance in coupled oscillators

I am looking for any references (books, articles, lecture slides) or Answers regarding Resonance in Coupled oscillators. I would like to know what resonance means in coupled oscillators and the ...
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Diagonalizing eigensystem to find normal modes of coupled oscillator [closed]

I've two equations of motion that arose in a coupled oscillators in a magnetic field $\rightarrow$ continuum problem in classical mechanics: \begin{eqnarray*} -\omega^2 X &=& - \omega_0^2 X (2 ...
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Question about coupled pendula

Consider two pendula, as pictured below, consisting of point masses $m_1, m_2$ and massless rods of length $L_1, L_2$, with $L_2 = 2 L_1$ and $m_1 = 2 m_2$. The two pendula are connected by a spring ...
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Hamilonian as a sum of quantum oscillators with symmetric matrices

I'm watching the Introduction to Quantum Field Theory course by Tobias Osborne (the lecture notes for which can be found here https://raw.githubusercontent.com/avstjohn/qft/master/QFT.pdf). In the ...
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Definition of Phase Shifts (Coupled Oscillators)

I was wondering if maybe someone could look at this excerpt from a textbook (Attached). It states that “the displacement of the two masses will be in opposite directions (out of phase by pi)” but I ...
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Coupled oscillators - why is the order of $x_2 - x_1$ or vice versa the way it is? [closed]

In my physics class, we have been working on the two mass, three spring (coupled oscillator) problem and I found this great video that helps explain how to set up one of these problems. (Link: Coupled ...
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Need help creating the Lagrangian for a coupled pendulum [closed]

I know that for 2 separate single pendulums, the kinetic and potential energies are: $$KE = \frac{1}{2}I(\dot\theta_1^2 + \dot\theta_2^2)$$ $$PE = 2mgl - mgl(\cos\theta_1 + \cos\theta_2)$$ But I don't ...
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Symmetry and Finite Coupled Oscillators

For an infinite system of coupled oscillators of identical mass and spring constant k. The matrix equation of motion is $\ddot{X}=M^{-1}KX$. and the eigenvectors of the solutions are those of the ...
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Good basis for coupled modes

Suppose, there is an electro-optical modulator that can couple the neighboring modes in an optical ring resonator. The Hamiltonian for the system looks something like this: $$H=-\frac{J}{2} \sum_{m}b_{...
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Questions About Highly Coupled Magnetic Resonance

This is one of very few in-depth sources of information I can find online about Highly Coupled Magnetic Resonance: The last sentence of the third-to-last paragraph says that research is underway that ...
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Confusion between superposition of SHMs

I am a high school student and i am little confused between superposition of Simple harmonic motions{SHM's}, suppose a spring of spring constant $k_1$ has time period $T_1$ and another spring of ...
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Coupled oscilators equation of motion

There is something physically I do not understand, consider this situation: Lets assume there is no friction. if we defind $x_1$ and $x_2$ the displacement of mass $M_1$ and $M_2$ respectively, we ...
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Slow parameter oscillations in coupled simple harmonic oscillator

In a paper by Turaev, he studies systems with a slow variation of parameters. The following is on page two, right column: He first discusses the one dimensional particle in a box, with ends at $x=-1$ ...
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Interpretation of normal modes from the mathematical formula

In the topic of small oscillations, the system below has a normal mode described by: $$n_{1} = \frac{x1+x2}{2}.$$ This normal mode is represented as the symmetric mode: In that case, the center of ...
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Where do we observe the effect of quantum-level repulsion and what is its origin? [duplicate]

Wikipedia says that for a system of two coupled oscillators, as the coupling strength between the oscillators increases, the lower frequency decreases and the higher increases. This effect can be ...
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Quantum energy spectrum of coupled LC harmonic oscillators

For a LC harmonic oscillator, the energy spectrum is evenly spaced by $$ \Delta E = \hbar \omega \quad \omega = {1\over \sqrt{LC} } $$ For two inductively coupled LC harmonic oscillators with mutual ...
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What does it mean if a circuit has 2 resonances?

I was going through capacitive coupling of qubits, and someone said to me that if we couple qubits with a capacitor we have 2 resonances. I do not understand, what does it mean to have 2 resonances.
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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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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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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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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 $k_1=k_2$. ...