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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Interpreting physical meaning of normal modes
What really is a normal mode? Maybe it's because of my teachers but I find it really abstract. I know that "numerically" corresponds to the eigenvectors of the equation $\ddot{X}= -M^{-1}KX$ ...
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General wavefunction for a system of two coupled, quantum oscillators
Suppose we have two quantum harmonic oscillators, with different masses $m_{1},m_{2}$ and frequencies $\omega_{1,2}$. Then we can say particles are \emph{distinguishable}, in the sense that particle $...
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Quantum harmonic oscillators with momentum-position coupling
I have two coupled quantum harmonic oscillators given by the following Hamiltonian:
$$H=\frac{p_{x}^{2}}{2}+\frac{\omega^{2} x^{2}}{2}+\frac{p_{y}^{2}}{2}+\frac{\Omega^{2} y^{2}}{2}+\frac{C p_{x} y}{2}...
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Is there any effect that draws the oscillation frequencies of two particles together?
I'm looking for any sort of coupling that draws oscillation amplitudes together if one couples two (nearly) harmonic oscillators, basically the opposite of avoided crossing or level repulsion.
Is ...
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Trouble finding the matrix form of potential energy in small oscillations (Goldstein linear triatomic molecule example)
I'm currently trying to learn small oscillations, I kind of comprehend the general theory, but I'm having hard times finding the matrix forms of the potential and kinetic energy. I have been following ...
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Two-mass spring system in $x$-$y$ plane motion [closed]
I wrote a coupled differential system that describes two masses coupled by one spring. Both masses are free to move in x-y plane. To test the model, I plotted a position of two masses in x-...
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Why is there only partial energy exchange for coupled oscillators with different masses?
So I was playing around with this widget and I noticed that when the spring masses are the same, the energy is completely exchanged (note one has to set the "show graph" option to "...
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How to obtain the wave function of time-dependent coupled two harmonic oscillators?
The form of the Hamiltonian of this system is
\begin{equation}
H = \frac{p_1^2}{2} + \frac{p_2^2}{2} + \frac{x_1^2}{2} + \frac{1}{2}\omega^2(t)x_2^2 + \frac{q}{d}x_1x_2
\end{equation}
where $p_1$ and $...
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Forced coupled oscillators and antiresonance
I was looking at the wikipedia page for anti-resonance and have found a solution to the steady state equation, for a forced coupled oscillator, however there is no explanation on how this result is ...
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Steady state solution to a coupled oscillator equation where one of the oscillators are driven?
I am analysing the dynamics of quantum van der Pol oscillators in the classical limit and I found the following equation of motions for the complex amplitudes of oscillator A and B:
$$ \dot \alpha = (-...
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Relative Phase distribution for two coupled van der Pol oscillators where one has a Drive?
I was solving the dynamics for a driven coupled (inertial) van der Pol oscillators, where only one oscillator is driven. I started with the complex amplitudes $\alpha$ for both of the systems which ...
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Confusion with Noether's theorem: Time symmetric system with velocity-dependent terms?
Noether's theorem says that any system that is time-translation-symmetric displays energy conservation, and vice-versa. However, I'm not sure if this is the case.
Suppose we have a two particles (of ...
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How to calculate the damping coefficient using the tagent delta value? [closed]
The Problem
I have a mechanical system with multiple degrees of freedom like this one:
And I am given the following information:
The mass $m$ in (kg)
The spring constants $k$ in (N/m)
The damping ...
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Calculating the eigenenergies of two coupled quantum harmonic oscillators? [closed]
I am trying to calculate what the energy spectrum of two coupled quantum harmonic oscillators look like but didn't know the steps to take in order to do this, I have a Hamiltonian of the form:
$$H = \...
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Intuitive understanding of the resonance of a bridge
Dynamics of strides, walking and corresponding force cycles: In the civil engineering literature, it is known that the resonant frequency of a bridge should not be the same as that of the strides of ...
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Difference between reactive (coherent) and dissipative coupling in open quantum systems?
I am unsure about the physical interpretation of the different types of coupling, I understand that reactive coupling is manifested by a term within the Hamiltonian and dissipative is via a term in ...
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Linearising two coupled bosonic modes
In Sec. IV of this paper the authors consider the Hamiltonian (Eq. 10)
$$
H = \omega_a a^\dagger a + \omega_b b^\dagger b - g_0 a^\dagger a (b + b^\dagger)
$$
in the regime $\omega_a / \omega_b \...
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A mechanical system with coupled oscillators and dampers
I found here Coupled oscillators, the last one on this page, a system named by the author "Vehicle Suspension System". But I am not sure this modelizes a suspension for a vehicle.
I am ...
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Pendulum with flexible connecting rod: synchrony in nature
Consider a pendulum with a flexible connecting rod. When initializing the free motion, the bob would be released at a position such that the connecting rod is flexed as shown in the figure. There are ...
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1answer
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Double eigenfrequencies of normal modes
If I have 3 masses displayed along a ring connected by springs, the frequencies I found were:
$$\omega^2=\frac{3k}{m},0$$
I don´t understand why I have 2 double eigenfrequencies.
Is this possible, if ...
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Hamiltonian for closed system describing parametric excitation
I am trying to describe a system with parametric excitation.
Usually this is described as an open system where the time dependence of a parameter is explicitly included in this differential equation:
$...
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How can a rocking boat be modelled?
I am seeking an algoritm for how an unpropelled, unanchored boat at rest moves due to waves. Pitch, yaw, vertical and horizontal displacement are sought.
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Differential equations of a forced coupled spring-pendulum system
Currently working on a problem and I can really figure out how to write the differential equations for it. Here's the situation:
So we have a mass $m$ tied to the wall with a spring of constant $k$. ...
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2answers
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Simultaneous diagonalization of potential and kinetic energy
I am trying to prove that the matrix expression of the potential energy (Hessian matrix from a Taylor expansion in several variables of the potential) is diagonal considering small oscillations, when ...
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Infinite wavelength for the 1D harmonic chain of oscillators corresponds to a uniform translation, but why does it still have a finite phase velocity?
The dispersion for the 1d chain of harmonic oscillators is
$\nu = \sqrt{\frac{\alpha}{m}} \frac{|\sin(\pi k d)|}{\pi}$
Where I'm explicitly not using angular frequencies ($\nu = \frac{1}{T}, k = \frac{...
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Three masses with 2 springs in 1D
Consider three arbitrary masses attached by two different springs in vacuum, starting at arbitrary initial positions with no initial velocities. Is this system chaotic? Is this system analytically ...
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Forced Oscillation Explained with Violin String
In this lecture on Forced Oscillations, Normal Modes, Resonances, Musical Instruments, the professor says that by moving a bow over a violin string, you expose it to a lot of frequencies.
Is there a ...
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1answer
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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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2answers
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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 ...
