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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Motion of an $n$ mass $n$ spring system [closed]
While reading wave motion I encountered the problem of $n$ identical masses with $n$ identical springs in between them.
If we give a sudden push
to the wall attached to the first spring, what will ...
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Terminology question: in-phase or out-of-phase?
Suppose that in a chain of many coupled oscillators, the displacements of two consecutive particles, in a normal mode of oscillation with frequency $\omega$, are given by $$x_p(t)=A_pe^{i\omega t}$$ ...
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Why is any arbitrary motion of a coupled oscillator writable as a linear combination of its normal modes? [duplicate]
Consider the following example of a coupled oscillator. Let two identical pendulums, each of length $\ell$ and mass $m$ be connected by a spring of force constant $k$. The system has two normal modes ...
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How to describe a series of damped harmonic oscilators?
I am looking for textbooks or papers that provide an analysis for a series of damped springs. I am having a tricky time working out the details on my own. I know that if $F=-k\Delta x$ a series of ...
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Equations of motion only have a solution for very specific initial conditions
An exercise made me consider the following Lagrangian
$$L = \dot{x}_1^2+\dot{x}_2^2+2 \dot{x}_1 \dot{x}_2 + x_1^2+x_2^2.\tag{1}$$
If I didn't make a mistake the equations of motion should be given by:
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Estimation of number of states to be used to obtain Wigner-Dyson distribution in a chaotic coupled oscilllator
A coupled harmonic oscillator with quadratic coupling -
$$
H = \frac{1}{2}(p_x^2 + p_y^2) + \frac{1}{2}(x^2 + y^2) + g x^2y^2,
$$
is known to be non-integrable, hence chaotic (for reference look at
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How to perform a Gaussian functional integral?
I'm completely beginner to the quantum field theory and try to learn the basics of functional integrals. However, I could not understand clearly. Could someone please explain the idea with the help of ...
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Question about the frequency of normal modes in coupled oscillators and their derivation [closed]
Consider a two-mass system that is coupled by three springs, such that: $m_1=m_2=m$; $k_1=k_3=k; k_2=k_{12}$. It can be written in terms of the following coordinates: $\eta_1=x_1-l_1$ and $\eta_2=x_2-(...
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In a coupled pendulum system, how do pendulums *know* which one is giving its energy, and which one is receiving it?
Picture the system below:
If you give the pendulum on the left a push, it will slowly transfer its energy to the one on the right, until it stops completely and the right one is in full swing, then ...
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Self-synchronizing and -desynchronizing systems of oscillators
There are biological systems with adaptable frequencies that are able to synchronize their frequencies, mainly individuals (see e.g. reproductive synchrony). In this case, also the phase is typically ...
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Commutation relations for coupled systems
Consider $N$ coupled oscillators whose Hamiltonian is $$\hat{H}=\sum_j^N\frac{\hat{p}_j^2}{2m} +\frac{1}{2}K(\hat{x}_{j+1}-\hat{x}_j)^2,$$ which represents a system like ($K$ is the spring constant):
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Normal modes of coupled oscillators
For two pendulums of mass $m_1$ and $m_2$, coupled by a spring of constant k, both suspended by strings of length $l$, the following matrix equality results from their equations of motion:
$$ \omega^2
...
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Equations of motion of two bodies attached to three springs
I've been tasked with describing the equations of motion of two bodies attached via three springs, as visualized below.
Let $x_1(t)$ and $x_2(t)$ denote the $x$-displacements of boxes $m_1$ and $m_2$ ...
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A coupled nonlinear dynamical system in four dimensional phase space
I have come across a coupled nonlinear dynamical system given below
$$ r\, \ddot{x} + \dot{x} = \sin y~,$$
$$ r\, \ddot{y} + \dot{y} = \sin x~,$$
where $r$ is some real number and $\dot{x}$ denotes $\...
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What are normal modes?
In normal modes analysis the differential equations of the system are Fourier transformed and the Fourier monochromatics are found. I think these monochromatics are usually called normal modes of the ...
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Use Hamilton's principle to show expression for $L$ [closed]
I have following diagram
I have here to find the kinetic energy and the potential energy. I think that kinetic energy is:
$$T=\frac{1}{2} M(\dot{x_1}^2+\dot{x_2}^2)$$
and the potenitial energy must ...
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Why is synchronisation only possible for self-sustaining oscillators
A self sustained oscillator is any oscillator which obeys the following 3 key properties (Balanov 2009):
They do not damp
They are capable of oscillating without being driven by an external force.
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Deriving the parameter $\exp(\eta)$ in eq. (2.9) from "Illustrative example of Feynman’s rest of the universe"
I'm working on research on the Entanglement in Coupled Harmonic Oscillators when I stumble upon the research paper "Illustrative example of Feynman’s rest of the universe" https://doi.org/10....
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Normal Mode in a Vibration?
What exactly is the Normal Mode? According to me, it forms a basis functions (where Max transverse amplitude is fixed wrt position), and all arbitrary vibration in the string can be written in the ...
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Dispersion of finite 2D lattice
Following problem: I have the coupling matrix for an $N$-by-$N$ finite lattice of coupled masses (only nearest-neighbour coupling, periodic until terminated). I would like to numerically calculate its ...
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A confusion about position and momentum operators in a coupled system
On page 25 of "Quantum field theory for the gifted amateur" by To has written:
Consider a linear chain of $N$ atoms (see Fig. 2.5), each of mass m, and connected by springs of unstretched ...
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Invert mapping from physical to coupled resonator parameters
If we couple two LC resonators (named "a" and "b") through a mutual inductance $M$, Kirchhoff's laws take the following form
\begin{align}
C_a \ddot{V}_a + \frac{V_a}{L_a'} - \...
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Are there non-orthogonal "normal" modes for non-identical coupled oscillators?
The question is broad, I will specify an example to elaborate what I'm asking.
Suppose I have two different LC circuits inductively coupled (or capacitively, but the question I have will be relevant ...
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Distance Between the Point Masses in a Pair of Coupled Pendula [closed]
Here is an embarrassingly simple problem, which for some reason I can't figure out. You can also find my solution attempt here.
Two point particles of mass $m$, a pair of identical rigid rods of ...
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Coupled Oscillations: no beating effect
Consider the famous coupled oscillation problem of 2 spring pendulum:
In a special case the solution can be given as follows:
$x_1 = \displaystyle \frac{C_1}{2}\,\cos(\omega_1\cdot t)+\frac{C_1}{2}\,\...
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Hamiltonian of two coupled oscillators
Lets say I have this system:
Two different masses with three different springs.
It's not very nice to do, but I can find the eigenvalues of this system (It's not nice because the two masses are ...
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The equation of motion(EOM) of rotation of couple moment of magnet bar which rotation axis is fixed at middle and placed in uniform magnetic fields
The magnet bar which rotation axis itself is fixed at the middle of the bar and the magnet bar is placed in the uniform magnetic fields.
$$ I \left[ \text{kg} \cdot \text{m}^{2} \right] :=\text{...
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Deriving the wavefunctions of coupled QHO without diagonalizing the Hamiltonian
I have the following Hamiltonian for two coupled quantum harmonic oscillators
$$H=\frac{p_{x}^{2}}{2}+\frac{\omega^{2} x^{2}}{2}+\frac{p_{y}^{2}}{2}+\frac{\Omega^{2} y^{2}}{2}+\frac{C p_{x} y}{2}.$$
...
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How does time translational invariance and linearity imply irreducible solutions?
The author of the book THE PHYSICS OF WAVES has mentioned on page 69, at the start of second last paragraph, that
The point is worth repeating: Time translation invariance and linearity imply that we ...
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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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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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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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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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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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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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