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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Equations of motion for coupled harmonic oscillators
We just started QFT, and I'm following our professor's notes but there is a passage I do not understand. We are speaking about a system of $N$ coupled harmonic oscillators $y_j(t)$ for $j = 1, ..., N$ ...
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Coupled quantum harmonic oscillator: Decomposition in terms of number basis representation for numerical implementation
Consider the Hamiltonian of a coupled quantum harmonic oscillator
\begin{align}
\hat{H}&=\frac{1}{2m}(p_1^2+p_2^2)+\frac{m\omega^2}{2}(q_1^2+q_2^2)+\alpha(q_2-q_1)^2
\\&=\frac{1}{2m}(p_1^2+...
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Energy transfer between modes of linear chain of harmonic oscillators?
Consider a one dimensional chain of N classical point masses interacting with harmonic neighbor forces (with periodic boundaries for specificity). If the positions and velocities are prepared in a ...
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Does a chain of classical harmonic oscillators exhibit non-harmonic oscillations?
Consider a one dimensional chain of N classical point masses interacting with neighbor harmonic forces. Is it possible to find initial conditions (positions and velocities) such that non-periodic (...
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Normal modes of a circular array of interacting particles
I want to study the normal modes of an array of $N$ identical atoms placed in a circular lattice. The particles interact among them via Yukawa interaction potential,
$$\phi_Y(r)=\frac{A}{r}exp(-r/r_0)....
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Vibrational Modes and Imaginary Frequencies of a Three Spring System
This question is an extension of the one I posted a few days ago: Rigid Body and Two-Spring System and the Lagrangian.
I am attempting to find the vibrational modes and their frequencies of an ...
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How to find particular solution to this system of two masses connected by a spring with constant applied forces? Does an analytic solution exist?
This question comes from review I'm doing on my own, so it's not any homework question. I thought this question would be easy to solve, but I seem to be stuck and I am having second thoughts, so I'm ...
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Interpretation of this Hamiltonian
I'm studying a system with the following Hamiltonian $$ H = \frac{1}{2}P^TAP + \frac{1}{2}Q^T B Q$$ where $P,Q$ are canonical variables (4-vectors) and $A,B$ matrices such that $A = A^\dagger$ and $B =...
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Coupled-mode theory and slowly varying envelope approximation
I am facing a situation where I have the following coupled-system equation:
$ \dot{U}(z) = i \; M(z) \cdot U(z) \quad ,$
where U is a N-vector and M is a NxN matrix.
Now, the diagonal elements of M ...
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If two pendulums connected by a spring both following Simply Harmonic Motion - why do they need the same time dependence?
I was reading a section of Introduction to Mechanics by Kleppner and Kolenkow:
where it talks about the same time dependence. I'm not very familiar with this term but was wondering if there was some ...
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Diagonalizing the tridiagonal matrix for finding the normal mode [closed]
Suppose we have n particles connected by a string tied to fixed ends.
In matrix notation the equation of motion can be written as
$\begin{pmatrix}2\omega_o^2&-\omega_o^2&0&0&.&.&...
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Coupled pendulum question about equations of motion [closed]
I am working on problem number 2.3 of the Franklin, Powell, Naemi book Feedback Control of Dynamic Systems. The problem uses the simple coupled pendulum system below, where the two pendulum masses are ...
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Why do vibration MEMS gyroscopes need two oscillating modes?
Everywhere I look, I see that MEMS gyroscopes need to have two modes, that are orthogonal in direction. However, that puts a lot of significance to the detuning of these modes, and calls for highly ...
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Diagonalisation of two coupled Quantum Harmonic Oscillators with different frequencies
I was able to diagonalise $H=\hbar\omega a^{\dagger}a+\hbar\omega b^{\dagger} b+\hbar(u a^{\dagger}b+u b^{\dagger} a)$ using the Bogoliubov transformation \begin{equation}
a=\cos{\alpha}c_1+\sin{\...
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Two springs soldered together, equations of motion
I know there are already several questions about springs in series, but I think this one is different. It is from an exercise from a German book ("Physik mit Bleistift" by Hermann Schulz) ...
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Why do systems of $n$ coupled oscillators have $n$ normal modes?
Consider a linear system of $n$ differential equations with constant coefficients corresponding to a physical scenario where I have $n$ coupled oscillators (like $n$ masses attached by springs in ...
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Why is the product of the $L$ and $C$ matrices for coupled transmission lines diagonal?
Background - transmission line
$\newcommand{\ket}[1]{\left \lvert #1 \right \rangle}$
A transmission line can be modeled as an infinite sequence of inductors and capacitors:
...
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Derivative term in chain of LC coupled oscillators Hamiltonian
I am taking quantum superconducting circuits course and I cannot recover a formula provided by the lecturer.
I want to calculate the Hamiltonian of the following distributed element model of coplanar ...
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What is the relation between Chebyshev polynomials and coupled oscillators?
I have been told that Chebyshev polynomials are key for finding the normal modes of oscillations of a linear chain of coupled oscillators, since they are the eigenmodes of the system. However, I ...
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Nonlinear PDE from Chain of Oscillators
Some years ago, I was reviewing the calculation for the dynamics of limiting case for a chain of springs with transverse oscillations and found a partial differential equation for which I haven't been ...
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Mass-spring system linear equations: I don't get the last term, shouldn't it be $V=\frac{1}{2}k_3x_{\text{wall}}^2-2k_3x_{\text{wall}}x_2+k_3x_2^2$?
I don't understand the last term in setting up the linear system of equations for multiple mass-spring systems. It is about the last spring in the next example:
Source: https://math24.net/mass-spring-...
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Difference between Symmetric and Phase difference?
For a simple coupled oscillator system such as the one here, with equal spring constants and equal masses (with a displacement from equilibrium of $x_1$ and $x_2$), it follows that:
$(\ddot{x}_1+\ddot{...
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How to integrate a system with a hamiltonian with non-diagonal kinetic energy
I have the following classical Hamiltonian for two coupled oscillators in the same molecule:
$$H=T+V =\left(\frac{p_1^2}{2\mu}+\frac{p_2^2}{2\mu}+k_pp_1p_2\right)+
\left(\frac{1}{2}\mu\omega_1^{2}x_1^...
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Does Hooke's law explain classical wave behavior?
Will Hooke's law $F = -kx$ applied to a large mass-spring grid array such as:
provide the full and complete mechanistic explanation for classical wave behavior, including the 2nd order wave ...
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How to calculate the energy of a spring-mass system considering harmonic oscillation of the normal mode? [closed]
For a spring-mass system, we know that the potential and kinetic energy are
$$E_p = \frac{1}{2}ku^2 \text{ and } E_k = \frac{1}{2}m\dot{u}^2.$$
where $k$, $m$ and $u$ are the spring constant, mass and ...
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Transfer function of system of coupled 2nd order ODE
I am wondering how to calculate transfer function $H(s)$ of system described by 3 coupled differential equations. The pourpose of work is to calculate "Bodedx" diagram ($|H(i\omega)|(\omega)$...
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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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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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Coupled Oscillator Period [closed]
I was studying an example of a coupled oscillator the other day, namely two identical masses attached to three springs, the lateral ones of which with the same elastic constant, when I came across the ...
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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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