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].
254
questions
1
vote
0
answers
26
views
Coupled quantum harmonic oscillators in vacuum state
I'm having some problems in understanding something conceptually. I believe that I have some unphysical results (so I did something wrong in the calculations) or maybe the results that I have are not ...
- 1,472
1
vote
2
answers
43
views
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&.&.&...
- 440
0
votes
1
answer
20
views
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 ...
- 171
0
votes
0
answers
16
views
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 ...
- 11
3
votes
1
answer
156
views
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{\...
- 247
4
votes
1
answer
91
views
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) ...
- 203
1
vote
1
answer
112
views
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 ...
- 229
2
votes
1
answer
101
views
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:
...
- 23.7k
0
votes
1
answer
56
views
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 ...
- 848
1
vote
0
answers
62
views
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 ...
1
vote
1
answer
76
views
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 ...
- 111
0
votes
0
answers
16
views
Zero wavenumber and nonzero frequncy for a finite loaded string
TL;DR: How do I make physical sense of a state of zero wavenumber and nonzero frequency?
I was solving a problem of finding the stable wave states of a string with $n$ equidistant, but alternating ...
- 101
0
votes
3
answers
47
views
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-...
- 1,296
1
vote
1
answer
40
views
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{...
- 2,596
0
votes
0
answers
54
views
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:
$$H=T+V =\left(\frac{p_1^2}{2\mu}+\frac{p_2^2}{2\mu}+k_pp_1p_2\right)+
\left(\frac{1}{2}\mu\omega_1^{2}x_1^...
0
votes
0
answers
43
views
Interpreting quotient norms in context of coupled oscillators
For simplicity and concreteness, consider coupled harmonic oscillators given in the diagram below (copied from this question):
Our phase space is $V = \mathbb{R}^4$, which I'll decompose into $V = ...
- 151
1
vote
1
answer
120
views
Does Hooke's law explain classical wave behavior?
Will Hooke's law $F = -kx$ applied to a large mass-spring grid array such as:
provide the full and complete mechanistic explanation for classical wave behavior, including the 2nd order wave ...
- 537
1
vote
3
answers
62
views
How to calculate the energy of a spring-mass system considering harmonic oscillation of the normal mode? [closed]
For a spring-mass system, we know that the potential and kinetic energy are
$$E_p = \frac{1}{2}ku^2 \text{ and } E_k = \frac{1}{2}m\dot{u}^2.$$
where $k$, $m$ and $u$ are the spring constant, mass and ...
- 13
0
votes
0
answers
31
views
Conceptual Question of Small Oscillations of Coupled Harmonic Oscillators - Classical Mechanics
Is my following understanding of small vibrations correct?
Modal matrix is the transformation matrix that relates general coordinates and the normal coordinates
Normal coordinates are the linear ...
- 195
0
votes
1
answer
109
views
Transfer function of system of coupled 2nd order ODE
I am wondering how to calculate transfer function $H(s)$ of system described by 3 coupled differential equations. The pourpose of work is to calculate "Bodedx" diagram ($|H(i\omega)|(\omega)$...
- 956
1
vote
0
answers
85
views
Motion of an $n$ mass $n$ spring system [closed]
While reading wave motion I encountered the problem of $n$ identical masses with $n$ identical springs in between them.
If we give a sudden push
to the wall attached to the first spring, what will ...
- 43
0
votes
0
answers
30
views
Terminology question: in-phase or out-of-phase?
Suppose that in a chain of many coupled oscillators, the displacements of two consecutive particles, in a normal mode of oscillation with frequency $\omega$, are given by $$x_p(t)=A_pe^{i\omega t}$$ ...
- 10.4k
0
votes
0
answers
45
views
Why is any arbitrary motion of a coupled oscillator writable as a linear combination of its normal modes? [duplicate]
Consider the following example of a coupled oscillator. Let two identical pendulums, each of length $\ell$ and mass $m$ be connected by a spring of force constant $k$. The system has two normal modes ...
- 10.4k
0
votes
1
answer
64
views
How to describe a series of damped harmonic oscilators?
I am looking for textbooks or papers that provide an analysis for a series of damped springs. I am having a tricky time working out the details on my own. I know that if $F=-k\Delta x$ a series of ...
- 47
22
votes
3
answers
2k
views
Equations of motion only have a solution for very specific initial conditions
An exercise made me consider the following Lagrangian
$$L = \dot{x}_1^2+\dot{x}_2^2+2 \dot{x}_1 \dot{x}_2 + x_1^2+x_2^2.\tag{1}$$
If I didn't make a mistake the equations of motion should be given by:
...
- 533
1
vote
1
answer
57
views
Coupled Oscillator Period [closed]
I was studying an example of a coupled oscillator the other day, namely two identical masses attached to three springs, the lateral ones of which with the same elastic constant, when I came across the ...
- 356
1
vote
0
answers
22
views
Estimation of number of states to be used to obtain Wigner-Dyson distribution in a chaotic coupled oscilllator
A coupled harmonic oscillator with quadratic coupling -
$$
H = \frac{1}{2}(p_x^2 + p_y^2) + \frac{1}{2}(x^2 + y^2) + g x^2y^2,
$$
is known to be non-integrable, hence chaotic (for reference look at
...
- 826
1
vote
2
answers
515
views
How to perform a Gaussian functional integral?
I'm completely beginner to the quantum field theory and try to learn the basics of functional integrals. However, I could not understand clearly. Could someone please explain the idea with the help of ...
- 63
0
votes
1
answer
88
views
Question about the frequency of normal modes in coupled oscillators and their derivation [closed]
Consider a two-mass system that is coupled by three springs, such that: $m_1=m_2=m$; $k_1=k_3=k; k_2=k_{12}$. It can be written in terms of the following coordinates: $\eta_1=x_1-l_1$ and $\eta_2=x_2-(...
- 1,396
0
votes
1
answer
165
views
In a coupled pendulum system, how do pendulums *know* which one is giving its energy, and which one is receiving it?
Picture the system below:
If you give the pendulum on the left a push, it will slowly transfer its energy to the one on the right, until it stops completely and the right one is in full swing, then ...
1
vote
1
answer
55
views
Self-synchronizing and -desynchronizing systems of oscillators
There are biological systems with adaptable frequencies that are able to synchronize their frequencies, mainly individuals (see e.g. reproductive synchrony). In this case, also the phase is typically ...
1
vote
1
answer
72
views
Commutation relations for coupled systems
Consider $N$ coupled oscillators whose Hamiltonian is $$\hat{H}=\sum_j^N\frac{\hat{p}_j^2}{2m} +\frac{1}{2}K(\hat{x}_{j+1}-\hat{x}_j)^2,$$ which represents a system like ($K$ is the spring constant):
...
- 4,011
1
vote
1
answer
184
views
Normal modes of coupled oscillators
For two pendulums of mass $m_1$ and $m_2$, coupled by a spring of constant k, both suspended by strings of length $l$, the following matrix equality results from their equations of motion:
$$ \omega^2
...
2
votes
1
answer
454
views
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 $m_2$ ...
- 175
2
votes
1
answer
99
views
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 $\...
- 21
1
vote
0
answers
129
views
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 ...
- 1,680
-1
votes
1
answer
82
views
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 ...
- 99
1
vote
1
answer
68
views
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.
...
- 1,427
2
votes
0
answers
37
views
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....
- 23
1
vote
1
answer
111
views
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 ...
2
votes
1
answer
105
views
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 ...
- 27
0
votes
1
answer
43
views
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 ...
- 63
1
vote
2
answers
61
views
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'} - \...
- 23.7k
1
vote
2
answers
189
views
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 ...
2
votes
1
answer
59
views
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 ...
- 375
1
vote
1
answer
91
views
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}\,\...
- 406
1
vote
2
answers
2k
views
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 ...
- 11
0
votes
1
answer
87
views
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{...
0
votes
1
answer
106
views
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 ...
- 193
9
votes
3
answers
816
views
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 $\ddot{X}= -M^{-1}KX$ ...
- 137