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0answers
37 views

Resonance and energy flow

In this post, Alfred Centauri describes the resonance as a phenomenon which appears when 'the energy flow from the driving source is unidirectional' and then shows that this is the case for $\Omega=\...
1
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0answers
49 views

How to evaluate the period of a particle in a system with potential energy $U=-U_0/\cosh^2(\alpha x)$?

I am working through the textbook "Mechanics", from the series "Course of Theoretical Physics " by Landau and Lifshitz. In Chapter 3, where the authors talk about integrating the equation of motion $E=...
5
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1answer
104 views

How can dissipative/friction terms be incorporated into a Lagrangian?

I'm trying to find a suitable Lagrangian for a damped harmonic oscillator, a system that satisfies the following equation of motion: $$m \ddot{x} + \gamma \dot{x} + \frac{d\phi}{dx} = 0.$$ What I ...
1
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0answers
19 views

String conservation with springs

I was trying to solve this Problem that asked to find the period of small oscillations for this system. To do so I used the fact that for a massless pulley with strings around it, the sum of the ...
0
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3answers
66 views

Rotational Kinetic Energy of a Pendulum

By the parallel axis theorem, a pendulum that rotates around a point $P$ and a distance $l$ from it's center, has kinetic energy $E_{kin}= \frac{\omega^2}{2}(\frac{2mR^2}{5}+ml^2)$. Where R is the ...
3
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1answer
175 views

Applying Kramers-Kronig relation to a simple damped oscillator

I just discovered the Kramers-Kronig relation and am trying to apply it to a simple damped oscillator of the form subjected to an impulse at $t=0$, which is a causal system: $$m\ddot x + c\dot x + k ...
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2answers
40 views

Damped drive oscillating systems

I am currently looking at the theory of find the viscosity of and object through damped harmonic motion, and tho it can be done there is obviously a limitation with regrades to the medium. If the ...
1
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1answer
34 views

Understanding a reference (Cummins on 2d order ODE)

In the first page of The Impulse Response Function and Ship Motions (Cummins, 1962), it is written that: We can now write an equation, which has the appearance of a differential equation, relating ...
1
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0answers
40 views

Swing - time taken [duplicate]

I was thinking about how I would go about calculating the time taken for a swing to swing from one side to the other, assuming that there only exists a gravitational force and discarding all other ...
1
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0answers
66 views

Spring with oscillating support (Goldstein chapter 11, problem 2)

The problem: A point mass m hangs at one end of a vertically hung hooke-like spring of force constant k. The other end of the spring is oscillated up and down according to $z=a\cos(w_1t)$. By ...
2
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2answers
52 views

What is the difference between “monochromatic” and “impulse” force?

In a paper I am reading (linked below), the following is stated: The transient motions of the sphere and the gas bubble in the elastic, incompressible, inviscous medium are investigated in response ...
2
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3answers
78 views

Oscillator with decaying restoring force

Suppose a system that is described by the equation of motion: $$ \ddot{x} = -k\cdot x\cdot \exp\left(-\frac{t^2}{2\sigma^2}\right). $$ (For example a spring with decaying stiffness.) I'd like to ...
3
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1answer
49 views

Damped oscillations and generalized friction

I'm reading damped oscillations from the book Classical Mechanics by Landau and Lifshitz, quoting from the text - "There exists, however a class of problems where motion in medium can be ...
0
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1answer
30 views

Large Damped Harmonic Oscillator misunderstanding

So I'm confused, here with what is highlighted. When the book says of "order $1/y_-$" you will reduce the displacement by a factor of $1/e$. Does of order mean when the time is equal to $1/y_-$, if ...
0
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0answers
29 views

Breaking spaghetti and conservation of energy [duplicate]

For the past few days there have been news about scientists solving the old problem of bending a piece of spaghetti and breaking it into exactly two halves. Earlier it was already determined that the ...
-1
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1answer
122 views

Energy of classical Inverted Harmonic Oscillator

Quick one. Does the energy of inverted harmonic oscillator $$H(x, p) = \frac{p^2}{2} - \frac{x^2}{2},$$ remain conserved?
7
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3answers
305 views

How can a pendulum have amplitude greater than $\pi$?

How can a pendulum have amplitude angle greater than $\pi$? I've been reading about phase plots, which are graphs of the $\frac{d\theta}{dt}$ on the $y$ axis and $\theta$ on the $x$ axis, shown below. ...
10
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1answer
304 views

A damped harmonic oscillator is NOT a dissipative system?

I know this sounds rather insane, but it says so in my book. The argument is the following: Given a damped harmonic oscilator $$\ddot{q}+\frac{b}{m}\dot{q}+\frac{k}{m}q=0 \tag1 $$ this system can be ...
0
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1answer
31 views

The growth of the error when approximating a differential equation

When solving the differential equation $\ddot\theta + \omega^2 \sin{\theta} = 0$, my text book approximates that $\sin{\theta} = \theta$ and solves the approximated equation $\ddot{\theta} + \omega^2\...
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2answers
147 views

How can longitudinal and transverse waves be produced by hitting a vertical rod?

Would someone please give an intuitive explanation of this? I can still visualise an end of the rod getting compressed and thus transmitting a longitudinal wave, but how can a transverse wave be ...
0
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1answer
59 views

Linear momentum of atoms of a molecule and their frequencies

This exerpt on "Normal Modes of a Diatomic Molecule" is from Introduction to Mechanics Kleppner and Kolenkow: Suppose we have a polyatomic molecule model with N masses and several springs coupling ...
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0answers
164 views

Equations for inverted pendulum with a fixed base?

I'm researching how to control an inverted pendulum with a fixed base, using an actuated mass at the head of the pendulum to shift the center of mass in reaction to the tilt. I'm trying to research ...
1
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1answer
139 views

Is it possible to calculate a potential given period of oscillation as a function of energy?

Suppose I have a smooth potential $U:\mathbb R\to\mathbb R$ with $U(x)=U(-x)$, $U(0)=0$, and $U'(x)>0$ for $x>0$. A particle of mass $1$ at rest at position $x=x_0$ has total energy $U(x_0)$, ...
0
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1answer
28 views

In mechanical systems, it is easier to understand concept of natural frequency, could somebody explain natural frequency in electronic elements?

In mechanical systems, natural frequency simply means the frequency at which the body will oscillate when it is disturbed(assuming the body will suffer zero resistance in motion) but wihle studying ...
0
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1answer
60 views

Is it possible to approximate the motion under a V-shaped potential as harmonic motion?

Let's assume an $xy$ plane and let there be a force field defined by the potential $$V=F_0|x|$$ Though the potential is not differentiable still its a perfectly realisable system. If we solve the ...
0
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0answers
100 views

Deterministic non-harmonic motion of a single spring

Is there anything describing a spring's motion when e.g pulled and released, or pushed and released, with no mass-body attached to the spring. I.e not your usual "mass motion on a spring"-equation. I'...
0
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1answer
104 views

Modeling physical pendulum as a simple pendulum system

Before you continue, please excuse my English. It is not my native language.How can I change the physical pendulum system into the simple pendulum system?Should I just design the length of thread or ...
8
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6answers
2k views

Simple Pendulum Why Generalized Coordinate Always Angle?

When writing the equations of motion for the simple pendulum, why do textbooks always choose $\theta$ to be the generalized coordinate? The force of gravity is in the y-direction so wouldn't it be ...
1
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0answers
191 views

Equations of motion for a torqued spherical pendulum

I Want to simulate a spherical pendulum with a torquer on it, i.e. the angles of the pendulum change not only due to torque generated by gravity, but also by a torquer attached to the top of the ...
0
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5answers
8k views

How to find the frequency of small oscillation of a particle in a given potential? [closed]

A particle of mass $m$ is in a potential $$V(x)=-\frac12ax^2+\frac14bx^4$$ where $a$ and $b$ are positive constants. The equilibrium points occur when the potential $V$ is either minimum or maximum, i....
2
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2answers
721 views

How to find the period of small oscillations about this circular motion?

This is the question from Goldstein's Classical Mechanics, 2nd edition. Chapter 3 problem 1. A particle of mass $m$ is constrained to move under gravity without friction on the inside of a ...
1
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1answer
454 views

Horizontally driven inverted pendulum

I have came across this situation, where a cart of mass $M$ moves along the (horizontal) $x$ axis and a second mass m is suspended at the end of a rigid, massless rod of length $L$ (the rod is ...
2
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2answers
783 views

Approximations for a spring pendulum's equations of motion in 2D

I'm working on Exercise 24 in Classical Mechanics, 3rd ed by Goldstein, Poole, and Safko. It concerns the spring pendulum and approximations to its equations of motion. I'm trained more in pure ...
0
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1answer
55 views

Constrained oscillator on an $n$-sphere

I have a particle in $n + 1$ dimensional space, whose components satisfy the equations $$\ddot{x}_i+\omega^2_ix_i =0. $$ and I want to calculate the constraining force $F(\vec{x})$ that holds the ...
1
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2answers
191 views

Does fluid motion follow some periodic function?

I have heard of oscillations (i.e. simple harmonic motions) where a particle repeats its motion after a period of time, due to the restoring force acting opposite to the displacement and proportional ...
1
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0answers
404 views

Phase plane of simple pendulum

I'm trying to create a phase plane of simple pendulum motion by plotting $\dot\theta$ against $\theta$ in Matlab. I have the equation $\ddot\theta + \sin\theta = 0$, then by integrating I come to the ...
1
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2answers
189 views

The $r$-component of the Total Force on a Simple Pendulum

Let us consider a simple pendulum in vacuum. Its bob has a mass $m$. There are two forces acting on the bob: the tension of the string $\textbf{T}$ and the uniform gravitational force, $\textbf{W} = m ...
0
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1answer
280 views

Dependence of Average energy of a Driven Damped Oscillator?

What I don't understand is - How they concluded that the average energy should be zero except near resonance - and how that implies that $\omega$ can be replaced by $\omega_{0}$ in this expression. Am ...
0
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1answer
1k views

How to find formula for the period of a compound pendulum?

So I had a lab yesterday concerning finding a general mathematical expression for the period of compound pendulum. The pendulum let swing freely and time was recorded as well as the length from the ...
1
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3answers
861 views

Difference between “Periodic motion” and “Oscillating Motion”

So far I know one of them is a special case of the other: The Oscillating motion being the special case of Periodic motion. But I don't know the precise "Kinematical definition" of each one. I mean ...
0
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1answer
214 views

Velocity of small oscillations

We consider that the displacements about equilibrium are small, in small oscillation. Do we also tacitly assume that the velocity (rather, its magnitude) remains small as well? In Goldstein, it's ...
0
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3answers
3k views

Why does the period of a pendulum decrease in an accelerating frame? [duplicate]

If there is a simple pendulum in a non-accelerating frame with period $T_1$, it will have period $T_2 < T_1$ when placed in a frame accelerating perpendicularly to the direction of gravity. Why?
1
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1answer
211 views

Lyapunov exponents of a damped, driven harmonic oscillator

I am supposed to calculate Lyapunov exponent of a damped, driven harmonic oscillator given by $\ddot{x} + 2\beta \dot{x} + \omega_0^2 x = f\cos(\omega t)$ Lyapunov exponent is $\lambda$ in $\delta x(...
6
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0answers
103 views

Why are vibrations so common? [closed]

Why are vibrations so common? We all know, or pretend to know, that symmetries and the least action principle lead to conservation laws.Is there something more fundamental behind the fact that ...
0
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0answers
78 views

Pendulum in radial gravity field

All I could find about pendulums assumes that the force on the pendulum mass m is mg directed downwards. The case of m being attracted only by the radial gravity pull (thus replacing the "plane" ...
0
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1answer
559 views

An overdamped oscillator with natural frequency ω and damping coefficient γ starts out at position x0 > 0 [closed]

An overdamped oscillator with natural frequency ω and damping coefficient γ starts out at position x0 > 0. What is the maximum initial speed (directed toward the origin) it can have and not cross the ...
0
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2answers
143 views

Energy of driven dampened oscillator

Given the oscillator described by: $$m\ddot{x}+\gamma \dot{x}+kx=F_0\cos(\omega t)$$ And supposing the system is at it's stable state, I wish to calculate the following: 1) The system's energy at any ...
2
votes
1answer
203 views

Simulation of oscillator with frequency dependent damping

What would be the equation for the frequency dependent damping of harmonic oscillator? Is there something like: $$ \ddot{x}+2\delta\dot{x}+\omega_0^2x = \frac{F}{m}f(t) $$ with frequency dependent ...
3
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4answers
2k views

What is the time period of an oscillator with varying spring constant?

It is well known that the time period of a harmonic oscillator when mass $m$ and spring constant $k$ are constant is $T=2\pi\sqrt{m/k}$. However, I would be interested to know what the time period ...
7
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2answers
516 views

Are there any fully analytically solvable nonlinear oscillators?

I'm trying to find a simple one-dimensional problem, in which a particle would oscillate with some energy, and the period of oscillation would depend on particle energy (unlike in harmonic oscillator)....