Ratio of amplitudes in a damped oscillator

In summary, the conversation discusses an exercise in analytical mechanics involving a damped harmonic oscillator and the ratio of two successive maxima in displacement. The note states that the maxima do not occur at the points of contact between the displacement curve and the curve Ae^(-yt), causing confusion for the person attempting the exercise. The conversation suggests referring to relevant equations and graphs to better understand why the maxima do not occur at the points of contact. The person is encouraged to consider the slopes at these points to better understand the concept.
  • #1
tiago23
4
1

Homework Statement


Show that the ratio of two successive maxima in the displacement of a damped harmonic oscillator is constant.(Note: The maxima do not occur at the points of contact of the displacement curve with the curve Aeˆ(-yt) where y is supposed to be gamma.

2. Homework Equations

The Attempt at a Solution


I was going through Fowles' Analytical Mechanics and found this exercise in the oscillations chapter. My problem isn't so much with doing the exercise as it is with the note of the authors in the end about the maxima not being in the points of contact between the two curves: Why not? Where else would the maxima of displacement be? I don't get that. Where is my mistake?
 
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  • #2
tiago23 said:

Homework Statement


Show that the ratio of two successive maxima in the displacement of a damped harmonic oscillator is constant.(Note: The maxima do not occur at the points of contact of the displacement curve with the curve Aeˆ(-yt) where y is supposed to be gamma.

2. Homework Equations

The Attempt at a Solution


I was going through Fowles' Analytical Mechanics and found this exercise in the oscillations chapter. My problem isn't so much with doing the exercise as it is with the note of the authors in the end about the maxima not being in the points of contact between the two curves: Why not? Where else would the maxima of displacement be? I don't get that. Where is my mistake?
This is one of the reasons we have a "Relevant Equations" section in the Template. It would help us a lot if you posted the Relevant Equations for damped harmonic motion, showed some graphs (use Google Images with attribution if necessary), and described how your perception is different from the Relevant Equations and those graphs. Can you do that for us please? Thanks. :smile:
 
  • #3
Have you got this now? Well you might well, remembering what these curves look like, think for a moment that the points of contact were at the maxima. But then as soon as somebody tells you they aren't you think a few seconds and say doh, oh yes, right! So I almost don't like to spell anything further out. What can you say about slopes at the points of contact between the two curves?
 
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What is a damped oscillator and how does it work?

A damped oscillator is a type of mechanical or electronic system that exhibits periodic motion, where the amplitude of the motion decreases over time due to the presence of a damping force. This damping force is usually friction or resistance, which acts to dissipate the energy of the oscillator. The motion of a damped oscillator can be described using the differential equation: x'' + 2βx' + ω2x = 0, where x is the displacement from equilibrium, β is the damping coefficient, and ω is the natural frequency of the oscillator.

What is the ratio of amplitudes in a damped oscillator and how is it calculated?

The ratio of amplitudes in a damped oscillator refers to the relationship between the amplitude of the motion at two different points in time. It is commonly denoted as At/At+T, where At is the amplitude at time t and At+T is the amplitude at time t+T, where T is the period of the oscillator. This ratio can be calculated using the formula: At/At+T = e-βT/2, where e is the base of the natural logarithm.

How does the damping coefficient affect the ratio of amplitudes in a damped oscillator?

The damping coefficient, β, plays a crucial role in determining the behavior of a damped oscillator. As the damping coefficient increases, the ratio of amplitudes decreases. This means that the amplitude of the motion decreases more rapidly over time, resulting in a shorter period and a quicker approach to equilibrium. Conversely, as the damping coefficient decreases, the ratio of amplitudes increases, indicating a slower approach to equilibrium and a longer period of oscillation.

What is the relationship between the natural frequency and the ratio of amplitudes in a damped oscillator?

The natural frequency, ω, is a measure of how quickly the damped oscillator will oscillate when there is no damping present. It is directly related to the ratio of amplitudes, with a higher natural frequency resulting in a smaller ratio of amplitudes and a lower natural frequency resulting in a larger ratio. This relationship can be seen in the formula for the ratio of amplitudes, where ω is present in the exponent, indicating its influence on the value of the ratio.

What factors can affect the ratio of amplitudes in a damped oscillator?

Aside from the damping coefficient and the natural frequency, there are other factors that can affect the ratio of amplitudes in a damped oscillator. These include the initial conditions of the oscillator, such as the initial amplitude and velocity, as well as external forces acting on the oscillator. Additionally, changes in the damping force or the natural frequency over time can also impact the ratio of amplitudes in a damped oscillator.

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