What Is the Relationship Between Damping and Resonance in Driven Oscillators?

[swuster]

Homework Statement

A driven oscillator with mass m, spring constant k, and damping coefficient b is is driven by a
force $$F_{o}cos(\omega t)$$. The resulting steady-state oscillations are described by $$x(t) = Re{\underline{A}e^{i\omega t}}$$ where:

$$\underline{A} = \frac{F_{0}/m}{(\omega_{o}^{2} - \omega^{2}) + i(2\omega/\tau)} = Ae^{i\varphi} , \omega_{o} = \sqrt{k/m} , \tau \equiv 2m/b$$

Show that for light damping ($$Q = \tau \omega_{o} / 2 >> 1$$), the maximum amplitude occurs at approximately $$\omega = \omega_{o}$$ and that the maximum amplitude is approximately Q times the amplitude for very low driving frequencies.

Homework Equations

n/a

The Attempt at a Solution

The amplitude is $$Ae^{i\varphi}$$ which has its maximum magnitude when $$\varphi = -\pi/2$$. Therefore, the driving frequency must be near the resonant frequency so that the term is completely imaginary and negative, creating a phase angle of -90 degrees. I don't understand what the light damping and quality factor has to do with this, however, nor how to prove the value of the amplitude. The amplitude for low driving frequencies is $$\frac{F_{0}/m}{\omega_{o}^{2}}$$...

 

[jbsteele]

so I'm guessing that the maximum amplitude is Q times this? I'm unsure how to prove it, however.Any help is much appreciated! Thank you.

 

[tomcue]

which is not very helpful.

I would like to clarify that the given statement is not a complete problem or question. It only provides information about a damped driven oscillator and asks to show certain relationships. Therefore, I would suggest that before attempting to solve the problem, it is important to understand the concept of a damped driven oscillator and the equations given.

A damped driven oscillator is a system where an external force is applied to an oscillator that is also experiencing damping. This results in steady-state oscillations with a specific amplitude and phase angle. The equation given, x(t) = Re{Ae^(iωt)}, describes the displacement of the oscillator at any given time, where A is the complex amplitude and ω is the driving frequency.

To solve the given problem, we need to understand the concept of quality factor (Q) and its relationship with damping. Quality factor is a measure of the efficiency of an oscillator, and it is defined as the ratio of the resonant frequency (ω_o) to the width of the resonance curve (2ω/τ). In other words, it represents the number of oscillations an oscillator can make before losing its energy due to damping. Therefore, a higher Q value indicates a lower damping, and vice versa.

Now, let's look at the given equation for A. We can see that it has two terms in the denominator, one for the resonant frequency and one for the damping. For light damping (Q >> 1), the second term (2ω/τ) becomes negligible compared to the first term (ω_o^2 - ω^2). As a result, the amplitude becomes maximum when the driving frequency (ω) is close to the resonant frequency (ω_o). This is because when ω = ω_o, the first term becomes zero, and the amplitude becomes infinite. In other words, the oscillator is able to resonate at its natural frequency without losing much energy due to damping.

Moreover, we can also see that the amplitude is inversely proportional to the damping coefficient (τ). This means that for light damping (high Q value), the amplitude will be larger compared to heavy damping (low Q value). This explains the second part of the statement, where the maximum amplitude is Q times the amplitude for very low driving frequencies. This can be seen by comparing the amplitude at ω = 0, which is A = F_0/mω_o^2, with