Harmonic Oscillators: Resonance Bandwidth & Frequency Range

[fifteenover4]

Show that the resonance bandwidth corresponds to the frequency range for
which –1 < tan χ < +1. (The resonance bandwidth is the range for which the
average power is greater than 0.5 times the peak power.)

I'm pretty damn stumped with this.

 

[BvU]

Hello 4:15,
Could you use the template so that others know what this is about?
And that they can also see what $\chi$ stands for ?
And what you already know about these peak and average powers ?

 

[fifteenover4]

I've re-written it exactly as it was handed to me, hence, being stumped.

We know nothing about the peak and average powers, Ki is, I assume the impedance angle.

 

[BvU]

Hello again. Still wondering if this is mechanics or electronics. Never mind, some things are universal. Google "forced harmonic oscillator" or "forced harmonic oscillator solution" to get going.

I don't feel like re-hashing the formulas. They are ubiquitous, e.g http://www.physics.oregonstate.edu/~tate/COURSES/ph421/lectures/L9.pdf (Prof. J Tate), UTexas (prof. R. Fitzpatrick), etc. etc.
Basically what you should have collected under 2. of the template.

Being stumped when confronted with something new (? isn't this in some curriculum context?) isn't good enough. Do something and get stuck somewhere; then describe the situation you are in and ask for help. You are not being helped at all if someone else does that for you. Read the guidelines.

Once you have understood the derivation and the expressions (the investment is well worth it; lasts you a lifetime) it is a piece of cake to show what is asked of you.

All this is well meant and certainly not intended offensively.

 

[paranormal]

I can provide an explanation for the relationship between the resonance bandwidth and the frequency range suggested in the content.

First, let's define some terms. A harmonic oscillator is a system that exhibits periodic motion, such as a pendulum or a vibrating string. Resonance is the phenomenon in which a system responds most strongly to an external force when the frequency of the force matches the natural frequency of the system. The resonance bandwidth is the range of frequencies for which the system exhibits a significant response.

Now, let's consider the expression -1 < tan χ < +1. This is the range of values for the tangent of an angle that fall between -1 and +1. In terms of resonance, this represents the angle of phase difference between the driving force and the response of the system. When the angle is between -1 and +1, it indicates that the driving force and the response are in phase, meaning they are in sync and reinforcing each other.

Now, let's connect this to the resonance bandwidth. When the driving force and the response are in phase, it means that the system is responding strongly to the force, resulting in a higher average power. This is because the force is being applied at the optimal frequency for the system. When the angle is outside of this range, the force and response are out of phase, resulting in a weaker response and lower average power.

Therefore, the resonance bandwidth, which is the range for which the average power is greater than 0.5 times the peak power, corresponds to the frequency range for which -1 < tan χ < +1. This means that for frequencies within this range, the system will exhibit a strong response and have a high average power, making it ideal for resonance. Outside of this range, the system will have a weaker response and lower average power, making it less suitable for resonance.

In conclusion, the relationship between the resonance bandwidth and the frequency range for which -1 < tan χ < +1 can be explained by the in-phase relationship between the driving force and the response of the system, resulting in a higher average power within the resonance bandwidth. I hope this explanation helps to clarify the concept of resonance and its relationship to the tangent of the phase angle.