Waves and Vibrations -- Vector Diagram

[Blue Kangaroo]

Homework Statement

A lightly damped oscillator (mass, m, and spring constant,s, damping constant, b, with one end fixed) under
external harmonic force (F0cosωt).(i.e., Standard mass-spring system with lubricated damper, and driven by an external harmonic force.) Since it is lightly damped,γ<2ω0.
a.
Express the equation of motion (2nd order differential equation) for this forced oscillator

b.
Draw a vector diagram representing the equation of motion in general. Then, derive and express the resultant amplitude A and phase shift Φ. Make sure to draw arrows for vectors and the direction of the angle, φ.
.

Homework Equations

F_o/mcosωt = ψ +γψ + ωo²ψ

I'm not sure how to put dots to signify derivatives, but the first ψ is the 2nd derivative and the second one is the first derivative.

The Attempt at a Solution

I think the equation of motion is just as I wrote it previously. I need a little help with the vector diagram. I'm just having a little trouble understanding them. My professor is jusr OK and I don't really care for the book. I've uploaded some work I did on scratch paper, so if anyone could point out mistakes or give me some tips, I would greatly appreciate it. Thanks in advance.

 

[TSny]

Your diagram looks OK to me. It would be nice to include arrow heads on all the vectors, and the diagram should be closed (the ω²A vector should meet up with the tip of the F_o/m vector).

Use simple geometry or trig on your diagram to express the length of the vector ω_o²A in terms of the length of the vector ω²A and a projection of the F_o/m vector. Likewise, try to find a way to express the length of the γωA vector in terms of the F_o/m vector.

[EDIT: It looks like you already have equations that you can solve for the amplitude A and phase angle. For the phase angle, it might be easier to work with the tangent of the angle rather than the cosine. Anyway, it looks like you have already essentially solved it. Do you have a specific question regarding the solution?]

 

[Blue Kangaroo]

So I have (F_o/m)sinΦ=γωA and tanΦ=γω/(ω^2_o-ω²) It seems to me like I'm missing some final piece to the puzzle. I think I may be too tired and just need to sleep on it.

 

[TSny]

tanΦ=γω/(ω^2_o-ω²)

This is essentially the solution for the phase angle.

To get the amplitude, solve your equation

 

[Blue Kangaroo]

So I came up with A=(F_o/m)/(ω^2₀-ω²+ωγ). Thanks for the help, TSny. There is a part C and more to this, but hopefully I can get them on my own now.

 

[TSny]

Check your result for A. It’s not quite right.

 

[Blue Kangaroo]

I noticed my mistake. A should be (F_o/m)(1/(ω²_o-ω²+γ²ω²)^0.5

 

[TSny]

I noticed my mistake. A should be (F_o/m)(1/(ω²_o-ω²+γ²ω²)^0.5

Very close, but still not quite it.

 

[Blue Kangaroo]

Got it now, A=(F_o/m)((ω²_o-ω²)²+γ²ω²)^0.5

Now just one more question: How can the diagram, along with A and Φ. I know in light damping γ<2ω_o, but I'm not quite sure how that helps me here.

 

[TSny]

Got it now, A=(F_o/m)((ω²_o-ω²)²+γ²ω²)^0.5

Yes, looks good.

Now just one more question: How can the diagram, along with A and Φ. I know in light damping γ<2ω_o, but I'm not quite sure how that helps me here.

I'm sorry, I do not understand the question. Can you reword it?

 

[Blue Kangaroo]

Sorry about that, I somehow left out a bunch of words, I'm not quite sure how I did that. Reading it now, it doesn't make a lick of sense.

How can the diagram, along with the resultant A and Φ, be approximated if this is a very lightly damped system?

I know that for light damping γ<2ω_o

 

[TSny]

I don't feel confident in how to approach part c. It seems to me that even if the damping is small, you would need to distinguish between three cases:
(1) driving frequency ω very near the resonant frequency
(2) driving frequency ω far from resonance
(3) driving frequency intermediate between cases (1) and (2)

The only simplification that I see off hand is that the resonant frequency is close to the natural frequency ω_o when the damping is small.

Maybe someone else will be able to help here.

Just to make sure we understanding the question, it might help to type out the question for part c exactly as it was given to you (word for word).

 

[Blue Kangaroo]

OK, here is the problem word for word.

If this is a very lightly damped system, how can the above diagram and resultant amplitude A and phase shift φ be approximated? Then, modify the above vector diagram in (b)

I should have read further. This was the next line

Draw a vector diagram when ω<<ω_o. Then derive and express the resultant amplitude A and phase shift φ.
I've uploaded some scratch work I did. I know my writing isn't great, but as a quadriplegic it's the best I can do.
.
.

 

[TSny]

I should have read further. This was the next line

Draw a vector diagram when ω<<ω_o. Then derive and express the resultant amplitude A and phase shift φ.

OK, so you have the situation where you have weak damping and the driving frequency is well below resonance.

Before thinking about the approximation, there is something you can do that simplifies the vector diagram even when you are not making an approximation. Note that the two vectors ω²A and ω₀²A are always in opposite directions. So, you can always combine them into one vector as shown:

And so the diagram reduces to a right triangle:

For the case where ω<< ω₀, how would you approximate the blue vector (ω₀² - ω²)A?

For weak damping, how should the length of ωγA compare to the length of the blue vector?

 

[Blue Kangaroo]

OK, so you have the situation where you have weak damping and the driving frequency is well below resonance.

Before thinking about the approximation, there is something you can do that simplifies the vector diagram even when you are not making an approximation. Note that the two vectors ω²A and ω₀²A are always in opposite directions. So, you can always combine them into one vector as shown:

View attachment 214056

And so the diagram reduces to a right triangle:

View attachment 214057

For the case where ω<< ω₀, how would you approximate the blue vector (ω₀² - ω²)A?

For weak damping, how should the length of ωγA compare to the length of the blue vector?

In that case, I believe I would approximate (ω^2_o-ω²) as just ω^2_oA. Also ωγA should be much shorter than the blue vector.

 

[TSny]

In that case, I believe I would approximate (ω^2_o-ω²) as just ω^2_oA. Also ωγA should be much shorter than the blue vector.

Yes. From a sketch of the figure corresponding to these two conditions, you can use a small angle approximation to get simple expressions for A and ∅.

 

[Blue Kangaroo]

I came up with Φ=γω/(ω^2_o-ω²) and A=(F_o/m)/(ω^2_o-ω²).

 

[TSny]

Yes. That’s what I get, too. Hope that’s what they wanted.
You can simplify the denominator for A and Φ

 

[Blue Kangaroo]

Because ω<<ω_o, can I say the denominator is just ω^2_o?

 

[TSny]

Because ω<<ω_o, can I say the denominator is just ω^2_o?

I think so. By dropping ω² compared to ω₀², your answers are still accurate to first order in the small quantity ω/ω₀.

 

[Blue Kangaroo]

Alright, thanks for all your help. There are more parts to this, but I will get them on my own.

 

[TSny]

OK. Sounds good. I enjoyed this problem as I had never seen these vector diagrams in the context of driven harmonic motion before.

Best.

 

[Blue Kangaroo]

Just out of curiosity, where else may I see them in the future?

 

[TSny]

Just out of curiosity, where else may I see them in the future?

These types of vector diagrams are widely used in analyzing alternating electrical circuits and in the study of wave interference. The vectors are often called "phasors". If you do a web search on phasors, you will find a lot of material.