Calculate Q-Value & Energy Loss in a Mass-Spring System | Rearranging Equations

[Ryan McDonald]

Homework Statement

The rate at which the mass–spring system loses energy to its surroundings is referred
to as the Q-value for the oscillator. The Q-value is defined as:

Q=2π(E/ΔE)

I need to find ΔE/E, which is the fractional energy loss per cycle of the oscillation.

How would I rearrange this so that ΔE/E = 2π/Q

Homework Equations

Q=2π(E/ΔE)

The Attempt at a Solution

I've no idea how to rearrange it so that E/ΔE is swapped round

 

[Maged Saeed]

$$Q=2\pi(\frac{E}{\triangle E})$$
then
$$\frac{Q}{2\pi} =(\frac{E}{\triangle E})$$
$$\frac{2\pi}{Q}=(\frac{ \triangle E}{ E})$$

 

[Ryan McDonald]

$$Q=2\pi(\frac{E}{\triangle E})$$
then
$$\frac{Q}{2\pi} =(\frac{E}\{\triangle E})$$
$$\frac{2\pi}{Q}=(\frac{ \triangle E}\{ E})$$

Hi Maged, thanks for your reply. Apologies, this is all new to me. What do you mean when you say /frac?

 

[Maged Saeed]

Hi Maged, thanks for your reply. Apologies, this is all new to me. What do you mean when you say /frac?

I have tried to write my answer in Latex preview
to make those staff clear for you ,,
But it seems there is something went wrong

any way ,,

I have written it again

$$Q=2\pi(\frac{E}{\triangle E})$$
$$\frac{Q}{2\pi}=(\frac{E}{\triangle E})$$
$$\frac{2\pi}{Q}=(\frac{\triangle E}{E})$$

 

[Maged Saeed]

What it looks like now
is it understandable ?
(:

 

[Maged Saeed]

I have fixed my problem though,
Reload the page ,

 

[Ryan McDonald]

What it looks like now
is it understandable ?
:)

Indeed it is, thanks very much. Appreciate your help.

 

[NascentOxygen]

I have tried to write my answer in Latex preview
to make those staff clear for you ,,
But it seems there is something went wrong

any way ,,

I have written it again

$$Q=2\pi(\frac{E}{\triangle E})$$
$$\frac{Q}{2\pi}=(\frac{E}{\triangle E})$$

Now, take the reciprocal of each side ...

$$\frac{2\pi}{Q}=(\frac{\triangle E}{E})$$

 

[Maged Saeed]

Now, take the reciprocal of each side ...

I have missed this point ,, it should be there ,,,

:)

 

[BvU]

Did you read the later postings in your thread 79065 ?

Point is that the 'definition' doesn't hold unless Q is very large.
Energy in an oscillator has to do with amplitude squared (e.g. with a spring: E = 1/2 k x²).

A damped oscillation is represented (see here ) by $$ z(t) = A e^{-\zeta\omega_0t}\; sin \left ( \sqrt{1-\zeta^2}\; \omega_0t + \phi \right )$$ and with

$\zeta \ll 1$ so $T \approx {2\pi\over \omega_0}$ for $\ \left( z(t+T)\right )^2 / \left( z(t)\right )^2\$

you get $\ e^{-2 \zeta\; 2\pi} = e^{-2\pi / Q}\$ if $\ Q={1\over 2\zeta}$.

Again with Q large enough, this is $1-{2\pi \over Q}\ .\ \$ So ${\Delta E\over E} = {2\pi \over Q}$


Now to your question: how to rearrange Q = 2π(E/ΔE) to get ΔE/E = 2π/Q :

multiply left and right by ΔE
divide left and right by E
divide left and right by Q

but is this really what you were asking ?

 

[Ryan McDonald]

I have missed this point ,, it should be there ,,,

:)

Thanks

Did you read the later postings in your thread 79065 ?

Point is that the 'definition' doesn't hold unless Q is very large.
Energy in an oscillator has to do with amplitude squared (e.g. with a spring: E = 1/2 k x²).

A damped oscillation is represented (see here ) by $$ z(t) = A e^{-\zeta\omega_0t}\; sin \left ( \sqrt{1-\zeta^2}\; \omega_0t + \phi \right )$$ and with

$\zeta \ll 1$ so $T \approx {2\pi\over \omega_0}$ for $\ \left( z(t+T)\right )^2 / \left( z(t)\right )^2\$

you get $\ e^{-2 \zeta\; 2\pi} = e^{-2\pi / Q}\$ if $\ Q={1\over 2\zeta}$.

Again with Q large enough, this is $1-{2\pi \over Q}\ .\ \$ So ${\Delta E\over E} = {2\pi \over Q}$


Now to your question: how to rearrange Q = 2π(E/ΔE) to get ΔE/E = 2π/Q :

multiply left and right by ΔE
divide left and right by E
divide left and right by Q

but is this really what you were asking ?

Indeed, it was what I was asking. With reference to the earlier post, it turns out the graph was difficult to read and I was advised to read it so that it fitted the equation.

Thanks for your reply though, appreciated,

 

[NascentOxygen]

Did you read the later postings in your thread 79065 ?

Good catch! Ryan M^cDonald has yet to answer my question in my last post there. :(

Hopefully he will find time to do so shortly.

Point is that the 'definition' doesn't hold unless Q is very large.

That restriction wasn't clear to me in that thread. Discussion wasn't exhaustive, but I concluded that in a system showing stable oscillation, ΔE would represent the energy deficit the system must add during each cycle to maintain constant amplitude of oscillation, regardless of Q. Are you saying this would not be correct unless Q is "large"?

 

[BvU]

@NascentOxygen: I've been looking at this thing as an undriven oscillator all the time. Mainly because of the way the OP in thread Calculating energy left in oscillator after 3 cycles was formulated.

However, in post #5 there it becomes clear that Ryan found his Q from the resonance curves, so what rude man states in his post #15 applies.

I must admit I have a hard time (=lazy me) coming up with a simple and elegant derivation for ΔE/E in the case of a driven damped oscillator.

@Ryan McDonald: 790685 post #13 should still be OK


@moderator: this whole thread should be appended to (or referred to in) thread 790685

 

[Ryan McDonald]

@NascentOxygen: I've been looking at this thing as an undriven oscillator all the time. Mainly because of the way the OP in thread Calculating energy left in oscillator after 3 cycles was formulated.

However, in post #5 there it becomes clear that Ryan found his Q from the resonance curves, so what rude man states in his post #15 applies.

I must admit I have a hard time (=lazy me) coming up with a simple and elegant derivation for ΔE/E in the case of a driven damped oscillator.

@Ryan McDonald: 790685 post #13 should still be OK


@moderator: this whole thread should be appended to (or referred to in) thread 790685

Thanks BvU. I found the question to be irritating and extremely ambiguous. Anyways, hopefully I've arrived at the correct solution now.

 

[BvU]

If you share it we can look at it -- and perhaps comment :)

There's no reason to find the question irritating or ambiguous if there is a context.

And: any investment in this subject pays off for a lifetime in a vast number of disciplines, so cheer up!

 

[NascentOxygen]

in post #5 there it becomes clear that Ryan found his Q from the resonance curves

I'm unsure whether f_o/b.w. can be equated to Q for low Q's. I'd like to see that demonstrated as being true; meanwhile I'm regarding it as an approximation which holds for high Q's. After all, I expect it's only going to be exact for 2^nd order systems, in any case.

 

[Ryan McDonald]

If you share it we can look at it -- and perhaps comment :)

Ok, here is my attempt.

I've taken Q to be 7 for plot a and 10 for plot b.

Rearranging Q=2π (E/ΔE)

ΔE/E = 2π/Q

For plot a:

After one cycle 2π/Q = 0.1

After two cycles (0.9*0.9) = 0.01

After three cycles = (0.81*0.9) = 0.001


For plot b:

After one cycle 2π/Q = 0.37

After two cycles = (0.63*0.63) = 0.23

After three cycles = (0.4*0.63) = 0.15

I'm sure there must be a more elegant way of calculating the energy left after 3 cycles but unfortunately, it evades me.

Thoughts welcome

Would it be 1-(2π/7)ⁿ for plot a? Which would show the amount of energy lost after n cycles

 

[BvU]

Q to be 7 for plot a and 10 for plot b

Are these still from the picture 790685 post #5 ?
In that case I expect a to have a higher Q than b !

And what is $\Delta\omega/\omega$ ?

This one says "the bandwidth over which the power of vibration is greater than half the power at the resonant frequency, ωr = 2πfr is the angular resonant frequency, and Δω is the angular half-power bandwidth" but you are looking at an amplitude plot !

Then: your calculation: if q = 7, 2π/Q = 0.9, so after 1 cycle there is only 0.1 left. After 2 cycles 0.01 and after 3 cycles 0.001 !

 

[Ryan McDonald]

Are these still from the picture 790685 post #5 ?
In that case I expect a to have a higher Q than b !

And what is $\Delta\omega/\omega$ ?

This one says "the bandwidth over which the power of vibration is greater than half the power at the resonant frequency, ωr = 2πfr is the angular resonant frequency, and Δω is the angular half-power bandwidth" but you are looking at an amplitude plot !

Then: your calculation: if q = 7, 2π/Q = 0.9, so after 1 cycle there is only 0.1 left. After 2 cycles 0.01 and after 3 cycles 0.001 !

 

[Ryan McDonald]

Ah, silly me! I see my error now.

Yes they are from picture 790685 post #5

Q can be calculated from ω/Δω

where ω is the value at the peak of the curve

and Δω is the width of its peak at the halfway point

The graph is difficult to read but for plot b i originally took that to be 50/(55-45) which didn't work as q was less than 2π. I was advised to use a value of 7 to make the calculation work.

 

[Ryan McDonald]

If you share it we can look at it -- and perhaps comment :)

Are these still from the picture 790685 post #5 ?
In that case I expect a to have a higher Q than b !

And what is $\Delta\omega/\omega$ ?

This one says "the bandwidth over which the power of vibration is greater than half the power at the resonant frequency, ωr = 2πfr is the angular resonant frequency, and Δω is the angular half-power bandwidth" but you are looking at an amplitude plot !

Then: your calculation: if q = 7, 2π/Q = 0.9, so after 1 cycle there is only 0.1 left. After 2 cycles 0.01 and after 3 cycles 0.001 !

Sorry BvU, I've got myself a little confused! You are correct with regards to the values of the plots.

Plot a is 10 and plot b is 7.

It seems I'm a bit of a numpty today!

 

[BvU]

Q can be calculated from ω/Δω
where ω is the value at the peak of the curve

Of the power curve. But your curve says amplitude at the vertical axis ! (is it amplitude ? What a coincidence both are same height !)

 

[Ryan McDonald]

Of the power curve. But your curve says amplitude at the vertical axis ! (is it amplitude ? What a coincidence both are same height !)

I'm having a nightmare with this question aren't I?!

We are advised that Q=(ω/Δω)

or Q=(f/Δf)

Can I work out Q using either of these equations from the information shown in the graph?

I'm so confused..

 

[BvU]

You have a plot for the amplitude. The power is proportional to amplitude squared. So if the power is 1/2, the amplitude is 1/√2.

You want to take the width at maximum/√2

 

[Ryan McDonald]

You have a plot for the amplitude. The power is proportional to amplitude squared. So if the power is 1/2, the amplitude is 1/√2.

You want to take the width at maximum/√2

 

[Ryan McDonald]

Thanks, as plot a and b have the same height, does that mean they will have the same answer?

 

[BvU]

Answer to what ? I thought you wanted to find a ratio !

 

[NascentOxygen]

I was advised to use a value of 7 to make the calculation work.

Advised by whom, may I ask?

 

[NascentOxygen]

Q can be calculated from ω/Δω

where ω is the value at the peak of the curve

and Δω is the width of its peak at the halfway point

The "half-way" level being the $\frac 1 {\sqrt 2}$ level because of the power versus amplitude relationship, as BvU reminded you. (The $\frac 1 {\sqrt 2}$ level is often referred to as the 0.707 or -3 dB level.)