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

• Ryan McDonald
In summary, 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)

## 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

Q=2π(E/ΔE)

## The Attempt at a Solution

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

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

Last edited:
Maged Saeed said:
$$Q=2\pi(\frac{E}{\triangle E})$$
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
Ryan McDonald said:
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

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

• NascentOxygen
What it looks like now
is it understandable ?
(:

I have fixed my problem though,

Maged Saeed said:
What it looks like now
is it understandable ?
:)

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

• Maged Saeed
Maged Saeed said:
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 ...
$$\frac{2\pi}{Q}=(\frac{\triangle E}{E})$$

• Ryan McDonald and Maged Saeed
NascentOxygen said:
Now, take the reciprocal of each side ...

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

:)

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 and Maged Saeed
Maged Saeed said:
I have missed this point ,, it should be there ,,,

:)

Thanks
BvU said:

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.

• Maged Saeed
BvU said:
Good catch! Ryan McDonald 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"?

@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

BvU said:
@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.

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!

BvU said:
in post #5 there it becomes clear that Ryan found his Q from the resonance curves
I'm unsure whether fo/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 2nd order systems, in any case.

BvU said:
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

Last edited:
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 !

BvU said:
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 !

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.

Last edited:
BvU said:
If you share it we can look at it -- and perhaps comment :)

BvU said:
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!

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 !)

BvU said:
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?!

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..

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

BvU said:
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

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

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

Ryan McDonald said:
I was advised to use a value of 7 to make the calculation work.