Cone movement vs. frequency

[Cal Weldon]

Originally posted by AndrewT
I imagine a DC of 1Volt will cause the speaker Voice Coil to move to a new position. Let's suppose for a moment it moves 5mm.

One of the concerns might be that the cone does not move a certain amount with DC. It attempts to move the coil out of the gap and only comes to a halt because of mechanical restrictions of the suspension.

[Ron E]

Originally posted by Nichol1997
I am looking for the mathematical relationship between cone movement versus frequency.

It may be indirect, but go to
http://www.diysubwoofers.org/prt/ported3.htm

The SPLmax is the SPL at Xmax+15% assuming that the suspension is linear. Excursion and SPL will actually be slightly less than predicted due to nonlinearities.

In the simple formulas on that page, you can use Fb=0 to simulate a sealed box.

Have fun!

[maxro]

Originally posted by Cal Weldon
One of the concerns might be that the cone does not move a certain amount with DC. It attempts to move the coil out of the gap and only comes to a halt because of mechanical restrictions of the suspension.

Just the musings of an idiot here (me, not Cal), but could it be that there is a direct relationship between wave length and excursion for a given SPL?

As Cal points out, a DC signal just pushes the cone until it hits some sort of physical restriction. I suppose there is no specific SPL per se in a DC thud.

And a sine wave pushes out to its apex, then reverses until its nadir. So, if the cone were to move at a constant speed (constant voltage?), a low frequency wave would have more time, and thus distance, to travel before reversing direction than a higher frequency.

But, is this relationship linear with frequency for a given voltage? Squared for a given SPL?

max

[AndrewT]

Originally posted by sreten
My assumption is he meant constant SPL with a theorectical
driver that has a flat response and constant voltage drive.

yes, all three of you forgot to read the question.

Originally posted by Cal Weldon
One of the concerns might be that the cone does not move a certain amount with DC. It attempts to move the coil out of the gap and only comes to a halt because of mechanical restrictions of the suspension.

That is almost how a speaker works in a linear fashion (+- small non linearites). Apply 1mV and it moves 1unit. Apply 10mV and it moves 10units. Apply 1V and it moves 1000units. Apply a battery (1v5) and it moves 1500units. Try it!!!

Originally posted by maxro
Just the musings of an idiot here (me, not Cal), but could it be that there is a direct relationship between wave length and excursion for a given SPL?

That is exactly the relationship that Sreten & Co. were quoting.

As Cal points out, a DC signal just pushes the cone until it hits some sort of physical restriction. I suppose there is no specific SPL per se in a DC thud.

not quite, it doesn't hit a stop. The force generated by the coil reacting against the magnetic field BALANCES the force from the surround and spider. increase the drive force (more current) and the suspension has to displace farther to generate more restoring force to maintain the balance (for every force there is an equal and opposite reaction, some quote by a long dead Englishmen, Yeah I know sometimes the English get it right).

And a sine wave pushes out to its apex, then reverses until its nadir. So, if the cone were to move at a constant speed (constant voltage?) [NO NO NO] a low frequency wave would have more time, and thus distance, to travel before reversing direction than a higher frequency.

a low frequency at constant voltage drive moves at low speed and a high frequency moves at high speed. Now change it to constant power input (nearly constant SPL) and the SPEED of cone movement is substantially constant for constant power (that is my understanding of constant SPL, am I wrong?).

But, is this relationship linear with frequency for a given voltage? Squared for a given SPL?

Good question, Clarify it for all of us?

[Svante]

Originally posted by Nichol1997
I am looking for the mathematical relationship between cone movement versus frequency.

Actually I missed this very clear question in the very first post...

The complex formulation of the cone amplitude in a closed box is:

x(w) = K / (1 + jw/(Qtc*wc) + (jw/wc)²)

Neglecting the phase, by taking the absolute value this turns into:

|x(w)| = K / sqrt( (1-(w/wc)²)² + (w/(Qtc*wc))² )

...where wc is 2*pi*fc, fc is the resonant frequency in the box, Qtc is the Q of the driver mounted in the box.

It can be seen that at low frequencies this turns into

|x(wlow)|=K

ie independent of frequency, and for hight frequencies:

|x(whigh)|=K / (w/wc)²

ie sloping by -12dB/octave (due to the 1/w² relation).

Hope I got it right...

[facundonu]

a low frequency at constant voltage drive moves at low speed and a high frequency moves at high speed. Now change it to constant power input (nearly constant SPL) and the SPEED of cone movement is substantially constant for constant power (that is my understanding of constant SPL, am I wrong?).

the power of a voltage sine wave doesn't depend on its frequency, at a given constant amplitude. unless the load the wave is applied to is frequency dependant, of course.
constant SPL is constant power (power of the electric signal)(with a flat response speaker), and constant power is constant amplitude (amplitude of the electric signal) (with the same flat response speaker, wich also has a constant impedance). in order to fullfill all these conditions, this speaker has to increase its excursion to continue dlivering constant SPL, as frequency decreases.


having said all that, this is wrong:
"a low frequency at constant voltage drive moves at low speed and a high frequency moves at high speed"

[AndrewT]

Hi,
sounds like my thinking is **** for elbow. (damned dictionary won't let me use a collocquial phrase).

Originally posted by Facundonu.

having said all that, this is wrong:
"a low frequency at constant voltage drive moves at low speed and a high frequency moves at high speed"

I need to re-teach my brain about this SPL/excursion thing.

Originally posted by Ron E.
At DC, the only thing that matters is stiffness, at ~3x resonance and up all that matters is mass, neglecting inductance. At resonance, the contributions of stiffness and mass are equal, and damping dominates.

1/f^2 is correct - every time you halve frequency, excursion increases by 4 for the same SPL. The reason sealed boxes don't double with each halving below resonance is because the 12dB/oct rolloff reduces excursion requirements.

A speaker that had constant excursion with frequency would have a 12dB/oct rising response....

The excursion formula depends on whether the speaker is enclosed or not and what sort of enclosure there is. In free air, it acts the same as if it were in a very large box - like 10000 cubic feet or so so you can enter that into a program that calculates escursion.

seems to be saying that the constant amplitude I have been arguing ONLY applies below resonance.
Is that right?

[Svante]

I think this needs clarification:

At low frequencies, below resonance fs, this is true:

-The movement is spring controlled ie x=F*Cms *
-The cone excursion is proportional to the applied voltage.
-The cone velocity (speed) is iproportional to f.
-The cone acceleration is proportional to f².

At higher frequencies, above the resonance fs, this is true:

-The movement is mass controlled ie a=F/Mms **
-The cone acceleration is proportional to the applied voltage.
-The cone velocity (speed) is proportional to 1/f.
-The cone excursion is proportional to 1/f².

*Cms is the compliance of the driver, x is the excursion, F is the driving force (infinite baffle assumed)

** Mms is the moving mass of the driver, a is cone acceleration.

[AndrewT]

Thanks Svante.
That is very clear.

Is it possible to give such clarity when mounted in either a sealed box or a vented box or a transmission line?

[Svante]

Originally posted by AndrewT
Thanks Svante.
That is very clear.

Is it possible to give such clarity when mounted in either a sealed box or a vented box or a transmission line?

Sealed box is principally identical, it is just that the resonace frequency is slightly shifted upwards.

Bass reflex box is similar except around the helmholtz (box) resonance where the cone amplitude has a local minimum.

Tranmission line is similar to the bass-reflex box, in turn.

So, for the understanding, they are pretty much the same, at least a bit away from the system resonances.