I've kept out of this thread but now that we are getting contributions from people who actually understand tonearm design I thought I might contribute a little.
Frank it is true that VTF will increase with record thickness but by my calculations the effect is very small indeed. Taking a dynamically balanced arm and assuming the VTF adjustment allows say 30mN range for a full 360 degree rotation gives us an effective compliance around 47,000 um/mN assuming 225 mm arm length. A typical record thickness differential of 1mm will cause a change of about 0.02mN which is around 0.1% of typical tracking force.
Mark Kelly
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Next a look at the effect of "warp riding". Assume a standard "taco warp" so the warp frequency is 7 rad.s^-1. This gives a maximal velocity of .007 m.s^-1 in the vertical plane for each mm of vertical warp and a maximal acceleration of .049 m.s^-2 (again for each mm of vertical warp).
The product of this acceleration and the effective inertial mass of the arm / cart combination gives a maximal VTF variation when riding the warp. If the inertial mass were 25 g (say 15g arm plus 10g cartridge) and the warp were 5mm high, the maximal variation would be around 6.2 mN. A similar calculation allows a maximal warp tolerance to be derived for any arm / cart combination as a function of VTF.
The important point is that it has nothing to do with the balance of the arm but is stictly related to the moment of inertia (and the mass of the cart).
Mark Kelly
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Completely wrong.
Your first paragraph makes no sense: the effective mass of an arm is simply the moment of inertia divided by the square of the effective length.
In the second para you present a supposition which I have already shown to be wrong but you do not support it with evidence.
Mark Kelly
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T-Bone
Yes, that is what I am saying.
It follows from a simple torque balance on the arm according to D'Alembert's principle. Rather than boring everyone by converting forces to torques and computing moments of inertia, I used the concept of equivalent mass.
BTW the figure given only applies to the example given, obviously different constraints will result in different figures.
Note that this is a force variation, the way the cart responds to the force variation will depend on the compliance. Also note the assumption of equal inertia is not equivalent to an assumption of equal structure but the differences are so small as to be immaterial to the argument.
Mark Kelly
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Axelwahl
Your suppositions are correct and the figures are reasonable.
If we model a 100 g counterweight positioned 50mm behind the pivot, its contribution to the effective mass of a 225mm arm is 4.94 g*. If this is the position for VTF of 20mN, it needs to be moved 4.6 mm to come into neutral balance. The new position will indeed make a higher contribution to effective mass, it becomes 5.89 g an increase of 0.95g.
This would increase the maximal tracking force deviation quoted above from 6.2 mN to about 6.4 mN, about a 3% increase.
* this ignores the moment of inertia of the counterwight about its own centre of mass but since this doesn't change with position it isn't important so I left it out.
Mark Kelly
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Kirkus
You make a good point so I'll answer that first. The answer should make it plain that everything Dertonarm has said in response is wrong.
I chose the example I used because the frequency was far enough away from practical cart/arm resonances for that factor to be safely ignored. As the frequency of the warp increases two things happen. The first is that the maximal acceleration (per mm of warp) increases with the square of frequency so the effect becomes more and more pronounced.
The second is that the phase of the arm's response to the warp becomes important. The easiest way to analyse this is to convert to electrical analogy. We can use either a force current or a force voltage analogy, the first is more elegant mathematically so I'll use that. In this analogy the moment of inertia of the arm (with the cartridge attached) becomes an inductance. The rotational compliance of the cartridge (which is the actual compliance times the square of the effective length) becomes a capacitance.
These two form an LC low pass filter and it is obvious by inspection that the product is identical to the usual product used in resonance calculations (effective mass x compliance) so the f0 of the filter is the same as the resonant frequency.
The other thing we need is to know the Q of this filter, which is determined by the hysteresis loss in the cartridge suspension. Since this will also affect the high frequency response of the cartridge way may assume that it is low enough that the Q of the filter is quite high. Perhaps JCarr can chime in here with some accurate values, if not well just assume a range of values greater than 5.
The phase angle of the response is given by :
Tan^-1(Q(2.f/f0 + SQRT(4-1/Q^2)) - Tan^-1(Q(2.f/f0 + SQRT(4+1/Q^2))
From which is may be seen that if Q is greater 5 and f/f0 is less than 0.5 then the phase error is less than 7.5 degrees.
Similarly the amplitude response may be calculated from the formula
1/(1 (f/f0)^2 +j.f/Q.f0)
From which it can be seen that there is some amplitude peaking: about 30% for Q = 5 and f/f0 = 0.5, reducing to a few percent when f/f0 is less than 0.2. If we take 10% as an acceptable error then as long as the warp frequency is less than 0.3 times the resonant frequency of the arm / cart combination the calculation I gave in my post above is good enough for Jazz. Using your range of warp frequencies this is equivalent to arm cart resonance being around 10.
The point to note is that all this is dependent on the moment of inertia of the arm and the compliance of the cartridge. It has nothing to do with the method by which VTF is applied. |
Kirkus
I composed the above offline before you posted again, it is in response to your post from last night (my time).
Since the issue of efective mass is causing confusion, I can take it out of the argument by reverting to rotational units. If we assume the arm is 225mm long then we have an angular deviation of 4.4 mrad per mm of warp so a 5mm warp will be 22mrad. At the frequency given this is a maximal acceleration of 1.1 rad.s^-2. If the moment of inertia of the arm and cartridge combination is 1.26 x 10^-3 kg.m^2 the maximal torque transmitted to the arm is 1.38 x mNm which is equivalent to a force of 6.2 mN acting at a distance of 225mm. This is exactly equivalent to the previous calculation.
Mark Kelly
BTW I meant to write "the phase and amplitude of the response becomes important" in the post above.
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DIVIDED BY
Rotational compliance is linear compliance DIVIDED BY the square of the effective length. Sorry I wrote it the wrong way around.
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Kirkus
We are indeed "singing from sheet".
The point that seems to be lost on the other participants is that this is a dynamic analysis. This follows directly from D'Alemberts principle.
Mark Kelly
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Dertonarm
That is completely wrong (again). The moment of inertia of a rigid body does not change with movement.
Yes we are talking differnt models. Mine is a model of what is happening, yours is a fiction. |
Dertonarm
That is completely wrong (again). The resonant frequency changes because the moment of inertia changes. The effective mass is simply the moment of inertia divided by the square of the effective length.
The total mass is irrelevant to the argument.
As far as I can see my model is complete according to D'Alembert principle. If you can show me something I have left out and provide a reasonable basis for the claim I'm listening.
Mark Kelly
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Axelwahl your last point is where things get interesting.
Like everything else it is a matter of compromise. As you have noted, a low inertia arm reduces the maximal VTF variation. The compliance required to keep the resonant frequency in the right range changes at the same time, so the effect of a given warp in terms of displacement of the cantilever suspension depends on the resonant frequency: the higher the resonant frequency the smaller the effect.
Unfortunately we're not free to move here. As previously noted the product of inertia and (rotational) compliance forms a low pass filter. As the equation previously given shows, the attenuation and phase response of this filter depends on the ratio of f/f0, so as the resonant frequency moves towards the audio band the effects of these become more and more pronounced. That is why resonant frequency is optimised over such a narrow range.
Mark Kelly
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Axelwahl
The only advantage I can quantify for spring applied VTF is one that seems to have have been ignored in this thread, namely immunity to movement in response to externally applied acceleration (read noise). As I see it, get rid of the noise and there's no point.
Perrew
Not going to go there.
Mark Kelly
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Ahh, so all along you were putting us off the scent in our quest for understanding. Silly me, I thought it was because you had a completely inadequate grasp of the fundamental physics underlying the phenomena we were discussing.
Thank you so much for setting me right.
Mark Kelly
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