Tone arm resonance and cartridge compliance: How do they interact??

Haven't seen a telegram in maybe 50 years except while watching an old movie on TCM.

It's essentially a problem in integral calculus.I have and will continue to align myself with what Raul and cleeds have said, which is what I also have written, in response to BR3098.  Adding mass at the headshell has the largest possible relative effect on effective mass, compared to adding mass anywhere closer to the fulcrum or pivot.  The effect of added mass on total effective mass is lessened as one approaches the pivot.  As we have already noted, adding mass, because it increases effective mass and would have the tendency to reduce the resonant frequency in magnitude, but not necessarily in amplitude, which would depend more on damping, stiffness, materials used, etc.

BR3098, forgive me for being overly sensitive and for misinterpreting your earlier remark.  In any case, I was not in any way upset or angry, just confused.

Raul, I guess we are all guilty of what we call in English "talking past each other", which means not paying attention to what the other guy said.  I fully agree that no calculation based on pure physics will really predict how a given combo of tonearm and cartridge will behave in practice.  The numbers are only a starting point.  This is why I pay no attention to the notion that if your calculated resonant frequency is not between 8Hz and 12Hz, the match is poor.  It's well and good to use the numbers as a guide, but that's all they're good for.  So, I think we agree.
I AM kind of a math guy, so it is interesting to me to note that the resonant frequency depends upon the square root of the quantity M*C, where M = effective mass, and C= Cartridge compliance (at 10Hz).  Because of the dependence upon the square root, there are actually a very wide range of values for both quantities that will work to fulfill the 8 to 12Hz desired endpoint.  For fun, one can play with one of the on-line calculators to see what extremes you can get away with.  But the fact is that few of us really know what the effective mass of our tonearm plus cartridge plus hardware actually is, due to all that was already discussed, and even cartridge compliance probably varies from sample to sample and according to the age of the cartridge.

br3098, If you wanted to insult me, you could not have done a better job than to accuse me of cutting and pasting from a text. ( You wrote, "I agree and disagree with different parts of your statement. Rather than cut and paste a text on mechanical amplifiers and resonance,...")  Whatever I wrote anywhere on this thread came out of my own head, and I take full responsibility for it, right or wrong.  I also am happy and willing to be corrected, if wrong.
Now, as to the second part of your inquiry, "please tell me how adding weight at the headshell is beneficial to the resonance of the lever", I'd be happy to respond if I could figure out what you are talking about.  Where did I claim that adding weight to a headshell was "beneficial" to resonance?  What I did say in response to Raul is that exchanging a heavy headshell for a lightweight one (or for that matter exchanging a lightweight headshell for a heavy one) has an "inordinate" effect on effective mass. By using the word "inordinate", I mean the effect on effective mass is nearly 1:1, since the headshell is right over top of the cantilever.  Whereas, adding mass anywhere else on the arm going back to the pivot point is of lesser consequence due to its distribution on the lever.  Effective mass is a quantity related to the distribution of the mass of the tonearm, not merely to its weight if you put the whole shebang on a scale. Do you disagree with that?  If so, state your case.

Dear Raul, I have to differ with you only on what you wrote in your first paragraph. The effective mass of the tonearm and the compliance of the cartridge determine the resonant frequency according to the formula that you know and I know too. You are probably right that in the real world resonance  is not so easy to figure out just from numbers. The first of all problem is that we really usually don’t know the effective mass of the tonearm/cartridge we’re using. For example, the Fidelity research has a very high effective mass when used with its factory supplied head shell, which in itself is heavy. And head shells have an inordinate affect on effective mass, because they are out there at the end of the tonearm. I use the Fidelity research tonearm with a high compliance cartridge, but I use a very lightweight head shell when I do that. And yes, from a listener standpoint the combination is very successful. But I have to freely admit that I really don’t know exactly what is the effective mass of the tone arm and cartridge combination that I ended up with. The Acutex cartridge that I used with the Fidelity research tonearm is also an extremely lightweight cartridge body in and of itself. Probably many grams lighter than a typical fidelity research cartridge of the vintage era. I can only tell you there is no audible sense that the resonant  frequency is interfering with bass response or anything else. The best way to work out what is going on with the tonearm and cartridge and resonance  is probably to figure out the resonant frequency by using a test LP or a very novel test that I read about on vinyl asylum. So then you would hopefully know the compliance of the cartridge and the resonant frequency and from that you can work backwards to get some idea of effective mass.

My Triplanar and I suppose several other tonearms achieve the same end by supplying a set of CWs that vary in mass. You can slide them on or off the rear end of the arm. Further the Triplanar CW is physically decoupled from the pivot by a nonrigid damped joint.

br3098, You are confusing two different things.  roberjermain is correct, and I wrote the same thing, a few posts above yours.  BOTH increasing tonearm effective mass and increasing cartridge compliance will decrease the resonant frequency of the system.  If you will look at the equation, you will see that the product of the two quantities is inversely proportional to Fres.  Now, what bdp and you are talking about does not per se alter the resonant frequency, as long as you don't change effective mass and compliance. What damping and choice of materials does, if done smartly, is take the energy of the resonance that will occur at the calculated frequency and spread out or broaden the resonant peak, the range of frequencies at which the energy of resonance is dissipated, which is usually a good thing that can mitigate an otherwise bad pairing of cartridge and tonearm.  And yes, the (distance from the pivot to the center of mass of the CW)-squared times the mass of the CW, adds to effective mass.  That's why, if you want to minimize effective mass, you want a heavy CW that can be moved as close as possible to the pivot. (Because the em will vary as the distance-squared but only in direct proportion to CW mass.)

I am surprised no one has mentioned yet that the tonearm has no intrinsic resonance, because the only resonance it has that you care about is the resonance  with the cartridge bolted onto it. The effective mass of the tonearm is equal to the basic effective mass plus contributions from the cartridge, the head shell, the cartridge screws, and where the counterweight is located. That’s why in the end few of us really know exactly what is the effective mass to which we are coupling our cartridges. Finally, the resonant frequency is a product of the effective mass multiplied by the compliance of the cartridge after you take the square root of that product and invert it.So, high mass and high compliance both have a tendency to reduce the resonant frequency. There is an inverse relationship.