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

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.

lewm, I agree and disagree with different parts of your statement. Rather than cut and paste a text on mechanical amplifiers and resonance, please tell me how adding weight at the headshell is beneficial to the resonance of the lever?

I agree that increasing the mass of the lever (arm tube) will dampen a mechanical amplifier. But increasing the resistant load will have the opposite effect, decreasing stability and increasing secondary effects (hysteresis, etc.) Ideally, you would change the mass of the tonearm buy using arm tubes of differing weight and matched headshells. Some manufacturers used to do this.

If you are going to stabilizing mass to most conventional tonearms, adding it as close as possible to the pivot point will both increase stability and lower resonance.

I’ve used my Stanton 881s on my Ikeda 407 with a light headshell. And with the counterweight all the way in I can’t balance it with the dynamic balance off. When I add dynamic weight I can only add a half gram to be correct.   Even though this is wrong I agree with  @lewm  and really like how this cartridge sounds on the Ikeda than on my lighter  tonearm. My speakers do reach very low bass response and I do get some woofer pumping with this setup. My woofers face backwards so I didn’t notice at first. I don’t think this is very good for the speakers and no longer use this setup. Do you @lewm get this woofer action at all using your FR arm in this way. 

The different size counterweights are supplied so you can counterbalance different weight (mass) cartridges. You can not control resonance with different mass counterweights because once the cartridge is balanced you wind up with the same effective mass regardless of counter weight mass.
Now Raul, unless you are blind measuring resonance points with a test record is painfully easy. Your comment would leave me to believe that you have never set up a cartridge correctly.

The problem with calculations is that there is enough variation in tonearms and cartridges that the published specs may be off a little.
The record tells you what is really happening. But the resonance is not a sharp peak. It is a bell curve. The point where the cartridge starts bouncing is what you pay attention to. You know you are close when you hear the warble. Using the test record also gives you an idea of how severe the resonance point is. Some combinations barely move at the resonance point while with others the cartridge almost goes air born in which case it might have to be changed. I suspect it is a matter of damping.
Moment of inertia is different then effective mass. Longer heavier arms have higher moments of inertia. This really only creates issues following warps. If records are stored correctly very few should be significantly warped. Vacuum hold down and reflex clamping solve the problem under any circumstance.