EFFECTIVE MASS: Here is a cogent, scientifically based and rational explanation

Howard Popeck asked Geoffrey Owen, the driving force behind Helius Designs if he could and indeed would find the time to explain the reality behind effective mass and - in so doing - confound the misinformation, disinformation and scientific ignorance that clouds so many discussions of this topic.

As an aside and somewhat mischievously, Howard (who prior to this article couldn’t explain Eff.M to save his life, and neither could I) if Eff.M was actually important and if so, how could a ‘civilian’ make use of this knowledge. Mr. Owen graciously replied and below is what he told us. We are, as is always the case with his contributions, both enlightened and appreciative.

Thank you Geoffrey

Neil McCauley / editor in chief

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When asked to explain meaning of Effective Mass, I feel a certain affinity with Stephen Hawking when he talks of Schrödinger’s Cat – ‘I reach for my gun’, he famously said.

Erwin Schrödinger’s wave equation has one thing in common with Effective Mass - its significance is just as obscure when transposed to the real world. The cat story is something you can look up on the internet if you’re interested; suffice it to say the scenario was created to illustrate the absurdity of scaling up quantum phenomena to human size. Schrödinger’s ‘thought experiment’ remains as entertaining and bizarre today as it was in the 1930’s, yet it still remains mathematically correct to describe the cat as being simultaneously half alive and half dead.

The context is all important

Effective Mass is a parameter considered important by enthusiasts, yet is actually meaningless when taken out of context with cartridge compliance. Including the word ‘mass’ in the phrase is somewhat misleading as there exists only an abstract association between the two.

What people tend to mean when they talk of Effective Mass is ‘how heavy the arm appears to be, to the cartridge’.

Avoiding oversimplification

‘Mass’ is certainly used when calculating Eff.M but it’s a variable in a rather simple equation that doesn’t tell you much of any use. The parameter that really matters is the fundamental resonance betwixt cartridge and tonearm – in other words, it’s about the working relationship between the inertial mass of the arm and the cartridge compliance.

At best, Eff.M should only be seen by the end user as a guideline for compatibility. It’s little used by audio engineers and is a bit old-fashioned in an age where modern design and manufacturing technology have circumvented most of the historical limitations associated with mismatch.

Aside of extreme examples, it’s in the interest of cartridge manufacturers to ensure their products are compatible with the widest range of tonearms. Today’s engineers are making the most of new materials and taking advantage of new techniques in miniaturisation that drives the industry towards ever higher standards of performance.

Try to visualise this

The traditional mantra advises that using a heavyweight, low compliant cartridge with a low Eff.M tonearm will lead to resonance ( see link to animation below ) - conversely, the mounting of an ultra-light, high-compliant cartridge into a ‘heavy’ arm means the delicate cantilever suspension is always skewed as it tries to pull the heavy arm across the record.

Okay, but are you any the wiser?

But if I said to you ‘this arm has an Eff.M of 8g and this other one stands at 12g’, would you be any the wiser when choosing a cartridge to match? Would you reject a cartridge you really love the sound of, in preference for your second or third choice just because the Eff.M was slightly out of range? I would hope not.

The (usually) unrealised impact on amplifier power

So, in respect of Eff.M – what does it mean to ask if a cartridge is compatible with an arm? Mostly, it’s the need to keep the arm/cartridge resonance within the 8Hz to 15Hz range – any higher and you’re in the audio band, any lower and the arm is likely to skip a few grooves. Either will consume a large percentage of your amplifier power as low frequencies consume considerably more power than higher ones.

In other words, it’s not Eff.M that’s the issue per sec, but the danger that a mismatched combination of components will lead to either power-sapping resonance or stress on the cartridge suspension.

In objective scientific terms

Mathematically, a tonearm/cartridge combination can be described as a system in which a probe

( the stylus ) is subject to an oscillating force of variable frequency and amplitude. Kinetic energy is transferred, through a damped, pivoted beam (cantilever) into a minimally damped instable horizontal pendulum with two degrees of spherical freedom. ( tonearms works in polar co-ordinates, not Cartesian )

It’s a constrained, damped, double horizontal pendulum, system.

That last bit wasn’t very helpful, was it?

The problem is how to turn the mathematics into something you can visualise...and after trawling the interweb, I’ve found a suitable animation that demonstrates the scenario in a non-mathematical but effective way.

Before you click on link below let me describe the situation - the block you see oscillating is analogous to the cartridge when observed looking down the length of the tonearm from the front.

The resonance demonstrated is what happens when both the cartridge compliance, and Eff.M, are unacceptably low. ( very stiff cantilever, weak tonearm ) - It doesn’t take much imagination to see why you’ll be jumping grooves.

The two large springs in this model are analogous to the cartridge suspension.

When you click on this, be prepared to then:-

  1. Click on the arrow
  2. Click on ‘Experiment’.
  3. Click and hold on the blue block and pull it to one side.
  4. Click on start.

http://www.physics-chemistry-interactive-flash-animation.com/mechanics_forces_gravitation_energy_interactive/oscillations_horizontal_pendulum_mass_spring.htm

This demonstration doesn’t take into account phase differentials, damping or the Fourier series associated with relative arm/stylus motion but it serves to illustrate my point. Very low Eff.M tonearms will never give you true rendition of what’s happening in the groove as the arm tend to move too much relative to the stylus, often sounding a little ‘lively’ (they ‘reveal’ detail in the music that’s not really there.)

Long, thin walled, parallel arm tubes are subject other harmonic resonances that get added to the musical signal. If you’ve ever listened to Mike Oldfield’s Tubular bells, or have any wind chimes in your garden you’ll understand what generates the ‘tone’ in your tonearm.

Some of you will smile at this and ….

You might think you’ve avoided the problem by using an arm with a carbon fibre tube – sorry - anywhere between 35% and 65% of the material you think is carbon fibre, is actually plastic resin. Moreover, the velocity of sound in CF is literally half that through aluminium so the material is very ‘slow’. Its ability to transmit audio frequencies is nowhere near as efficient as aluminium. It’s a very ‘lossy’ material better suited to the car industry as it dissipates energy through radiation.

I mentioned that they were unstable horizontal pendulums. A true pendulum has only one freedom, it should only swing backwards and forwards. In our case the tonearm is badly shaken as it moves and, as a result, the signal that goes to your amplifier contains Fourier components of contra-movement.

The ‘sweet zone’

This is where we come back to the point of this article – as audio engineers we’re more interested in what affects the sound than Eff.M. As long as arm/cartridge resonance falls within that ‘sweet zone’ of 8Hz to 15Hz, it’s not something we worry about.

For those interested in the hard-core physics …..

Then I suggest you read the following articles and note the mathematical similarities with tonearms. Please note these models are analogous to the cartridge only.

http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/viewFile/5153/1618

Fig 3 in this shows the dangers of the tonearm moving one way and the stylus moving the other – the amplitude of the signal fed to your amplifier is twice what it should be.

Fig 4 shows a more realistic scenario where the tonearm is slightly out of phase with respect to the stylus and where the arm’s inertial mass damps the resultant signal.

When the system complexity includes the tonearm ( adding a second horizontal pendulum ) the maths becomes more challenging –

http://www-nonlinear.physik.uni-bremen.de/download/ForcesDoublePendulum.pdf

And when you add co-efficients of damping – the equations become quite unwieldy!

In truth, the problem with the maths is that every cartridge is different

So having spent hours deriving this stuff, you have no idea what numbers to put in. Consider, for instance, that cantilevers are pre-loaded, so in free air, they hang lower than when they have 2g of downforce acting on them – ergo the damping is different for each of the two operational planes.

The only realistic way to resolve this is to enter a range of variables into a computer model and plot the relative movement of stylus to tonearm.

You don’t need to understand all this, just take my word for it that when the stylus wants to go one way, the tonearm ( because its pivoted and carries inertia ) is usually inclined to go another....and the signal that’s generated is what goes to your amplifier, is a combination of the two motions – what the stylus sees in the groove, and how the tonearm reacts.

Parting company with conventional thinking

This is where Eff.M comes in and where I part company from conventional thinking. If you look at all the Helius super-arms ( from the Orion and Cyalene to the current Omega and immanent Phaedra ) none could be described as being weedy.

Obviously the arm has to be free to tack across the record and cope with any non-linearities due to warps, lead-in spaces ( to the next track), mechanical wow due to non-centrality of the centre hole, etc. - but, the key point remains that the tonearm must not contribute to the signal generated.

And to achieve that, cartridges need sufficient Effective Mass behind them to allow them to work properly – but not too much to cause problems.

Thank you

Geoffrey Owen / Helius Designs

http://www.heliusdesigns.co.uk/