# An Excel spreadsheet for DIY Tonearm's Effective Mass Calculation.

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Hi all,

I made an Excel page for the calculation of a Tonearm's Effective Mass and I would like to share it with you.

The method is pretty simple. All you need is a cartrirdge of known compliance and the use of the HFNRR Test Record (Low frequency horizontal resonance test).

Once the Cartidge is correctly setuped (VTF, VTA, Offset angle, Antiskating etc.) on the tonearm you let it ''read'' the low frequency horizontal resonance bands, and you find out the frequency at which the cantilever will become excited and vibrate out of control. Then you throw the results (cartridge's compliance, resonance frequency, weight of the cartridge and fasteners) into the Excel sheet and you have your Tonearm's effective mass.

After experimenting a little with various tonearms and cartridges I have found the results pretty accurate.

Regards

#### Attachments

• tonearm's effective mass calculator.zip
2.3 KB · Views: 1,401

Some Improvements

Hello all,

I had a discussion with a friend yesterday about this Excel sheet and he mention that it will be useful to also add an option for calculating the Resonance Frequency and a cartridge's compliance.

After some thought I made some improvements on the sheet, so it is now capable of calculating the resonance frequency of a given cartridge-tonearm combination which I think is very useful if you want to know if a cartridge is a good match for a tonearm or vice-versa. The optimum cartridge-tonearm resonance frequency should be between 8-12Hz.

The other improvement is the calculation of a cartridge's compliance which is useful if you own a vintage cartridge with unknown compliance.

What I found very interesting is that with this sheet you can know what is happening to the resonance frequency if for instance use heavier screws or some short of a weight between the cartridge and the head shell. Many of you may remember some cartridge manufacturers that supplied a small weight for that reason.

Regards

#### Attachments

• tonearm's effective mass calculator v2.zip
2.9 KB · Views: 857

#### codyLEE

Dear Sir,

As i need to know how to get the cartridge's compliance and resonance frequency,
could you please let me know if there have any illustration to explain these two more simple
for my easily understanding ?

Thank you for your kind assistance.

codyLEE said:
Dear Sir,

As i need to know how to get the cartridge's compliance and resonance frequency,
could you please let me know if there have any illustration to explain these two more simple
for my easily understanding ?

Thank you for your kind assistance.

Hello,

There are four (4) parameters in a cartridge-tonearm system, Cartridge's Compliance, Cartridge's Weight (+ screws, nuts etc.)Tonearm's Effective Mass and the systems resonance frequency.
We want the system's resonance frequency to be anywhere between 8-12Hz.

So if you need to calculate you cartridge's complince (if this parameter is NOT included in the manufacturers specification sheet sheet that came with your cartridge, or because you own a vintage cartridge) you must at least know the other three parameters.

Check out your tonearm's specification sheet for the tonearms effective mass and use the HFNRR Test Record (Low frequency horizontal resonance test) to find the resonace frequency (See post #1).

Put the results in the Excel Sheet (Version 2) and you'll get the results.

Hope that helps.
Regards
P

#### codyLEE

But I still have one question, as I do not have HFNRR test record(Low frequency horizontal resonance test)
to find the resonance frequency. So I wonder if I can use another test record to get the answer, or I must have
to buy one HFNRR test record ?

Once I have one HFNRR test record, do I still need any other instruments to do the test to get the resonance frequency ?

Thank you again for your help.

codyLEE said:

But I still have one question, as I do not have HFNRR test record(Low frequency horizontal resonance test)
to find the resonance frequency. So I wonder if I can use another test record to get the answer, or I must have
to buy one HFNRR test record ?

Once I have one HFNRR test record, do I still need any other instruments to do the test to get the resonance frequency ?

Thank you again for your help.

If I remember correct the only record that contained Low frequency horizontal resonance test tones was from Ortofon, but this is out of production long time ago.

The only Test Record that I know with these tones in current production is the HFNRR. You do not need any kind of measuring instruments for these test.

Regards
P

#### codyLEE

Noted for Resonance frequency issue.

But I checked with the Japan cartridge's manufacturer for cartridge's
compliance who replied to me that they do not know what it is. Therefore,

Thank you so much.

codyLEE said:
Noted for Resonance frequency issue.

But I checked with the Japan cartridge's manufacturer for cartridge's
compliance who replied to me that they do not know what it is. Therefore,

Cartridge Compliance (Elasticity) : Let's take a look at a spring (any spring) designed to carry a given load. Place too much weight on this spring and it collapses. Place not enough weight on the spring and the spring won't compress at all and remains rigid. Think of the cantilever as a spring.

In the case of a phono cartridge, the cantilever is a rigid arm connected to a springing medium mounted up within the body of the cartridge. This springing element may be as simple as a rubber donut that holds enough tension against the cantilever mounting to maintain relative position of the cantilever. Most importantly, the suspension must control the attitude of the stylus fitted at the cantilever's end.

The amount of distance that a cantilever deflects under a given force load is referred to as cantilever compliance. Higher compliance cantilevers deflect a greater distance when a given amount of force is applied. Lower compliance cantilevers deflect lesser distances when the same force is applied. In other words, high compliance = softer suspension , low compliance = stiffer suspension.

Tonearms Effective Mass: The amount of force felt at the stylus under dynamic conditions in any (xyz) arcing vector about the pivot. This differs from Verical Tracking Force which is set static and remains constant only under 'peaceful' conditions while the record is in play. Effective mass is influenced by the weight of the various appendages of the tonearm assembly in ratio to the distance from the pivot. Weight that is further from the pivot center will account for higher effective mass than the same weight if positioned closer to the pivot. Said slightly differently, the heavy bits on the tonearm need to be closer to the tonearm pivot or excessive effective mass will be the result.

Phono cartridges have different weights from one product to the next, therefore tonearm makers rate their arms in terms of effective mass before a cartridge is mounted.

Resonant Frequency (of the cantilever): The acoustic frequency at which the cantilever will become excited and vibrate out of control. This frequency is measured in cycles per second (Hz/sec). Resonant frequency of a cantilever is regarded as inescapable and the effect is controlled by manipulating this frequency to exist in a range below human hearing but not so low that it will become excited by external vibrations such as foot fall disturbance or that of a warped record. This ideal frequency range is 8 to 12 hz. The lowest of low organ notes rarely go below 20 hz. Footfall and record warps happen below 6 hz.
Some Phono Preamplifier manufacturers for that reason placed a subsonic filter cutting everything below 25Hz but this ''technique'' introdused other problems.

The effective mass of a tonearm in combination with the compliance of the cartridge cantilever serves to determine where the resonant frequency of a given tonearm/cartridge match up will be. In general terms, arms with high effective mass fitted with cartridges of high compliance result in resonant frequencies that fall below the ideal range. At the opposite end, arms with low effective mass mated to cartridges of low compliance result in resonant frequencies above the desired range. Both extremes are to be avoided. In other words, Resonant frequency of the cantilever is used as a guide to match cartridge's compliance to the tonearm's effective mass.

Regards
P

#### gpapag

Paid Member
Excellent write-up.
Allow me to note a small (scemantics) error that sliped through:
Resonant Frequency (of the cantilever): The acoustic frequency at which the cantilever will become excited and vibrate out of control. This frequency is measured in cycles per second (Hz/sec).
Should read:"...This frequency is measured in cycles per second (Hz ) "
Regards
George

This frequency is measured in cycles per second (Hz )

You're right. Error due to high speed...............thinking

Regards
P

#### codyLEE

But sorry as my English is poor, before getting your reply I have gone to some website to see such information. However, I can't understand them well, that's why I want to see if there can anyone helps me to know what is cartridge's compliance in simple way, like some drawings to show how to get the cartridge's compliance. So could you please let me know if it is possible to let the words become illustration or any website you suggest me to check with ?

Thank you so much for your help.

#### codyLEE

------------------------------------------------------
The Test LP is specific to the arm/cartridge/turntable alignment and setup.
Cannot see how it would help in establishing the resonant frequency of your turntable
For further information on content, see our website www.mothgroup.com
Mike Harris
Moth Group
-------------------------------------------------------
SO...... I 'M CONFUSED........

codyLEE said:
------------------------------------------------------
The Test LP is specific to the arm/cartridge/turntable alignment and setup.
Cannot see how it would help in establishing the resonant frequency of your turntable For further information on content, see our website www.mothgroup.com
Mike Harris
Moth Group
-------------------------------------------------------
SO...... I 'M CONFUSED........

Hi,

And yes, Mike from Moth Audio is absolutely RIGHT.
But that's because you asked him a wrong question, you see, the resonant frequency of the cartridge-arm system is NOT the same thing as the turntables resonance, (a very very complicated task to analyze).

With the HFNRR test record you can observe the cartridge-arm combination resonance frequency.

I suggest you re-read all the previous posts in this thread and try to understand my explanations.

Regards
P

#### nathanjh13

I'm interested in this but I wondered if someone could provide a worked example using the following perameters please. These are imaginary numbers plucked from the top of my head but I hope they provide a basis for explanation.

What is the effective mass if:

The arm is 300mm total length.

Distance from end to pivot point (counterweight side) is 60mm.

Counterweight mass is 50g the centroid of which acts 30mm from the pivot.

The arm tube is 10mm dia made from (say) Aluminium at 2600kg/m3, uniform circular cross section throughout ((if you like, assume a solid tube for this example) the cross sectional area of this will then be 78.540mm2).

Cartridge effective length from pivot is 239mm.

Cartridge mass is 6g, and compliance is 18mm/N

Can anyone oblige me? Is there anything missing from this?

I can calculate the Moments Of Inertia (mr^2 etc) but I want to double check the process, especially at the end to get an answer in grams.

This value would then be used to calculate the resonant frequency?

thanks
nathan

#### marekst

If you read Polish, the formula for effective mass and explanation is at (third page, second part):

audioproject

To roughly calculate effective mass (M) of the tonearm you need to know tonearm effective length (L), weight of counterweight balancing tonearm without cartridge (w) and horizontal distance from pivot to center of gravity of counterweigh when arm is balanced (r).
I said “roughly” but it is far more precise than using test record method. If you understand that formula you may be able to modify it for more precise calculation for tapered tonearms etc…

#### nathanjh13

Many thanks for that! My Polish is awful but Google Translate helped..

"Slightly simplified but very useful formula for calculating the effective mass of the arm:

M = [(w*r^2)/L^2]+[(2wr)/3L]

M - mass effective
w - mass counterweight balancing arm without padding
r - distance from the point of suspension of a counterweight (the best counterweight to hang by a thread, then you will see a counterweight suspension point) from the pivot point of the beam.
L - effective length of arm

This formula is simplified - assumes equal distribution of weight on the shoulder pads and leaves a negligible effect on the weight of the counterbalance beam. This formula does not require the weighing arm enough to take a counterbalance to the store and weigh. If someone has an arm with a variable diameter or want a more accurate model can make changes. That is why I've included departure.

Moment of inertia on the side of a counterweight is equal [to]

w*r^2

Moment of inertia on the side of the insert is equal to

(m*L^2) / 3

where m is the mass of the beam arm on the side of the insert.

In the balanced arm (without pad) moments of forces on both sides of the bearing are equal, so:

m * (L/2) = w * r

ie

m = (2wr)/L

Moment of inertia caused by the effective mass is equal to the sum of the moments of inertia of the arm on both sides, so:

M * L^2 = [w * r^2] + [(m * L^2) / 3] = [w * r^2] + [(2wrL) / 3]

M = [(w * r^2) / L^2] + [(2wr) / 3L]

III.
Reduced model of the relationship between the total effective mass of the arm (ie the sum of the effective mass of the arm, mass inserts and screws) and the resonant frequency of the arm pad fz susceptibility c:

Ms = 25356 / (c*f^2)

The above formula can also calculate the frequency f or the susceptibility c as you know the rest of the data."

#### Lilith_K

I follow the basic concept and math, but am interested in how to apply this to an arm that utalizes a variable mass such as the Mayware Formula 4 and similar arms that have a weight that can be positioned along the arm length. Doesn't this move the arm mass further from the pivot thereby changing the effective mass of the arm at the cantiliver. With this type of geometry, how do you calculate to determine optimum cartridge compliance and associated resonate frequencies?

Lilith K

"Life, don't talk to ME about life..." Marvin the Robot, Hitchhikers Guide to the Galaxy.

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