Measuring Tonearm Effective Mass

 senior member
 Posts: 258
 Joined: 15 Sep 2002 22:15
 Location: Norwich
Measuring Tonearm Effective Mass
I'm curious  how is tonearm effective mass measured or assessed? Has anyone here had to do it?

 member
 Posts: 108
 Joined: 28 Jan 2011 23:42
 Location: Scotland
Re: Measuring Tonearm Effective Mass
Using cartridge weight scales remove the counterweight from your tonearm and place the arm with cartridge fixed onto the scales. From this weight deduct the weight of the cartridge and you have the tonearm effective mass.

 long player
 Posts: 4741
 Joined: 06 Feb 2007 22:58
 Location: Monroe NY USA
Re: Measuring Tonearm Effective Mass
Effective mass INCLUDES the influence of the counterweight. It is the resistance of the whole moving parts to motion, and the resistance to the moving parts being brought to a stop once in motion. It is also known as inertia. The best source of information is the specification provided by the manufacturer of the arm. Failing that an approximation can be arrived at by weighing measuring and calculation.
This is interesting physics :) The concept of effective mass is this. The tonearm seems to be a complicated arrangement of moving parts however so far as the stylus is concerned the influence of all the parts of the arm can be simplified down to the moment of inertia (resistance to motion) that is present at the stylus tip, that resists the forces trying to move the stylus. The down force that is deliberately applied so that the stylus follows the groove is excluded from the analysis of effective mass. In other words effective mass is the inertia of the mass of the entire arm and cartridge that would be present if the inertia of all the moving parts was condensed into a pinpoint of space at the stylus.
Moment of inertia is the mass multiplied by the square of the distance between the mass and the pivot point. In a tonearm most of the inertia is in the mass of the cartridge and the headshell, plus the inertia of the counterweight. Because the cartridge is located much further from the pivot than the counterweight is from the pivot, the cartridge (and headshell) mass is the major component of the inertia. The arm wand, being relatively light, adds little inertia.
This is how to make an approximate calculation from an arm with no data. This is copied without permission from a post by an absent forum member in 2009.
's vinylforum post about tonearm effective mass 31st Dec 2009 in turntables and tonearms forum. he posted as ldg
quote
"I used missan's method to find the mass of the cartridge side of the tonearm. Then used I = m*(L^2)/3 to determine the MOI of that side of the arm, the RB250 being a straight tube (approx). (ed: MOI is Moment Of Inertia)
Then worked out the effective mass of my RB250, as per the original post, and obtained a new answer, 11.1g. Published figure is usually 12g, sometimes 11g. So that's a good result  thanks guys !
So here's a (revised) method for measuring effective mass of a tonearm. Good for tonearms that are straight, short stub, constant mass/unit length (like a nontapered tube), and where the headshell mass is light (about the same mass/unit length as the arm  see notes below).
Principle: To measure actual tonearm effective mass, all one needs to do is determine the moment of inertia of the tonearm about the pivot, then calculate the equivalent mass required at the effective tonearm length to provide the same moment of inertia, and that mass is then the effective mass of the tonearm.
Step1 The tonearm is a lever balanced about the pivot. The vast majority of mass on one side of the lever is a lump mass in the form of the counterweight. So weigh the counterweight (mass m [kg]) measure the distance from the centre of the balanced counterweight to the pivot with a ruler (r [m]), and then calculate moment of inertia from I=m*r^2 [kgm^2]
Step2 To evaluate MOI of the cartridge side of the tonearm, remove the counterbalance and cartridge (inc mountings), then use a weighing scale to measure the weight W of the tonearm at the headshell end, with the tonearm parallel to the platter. W is half the weight of the cartridge side of the tonearam (less a small bit for the stub  ignore), so the mass Z of the cartridge side of the tonearm Z = 2*W (kgf), and since it is vertical Z is also the mass in kg. The effective length L can either be measured (between stylus tip and pivot) or looked up from published figures for the tonearm. Then calculate moment of inertia from I = Z*(L^2)/3 [kgm^2]
Step 3 Calculate the total moment of inertia I(tot)
I(tot) = [m*(r^2)] + [Z*(L^2)/3] kgm^2
Then effective mass M at effective length L is given by
M*L^2 = [m*(r^2)] + [Z*(L^2)/3] kgm^2
So M = ([m*(r^2)] + [Z*(L^2)/3])/(L^2) kg
which reduces to
M = [m*(r^2/L^2)] + [Z/3] kg
In itself, this is an interesting result. It shows the contribution to effective mass from each side of the tonearm, mostly it comes from the cartridge side. It shows what to vary if one seeks to increase/decrease effective mass, principally the mass of the cartridge side of the tonearm, Z. But some influence is also possible from a heavier counterweight, and in a nonintuitive direction perhaps (heavier = lower M because balancing distance r influences M as power of 2).
For S shaped tonearms, would need to evaluate the MOI differently. Same for tapering mass/length arms. For tubular arms with detachable headshells, MOI of arm and headshell can be evaluated seperately and added together, that is a principle of MOI, contributions of coupled parts can simply be added. The stub is relative low mass and close to the pivot. One could correct, but i think it only makes a few % difference and is OK to ignore. All of these measurements/ calcs are just for tonearm, no cartridge or fixtures fitted. Add the cartridge/fixture mass in the normal way to obtain total effective mass.
Thanks again, guys. Comments ?
PS  Happy New Year ! I can have a drink now
Hi Red_One, yes ^2 means squared, raised to power of 2, and kgf is just a way of distinguishing between weight and mass.......1kgf = 1kg weight effectively, and kgm^2 is the unit kilogram*metressquared, which is the unit of I, the moment of inertia, useful in rotational physics, and there is a standard moment of inertia for a straight rod of length L and uniform mass per unit length total mass Z, I =Z*(L^2)/3 which someone derived long ago to save those of us who've already had a new year's eve drink from the embarassment of doing calculus to prove HTH, happy new year , and any q's just ask
_________________
Yes, time flies like an arrow, but fruit flies like a banana
"
end quote
This is interesting physics :) The concept of effective mass is this. The tonearm seems to be a complicated arrangement of moving parts however so far as the stylus is concerned the influence of all the parts of the arm can be simplified down to the moment of inertia (resistance to motion) that is present at the stylus tip, that resists the forces trying to move the stylus. The down force that is deliberately applied so that the stylus follows the groove is excluded from the analysis of effective mass. In other words effective mass is the inertia of the mass of the entire arm and cartridge that would be present if the inertia of all the moving parts was condensed into a pinpoint of space at the stylus.
Moment of inertia is the mass multiplied by the square of the distance between the mass and the pivot point. In a tonearm most of the inertia is in the mass of the cartridge and the headshell, plus the inertia of the counterweight. Because the cartridge is located much further from the pivot than the counterweight is from the pivot, the cartridge (and headshell) mass is the major component of the inertia. The arm wand, being relatively light, adds little inertia.
This is how to make an approximate calculation from an arm with no data. This is copied without permission from a post by an absent forum member in 2009.
's vinylforum post about tonearm effective mass 31st Dec 2009 in turntables and tonearms forum. he posted as ldg
quote
"I used missan's method to find the mass of the cartridge side of the tonearm. Then used I = m*(L^2)/3 to determine the MOI of that side of the arm, the RB250 being a straight tube (approx). (ed: MOI is Moment Of Inertia)
Then worked out the effective mass of my RB250, as per the original post, and obtained a new answer, 11.1g. Published figure is usually 12g, sometimes 11g. So that's a good result  thanks guys !
So here's a (revised) method for measuring effective mass of a tonearm. Good for tonearms that are straight, short stub, constant mass/unit length (like a nontapered tube), and where the headshell mass is light (about the same mass/unit length as the arm  see notes below).
Principle: To measure actual tonearm effective mass, all one needs to do is determine the moment of inertia of the tonearm about the pivot, then calculate the equivalent mass required at the effective tonearm length to provide the same moment of inertia, and that mass is then the effective mass of the tonearm.
Step1 The tonearm is a lever balanced about the pivot. The vast majority of mass on one side of the lever is a lump mass in the form of the counterweight. So weigh the counterweight (mass m [kg]) measure the distance from the centre of the balanced counterweight to the pivot with a ruler (r [m]), and then calculate moment of inertia from I=m*r^2 [kgm^2]
Step2 To evaluate MOI of the cartridge side of the tonearm, remove the counterbalance and cartridge (inc mountings), then use a weighing scale to measure the weight W of the tonearm at the headshell end, with the tonearm parallel to the platter. W is half the weight of the cartridge side of the tonearam (less a small bit for the stub  ignore), so the mass Z of the cartridge side of the tonearm Z = 2*W (kgf), and since it is vertical Z is also the mass in kg. The effective length L can either be measured (between stylus tip and pivot) or looked up from published figures for the tonearm. Then calculate moment of inertia from I = Z*(L^2)/3 [kgm^2]
Step 3 Calculate the total moment of inertia I(tot)
I(tot) = [m*(r^2)] + [Z*(L^2)/3] kgm^2
Then effective mass M at effective length L is given by
M*L^2 = [m*(r^2)] + [Z*(L^2)/3] kgm^2
So M = ([m*(r^2)] + [Z*(L^2)/3])/(L^2) kg
which reduces to
M = [m*(r^2/L^2)] + [Z/3] kg
In itself, this is an interesting result. It shows the contribution to effective mass from each side of the tonearm, mostly it comes from the cartridge side. It shows what to vary if one seeks to increase/decrease effective mass, principally the mass of the cartridge side of the tonearm, Z. But some influence is also possible from a heavier counterweight, and in a nonintuitive direction perhaps (heavier = lower M because balancing distance r influences M as power of 2).
For S shaped tonearms, would need to evaluate the MOI differently. Same for tapering mass/length arms. For tubular arms with detachable headshells, MOI of arm and headshell can be evaluated seperately and added together, that is a principle of MOI, contributions of coupled parts can simply be added. The stub is relative low mass and close to the pivot. One could correct, but i think it only makes a few % difference and is OK to ignore. All of these measurements/ calcs are just for tonearm, no cartridge or fixtures fitted. Add the cartridge/fixture mass in the normal way to obtain total effective mass.
Thanks again, guys. Comments ?
PS  Happy New Year ! I can have a drink now
Hi Red_One, yes ^2 means squared, raised to power of 2, and kgf is just a way of distinguishing between weight and mass.......1kgf = 1kg weight effectively, and kgm^2 is the unit kilogram*metressquared, which is the unit of I, the moment of inertia, useful in rotational physics, and there is a standard moment of inertia for a straight rod of length L and uniform mass per unit length total mass Z, I =Z*(L^2)/3 which someone derived long ago to save those of us who've already had a new year's eve drink from the embarassment of doing calculus to prove HTH, happy new year , and any q's just ask
_________________
Yes, time flies like an arrow, but fruit flies like a banana
"
end quote

 senior member
 Posts: 258
 Joined: 15 Sep 2002 22:15
 Location: Norwich
Re: Measuring Tonearm Effective Mass
Thanks for that! I think I can follow those steps. The reason for asking is that I've made a new onepiece CNCmachined alloy armtube for my Linn Ittok and clearly it has greater overall mass than the original Linn 3piece design, but I'm curious to know how this translates into overall effective mass.

 junior member
 Posts: 15
 Joined: 06 Mar 2017 05:06
Re: Measuring Tonearm Effective Mass
I have a question regarding Step 1 of this process. If the measurement from the center of the counterweight to the pivot is dependent on the system being balanced, then does not the mass of the installed cartridge and its hardware directly influence the calculated result? In my instance, the cartridge used to balance the system has a mass of 3.3g. This yields a calculated tonearm effective mass of 8g using the above method. When using a different cartridge in my lineup having a mass of 8.3g, upon balancing the system and measuring the distance from the center of the counterweight to the pivot, the counterweight is several millimeters farther away than the previous calculation. This is enough to change the calculated tonearm effective mass to 9g, a full gram heavier.analogaudio wrote:Effective mass INCLUDES the influence of the counterweight.
Step1 The tonearm is a lever balanced about the pivot. The vast majority of mass on one side of the lever is a lump mass in the form of the counterweight. So weigh the counterweight (mass m [kg]) measure the distance from the centre of the balanced counterweight to the pivot with a ruler (r [m]), and then calculate moment of inertia from I=m*r^2 [kgm^2]
end quote
Does this mean that this method should properly be applied to each and every cartridge one wishes to "match" to one's tonearm? Granted, the two cartridges I tried on my own tonearm are at the lower and upper limits of appropriate mass as stated by the manufacturer of the tonearm. So, if anything, it is clear that I may expect the effective mass of my particular tonearm to range between 8 and 9 grams. This is, of course, without any additional modifications like blutack or extra washers to counterbalance, etc. It is fascinating how dynamic the physics of a simple tonearm and cartridge actually are.
I guess my question then is this. Are the tonearm effective mass values provided by the manufacturer intended to be used as a general guideline? I was always under the impression that this value was constant. But, the calculation method demonstrates that every time the counterweight is "zeroed" with a new cartridge it is potentially in a different position which necessarily changes the effective mass of the tonearm system.
Or, did I do something incorrectly? It has been known to happen occasionally.

 vinyl engineer
 Posts: 22983
 Joined: 28 Oct 2002 03:24
 Location: North of Toronto, Canada
Re: Measuring Tonearm Effective Mass
You could probably work backward. With a test record that has resonance test tracks, and a cartridge of known compliance, it should be fairly simple to get a ballpark figure.
Cheers,
Alec
Cheers,
Alec

 Posts: 1
 Joined: 02 Mar 2016 18:21
Re: Measuring Tonearm Effective Mass
Probablement que la meilleure solution est de choisir son bras en fonction de la cellule souhaitée. car au final c'est bien elle qui capte la musique.
Personnellement je fabrique mes propres bras de lecture pour chacune de mes cellules qui ont des compliances qui varient de 5.106 (koetsu/DL103 par exemple) à >20.106( bobine MM, V15 par exemple)
je fabrique mes propres bras pour la raison essentielle d'utiliser la masses rajoutées, non pas pour faire bêtement du poids (ce qui ne fait que compliquer la tache de la cellule en augmentant considérablement le module d'inertie), mais pour faire les plus grands bras de lecture possibles.
Ainsi les longueurs des bras varient de 22'' à 27''. et au final la résonance pour chaque bras est bien idéalement de 10hz. (les bras sont calculés au mm et au gramme près pour obtenir 10hz)
Dans l'absolu ces bras sont très lourd (certains dépassent les 400g de masse totale) mais leur masse spécifique, qui tient compte du module d'inertie en pointe de stylet est bien dans la norme de 14 à 18g (suivant leur compliance respective. une compliance faible pour un bras "lourd"),
et suivant la formule très pertinente:
"calcul de la masse effective du bras à partir de son moment d'inertie calculé à la pointe du stylet"
ME = (MCP*(DCP^2) / (DPC^2)) + (2*MPC/3)
 exemple pour une denon DL S1:
 Distance PC/pointepivot (cm) DPC 60.10
 Masse du bras coté PC+cellules (g) MPC 19.48
+ vis + câblage+....
 Distance du contrepoids au pivot (cm) DCP 3.90
 Masse du contrepoids (g) MCP 300.20
 Masse Effective (g) ME 14.3
 cette masse effective de 14.3g est bien adaptée à la denon DLS1 et la résonance mesurée à l'usage est bien
idéalement de 10Hz
Bien sûr pour éviter tout effet de résonance ondulatoire, lié à leur très grandes longueurs,les bras sont équipés d'un double système d'amortissement en latéral et en vertical. Au final une cellule bien accordée n'a pas d'équivalent .
merci de m'avoir lu
Personnellement je fabrique mes propres bras de lecture pour chacune de mes cellules qui ont des compliances qui varient de 5.106 (koetsu/DL103 par exemple) à >20.106( bobine MM, V15 par exemple)
je fabrique mes propres bras pour la raison essentielle d'utiliser la masses rajoutées, non pas pour faire bêtement du poids (ce qui ne fait que compliquer la tache de la cellule en augmentant considérablement le module d'inertie), mais pour faire les plus grands bras de lecture possibles.
Ainsi les longueurs des bras varient de 22'' à 27''. et au final la résonance pour chaque bras est bien idéalement de 10hz. (les bras sont calculés au mm et au gramme près pour obtenir 10hz)
Dans l'absolu ces bras sont très lourd (certains dépassent les 400g de masse totale) mais leur masse spécifique, qui tient compte du module d'inertie en pointe de stylet est bien dans la norme de 14 à 18g (suivant leur compliance respective. une compliance faible pour un bras "lourd"),
et suivant la formule très pertinente:
"calcul de la masse effective du bras à partir de son moment d'inertie calculé à la pointe du stylet"
ME = (MCP*(DCP^2) / (DPC^2)) + (2*MPC/3)
 exemple pour une denon DL S1:
 Distance PC/pointepivot (cm) DPC 60.10
 Masse du bras coté PC+cellules (g) MPC 19.48
+ vis + câblage+....
 Distance du contrepoids au pivot (cm) DCP 3.90
 Masse du contrepoids (g) MCP 300.20
 Masse Effective (g) ME 14.3
 cette masse effective de 14.3g est bien adaptée à la denon DLS1 et la résonance mesurée à l'usage est bien
idéalement de 10Hz
Bien sûr pour éviter tout effet de résonance ondulatoire, lié à leur très grandes longueurs,les bras sont équipés d'un double système d'amortissement en latéral et en vertical. Au final une cellule bien accordée n'a pas d'équivalent .
merci de m'avoir lu