Re: Measuring Tonearm Effective Mass
Unread postby analogaudio » 25 Apr 2015 17:27
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 pin-point 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 non-tapered 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 non-intuitive 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.