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Cartridge / Arm Matching....or things you didn't really want to know about resonant frequency of a system tonearm/cartridge which is coupled to the record by the compliance of the cantilever suspension. ... 

1)

above photo #1: showing the cantilever and location of suspension.  The suspension element may be as simple as a tight fitting elastic grommet that grips both the shaft and the cartridge body.

 

2)

above photo #2: Showing stylus mounting in the cantilever.  This is a nude mounting that features a precise 'force fit' between stylus body and the bore in the cantilever.  Additionally, a small bit of adhesive is applied over the joint to ensure a lasting fit. 

side note: Whoever said "diamonds are forever" was not referring to phono cartridge stylii. They wear out. Yet, like their jewelry counterparts, they are expensive nonetheless.

Matching a cartridge to a tonearm (pivoting tone arms)

The focus here is upon elements of the cartridge design, and elements of the tonearm design which affect how a vinyl addict might go about choosing a given cartridge to fit a given tonearm.  First, some definitions.

Effective mass versus compliance results in a Resonant Frequency (of the cantilever)

Compliance: 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 phonograph 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, low compliance = stiffer.

Vertical Tracking Force (VTF): The amount of vertical force placed upon the cantilever is controlled by a careful balance between the weights of the arm on either end of the pivot.  Typically, the tonearm will have a long tube leading to a head shell with a phonograph cartridge and stylus at that end.  The other end of the tonearm will have weights intended to balance the mass of the long end.  By careful manipulation of the counterweight position, relative to the distance from the pivot bearing, a precisely measured amount of vertical force may be applied at the stylus end.

Effective Mass: The amount of force felt at the stylus under dynamic conditions in any (xyz) arcing vector about the pivot. This differs from VTF 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.  

Phonograph 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.  In the case of the SME 3009 Improved with fixed head shell, the arm is rated to have an effective mass of 6.5 grams.  Compare this to the Thorens TP16 Mk 1 tonearm  which has a rated effective mass of 16.5grams.

Resonant frequency of the cantilever is used as a guide to match suspension stiffness (compliance) of the cartridge to the tone arm's (effective) mass.

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.  Also referred to as 'hz'.  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.  

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. 

So....wouldn't it be useful if you could calculate a resonant frequency between a given arm and cart...?   You can,  just plug in some vital statistics into the short formula below.  Keep in mind the formula is intended as a rule of thumb.  The test record will be the final say on what a given arm cart combo can give in terms of cantilever resonant frequency.  Think of the formula as a 'on paper test' and the test record as a 'real world actual test'.

rf = 159 / sqrt ((eff. mass + cart weight + fastener weight) * (compliance))

rf: resonant frequency in hz
eff. mass: rated by tonearm manufacturer
cart weight: rated by cartridge manufacturer, but if accurate scales exist, an actual weight value could be used
fastener weight: screws, nuts, spacers, washers, shims.  They have weight and add to the mass over the stylus
compliance: rated by cartridge manufacturer

Freek's Resonant Frequency Calculator a short Excel spreadsheet that simplifies using the above formula. MS Excel required.

 

 

Example No. 1: Now let us think about matching a Shure V15VxMR to the above pictured TP16 Mk 1 tonearm.

 

effective mass rating: 16.5 grams
cartridge weight: 6.6 grams
fastener weight: .5 grams
compliance: 25

(16.5 + 6.6 + .5) * 25 = 590

sqrt 590 = 24.2899

159 / 24.2899 = 6.5459 hz calculated

The above figure was approximately verified with the HFNRR test record getting a test result value of 6 hz

I'll admit that I have lived with the above combination for a time.  I found the setup susceptible to footfall but it seemed to track record warps just fine.  A marginal matching.  It is the wrong side of marginal, too.  Another visual note about this match-up is that the high compliance of the Shure cantilever  was quite obvious when dropping the stylus into the lead in groove.  Considerable sideways deflection was evident.  This is not the same case with the above match up between the Shure and the much lighter SME tonearm.  For more notes about The Shure cartridge and it's dynamic stabilizer  when fitted to the TP16 mk 1 see the ** below.

 

Example No. 2: with the TP16 Mk 1 tonearm.  This time with the above pictured Blue Point Special cartridge...?  Let's crunch the numbers and see.

effective mass rating: 16.5 grams
cartridge weight: 9.0 grams
fastener weight: .5 grams
compliance: 12

(16.5 + 9.0 + .5) * 12 = 312

sqrt 312 = 17.6635

159 / 17.6635 = 9.0016 hz.

Example No. 3: let's calculate the rf for my SME 3009 Improved Fixed Head shell tonearm and Shure V15VxMR cartridge. 

The effective mass rating for the arm is 6.5 grams
The weight of the cartridge is rated at 6.6grams
fastener weight: .5 grams 
Compliance is rated at 25 (x 106   cm/dyne)*

(6.5 + 6.6 + .5) * 25 = 340

sqrt 340 = 18.4391

159 / 18.4391 = 8.6230 hz calculated

HFNRR test result: side 2 band 2 low freq. horizontal resonance test......10 hz 

Notes about the discrepancy between the test record figure and the calculation.  The test record value is considered to be the valid reference.  One of the variables in the calculation must be incorrect.  Perhaps effective mass rating of the tonearm may actually be lighter in the real world situation.  The finger lift was not used and so does not contribute to the effective mass.  I have no accurate means to measure the weight of the finger lift.  I used the shortest possible screws in the cartridge mounting.  Said screws and nuts are aluminum.

How to calculate Compliance based on a Test record result for resonant frequency:

C = 25330 / ((eff mass + cart mass + fastener weight) x (rf squared)

Example: Denon DL103R cartridge mounted to an Expressimo modified RB250

approximate effective mass of the Expressimo RB250: 10g
cartridge weight: 8.5g
fastenener weight: .5g
test record lateral rf result: 11 hz

((10 + 8.5 + .5) x 11^) or 19 * 121 = 2299  

then 

25330 / 2299 = 11.0178 (compliance)

 

 

*Shure does not publish compliance figures for this cartridge in the owners manual that comes with it.  A search of the knowledge base at the Shure website turned up the following information on the V15VxMR:

Compliance rating (dynamic): 25 x 10^6 cm/dyne
recommended (for best performance) tonearm effective mass: 6 to 12 grams
Shure Knowledge Base (link for more info)

**  :  The Shure V15VxMR phono cartridge uses a device that Shure calls a "dynamic stabilizer".  They (Shure) describe this device as a shock absorber for their cantilever.  If this device is put to use, it is claimed that a much wider range of effective mass tone arms may be used with this cartridge.  I have used this 'dynamic stabilizer' when the cartridge was fitted to the heavy TP16 mk1 and found that the cartridge tracked all records without any apparent fault.  When the device was parked in it's 'up' position (taken out of use) a much greater amount of cantilever deflection could be witnessed and the tonearm was more susceptible to external disturbance.  The 'device' was not put to use with the SME tonearm as the resonant frequency of the cantilever falls within the optimal range when fitted to this arm.

 

*** : All math/physics formulas and explanations offered on this page were either collected from or donated by persons more capable and knowledgeable than myself.  I just operate them.  If you see an error or have another formula useful to this topic please contact me.