-edible zone-
Cartridge / Arm Matching....or...what you'll need 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.
2)
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 being on 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 a
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 10¯6 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
: 25 x 10^6 cm/dyne
recommended (for best performance) tonearm
effective mass: 6 to 12 grams
** : 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
let me know.