On Jun 23, 7:25*pm, "Arny Krueger" wrote:
"Dick Pierce" wrote in message
...
On Jun 22, 9:13 pm, "Arny Krueger" wrote:
Well there is. The high frequency resonance is
dependent on inertia and compliance of the groove.
The low frequency resonance is dependent on
compliance and tone arm inertia (not tone arm
mass, as is commonly claimed.)
Wrong, it is dependent upon effective mass, not inertia.
You changed the rules of the gain by saying
"effective mass" as opposed to
what I said, which is "mass".
No, I did not. Might I suggest you consult a physical
mechanics text for a definition and exposition of
the subject?
Do a dimensional analysis of the formula for resonance
using inertia instead of mass and you come up with a
nonsensical result. Do it with mass, and your result is
in units of reciprocal time, which is frequency.
Sorry Dick, but if you get your math right, it all works. *
It's all about knowing which of the six dimensions you
are doing your math in.
Whether its one of the 3 linear dimensions (x,y,z) or
one of the 3 rotational dimensions, the results for a
calculation of resonance always come out in Hz. *
Note that a simple suspended body can be oscillating
at six different frequencies at the same time because
the six dimensions are orthogonal.
My goodness gracious, Arny, do you even know what
you are talking about? Do you even know what the
term "dimensional analysis" means? Do you know
the difference between "dimensions" and "degrees of
freedom? Do you understand what kinematical
analysis is? Your comments here would strongly
suggest you do not.
It's not clear that you are any longer interested in
a technical discussion based on the technical
merits or shortcomings of the topic, so,
For everyone else that might be interested:
dimensional analysis is a mathematical technique
invented by Fourier to, among other things, determine
the plausibility of equations involving physical quantities
or dimensions. It involves checking the consistency of
each term in an equation to make sure that they are
consistent. Let's use the formula for mechanical
resonant frequency as an illustration
As someone pointed out, the resonance of a simple
mechanical harmonic system is:
F = 1 / ( 2 pi sqrt(M/k) )
where F is frequency, m is mass, and k is stiffness.
If you want, instead, to use compliance (the
reciprocal of stiffness), it becomes:
F = 1 / ( 2 pi sqrt(MC) )
Now, the units or "dimensions" of each term are of
great significance here. I'm not talking dimensions
as in x, y, z, or positions in space, as Arny
misconstrued.
Rather, in what dimensions are each of the terms
of the equations expressed. Mass M, for example,
is expressed in dimensions of kilograms. Compliance
C could be expressed in dimensions of meters per
Newton. Frequency F is in units of reciprocal time:
F [s-1] = 1 / ( 2 pi sqrt( M [kg] C [m/N]) )
A dimensional analysis of the equation would involve
making sure that the units required on the left side of
the are a direct result of the units on the right side.
Let's look and see if that's the case. First, let's break
down the the units of compliance in to their fundamental
parts.
Compliance, as said, is in units of meters per Newton.
A Newton is a unit of force. From:
F = ma
or mass times acceleration, and acceleration is in
units meters per second squared. Substituting,
we have:
C = m / N
and since a Newton is a kg m/s^2, then
C = m / (kg m/s^2)
Let's start checking the consistency and see
what we end up with.
Starting with inside the radical:
M * C
put the units in, M = kg and C = m/(kg m/s^2),
and we get
kg * m / kg m/s^2
Eliminate like terms in the numerator and
denominator: since kg/kg = 1 and m/m = 1,
then we are left with:
1/1/s^2
which simplifies to
s^2
So inside the radical we have dimensions of
seconds squared.
The square root of that will be in seconds. So
our original formula is now reduces to:
F = 1 / 2 pi s
and since pi is a dimensionless quantity, the
result is that the right hand side of the equation
has dimensions of reciprocal seconds, and
frequency itself is in terms of reciprocal time.
Thus dimensional analysis shows that our formula
based on mass and mechanical compliance is
consistent and plausible.
Now, do the same, instead substitute inertia,
with dimensions of kg m^2, for mass, with
dimensions of kg, and see what you get.
Skipping the detailed derivation, the right hand
side of the equation ends up in units of meters
per second, which is velocity and very DEFINITELY
not frequency. Dimensional analysis shows that
an equation for frequency using inertia is
dimensionally inconsistent and thus not plausible.
Well, aside from a common simplifying assumption,
tone arms exist in six dimensions, like the rest of
the universe.
You have clear confused the fact that an unrestrained
body in 3-dimensional space exhibits 6 DEGREES
OF FREEDOM of motion, 3 translational, 3 rotational.
That's totally different than claiming tone arms exist in
6 dimensions.
And that's an UNRESTRAINED body. Tone arms aren't
unrestrained: they have bearings, pivots and such that
contrain several of those degrees of freedom. This is
where kinematics comes in to play. The simple fact is
the motion of a tone arm is restricted to only two
degrees of freedom of motion, hopefully, two rotationally
about two mutually perpendicular axes.
He neglected the
final step in the process which is that all of that moment
of inertia manifests itself as simple mass at any point
from the pivot equal to the moment of inertia divided
by the distance to the pivot point squared.
That's the simplifying assumption I was talking about.
That's NOT a simplifying assumption: it is a physical
fact.
Since the displacements are small, its a pretty good
assumption. It's also a difference between mass and
effective mass.
It has absolutely nothing whatsoever to do with the
size of the displacement. The effective mass is simply
the total moment of inertia about the axis, which
in dimensions of kg m^2, divided by the distance from
the point of interest to the axis of rotation squared,
in units of m^2. The result, again subject to dimensional
analysis:
M [kg] = R [kg m^2] / d [m] ^2
m [kg] = kg
is completely consistent.
Recal that to take a rotating mass to begin with and
turn it into moment of inertia is:
R [kg m^2] = M [kg] * d [m] ^2
that is, a point mass M at a distance D from the axis
of rotation d has a moment of inertia of M d^2. The
equation is perfectly symmetrical: it works perfectly
fine in both directions.
Arny, sorry, but I have to absolutely agree with the
good Mr. Lavo on one point: in what appears now to
be your attempt to score debating points, you have
made ridiculous, physically nonsensical assertions.
You have confused "dimensions" with "degrees of
freedom," you have completely ignored the kinematical
properties of physical bodies in general and tonearms
in particular, and you have made a mess of trying to
work within the well-defined and widely known and used
methodology of dimensional analysis.
If you want to respond to my post, might I suggest
you confine yourself to the technical points and
their technical merits or shortcomings.
I am not saying you yourself are a fool, this post,
in the context of a discussion of fundamental
physical mechanics, is foolish. Please do recognize
the difference, and provide us with renewed evidence
to the contrary.
For anyone who's actually interested in a more
detailed analysis of the fundamental mechanics
of tonearms, you are welcome to peruse an article
I wrote some years ago, which can be found at
http://www.cartchunk.org/audiotopics...mMechanics.pdf
There are a few other articles on other topics at:
http://www.cartchunk.org/audiotopics
For anyone else interested in understanding what's
really going on, do searches for "degrees of freedom,"
"kinimatics", "dimensional analysis." All of them lead
to reasonable definitions and expositions of the subjects,
though some may seem, due to the math involved, a
little obscure.