John,
Sorry for the delay in responding to your post. Certain matters
have
preoccupied me heavily and left little time for much else, of late. I
must say that Cremer's book "Structure-borne Sound" sounds enticingly
interesting although as you say the price is prohibitive. I had a
similar
experience when I considered acquiring "Shock and Vibration Handbook" by
Harris. It was only $150.00. I decided to pass on it. I've not
contacted DenHartog although I hope to shortly.
With regard again to the subject of string-driven motion at the
bridge, and its degree of linearity, if any, it may, perhaps, be
easier
to comprehend the controversy we have seen here and detect in academia
when the history of the study of the whole subject of mechanical
response
of systems under forcing is taken into account. The classical view
has
taken the approach that the response of a system will be linear and this
view has had a predominance in the West, although, always, "troublesome
questions" have been, in fact, noted and commented upon by various
analysts. In the last fifty years, however, the field of mechanics
and
vibration has developed a new corpus of material in this regard and this
is the material of non-linear mechanics and non-linear vibration. It
is
easy to understand the conflicted and non-critical approach of some,
both
in and out of academia, and, certainly, in the world of piano
technology, as the classical view has assumed a certain degree of
linearity of system reponse in situations similar to those we have
analyzed in the last two months or so, and taken this assumption as a
point of departure for subsequent analysis. Many classically oriented
authors, however, understand the difficulty in the analysis of motion in
problems such as that at the string bridge interface, take note of it,
suggest the suitability of energy methods at this point and pass on.
For
example in "Analytical Mechanics for Engineers" by Seely and Ensign can
be
found in reference to a point similar to that under discussion the
phrase,
"where the law of variation is not known". Hartog, if memory serves,
takes the approach that the calculation at such a point is "too
difficult
to be practicable" to paraphrase him.
However, in Russia the subject of non-linear mechanics and
non-linear vibration has a much longer and more developed history,
beginning in the 1890's. Russia has, indeed, at least until the
nineteen
- fifties or sixties been substantially ahead of the West in this
regard.
Beginning with the memoir of the Russian, M. A. Liapounoff in 1892
entitled "The General Problem of the Stability of Motion" a substantial
body of work ensued on the problem of non-linear phenomena. Important
contributions were made also by Poincare but the major body of the work
was carried out in Russia until the middle of the century when the
subject
was taken up in the West. However, this subject is still substantially
confused, in the minds of many,
by the previous ubiquity of the classical approach with its assumption
of
linearity of response. Some interesting titles on the subject are: N.
Minorsky, "Introduction to Non-Linear Mechanics"; A. Andronow and S.
Chaikin, "Theory of Oscillations"; N. Kryloff and N. Bogoliuboff,
"Introduction to Non-Linear Mechanics"; and J. J. Stoker's "Nonlinear
Vibrations" . Most of these were published around 1950.
It is, perhaps, easy to understand that the entrenched view of
many
in the world of piano technology, resting as they think they are
confidently in the bosom of authority and being caressed and sanctified
by
the murmurs of conventional but outdated academic opinion, renders its
owners unfortunately incapable of taking note of the many experimental
anomalies arising from their point of view which their complete
indifference to the numerous models and experiments suggested both by
you
and myself demonstrate. Their familiarity with pianos puts them in a
position to make pertinent observations which few professors have the
capability for and renders less excusable their unwillingess to do so.
Those in the academic world have, perhaps, some justification for their
misconceptions as they are ignorant of the numerous observations made
readily available to piano technicans and rebuilders by virtue of
merely
working on and servicing the instrument. However, the view which
chronically presumes linearity is indeed passing in the academic
world,
as it should, and, apparently then and only then will it be safe for new
"judgements" to be formed that conform more accurately to the facts at
hand by many outside the academic world and, perhaps, piano technicians
in
particular. There are many interesting facts to be developed for those
at the moment unawares of them by searching on the terms 'non-linear
mechanics and non-linear vibration" although, once again, I am sure we
will be told that this, in fact, has nothing to do with pianos, or
represents just some stodgy, abstract, theory.
Quoting from "The Response of Physical Systems" by John Trimmer, a
book written itself about 1950 when the advocacy of non-linear analysis
was just developing momentum in the West: p. 226 art. 11.2:
"Nonlinear
phenomena. Strictly speaking, all physical phenomena are nonlinear. It
is just that the approximation of linearity is more or less accurately
valid from one situation to another. In graphical terms, any curve is
approximated at a given point by the tangent at that point. Depending
on
the curve, however, the tangent approximation will be more or less
accurate over a larger or smaller range. This mode of thinking is
useful
in proceeding from linearity toward nonlinearity. The phenomenon
becomes
progressively less linear, or more nonlinear, as the range of variation
is
increased. In contrast to this progressive nonlinearity, however there
are many instances of essential nonlinearity, in which the phenomenon
must
be understood entirely in nonlinear terms, since there is no meaningful
linear approximation for the effect in question." This is precisely
what
I advocate as an accurate expression of the complexities of the
bridge/string interaction, and indeed, possibly, the entire soundboard
function.
Quoting again from Trimmer p. 227, to show the effect of loading:
" Linear fluid damping is frequently called viscous damping because the
viscosity of the fluid is the property which keeps the flow laminar. As
the relative velocity is increased, (the rate of loading, rh), however,
the laminar nature of the flow is lost and the friction force becomes
proportional to a higher power, roughly the square, of the velocity.
The
friction due to such turbulent flow is commonly called square-law
damping. Fluid friction thus provides an example of progressive
nonlinearity, as it is characterized by a GRADUAL change from the linear
to a noticeably nonlinear regime. Dry friction (often called coulomb
riction) is, however, essentially nonlinear, since it gives a force
which
is a DISCONTINOUS function of velocity."
Another quote from Trimmer, p. 229: art 11.3 "Oscillators. The
technologically important group of devices called oscillators
constitutes
one of the most significant fields for study of nonlinearity. This is
because all oscillators depend for thier operation on some nonlinear
action. ...... Feedback is indeed important in oscillators. One may
in
fact say that there are two principal kinds of oscillators: feedback
oscillators and relaxation oscillators:. the distinction between the
two
is similar to the distinction between progressive and essential
nonlinearity, in that relaxation oscillators operate by virtue of some
discontinuous, or quasi-discontinuous, effect. The bowing of a violin
string is an example of relaxation oscillations; here coulomb friction
is
the discontinuous effect."
Quoting again from Trimmer as to the difference between linear and
non-linear systems, p. 231: "As the above example shows, the fact that
e(a factor in the complicated expression for the output of the electric
circuit used as an example, rh) is small and that the equation might be
called quasi-linear does not mean the effect of e is small. The
limitation of amplitude to the value given in Eq. 11.6 is a phenomenon
which has no parallel whatever in linear systems. Thus, although
equations of the form 11.7 do indeed show different properties as e
becomes larger, the properties of such equations even for small e-values
inclues features which are ESSENTIALLY (caps mine, rh) different from
those of linear systems."
From Trimmer, p. 231 "....the final section of this book should be
devoted to a problem in particle dynamics. The methods and the point of
view of this book are based most directly on this branch of physics, as
expanded and developed first by Lagrange, later by E. J. Routh and Lord
Rayleigh, and more recently by Henri Poincare and the modern Russian
school. It is true that the growth of this science has been greatly
aided by many men whose primary interests were in other fields,
particularly in communications and other aspects of electrical
engineering. .... The facts remain, however, that the basic advances
have been made by men whose interests were nourished on the problems of
classical mechanics, and that the prototype problem is the dynamical
problem, particularly the dynamical problem of the particle."
With regard to the utility of the Laplace Transform mentioned
earlier
by one contributor to this discussion, this is proof positive of a
linear
approach, that is, one that is essentially incorrect. From Trimmer, p.
241; "Study of linear systems is almost a closed book; study of
nonlinear
systems is a book which is just being opened. According to this logic,
the necessary introductory point of view is that which, like the
classical
approach to differential equations, knits together the whole field of
linear and nonlinear systems; and any method RESTRICTED(caps mine, rh)
as
is the Laplace transformation to LINEAR(caps mine, rh) equations is
automatically relegated to a SECONDARY, (caps mine, rh), incidental
role."
It is readily evident to me that the entire question of string/
bridge/soundboard interaction is indeed a non-linear mechanical and
vibratory system requiring a different set of approaches than those
advocated by people believing in the pressurists model which is linear,
obsolete and flawed, and, indeed, these observations may also well apply
in the case of soundboard behavior itself, which, once charged with
strain energy radiates momentum away as sound and which needs further
investigation. It is also readily evident to me that the concepts of
physics, logic and mechanical relationships that some have as they say,
"grown up with" and are using in this discussion, are, in reality,
incapable of the subtlety of analysis necessary for a fuller exposition
of
these matters are they partake, indeed, of the same erroneous,
conceptual
flaw once so widely disseminated, namely, persistent linearity of system
response under forcing. Hence the mantra you refer to: "the string
moves the bridge and the bridge moves the soundboard."
Regards, Robin Hufford
John Delacour wrote:
> At 6:38 PM -0800 2/4/02, Robin Hufford wrote:
>
> > I am surprised to see DenHartog is still alive. He must be
> >quite old. In point of fact, he was Professor of Mechanical
> >Engineering and not a Professor of Physics. I will attempt to
> >contact him.
>
> At 8:36 PM -0800 1/13/02, Robin Hufford wrote:
> > A far better book, but one substantially more mathematical is
> >J.P.Den Hartog's MECHANICAL VIBRATIONS. This lucid book has a much
> >better treatment of the entire subject of vibrations, free and
> >forced and both damped and damped. It also includes analysis of
> >another, less encountered type known as self-excited vibrations. Of
> >particular relevance to the confused articulations of some on this
> >list regarding transverse and longititudinal waves and their
> >capabilities is the chapter entitled "Many Degrees of Freedom", that
> >is Chapter IV, in which will be found on p. 135 art 4.4:
>
> Robin,
>
> I haven't yet got Den Hartog's book, since Rayleigh and Morse are
> plenty to be getting on with. When it comes to dispersion in bending
> waves in bars and plates, both these authors deal with the question,
> though Morse seems to rely very much on Rayleigh. Rayleigh
> recommends Love's Mathematical Treatise on Elasticity (another Dover
> classic) for further reading and I have ordered this. One book which
> would certainly go into more detail and extend the treatment to
> orthotropic plates etc. would be Lothar Cremer's "Structure-Borne
> Sound", but this book is $175! A borrowing job.
>
> There seems to be a problem with presenting a simplified explanation
> of dispersion in physical terms, though the mathematical principles
> are not in dispute. Dan Russell's site gives a simple demonstration
> of dispersion in a bar (which is the same as in a plate, except that
> the thickness of the plate needs to be factored in) but to actually
> quantify the wave speeds in a piano soundboard at different
> frequencies would probably be impossible at present. I don't really
> see how it would benefit us to know in any case. The principle is
> clear and well documented for over 100 years and is much used in
> practical work involving ultrasound - more so probably than in the
> consideration of audio frequencies.
>
> At 10:50 PM +0100 2/3/02, Richard Brekne wrote:
> >This would seem to present a situation where essentially the higher
> >the string partial, the quicker the corresponding soundboard wave
> >runs through the SB system... which my unschooled mind wants to
> >jump at wondering why I dont hear the high frequencies before I hear
> >the fundementals.... to put a point on it.
>
> Benade does not deal extensively with dispersion, but writes (p.348)
> "...as we go from one part of the scale to the other, the
> predominantly excited components have different frequencies and
> therefore different 'spreading times' across the soundboard." Since
> the waves are travelling at several kilometres per second, we're not
> going to notice the high notes arriving first, though they will decay
> sooner for this reason and others.
>
> Dispersion is also observable in water waves and a visual
> demonstration could probably be set up, though the nature of the
> waves is completely different from flexural waves.
>
> At 1:38 AM -0800 2/3/02, Robin Hufford wrote:
> >This incidence device...clearly shows substantial motion of the
> >bridge when the string has been struck and at the same time shows
> >plainly that deflecting the string mechanically by applying pressure
> >and causing it to be displaced statically, does not have an effect
> >of any similarity to that when the string is struck and vibrates
> >harmonically.
>
> Absolutely. And this relationship is dealt with at great length in
> all books dealing with vibrations. Rayleigh writes "That the
> excursion should be at its maximum in one direction while the
> generating force is at its maximum in the opposite direction ... is
> sometimes considered a paradox..." The only way to resolve the
> paradox is to read and study how such a situation can arise, and
> simple examples exist as an introduction to this study.
>
> The relationship between the phase and amplitude of the vibrations at
> the string termination and those at a point on the soundboard just
> below has no similarity to the static "equivalent", and that is basic
> to the whole consideration of vibrations. When in addition we factor
> in all the other vibrations both natural and forced to which the
> point is subject, it is seen to be absurd to consider that the
> excursions of the soundboard, even at the point in question, in any
> way follow the excursions of the string point for point and angle for
> angle moment by moment.
>
> The frequencies of the string are, it goes without saying,
> transmitted to the soundboard and radiated into the air without any
> alteration in frequency, but the mechanisms by which this happens are
> what we have been trying to discuss. My rubber bridge experiment was
> designed to present a practically visual demonstration of the
> principles without straying too far away from the piano into such
> simple (but relevant and useful) pendulum demonstrations as Benade
> begins with. There has been no response to this demonstration and
> all we hear is the same tired mantra "The string moves the bridge and
> the bridge moves the soundboard", as though that empty statement were
> supposed to provide all the enlightenment nedded for an understanding
> of the acoustics of the piano.
>
> JD
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