Musical saws?

[Salmon Egg]

I may have asked this question before.

If you strike a saw with a wood mallet or the like, you get a dull thud.
If you bend the saw into an S curve, you get ringing. The same is true
if the vibration is excited by bowing. Is there a simple intuitive or
heuristic way to understand how bending the saw affects the Q of the
ringing?

I have seen the analysis by Scott and Woodhouse. I probaqbly could go
through it in detail if I tried hard enough. For something that
dramatic, however, I would hope that the main idea could be understood
without having to understand the mathematics in detail.

Bill

[Peter Larsen]

Salmon Egg wrote:

|| I may have asked this question before.
||
|| If you strike a saw with a wood mallet or the like, you get a dull
|| thud. If you bend the saw into an S curve, you get ringing. The same
|| is true if the vibration is excited by bowing. Is there a simple
|| intuitive or heuristic way to understand how bending the saw affects
|| the Q of the ringing?

Torsion resonance, also what makes a good guitarist sound good, requires
that the strings are operated with the skin on the "corner" of the fingertip
instead of with the nails. Very few guitarists do this .... torsion
resonance is also a vital component in the sound of other bowed instruments.

|| Bill

Kind regards

Peter Larsen

[Angelo Campanella]

Post by Salmon Egg
If you strike a saw with a wood mallet or the like, you get a dull thud.
If you bend the saw into an S curve, you get ringing. The same is true
if the vibration is excited by bowing. Is there a simple intuitive or
heuristic way to understand how bending the saw affects the Q of the
ringing?

The damping of a vibrating body depends on its "End (edge) Conditions".
Naturaly, when you touch or damp a loop, it will interfer with the
vibration and damp it. If you touch a node, little or no damping happens
because there is no motion there.

(Loops and nodes! Semantics alweays got in the way for me... Example:
For a sine wave motion, the node is the place where it stands still,
while the loop is where all the motion occurs.)

When striking an edge of a saw, the impact direction suggests that
mainly or only bending vibration where the tips should vibrate in along
the plane in the edge direction. The saw blade is extremely stiff in the
direction, so the frequency of tht vibration will be very high, perhaps
ulrtasonic. A mallet is somewhat soft, so the impact may not contain
much ultrasonic energy to impart. The "thud" is what is heard when some
ultrasound is conained that you can't hear as a tone. Try a small steel
hammer which will induce high frequency sound energy... you might be
abble to hear some sound at those high frequencies.

When the side of the saw is strucck or bowed, sudible sound will occur.
The frequency depends on the thickness of the saw sheet metal, its shape
(flat or bent) and the stiffness of the edge supports (your hand). By
itself, the saw blade sheet, being flimsy, will define an extreely low
vibration frequency.. essentally useless for music.

If you bend the saw sheet into a cylinder, the cylindrical sheet
shape will have a high stiffness and will now contain vibrational modes
("eigenmode") in the audio frequency range. You control the stiffness of
that bent sheet by its radius of curvature. There emerges a mode shape
where the nodes are handy to hold, and you may produce nice tones by
strumming the edge while you hold it only by the eigenmode nodes.

Post by Salmon Egg
I have seen the analysis by Scott and Woodhouse. I probaqbly could go
through it in detail if I tried hard enough. For something that
dramatic, however, I would hope that the main idea could be understood
without having to understand the mathematics in detail.

Mathemeticians are often historians that describe, in their own way,
the physical phenomena that are previously discovered. The discoveries
occur by accident, or by intense trial and error (e.g. Edison). Mother
nature tolerates a lot of tinkering.


Ange

[Peter Larsen]

Angelo Campanella wrote:

|| If you bend the saw sheet into a cylinder, the cylindrical sheet
|| shape will have a high stiffness and will now contain vibrational
|| modes ("eigenmode") in the audio frequency range. You control the
|| stiffness of that bent sheet by its radius of curvature. There
|| emerges a mode shape where the nodes are handy to hold, and you may
|| produce nice tones by strumming the edge while you hold it only by
|| the eigenmode nodes.

It is a bend AND a twist.

|| Ange

Kind regards

Peter Larsen

[Salmon Egg]

In article

Post by Angelo Campanella
If you bend the saw sheet into a cylinder, the cylindrical sheet
shape will have a high stiffness and will now contain vibrational modes
("eigenmode") in the audio frequency range. You control the stiffness of
that bent sheet by its radius of curvature. There emerges a mode shape
where the nodes are handy to hold, and you may produce nice tones by
strumming the edge while you hold it only by the eigenmode nodes.

Of what I have seen so far, this snippet seems to hold the greatest
promise for intuitive understanding. Apparently, the character of the
blade changes with the bend in some nonlinear way. What I need to
understand further is how the bending affects the low amplitude (linear)
vibration properties of the saw. I realize that such an approach may not
work,

A simpler task may be to understand the speed of sound in a slender
beam. For a thin string, the speed of sound is independent of frequency.
For a slender transversely vibrating beam, the speed of sound seems to
be inversely proportional to the square of the frequency. I have solved
the PDE for such a bean using linear mathematical physics. The resonant
frequency seems to be inversely proportional to the square of the beam's
length. That seems to be matched approximately by the vibration of a
clamped steel ruler. I do not yet have an intuitive understanding of
that phenomenon.

Bill

[Angelo Campanella]

Post by Salmon Egg
In article

Post by Angelo Campanella
If you bend the saw sheet into a cylinder, the cylindrical sheet
shape will have a high stiffness and will now contain vibrational modes
("eigenmode") in the audio frequency range. You control the stiffness of
that bent sheet by its radius of curvature. There emerges a mode shape
where the nodes are handy to hold, and you may produce nice tones by
strumming the edge while you hold it only by the eigenmode nodes.

Of what I have seen so far, this snippet seems to hold the greatest
promise for intuitive understanding. Apparently, the character of the
blade changes with the bend in some nonlinear way. What I need to
understand further is how the bending affects the low amplitude (linear)
vibration properties of the saw. I realize that such an approach may not
work,

As Peter Larson points out, it's both a bend AND a twist. The teist
defines new nodal lines, likely defining a closed polygon on he saw
blade , perhaps handisized. Furthermore the resonat frequncy of that
polygon in those edge conditions also depends n the stress in the sheet.
I have oserved that the resont frequency of a trapped sheet (both stiff
and containing impressed stresses) increases with the applied stress.
The twisting torque also imparts a high stress into the sheet steel, so
one nicely controls the resonance frequency of the polygon with the
degree of torque applied to the blade end.

Post by Salmon Egg
A simpler task may be to understand the speed of sound in a slender
beam. For a thin string, the speed of sound is independent of frequency.
For a slender transversely vibrating beam, the speed of sound seems to
be inversely proportional to the square of the frequency. I have solved
the PDE for such a bean using linear mathematical physics. The resonant
frequency seems to be inversely proportional to the square of the beam's
length. That seems to be matched approximately by the vibration of a
clamped steel ruler. I do not yet have an intuitive understanding of
that phenomenon.

I believe that the differential equation used to describe vibration of
a bar is 4th order, and the right hand side would not be zero when a
stress is impressed.


Ange

[Salmon Egg]

In article

Post by Angelo Campanella
I believe that the differential equation used to describe vibration of
a bar is 4th order, and the right hand side would not be zero when a
stress is impressed.

The PDE for the thin transversely vibrating bean indeed is a fourth
order (in position usually designated by x) LINEAR equation. It leads to
bending eigenfunctions using sin, sinh, cos, and cosh. While I
understand the mathematical basis dispersive wave wave speed, I do not
have an intuitive grasp of why that should be. Being a simpler system
than the bent saw, I thought that understanding the dispersion might be
a stepping stone to understanding the saw. It may be that my
comprehension will just not stretch enough.

Bill