U.S. patent application number 11/198547 was filed with the patent office on 2006-02-09 for tensegrity musical structures.
Invention is credited to Ronald J. Barnett, Gregory W. Cherry.
Application Number | 20060027071 11/198547 |
Document ID | / |
Family ID | 35756111 |
Filed Date | 2006-02-09 |
United States Patent
Application |
20060027071 |
Kind Code |
A1 |
Barnett; Ronald J. ; et
al. |
February 9, 2006 |
Tensegrity musical structures
Abstract
The invention is a single and/or musical instrument based on
Tensegrity structures. It shows how a simple retying of a
traditional Tensegrity, comprising a plurality of single shaped
and/or multi-shaped struts, results in the creation of a single
and/or multi-toned musical device, in which the compression struts
of traditional Tensegrities become musical instruments. The simple
retying involves moving the tie point of the struts from their
usual place, at the strut ends, into one of the nodal minima's for
the desired strut resonance mode. Another way of viewing this
transformation is to imagine lengthening the strut beyond the ends
where the strut is tied until the tie points are at resonant nodal
minima's. What is left after the transformation is that all of the
structural properties of a Tensegrity are maintained, while any of
the struts tat were retied in the transformation become musical
chimes that sound oust a musical note, or combination of notes when
struck with a playing hammer.
Inventors: |
Barnett; Ronald J.; (Santa
Rosa, CA) ; Cherry; Gregory W.; (Sebastopol,
CA) |
Correspondence
Address: |
RON BARNETT
813 CHERRY STREET
SANTA ROSA
CA
95404
US
|
Family ID: |
35756111 |
Appl. No.: |
11/198547 |
Filed: |
August 4, 2005 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
60600151 |
Aug 6, 2004 |
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Current U.S.
Class: |
84/402 |
Current CPC
Class: |
G10D 13/08 20130101;
G10K 1/07 20130101 |
Class at
Publication: |
084/402 |
International
Class: |
G10D 13/08 20060101
G10D013/08 |
Claims
1) method of converting any tensegrity structure into a musical
instrument where one our more of the struts become the resonating
elements by tying the struts together at any one or more internal
strut nodal minimum of the desired resonate mode for that
strut.
2) The method of claim 1, where the Tensegrity being converted into
a musical instrument is a "twisted prism". Where the number of
struts n can be any integer three or greater.
3) Multi-mode resonating chimes created from one or more of a
Tensegrities' struts by choosing the strut to be any shape or
combination of shapes, and or material or combinations of materials
to create desired multi-modal resonance in Tensegrity struts.
4) Chime described in claim 3. where the strut is a solid rod or
hollow tube with one or more distinct radial symmetries and or
asymmetries.
5) Chime described in claim 3. where the strut is a solid rod and
or a hollow tube with one or more distinct longitudinal symmetries
and or asymmetries. Longitudinal symmetries have variations on long
the length of the strut, and can include changes in shape,
materials, or properties of the material such as thickness and or
density.
6) Chime described in claims 3, 4, & 5, where chime is of
triangular, and or different cross-sections including symmetric and
or asymmetric rotational modulations with and without symmetric and
asymmetric radial variations thereof. For example, one such
variation with 5-fold rotational symmetry would form a five-pointed
star cross-section whose points are of different lengths and base
widths.
7) Chime described in claims 3, 4, & 5, is of rectangular, and
or different cross-sections including symmetric and or asymmetric
rotational modulations with and without symmetric and asymmetric
radial variations thereof.
8) Chime described in claims 3, 4, & 5, is of an oval, and or
different cross-sections including symmetric and or asymmetric
rotational modulations with and without symmetric and asymmetric
radial variations thereof.
9) Chime described in claim 3, 4, & 5, where the asymmetry is
designed to be nearly the same so as to create a rich sounding
phasing of two or more tones created making fullness in the sound
that is often desired of musical instruments by musicians.
10) Chime described in claim 3, 4, & 5, where the asymmetry in
cross-section is set to provide a particular frequency offset of
the one or more tones created, such as a 2nd, 3rd, 4th, 5th, 6th,
7th, a full octave, or any other frequency offset desired.
11) Any single or multi-modal chime described in claim 3. made of
metal, wood, plastic, glass, ceramic, and/or any other materials or
combinations thereof.
12) Any single or multi-modal chime described in claim 3. of a
rectangular, oval triangular, and any other shape, size, or
cross-section where each variation, including longitudinal
variations, can be realized singularly, or in any combination
thereof.
13) A method where desired cross coupling of the resonators in a
multi-resonator Tensegrity is accomplished by exploiting the
Tensegrities struts inherent ability to share an applied stress
with the struts adjacent to it.
Description
FIELD OF THE INVENTION
[0001] The present invention pertains generally to the creation of
musical Instruments using Tensegrity structures, and more
particularly, to a technique which enables most known Tensegrity
structures to be constructed so that each of the compression struts
become high "Q*" resonators which are the key components of the new
musical instruments.
*Q is the ratio between the stored energy and the dissipated energy
over a cycle.
BACKGROUND OF THE INVENTION
[0002] Man has been making music and creating musical instruments
since the beginning of recorded history. The present invention uses
a novel approach in the building of a recently discovered
architecture called Tensegrity to create new percussion, string, or
wind instruments. The present invention specifically illustrates
how to build a chime set, which can be hammer driven like a
xylophone or wind driven as a wind chime.
[0003] It is the express purpose of this patent to add to the
growing body of functional uses for Tensegrity. Heretofore
Tensegrity has been primarily used for sculptures and art. Many art
examples can be found in the literature. Buckminster Fuller and
others, such as the faculty of the University of Florida, have
pioneered some functional uses of Tensegrity in the building of
both permanent and transportable living units. Still others are
making inroads into the use of Tensegrity for furniture and
lighting fixtures. However, by and large the use of Tensegrity has
been mostly for Art and education.
SUMMARY OF THE INVENTION
[0004] What is Tensegrity, and how can a careful retying of a
standard Tensegrity structure create a musical Instrument? The
introduction to this invention answers that question in three
parts: first by defining Tensegrity and giving a brief description
of its origin; then by examining the key components of a musical
instrument and giving it a concise functional definition; and
finally putting those two parts together to reveal what a
Tensegrity Musical instrument is.
BRIEF DESCRIPTION OF THE DRAWINGS
[0005] FIG. 1. T-Icosahedron, the original eight strut Tensegrity.
The structure is tied at the strut ends and all the resonant modes
of the struts are suppressed and the unit is not a Tensegrity
chime.
[0006] FIG. 2A. Is a three fold "Twisted prism", the simplest
Tensegrity, consisting of the minimum number of elements, three
struts and nine tensional elements. Here only compressive forces
are allowed on the struts. It is a standard Tensegrity tied at the
ends of the struts so all of the overtones including the
fundamental are suppressed and the Tensegrity is not a chime.
[0007] FIG. 2B. Shows detail of the strut end showing continuous
tensional elements. If the strut is only supported in two points,
this arrangement will not allow bending or torsional forces on the
struts
[0008] FIG. 2C. Shows detail of the strut end illustrating three
tensile units not attached to the same point, but attached to a
ring on the strut defined by a cross-sectional cut perpendicular to
the cylindrical axis of the strut. This multi-point tensile element
attachment acts almost like a single point attachment except that
it allows torsion forces in the strut, and is therefore said to be
functionally continuous.
[0009] FIG. 2D. Detail illustrates a discontinuous group of tensile
elements attaching to a strut end, which allows torsion and bending
forces on the strut.
[0010] FIG. 3. Is a Tensegrity cube with eight separate
functionally continuous tensile groups of nine elements each. These
tensile groups are called Tensegrity joints. Here again the ends of
the struts are constrained and all resonant modes of the strut are
dampened.
[0011] FIG. 4A. Resonating tubular strut tied at all four nodal
points of the struts third overtone, such that the third overtone
is allowed to oscillate freely, while all other overtones including
the fundamental are strongly suppressed.
[0012] FIG. 4B. Resonating tubular strut tied only to the outer two
nodal points of the resonators third overtone, behaves as resonator
in FIG. 4A only the overtones are not as well suppressed and the
strut is not as firmly supported mechanically.
[0013] FIG. 5A. Is a three fold "twisted prism" whose struts are
tied together at all four nodal points allowing the third overtone
to oscillate freely, while strongly dampening all other strut
overtones. This Tensegrity chime unit illustrates a musical chime
with redundant tensile elements added to facilitate overtone
dampening and increased structural integrity.
[0014] FIG. 5B. The three fold "twisted prism" shown here produces
the same musical note as FIG. 5B. It is an example of a Tensegrity
chime with a functionally continuous set of tensile elements tied
only at the outer pair of nodal points of the struts third
overtone. This again allows the struts third over tone to oscillate
freely while suppressing all other overtones. This arrangement,
while still suppressing all other strut overtones does so less
vigorously since the inner pair of nodal points are not constrained
as they are in FIG. 5A. It is also a mechanically weaker structure
then the Tensegrity chime in FIG. 5A.
[0015] FIG. 6. Illustrates a five fold "twisted prism" whose struts
are tied together with a continuous set of tensile elements. All of
the struts resonances are dampened and the Tensegrity is not a
chime set since it is standard Tensegrity tied together at the
endpoints of the struts.
[0016] FIG. 7. Shows a top view of a three fold "twisted prism"
Tensegrity tied as a chime unit. The tubes are tied together at the
fundamental modes nodal points allowing the lowest order to
resonate.
[0017] FIG. 8. Side view of a three fold "twisted prism" tied at
the nodal points of the tubes fundamental mode. Only the the lowest
order mode can oscillate freely in this chime unit as all other
overtones are suppressed
[0018] FIG. 9. Top view of a five fold "twisted prism" Tensegrity
chime. Here again, the struts are tied together at the nodal points
of the struts fundamental mode and all other overtones are
suppressed.
[0019] FIG. 10. Illustrates a five fold "twisted prism" Tensegrity
chime assembly. The tubes are tied together at the fundamental
modes nodal points, allowing the lowest order to resonate. All
other overtones are dampened, so that If one of the struts is
struck it rings out with its pure fundamental tone.
[0020] FIG. 11. Floating struts in a T-octahedron dome.
[0021] FIG. 12. Illustrates a three fold "twisted prism" tied at
the nodal points of the struts fundamental mode. The chime assembly
is attached to the base at the midpoints of the bottom tensile
elements. The floating nature of the struts is maintained as the
unit is attached to a stand. Since the struts still float in a
tension web they can still resonate freely when struck with a
mallet. This allows the structure to become a musical instrument
that can be set down on a table and played.
[0022] FIG. 13. a five fold "twisted prism" tied at the nodal
points of the struts fundamental mode. The chime assembly is
attached to the base at the midpoints of the bottom tensile
elements. The floating nature of the struts is maintained as the
unit is attached to a stand. Since the struts still float in a
tension web they can still resonate freely when struck with a
mallet. This allows the structure to become a musical instrument
that can be set down on a table and played.
[0023] FIG. 14. Is a three fold "twisted prism" Tensegrity chime
hung as a wind chime with a hanging spherical striker and wind
sail.
[0024] FIG. 15A Shows an isometric view of a resonating flat bar
suspended at the fundamental points.
[0025] FIG. 15B Side view of a resonating flat bar illustrating the
fundamental free beam mode in oscillation. First overtone
frequency=F1
[0026] FIG. 15C Side view of a resonating flat bar illustrating the
second free beam mode in oscillation. Second overtone
frequency=2.76*F1
[0027] FIG. 15D Side view of a resonating flat bar illustrating the
third free beam mode in oscillation. Third overtone
frequency=5.40*F1
[0028] FIG. 16. Shows a single tube illustrating the fundamental
mode in oscillation.
[0029] FIG. 17. Illustrates a five fold "twisted prism" Tensegrity
chime assembly hung as a wind chime with a hanging spherical
striker and wind sail.
[0030] FIG. 18. Is a three fold "twisted prism", as in FIG. 8,
whose struts are tied together at the nodal points of the tubes
fundamental mode, except that the tying is done with one continuous
tensile element instead of with a set of nine discrete tensile
elements. Here the tensile element is truly continuous, where as
the "twisted prism" shown in FIG. 8 is, as defined in this patent
application, to be functionally continuous. One of the purposes of
including this drawing in the invention is to illustrate another
method of Tensegrity tying. The invention is for all and any method
of Tensegrity tying.
[0031] FIG. 19. Shows a dual mode resonator with independent
horizontal and vertical oscillation in a single rectangular cross
section.
[0032] FIG. 20. Illustrates a Tensegrity cube with continuous
tensional elements.
[0033] FIG. 21. The Needle Tower shown here is permanently
exhibited in the Hirshhorn Galleries outdoor sculpture garden, in
the Smithsonian, in Washington D.C. It is a 60 foot Tensegrity
sculpture. The Needle Tower, like all Tensegrity structures defined
in this invention can be made into a musical instruments by retying
the struts at their nodal points. That is, the Needle Tower
Illustrates a Tensegrity that could, by applying the teaching of
this invention, be converted to a Tensegrity musical instrument.
Note here that the nature of this sculpture, with its progressively
decreasing strut size would naturally become multi-tonal during the
conversion process taught by this invention.
DETAILED DESCRIPTION
[0034] During the course of this detailed description the following
will be developed and presented: the history of Tensegrity; the
ambiguities in the definition of Tensegrity; a specific definition
of a generalized Tensegrity; and a concise functional definition of
a musical instrument. The invention is shown to be any musical
instrument formed from a described specific tying of any
generalized Tensegrity as defined here. Additionally, several
examples of the preferred embodiment will be described and
illustrated.
1) What is Tensegrity?
[0035] a) Tensegrity structures were invented by the well-known
artist/sculptor Kenneth Snelson in 1948 and later named Tensegrity
by the iconic inventor Buckminster Fuller in 1953. Both of these
Tensegrity pioneers took out early Patents: [0036] i) Kenneth
Snelson took out U.S. Pat. No. 3,169,611 "Continuous Tension,
discontinuous compression structure" Feb. 16, 1965 [0037] ii)
Buckminster Fuller took out U.S. Pat. No. 3,063,521 "Tensional
Integrities" Nov. 13, 1962. [0038] b) The word Tensegrity comes
from the combination of tension and integrity; Buckminster Fuller
defines Tensegrity as the balance between discrete compression
members and a continuous tension member, which joins them together.
[0039] c) A traditional Tensegrity structure is composed of struts
in compression connected at their ends with tendons tied in tension
in such away that the struts are held apart. [0040] i) The result
is a whole unified object that consists of an integrated system of
isolated struts held in suspension between discrete or continuous,
tensional, string like elements. [0041] ii) The struts are always
discontinuous and generally rigid. That is, none of the struts
touch each other and usually only support compressive loading.
Contrasting with the continuous tensional elements that only
support tension loading. The struts figuratively float in a sea of
string like tensional elements. [0042] iii) The whole object now
behaves like a cooperative unit rather than a collection of
separate parts that happen to be attached physically together.
Because of the inherent symmetry in construction of the structure
internal stresses are stored within and redistributed between the
wires and struts such that the struts are in compression only and
the stored energy serves to give shape to the structure and hold
the struts forcibly apart. [0043] iv) Additionally the said
cooperative nature of the Tensegrity unit provides load sharing
when one or more of the elements is externally stressed. Under such
an external stress the elements of the structure respond and
interact with each other to help share the load by redistributing
and absorbing the offending stress among themselves. Resulting in
an overall light weight, robust, and flexible structure. [0044] v)
This stress spreading property of Tensegrities is accomplished
through the tensile elements that the struts "float" in. When an
attempt is made to put compressive, bending, shear, or torsion load
on the Tensegrity the "floating" nature of those struts cause the
stressed struts to yield in the web of wires in which they are
contained. This web of wires transmits the bulk of the offending
stress to the adjacent struts as the wires attached to the stressed
struts pull on said adjacent struts. The adjacent struts, in turn,
transmit the stress they receive to their respective adjacent
struts. This process of passing on applied bending, shear,
compressive, and torsion strut stresses to adjacent struts through
the struts attached to tensile elements continues through the
entire Tensegrity in a wave-like manner until an equilibrium is
reached where all elements both struts and wires share the applied
stresses. [0045] vi) In a similar manner, when a tensile element of
a Tensegrity is pushed or pulled sideways the wire is not
stretched, as it would be if it were a part of a traditional
structure such as a spoke in a bicycle wheel. Rather than
stretching, the wire ends pull on the wires and struts they are
attached to. This pull relieves some of the stress applied, by
transferring that stress to those other elements it pulled on. In
turn, these secondarily stressed parts transmit the stress applied
to them to their own adjacent elements. Again, this stress
reduction process continues in a wave-like manner until equilibrium
is reached where all of the elements of the Tensegrity share the
initial offending stress. [0046] vii) This stress sharing property
is primarily, though not entirely, accomplished by the wires, the
tensile elements. The continuous nature of the tensile elements
facilitates this load sharing Tensegrity property. Tensegrities are
generally said to be structures that have a single continuous
tensile element, or more accurately a continuous set of discrete
tensile elements. To see this, take a look at the original
Tensegrity, a Tensegrity-Icosahedron (FIG. 1), where you will
notice that a set of tensile elements physically continuous, forms
a web. The tensional elements are made up of a single string or
many strings tied together. To further illustrate, imagine you were
as small and agile as ant, then the tension elements are physically
continuous if you could walk or climb over the entire length set of
the tensile elements without ever leaving them. On the other hand,
look at FIG. 3 and FIG. 5B which are not strictly Tensegrities as
defined by Kenneth Snelson, the founder of Tensegrity, because the
entire set of tensile elements are not continuous. FIG. 3 contains
eight separate groups ("Tensegrity Joints") of nine functionally
continuous tensional elements which are all required. FIG. 5A has
two separate groups of three continuous tensile elements which are
structurally redundant and not required to hold the struts apart.
Finally FIG. 8 and FIG. 10 show Tensegrities whose tensile elements
are physically discontinuous, yet are functionally continuous. Here
the tensile elements are separated for ease of manufacturing and
act nearly as if they were continuous. The break in continuity has
the effect of potentially introducing torsion and bending strain
into the rigid struts. The struts in a "pure Tensegrity" can only
hold compression strain due to the way they are suspended in two
points in the tensional element web. The larger body of
Tensegrities defined in this patent can be any of the Tensegrities
classes described above. They can have fully continuous,
functionally continuous, or even fully discontinuous tensile
elements. Furthermore the larger body of Tensegrities defined in
this patent can have struts of any material or shape, and are
specifically not limited to the usual elongated shaped struts.
Also, the Tensegrities described in this patent can have struts
with any number of tensile elements attached to them. [0047] viii)
The definition of a Tensegrity as defined by this invention
description is: is a tying of rigid or semi-rigid struts that can
support compression and/or bending and/or torsion, strung together
so said struts "float" separately from one another in a web of
tensional elements. [0048] ix) It is important to note here that
the said continuous element must also be physically attached to the
struts at the wire-strut intersection points (even if the
attachment is frictional) for the Tensegrity to remain together.
That is, if a single string representing the single tensional
element were threaded through all of the end-points in the
Tensegrity Icosahedron (FIG. 1), for example, to make all of the
appropriate interconnections between the struts, but were not
attached at said end-points the structure would self collapse. This
collapse releases the energy stored in the residual tension of the
wires and the residual compression, bending, or torsion stored in
the struts. In a Tensegrity the tensional element is usually though
not necessarily continuous. However in either case, it is required
to be affixed to the points where it contacts the struts. [0049] x)
As mentioned above, the Struts of a Tensegrity are always
discontinuous. However, not all Tensegrities have a physically
continuous tensional element. The tensional elements of some
Tensegrities are continuous while the tensional elements of other
Tensegrities are discrete pieces. In some cases the tensional
elements are discrete, yet functionally act as though they were
continuous (as described in secition "VII" above on pages 9 and
10). For example, if the tensional elements that attach to each end
of each of the three struts in the simplest Tensegrity, the
"Twisted Prism" (FIG. 2A), were attached separately to the radial
perimeter of each strut end (FIG. 2C), the structure would behave
as though the discrete tensional elements were piecewise
continuous. I define that is functionally continuous. If the same
Tensegrity shown in (FIG. 2A) were tied at the strut ends as shown
in (FIG. 2D), then structure is defined as having discontinuous
tensile elements. The Tensegrity Cube (FIG. 20) is made up of one
single continuous element attached firmly at each strut end and the
Tensegrity cube is defined as having continuous web of tensile
elements. In another case a Tensegrity could even have some of its
elements removed so that the structure would in general collapse
under gravity and the struts fall in together. However if the
structure is suspended from cables appropriately then the struts
would spring lightly back apart as gravitational forces make up for
the missing tensile elements when the structures. I call this a
"Semi-Tensegrity" and include it in the general class of
Tensegrities that can be converted to a musical instrument by the
method taught in this invention. That is, even this flaccid gravity
supported Tensegrity will make a Tensegrity Chime. [0050] xi) On
the other hand, well know Tensegrity mathematician, Robert
Burkhardt has shown a Tensegrity cube where each of the cube's
eight corners have their own separate independent continuous nine
element tensile unit. He calls this corner unit a "Tensegrity
Joint". The result, illustrated in (FIG. 3), is a Tensegrity
structure whose strut elements are made of twelve sticks that act
as discrete compression elements forming the cube's edges, while
the vertexes are formed by eight 9-element independent continuous
tensile units. At each of the cube's corners one of these
independent continuous tensile units bind the sticks together in a
manner in which the sticks don't touch each other. It is
interesting to note that it is possible to build a Tensegrity Cube
where the Tensional elements are continuous. These structures are
more common and have been built and displayed by others including
Buckminster Fuller and Robert Burkhardt (see FIG. 20 for
illustration of a Tensegrity Cube with a set of tensile elements
that are continuously connected). For the purposes of this
invention, both of these "Tensegrity Cubes" are considered
Tensegrities that can be restrung as a musical instrument as will
be described in this document. [0051] xii) In a similar manner a
newly created Tensegrity Musical Instrument may have a strut tied
in many (four for example) places in such away that the tensional
element ends up piecewise continuous as in the Tensegrity cube
shown in (FIG. 3). There can be musical reasons that additional
strut tie points are added and these additional tie points may,
while serving their musical purpose of nulling a particular
resonance mode of the strut, end up creating additional tensional
elements that are redundant from a structural point of view. These
redundant tensional elements, like the center two in (FIG. 4A) will
likely go from strut to strut without touching any of the other
tensional elements, as in (FIG. 5B), and thus make the group of
tensional elements for the Tensegrity discontinuous. In (FIG. 5B)
for an example, the simplest Tensegrity (FIG. 5A), with
functionally continuous tensile elements, is converted to a
Tensegrity with discontinuous tensile elements, by the addition of
two triangular groups of tensile elements to the top and bottom of
the structure. The single tubular strut shown in (FIG. 4a) is an
illustration of a resonant tube tied in four separate places
instead of the usual two places to assist in nulling out or
dampening the fundamental mode while allowing the 3.sup.rd overtone
of the of the tubular strut to exist unimpeded. When an elongated
strut of a Tensegrity is constrained, with tensional elements, at
the four nodal minima's of its 3.sup.rd overtone, as illustrated in
FIG. 4A, the fundamental tone and all other overtones are
suppressed. The two middle attachments on the strut, which may be
redundant from structural point of view, serves the function of
increasing the dampening of the fundamental mode and all overtones
except the 3.sup.rd. The purity and sound quality of the Tensegrity
chime is improved over a free hanging chime because of suppression
of non-harmonic overtones of the chime. It is the nature of a
percussive chime to have non harmonic overtones. For example, if
the fundamental tone of a bar of rectangular cross section is F1,
then the second overtone F2=2.76*F1, the third overtone F3=5.40*F1,
and the fourth overtone F3=8.93*F1. The tying of a chime with other
chimes into a Tensegrity improves the quality of its sound by
filtering out the offending overtones. This patent teaches that by
tying of the struts at the natural nodal minimums of the mode or
overtone desired, that the desired mode is enhanced while all other
modes are filtered out. This is one of the reasons the definition
of a Tensegrity has been broadened in this patent to include
discontinuous tensile element tying as well as continuous tying.
Also, a discontinuous tying such as, but not limited to, the Cube
in FIG. 3 may allow easier access to the individual chime strut
elements when playing them by striking them with a mallet. In
general the addition of discontinuous tying allows greater degrees
of freedom in the construction of Tensegrities, and this increased
freedom allows for the creation of optimum musical instruments for
sound quality, playing ability, ease of construction, increased
range of tones at an increased range of size, as well as increased
economy, and structural beauty. [0052] xiii) The functional
definition for Tensegrity, given in section "c part viii" on page
10 and in section "k" on page 19, allows the addition rendundant
tensile elements for purposes such as increased rigidity or mode
suppression on a chime. This functional definition also allows for
removal of a tensile element for the purpose of making a hanging
Tensegrity which is collapsible when it is taken down. The creation
of musical instruments taught in this invention are a retying of a
class of Tensegrities. These Tensegrities are defined specifically
by the definition given in section "c part viii" on page 10 and in
section k on page 19 of this document. The definition defined there
specifically does not require or preclude the tensional elements of
said Tensegrities from being continuous. It also does not require
or preclude, bending, shear, torsional, strains on the struts, as
the less general definition of Tensegrities given by Kenneth
Snelson does. [0053] d) An excellent example of Tensegrity is shown
in Snelson's Needle Tower (FIG. 21), which as mentioned before, is
permanently exhibited in the at the Smithsonian in Washington D.C.
This tower, consisting of progressively smaller aluminum tubes tied
apart with stainless steel cables so it rises 60 feet into the air.
The struts seem to
"float" in a sea of tensioning cables. In addition to showing what
Tensegrity is, some of the benefits of the use of Tensegrity are
illustrated. Here when the tower is loaded, for example in a high
wind, or by attaching a rope to its top and attempting to pull it
down, the whole tower will yield and absorb the incident stress
which is distributed or communicated over the whole structure by
the set of tensional elements, so that buckling associated with a
rigid conventionally built tower is avoided. Said another way;
attach a rope to the top of a conventional tower, of the same
height and mass as the Needle Tower, and pull with a force "F"
until the tower buckles and collapses. Next attach the rope in the
same manner to the needle tower and pull with the same force "F"
and observe that no buckling occurs. Instead the tower yields and
the bulk of the stress applied to the top of the structure is
effectively spread to all of its elements in with out damage to any
of those elements. When the stress is removed the tower simply
resumes its original unstressed position. The distribution of the
stress through throughout the whole Needle Tower is communicated
through the tensional elements. Thereby illustrating how the
tensional elements act in a continuous manner even when they are
composed of discrete elements. This is a key property of Tensegrity
structures. Another key Tensegrity property demonstrated with the
Needle Tower is flexibility. The tower like a weed in the wind
returns to its original position undamaged after the storm has
passed. Even this gigantic sculpture would become a percussion
musical instrument if its strut ends were extended past respective
tie points, such that tie points become the nodal points of the
fundamental mode of each of the struts. This conversion could
literally take place by welding on the appropriate length extension
tubes to the ends of all of the struts that are to become
resonators. [0054] e) Compare and contrast a child's latex balloon
with a 60 foot steel wire and aluminum pipe tower. As you can see,
the variation in appearance and use of Tensegrity structures
precludes giving it a definition based on its components such as is
done in other architectural forms where a board and nail, brick and
mortar, bamboo and reed, straw and mud, or glass and steel
definition can be used to describe traditional structures. Neither
can a Tensegrity structure be defined by its use as a gun,
airplane, boat, car, plow, and sewing machine are. We are left only
with a functional description of what Tensegrity is, and how it
behaves when stressed. Tensegrity structures are uniquely defined
by a functional description of how the composite components relate
to each other, and how as a collective the whole structure responds
to external stress. [0055] f) The father of Tensegrity, Kenneth
Snelson, now defines them as: "Tensegrity describes a closed
structural system composed of a set of three or more elongate
compression struts within a network of tension tendons, the
combined parts mutually supportive in such a way that the struts do
not touch one another, but press outwardly against nodal points in
the tension network to form a firm, triangulated, prestressed,
tension and compression unit" [0056] g) Because Tensegrity
represents a new, unfamiliar, and uniquely defined class of
structures that mimics nature they have been, and to some extent
still are, misunderstood by the general public. Also in the course
of his work Buckminster Fuller. Commingled some of his previous
work with Snelson's original structural idea adding further to the
confusion of what is meant by Tensegrity. This on going
misunderstanding has led, overtime, to a loss of rigor to the word
"Tensegrity", as it began to be applied to music scores, thought
systems, personal relationships, exercise regimes and so on. [0057]
h) It is instructive to look at the original Tensegrity structure
built by Snelson in 1948 and displayed by Buckminster Fuller in
1949. A drawing of a model of this "Tensegrity Icosahedron" is
shown in (FIG. 1). Some of the features and benefits seen here are
the same as we saw in the Needle Tower: 1) the struts are supported
by the tensile elements so they do not touch each other. 2) When a
stress is applied to any of the elements that element is displaced
and the wires holding it in place transmit the bulk of the applied
stress to the other components of the structure. [0058] i) Some
pneumatic examples of everyday Tensegrities are the balloon and the
automobile tire. Here the stretched elastic skin of the balloon and
the expanded rubber tire shell represent the continuous tensional
element, where as, the discrete compression elements are
represented by the air molecules contained inside the balloon and
tire. These pneumatic examples are a far stretch from Snelson's
original definition since the tensional elements are not connected
to said compression elements. This is an example of what has
happened to Snelson's original idea of Tensegrity. Furthermore,
since there are no actual struts in these so called "pneumatic
Tensegrities" they can not be included in the definition of
Tensegrities that can be made into musical instruments as taught by
this invention. [0059] j) As a result of indiscriminate use, the
meaning of the word Tensegrity has become increasingly vague. Since
its inception the meaning of Tensegrity has expanded to cover
everything from bicycle wheels, to anything that uses tension as
one of its main structural components, to the harmonious give and
take in the relationship between a student and teacher, to a
description of a certain type of music, to a description of nature
itself. [0060] k) For clarity in the description of the present
invention. The scope of the word Tensegrity will be narrowed to:
"physical structures consisting of rigid, or semi-rigid, struts of
arbitrary shape and material suspended in a network of tensional
elements, also of arbitrary shape and material, so the struts stand
apart from one another". In other words, the definition of
Tensegrity is: "struts tied together at two or more point such that
the struts are held apart either firmly or flaccidly apart, with or
without the use of gravity". The struts are held in a network of
tensile element such that the struts don't touch each other. This
definition of Tensegrity now more closely resembles the original
definition of Tensegrity based on Snelson's sculptures, and
includes them as a subset. This definition also, includes hanging
Tensegrities where one or more of the Tensional elements of a
Snelson Tensegrity have been removed so that the "semi-Tensegrity"
so formed, falls apart as the struts come together when the
Tensegrity is not hanging. When hanging the forces previously
provided by the missing tensile elements is supplied by gravity. It
notably does not include, pneumatic Tensegrities such as balloons
and automobile tires, as well any kind of strut and tensional
element as long as the struts support the compressive loads built
into Tensegrities, and the tensional elements can take the tensile
forces built into Tensegrities. The definition specifically does
not limit Tensegrity struts to compressive strains. The struts can
have any and all combinations of compressive, bending, tortional,
and/or shear strains when loaded by the Tensegrity assembly. [0061]
l) The definition does not limit in any fashion the shape or
composition of the struts. They could be glass spheres, bamboo hula
hoops, or ceramic coffee cups, or be of abstract geometric 2d or 3d
shapes such as flat triangular, square, pentagon, or completely
arbitrarily shaped plates, or solid cubes, pyramids, tetrahedrons,
rods, tubes, rectangular solids, asteroid shape, or of any other
shape including completely arbitrary ones. Further more the plates
and solids could be modified with arbitrary features such as holes,
serrations, or extrusions, of any kinds on them. Also the struts
can be semi-rigid as well as rigid. This would among other things
allow the struts to deform under the stress from the tensile
elements. The definitional also allows redundant tensional elements
not actually required for the struts to stand apart. [0062] m)
Likewise the tensional elements could be of any material and shape
including but not excluding anything else: rubber sheets, canvas
tarps, Kevlar straps, metal or wood chain, or carbon fiber cable.
[0063] n) The struts and tensional elements can be any shape or
materials. However, the struts are generally metal or wood, rods or
tubes, and the tensile elements are generally rope, cable, or wire.
[0064] o) A Tensegrity structure consists of a set of ridged
struts, suspended in a network of tensional elements, where the
tensional elements tie the struts tightly together, in such a way
that the struts stand forcibly apart. [0065] p) As mentioned
before, looking at physical Tensegrity structures is one of the
best ways to gain insight into what Tensegrity is. Lets examine the
simplest type of Tensegrities I call "twisted prisms" the class
which forms the preferred embodiment of this invention. The
simplest "twisted prism" (FIG. 2) requires three struts, has 3-fold
symmetry, and is built in a way such that its complexity can be
increased by adding any number of struts to the structure connected
in the same manner the original three were. A five strut "twisted
prism" exhibits 5-fold symmetry and is shown (FIG. 6). [0066] q) To
convert "twisted prism" Tensegrities into "Tensegrity Chimes" the
tension element tie points are simply moved in from the strut ends
to two of the of the nodal minima's of one of the struts free beam
resonant modes. The figures (FIG. 7) & (FIG. 8), show a 3-fold
symmetry version of the "twisted prism", while figures (FIG. 9)
& (FIG. 10) show a version with 5-fold symmetry. In the
examples above the fundamental mode is selected. In a similar
manner the 2.sup.nd, 3.sup.rd, 4.sup.th, 5.sup.th and so on,
overtones could have been selected by simply tying the Tensegrity
struts at their respective nodal points. In the case of the
overtones above the fundamental more than a single set of nodes are
possible. Accordingly any two of the nodes could be used to attach
the resonant strut to the Tensegrity. Also as described earlier,
any number of redundant tying to the other nodal minimums can be
employed for the variety of reasons mentioned earlier. [0067] r)
Snelson favored calling his Tensegrities by the more descriptive
name "floating compression" structures, which gives one a mental
image of the structures themselves. For example look again at the
Snelson's needle tower sculpture (FIG. 21) and the twisted prism
examples (FIG. 2) and (FIG. 6); all of these consist of three or
more struts in compression held tightly together in a network of
wires, in such a way that the struts stand forcibly apart from one
another. That is, the compression struts seem to float in a network
of tensional elements, and thus the name "floating compression".
This effect is even more pronounced when the number of elements
becomes large. Look at the Tensegrity sphere (FIG. 11), and noticed
that the multitude of compression struts seem to be literally
floating in a sea of high-tension wires. [0068] s) The Tensegrity
Musical Instruments described in this patent are Tensegrity
structures as defined above in sections k) through m), and are
generally the Snelson "Floating Compression" type Tensegrities.
[0069] t) Now for an idea of how a Tensegrity construction works,
look once more at (FIG. 2) the "twisted prism" the simplest
Tensegrity structure. This structure has three struts and nine
"tensional members" which are sometimes called "tendons". The
tendons are numbered one through nine, are the struts are numbered
one through three in roman numerals. Notice that tensile elements
1, 2, &3, form a triangle which tie the top of the struts
together, and that in the same manner tensile elements 4, 5, &
6, form a triangle which tie the bottom of the struts together. If
the struts are now held vertical a regular right triangular prism
is formed; where the three struts and six tensional elements define
the edges. Now imagine an axis "Z" through the center of the bottom
and top equilateral triangles formed by the tensile elements. Next
imagine that the bottom triangle is held fixed while the top
triangle is "twisted" counter clockwise (viewed from the top) about
the "Z" axis through 150 degrees while remaining parallel to the
bottom triangle. At this point the top of strut I will be nearly
directly over the bottom of strut II, and the top of strut II will
be over the bottom of strut III, and completing the symmetric loop
strut III will be over the bottom of strut I. Now as it works out
the distance between these symmetric top strut bottom strut pairs
is a local minimum. This means if the angle of the top triangle is
rotated with respect to the bottom triangle either clock wise or
counter-clockwise the distance will increase. Therefore as tendons
7, 8, & 9 are tied between these top-strut/bottom-strut pairs
the struts and strings will form a rigid structure. This structure
is called a Tensegrity. In an analogous manner the 5-fold "twisted
prism" shown in (FIG. 6) can be created. In general any n-fold
"twisted prism", where n is any integer three or greater, can be
created. [0070] u) Look again at the Needle Tower (FIG. 21) for a
more elegant example of a Tensegrity structure, which posses a more
complex symmetry. The 60 foot tall "Needle Tower" is displayed in
the sculpture garden just outside the Hirshhorn gallery in
Washington D.C.'s Smithsonian. [0071] v) See (FIG. 3) for a
variation on the typical cube Tensegrity (FIG. 20). Here a cube is
shown that consists of long struts that form the edges of the cube,
and which run between vertices that consist of localized
"Tensegrity Joints". This cube is also a Tensegrity since it, as
mentioned earlier, fits our working definition of Tensegrity, and
since it can be converted to a musical instrument in the same
manner as the "twisted prisms" have been in (FIG. 7), (FIG. 8),
(FIG. 9), and (FIG. 10). In both the Tensegrity cube and the
particular rigging of 3-fold & 5-fold "twisted prism"
Tensegrity Musical Chimes illustrated in (FIG. 7), (FIG. 8), and
(FIG. 9), notice that the tensional elements are not continuous,
but are rather only piecewise continuous. In the case of the
Tensegrity cube (FIG. 3) each "Tensegrity Joint" has a continuous
tensional element located in the eight corners of the cube, which
are discontinuous with each other. Here the cube's tensional
elements are not only physically discontinuous they are
functionally discontinuous in that some of the stress sharing
properties associated with a Tensegrity are lost. However in the
case of the particular Tensegrity Musical Chimes just mentioned,
the tensile elements are physically disconnected yet are
functionally continuous, in that the Tensegrity behaves the same as
if the tensile elements were physically connected.
2) Now that we have discussed what a Tensegrity is, and cleared
away some of the ambiguities surrounding Tensegrity, as well as
established a working definition for the Tensegrity nature of the
musical inventions this patent teaches how to create, we will turn
to establishing a working functional definition of what a musical
instrument is. [0072] a) In principle a musical instrument can be
described, to first order, as a set of resonating elements, a
method to excite the resonators, and a method to couple that
resonate energy into the air so the ear can appreciate it. [0073]
b) A first order, functional description of a musical instrument is
a set of resonating elements, a method to excite the resonators and
a method to couple that resonate energy into the air so the ear can
appreciate it. [0074] c) Put another way: a musical instrument is a
set of "resonators" or "resonating elements" such as stings of a
piano, the hollow tube of a trumpet, or the flat metal bars of a
Xylophone, whose modes can be excited with an "exciter", such as a
hammer, bow, or finger tip, and whose resonate energy can be
extracted into the air by a "coupling transducer", such as the bell
of a horn, the sounding board of a piano, or the bridge and ported
sounding box of a violin, so that it be heard by an ear in the
vicinity of the resonator. [0075] d) Sometimes the "coupling
transducer" is an integral part of the resonator and not a
physically separate part, such as the bell shape on the end of a
horn, the edge of a symbol, or the open end of hollow bamboo stick,
other times, and more commonly, it is a separate set of parts such
as the bridge and ported sound box of a cello, banjo, or bass, or
the magnetic pickup, amplifier, and speaker of an electric guitar.
In the Tensegrity musical instrument invented and described in this
patent the resonators have an integral "coupling transducer" much
like that of the hollow bamboo stick. [0076] e) Examples of the
"resonators", "exciters", and "coupling transducers" of common
musical instruments: [0077] i) In a piano the strings are the
resonating elements, the hammers excite the strings and the
sounding board serves to couple the energy of the resonators to the
air. [0078] ii) In a trumpet the air column contained within the
length of the horn, which forms a tube open on one end and closed
on the other, is the resonating element. The vibrating lips of the
person playing the trumpet serves to excite the resonator, and the
exponential bell shaped opening is the "coupling transducer" which
couples the resonating column of air to the air outside the tube by
matching the impedance of the air within the tube to that of the
surrounding free air. [0079] iii) In an acoustic guitar the strings
form the resonating elements as they did in the piano, the fingers
or picks of the guitar player excite the resonators as they are
plucked, and the bridge, and ported hollow guitar body and body
surface is the "coupling transducer" which couples energy from the
resonating strings to the free air so it can be heard effectively
by the listeners ear. [0080] iv) In a simple Xylophone like
instruments the "coupling transducers" are an integral part of the
resonator and do not require any external parts. The ends of the
resonators themselves are the transducers. This is distinct from
more elaborate Xylophones like instruments, which like the Marimba,
have an output coupling tube, which placed close to the middle of
each resonator to couple its resonate energy in to the free air.
The output coupling tube is the "coupling transducer" [0081] f) The
Tensegrity musical instruments described later in this patent, like
the xylophone use the surface of the resonators themselves for
"coupling transducers". [0082] g) The resonators for Xylophone like
instruments come in many forms. When the resonators are flat metal
bars the instrument is called a Xylophone. However many other
resonators are commonly used in this type of instrument, such as
wood flat bars, hollow metal or wood tubes, bamboo sticks, or
plastic or glass rods. [0083] h) These Xylophone like instruments
consist of a set of resonating chimes whose fundamental mode is
free to be excited when the chime is held at the two nodal points
about 22.4% of its length in from each end. When the resonating
bars/tubes are held at these nodes the ends of the chime are free
to flap in the air (see FIG. 15A). FIG. 15 illustrates the
fundamental mode FIG. 15A, the 2nd overtone FIG. 15B, and the third
overtone FIG. 15C. The free resonances of a beam create the
overtones which come about due to reinforcing reflections off the
beams ends. The fundamental resonance comes about when the round
trip phase difference is 360 degree's, and overtones come at
interger multiples of 360 degree's. The second overtone is at 2*360
degree's and the third overtone is at 3*360 degree's and so on. Due
to the complex nature of sound traveling in a solid the overtones
are non harmonic and none of the nodal minimum points for any of
the overtones are coincident. The non harmonic overtones sound
discordant and the chimes can make a clanking sound when struck.
The "exciter" is a hammer, which the musician strikes the resonator
with. Unlike most of the previously described instruments, the
coupling mechanism for the Xylophone chime is contained within the
chime itself. Here the chime is held or supported at the two
internal modes so as to force the ends of the chime to nodal
maximum's. The majority of the resonator energy that is coupled to
the air is now coupled at the chime's three nodal maximum's located
at the center and two free ends of the chime, where the resonator
has maximum velocity and displacement. 3) What is a Tensegrity
Musical Instrument? [0084] a) Most Tensegrity structures can be
tied so that they become musical instruments. A simple retying of
the struts so that the tensional elements tie points are moved into
one of nodal minima's of a desired free beam resonant mode. The
simplest and most common mode is the fundamental mode that is one
wavelength long and has two nodal minima located about 1/4
wavelength from each of the strut's ends. Here a musical instrument
is created by moving the tie points of the tensional elements in
from the end points to the two nodal minima on the strut. Compare
(FIG. 2) with (FIG. 8) and see the transformation of the three
strut twisted prism into a 3-resonator chime. A pleasing sounding
chime can be made by choosing the three lengths to be different and
so they form a major cord such as C, E, & G, or an intriguing
sound can be made by choosing the lengths so they form a minor
cord. Other variations such as cross coupling between the
resonators can be created so that when one chime in the set, say
one tuned to C, the other elements in the chime, say E, & G,
are sounded. This cross coupling can be controlled physically in
many ways. One of the ways is to designing the chime set so that
the tensile elements that normally only touch the struts where they
attach at the nodal minima's, touch at other points along the strut
and thereby utilize the Tensegrity nature of the chime to tug on
adjacent elements when stressed to couple energy from one resonator
to the next. The procedure just described is a part of what this
invention teaches. [0085] b) A Tensegrity Musical Instrument is a
Tensegrity structure where one or more of the strut members becomes
a Tensegrity chime by tying it to the rest of the Tensegrity,
"floating compression" structure by insisting that the two tie
points be at natural nodal minima's of the free beam resonance mode
that is desired see (FIG. 16). In most cases, and as done in the
preferred embodiment of the Tensegrity Musical Instruments of the
present invention, all of the struts are tied to each other such
that all of the struts in the structure become Tensegrity Musical
Chimes. Also, in most cases the struts are tied at two of the nodal
minima points as in the preferred embodiment of this invention.
However the struts can be tied at tied at each of the free beam
resonance nodal minima. In so doing a strut may be tied in 3, 4, 5
places as long as all of those places are nodal minima's for the
desired resonant mode, such as shown in (FIG. 4A). [0086] c) A
Tensegrity Musical instrument is a Tensegrity structure where the
junctions which separate the discrete struts with discrete
tensional elements or defined to be only at the internal nodal
minima's of the resonate modes chosen for that strut. [0087] d)
Where the struts of a Tensegrity structure are held only at nodal
minima such that the high Q of at least one of the free resonances
are retained, and such that the strut resonant energy can be
coupled effectively into the surrounding air to be appreciated by a
nearby ear. [0088] e) Where specifically, the traditional
Tensegrity structure struts, which usually are rods or hollow
tubes, are not to be held at their ends as they typically are. If
these struts are held at their ends all of the free beam resonant
modes of the strut are destroyed as the beam ends are forced to
become nodal minima's at the strut ends, where as the free beam
resonance of the strut is required to have a nodal maxima at the
beam end. In addition to destroying the Q of the resonators which
renders them unable store the energy of the musical tones, tying
these struts at their end points eliminates the previously
described integral "coupling transducer" preventing the strut end
from coupling the struts resonate energy into the surrounding air.
The new boundary conditions that restrict the movements of the end
of the beam by tying the tensional elements at the beam end
simultaneously kill the free beam resonances, and prevents energy
stored in the beam from coupling into the air. In addition to
killing the free beam resonances of the strut and destroying the
tubes natural built in "coupling transducer" the tensional elements
do not hold the ends of the beam firmly enough to create the new
resonance modes that would be created if the beams were held
rigidly in place. The Q of these potential new modes is
depressingly low since the ends move slightly coupling the energy
into lossy modes of the tensional elements. 4) Additional Detailed
Description [0089] a) Describe the general case of Tensegrity
structures into Tensegrity chimes: In general most know Tensegrity
structures can be retied at their free beam resonant nodal minima's
and thereby be converted to a Tensegrity musical instrument. The
present invention teaches that a Tensegrity musical instrument is
created by moving the tie point of most known Tensegrities, and new
yet to be created Tensegrities, from the strut ends (or wherever
they are initially tied) to the free beam nodal minima points on
the strut for the resonance mode desired. [0090] b) The "twisted
prism" subset Tensegrity Musical Instruments described earlier are
preferred embodiments of this invention. [0091] c) Some of these
preferred embodiments are illustrated by the following 3-fold and
5-fold Tensegrity Musical Instruments: [0092] i) (FIG. 12) shows a
3-Note Desk Chime [0093] ii) (FIG. 13) illustrates a 5-Note Desk
Chime [0094] iii) (FIG. 14) shows a 3-Note Wind Chime [0095] iv)
(FIG. 17) illustrates a 5-Note Wind Chime [0096] d) The structure
that forms the 3-Note Chime and 5-Note Chime Are "twisted prism"
Tensegrities, that can be generalized into and n-Note Chime with n
resonating struts. [0097] e) The Free beam strut resonant modes
with associated nodal minima and maxima points are shown in (FIG.
15A), (FIG. 15B), (FIG. 15C), & (FIG. 15D). If a chime is held
by elastic wires in tension anywhere other than one of its free
beam resonant nodal minima's it will not ring when struck with a
hammer, because the free beam resonances are damped severely or not
allowed depending on how far off the nodal minima's and how firmly
the struts are held. In addition, the wires do not hold the strut
firmly enough to set up a new set of resonances defined by the beam
holding points. Instead the energy induced in the strut by the
hammer strike is mostly lost to the holding wires. It is simply
carried away and dissipated as heat or acoustic radiation. Note
that for any given free beam mode to exist after it is tied into a
Tensegrity that the tie points must all be on one of the nodal
minima's for that mode. Notice that none of the free beam
resonances have nodal minima at either end of the strut. In fact
just the opposite is true. All of the free beam resonances have
nodal maximums at the ends. Therefore all of the free beam
resonances are thoroughly snuffed out once the strut is tied into a
Tensegrity in the traditional manner by attaching tensional
elements to both of its end points. [0098] f) As you can see from
the description just given of allowed strut resonances, that many
of the free beam resonances have different nodal minima points. So
when the nodal points to tie to are chosen all of the modes which
do not have nodal minima's there are filtered out. This permits the
Tensegrity Musical Instrument creator the freedom to choose which
free beam resonator modes to preserve by which nodal minima's he
chooses to attach the tensional elements to [0099] g) The musician
can further modulate which of the modes are sounded when the chime
is struck by choosing where to hit the chime and thereby which of
the allowed modes are excited. [0100] h) A dual mode Chime can be
created to get two tones or notes from one chime by choosing chime
of a rectangular cross-section in this way twice as many notes can
be obtained on a chime unit. For example a 4-chime set will now
have 8 notes instead of 4. This creates a whole 8-note scale (do,
ray, me, fa, so, la, te do) with only 4 dual mode chimes. The
resonant modes in a chime are formed by transverse mode traveling
waves that form resonant modes due to the finite length of the
chime tubes. The ends of the tubes form the boundaries that the
traveling waves reflect from, and these wave reflections
reinforcing the incident waves cause standing waves to form on the
tubes with nodal minima's and nodal maximum's. The fundamental free
beam transverse resonance mode for the chime occurs when it is one
wavelength long acoustically. That is when the round trip phase
change of a wave going from one end to the other, and back again to
the original end is 360 degrees. This mode contains three nodal
maximums and two nodal minima's. The maximum's are at both ends and
at half the length of the chime (in the middle), whereas the
minima's are approximately a quarter wave in from each end. If the
fundamental mode is desired then the chime is tied at these nodal
minima's. It is important to note that the nodal minima's for the
dual mode chimes described here are at the same point for both
modes. This means that when chime is tied into a Tensegrity
structure at these nodal minima's both notes are allowed.
[0101] i) Two orthogonal transverse waves can be supported on a
single rod or tube. Usually both of these modes travel at the same
velocity due to the radial symmetry of the tube. However if a
non-square rectangular or an oval tube is used for the chime the
velocities on the longer transverse axis Labeled "X" in (FIG. 19)
moves faster that the shorter transverse axis labeled "y" as the
traveling wave moves in the "z" direction. The reason for the speed
difference is that the velocity "V" is proportional to the
stiffness "S" divided by the linear density "D". (V.about.S/D).
That is to say the Velocity is directly proportional to the
stiffness. Also the frequency "F" of resonance of the tube is
directly proportional to the velocity "V" and inversely
proportional to the length of the tube "L". Therefore the Frequency
"F" is directly proportional to the Stiffness "S". Further more the
stiffness "S" goes up with the cube of the thickness "t". Putting
this altogether means the two orthogonal modes which now have
different thicknesses travel at different velocities and have
different fundamental resonant frequencies due to the increased
stiffness of in the "x" direction compared to the y direction of
the cross-section of the chime. Further more each note can be
excited with independently or both notes can be excited
simultaneously. If the chime is hit with a hammer in the stiffer
"x" direction the higher note will sound, and if hit the "y"
direction the lower note will sound. Also both notes can be made to
sound simultaneously with a single hammer stroke at 45 degrees from
the "X" axis. Still more interesting is that the relative amplitude
of each note, one to the other, can be controlled by the strike
angle. In the hands of a skilled percussionist a single dual mode
chime invented here can make a whole range of sounds with varying
amplitudes of the chimes two fundamentals and many harmonics. Also
if the cross-section of the tube is designed so that it is only
slightly asymmetrical (i.e. slightly oval or slightly off square),
then rich sounding phasing of the two tones occur adding a fullness
to the chime sound. This fuller sound is often desired in musical
instruments.
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