U.S. patent application number 12/926157 was filed with the patent office on 2011-05-12 for meta-material vibration energy harvester.
Invention is credited to John J. McCoy, Rajesh Patel.
Application Number | 20110109102 12/926157 |
Document ID | / |
Family ID | 43973599 |
Filed Date | 2011-05-12 |
United States Patent
Application |
20110109102 |
Kind Code |
A1 |
McCoy; John J. ; et
al. |
May 12, 2011 |
Meta-material vibration energy harvester
Abstract
A meta-material vibration energy harvester includes a housing
element encapsulating a multiplicity of oscillators capable of
harvesting a significant percentage of the total mechanical energy
diffusely distributed throughout the vibrating structure, the
harvester design resulting in a rapid transfer of mechanical energy
entering it via the housing element from the element to the
oscillators wherein the energy remains trapped while accumulating
over an extended time, the percentage of energy transfer primarily
depending on the ratios of the sum of the oscillator masses to the
sum of the housing mass and of the measure of the mass of the
vibrating structure and of the width of the band spanned by the
oscillators to its center frequency, both the relevant measure of
the vibrating structure mass and the values of the mass and
frequency ratios that maximize the percentage of internal energy
transfer depending on the harvesting scenario.
Inventors: |
McCoy; John J.; (Washington,
DC) ; Patel; Rajesh; (Brooklyn, NY) |
Family ID: |
43973599 |
Appl. No.: |
12/926157 |
Filed: |
October 28, 2010 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
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61272753 |
Oct 29, 2009 |
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Current U.S.
Class: |
290/1R |
Current CPC
Class: |
F03G 7/08 20130101 |
Class at
Publication: |
290/1.R |
International
Class: |
F03G 7/08 20060101
F03G007/08 |
Claims
1. A meta-material mechanical/electrical energy converter (MMMEEC)
for converting mechanical energy to an electrical current,
comprising a rigid housing element of mass M encapsulating N
oscillators each joined with a mechanical/electrical converter
element, and each having a mass m.sub.j and a resonance frequency
.omega..sub.j, together with electrical circuitry for collecting
individual electric currents from each of the combination
oscillator-mechanical/electrical-converters and outputting the sum
of currents via the housing element, wherein the source of
mechanical energy inputted to the MMMEEC is a time-varying external
force acting on the rigid housing element, and further wherein N is
substantially large and the distribution of .omega..sub.j is
substantially uniform across a frequency band of width .OMEGA. so
that the frequency band is densely filled, and wherein the masses
m.sub.j are substantially equal to one another, and further wherein
the ratios of the sum of the m.sub.j masses to M and the width of
the frequency band .OMEGA. to the center frequency of the band,
.omega..sub.c, have predetermined values, and further wherein the
value of .omega..sub.c equals the frequency for which the spectral
content of the time-varying external force is near its maximum,
whereby the MMMEEC has the capacity of converting a substantially
larger quantity of mechanical energy than would a device for which
the housing element of similar mass, comprising any material but no
internal oscillators, is combined with a mechanical/electrical
converter element.
2. A meta-material vibration energy harvester (MMVEH) for
harvesting mechanical energy from a vibrating structure (VS) that
is the source of a time-varying external force acting on the rigid
housing element, comprising the MMMEEC according to claim 1 adapted
to be connected to at least one local region of the vibrating
structure, wherein the predetermined value of the ratio of the sum
of the m.sub.j masses to M further depends on a measure of the
inertia of the VS and a measure of the resistance to deformation of
the VS, said measures depending on the geometry and composition of
the VS; the coherence of the vibrations contained in the VS; and,
the type connection of the MMVEH and the VS, whereby the MMVEH has
the capacity of harvesting an other than infinitesimal percentage
of the total mechanical energy diffusely distributed throughout the
vibrating structure.
3. A deformable meta-material mechanical/electrical energy
converter (DMMMEEC) for converting mechanical energy to an
electrical current, comprising a deformable housing element
containing a naturally occurring material with spatially varying
mechanical properties, which includes a spatially varying mass
density .rho., and encapsulating N oscillators each joined with a
mechanical/electrical converter element, and each having a mass
m.sub.j and a resonance frequency .omega..sub.j, together with
electrical circuitry for collecting individual electric currents
from each of the combination
oscillator-mechanical/electrical-converters and outputting the sum
of currents via the housing element, wherein the source of
mechanical energy inputted to the MMMEEC is a distribution, whether
discrete or continuous, of time-varying external forces of limited
spectral content acting over a region of the boundary of the
deforming housing element, and further wherein N is substantially
large and the distribution of internal elements is substantially
uniform so that the volume of the housing element is densely
filled, and further wherein the distribution of .omega..sub.j is
substantially uniform across a frequency band of width .OMEGA. so
that the frequency band is densely filled, and further wherein the
masses m.sub.j are substantially equal to one another, and further
wherein the distribution of the internal elements are such that the
ratios of the sum of those m.sub.j in a representative unit volume
element to .rho., and the width of the frequency band .OMEGA. to
the center frequency of the band .omega..sub.c, have predetermined
values, and further wherein the value of .omega..sub.c equals the
frequency for which the spectral content of the time-varying
external force is near its maximum, and further wherein the
spatially varying mechanical properties are predetermined so as to
maximize the mechanical energy inputted by the external forces
acting, whereby the MMMEEC has the capacity of converting a
substantially larger quantity of mechanical energy than would a
device for which the housing element of similar mass, comprising
any material found in nature but no internal oscillators, is
combined with an external distribution of mechanical/electrical
converter elements.
4. A deformable meta-material vibration energy harvester (DMMVEH)
for harvesting mechanical energy from a vibrating structure (VS)
that is the source of a time-varying external force distribution
acting on the deformable housing element, comprising the DMMMEEC
according to claim 3 adapted to be connected to at least one local
region of the vibrating structure, wherein the predetermined value
of the ratio of the sum of the m.sub.j masses in representative
unit volume to .rho. further depends on the geometry and
composition of the VS; the coherence of the vibrations contained in
the VS; and, the type connection of the DMMVEH and the VS, whereby
the DMMVEH has the capacity of harvesting an other than
infinitesimal percentage of the total mechanical energy diffusely
distributed throughout the vibrating structure.
5. The meta-material vibration energy harvester (MMVEH) according
to claim 2, wherein the MMVEH is directly connected to a local
region of a resonant vibrating structure (RVS), in which the
measure of inertia is the total mass of the VS and the measure of
resistance to deformation has a substantial linearly elastic
component, wherefore the spectral content of the time-varying force
acting on the rigid housing element is centered at a frequency
close to a resonant frequency of the structure, and wherein the
ratio of the sum of the m.sub.j masses to the sum of housing
element mass and the mass of the VS is about 0.01 and the ratio of
.OMEGA. to .omega..sub.c, is about 0.1, whereby the MMVEH has the
capacity of harvesting a substantial percentage of the total
mechanical energy diffusely distributed throughout the vibrating
structure
6. A MMVEH system comprising a multiplicity of MMMVEH's according
to claim 2, wherein each harvester is joined at a predetermined
location of the VS, such that the multiplicity is distributed
across the entirety of the VS, and further wherein the combined sum
of the internal oscillator masses of each of said MMVEH has a given
value as large as practical subject to the restriction that the
value is less than 1% of the total mass of the vibrating structure,
whereby the capacity for harvesting the mechanical energy diffusely
distributed throughout the vibrating structure is at a maximum.
7. The meta-material vibration energy harvester MMVEH according to
claim 2, wherein the MMVEH is directly connected to a local region
of a non-resonant vibrating structure (NRVS), in which the measure
of inertia is the mass of the region of the VS near the connected
MMVEH and the measure of resistance to deformation has a
substantial dissipative component, wherefore the spectral content
of the time-varying force acting on the rigid housing element
depends on the spectral content of the external forcing of the VS,
wherein the ratio of the sum of the m.sub.j masses to the sum of
the mass of the housing element and the mass of the local region of
the VS has a predetermined value that depends on the resistance of
deformation of the VS and on .omega..sub.c, whereby the MMVEH has
the capacity of harvesting an other than infinitesimal percentage
of the total mechanical energy diffusely distributed throughout the
vibrating structure
8. The DMMMEEC according to claim 3, wherein the housing element is
a rod, and further wherein the source of mechanical energy inputted
to the MRMEEC is a pair of equal and oppositely directed
time-varying forces of limited spectral content, acting at the two
end sections of the rod.
9. The DMMVEH according to claim 4, wherein the housing element is
a rod joined at two locations to a VS.
10. The DMMMEEC according to claim 3, wherein the housing element
is a beam.
11. The DMMVEH according to claim 4, wherein the housing element is
a beam joined, whether discretely or continuously, along a line on
a boundary surface of a VS.
12. The DMMMEEC according to claim 3, wherein the housing element
is a plate.
13. The DMMVEH according to claim 4, wherein the housing element is
a beam joined, where discretely or continuously, over a region in
the boundary surface of a VS
Description
FIELD OF THE INVENTION
[0001] The invention relates to a vibration energy harvester
comprised of a large number of oscillators encapsulated in a
housing element, combined with mechanical/electrical energy
converters, one for each oscillator, and the internal circuitry for
collecting the individual electric currents and outputting their
total, via the housing element. The invention is applicable to
virtually all structures and vibration fields. Moreover, when used
in multiplicity, the invention is capable of harvesting a
significant percentage of the total available energy in a vibrating
structure.
BACKGROUND OF THE INVENTION
[0002] In search of the solution to the world energy crisis,
researchers and industry have focused their efforts on harnessing
energy from alternative energy sources, most notably the sun, wind,
and ocean waves.
[0003] Technologies for harvesting energy from these sources are in
their infancy, though significant progress has been made. However,
the efficiency of existing technologies is limited, and resulting
environmental issues, such as noise and microclimate change, have
become increasingly problematic. Unless these technologies can be
dramatically improved, their energy harvesting potential will be
limited. High governmental priority has been assigned to improving
these technologies. The focus so far has been to engineer more
efficient devices (e.g. blades and turbines) that convert wind and
surface wave energy into electricity and to improve solar panel
design.
[0004] Little attention is being paid to harnessing the vibration
energy that is created by natural dynamic environmental processes
and by manmade processes, both of which represent vast pools of
renewable mechanical energy. Natural dynamic environmental
processes (wind, ocean waves, subsurface ocean currents, tectonic
movement) cause the vibration of structures with which they come
into contact. Vibration is also created in almost all manmade
processes. It results from all forms of transport, in the vehicles
themselves, and from vehicular contact with roads, bridges, train
tracks and other infrastructure. Technologies for harvesting these
alternative power sources are in their infancies. Vibration energy
harvesters (VEH) are a logical way to convert this vibration and
convert it to electrical energy.
[0005] The state-of-the-art of VEH, as measures by patent
applications and awards, are limited to micro-electro-mechanical
systems (MEMS) that are only capable of generating very small
amounts of power measures in the order of milliwatts. The
development of a technology based on a VEH that is capable of
harvesting a measurable percentage of the total energy in vibrating
structures (VS) deserves attention. Accepting the possibility of
such a device, its development would allow for harvesting a
substantial percentage of the total energy is VS, using a system
comprising a moderate number of devices integrated with and
distributed throughout the VS. Such a system would harness the vast
mechanical energy pools created in the natural world by first
converting the energy as vibration in structures designed for this
purpose. One advantage of this novel approach is that the
integration of the VEH as part of the VS isolates all moving parts
from the potentially hostile environment that is the source of
vibration. Structures can be designed to capture the mechanical
energy in subsurface ocean currents and in deep ocean internal
waves. Another advantage this novel approach is that the coherence
of the source of mechanical energy being tapped need not be as
great as that required by a wind turbine, for example. Structures
can be designed to capture energy from atmospheric turbulence.
[0006] The invention comprises a VEH capable of harvesting a
measurable percentage of the total energy in vibrating
structures.
[0007] The use of vibration energy harvesters (VEH) comprised of
multiple oscillators attached to a base that is in turn attached to
a vibrating structure (VS), is known in the art. The use of
vibration dampers (VD) comprised of multiple oscillators
encapsulated in a housing element that is in turn attached to a VS,
is also known in the art. Accepting that the base/housing-element
is rigid, its motion does not depend on the locations of the
oscillators; the geometric distinction is, therefore,
irrelevant.
[0008] The invention is distinguished over the existing arts by the
different objectives, numbers of oscillators, and the relative
massiveness of the devices. For example, VEH typically have a small
to moderate number of oscillators; say, less than 10, while VD of
the type related to the invention typically have a much larger
number; say, greater than several 100's, even 1000's. Another
distinction is known VEH have vastly smaller sizes and masses; U.S.
Pat. No. 6,858,970 B2, for example, describes a MEMS harvester with
mass- and size-ratios, relative to those of the VS, that approach
infinitesimals. By contrast, effective VD have mass- and
size-ratios, relative to those of the VS, which while still small
relative to 1, are finite.
[0009] Because of their small mass, known VEH are severely limited
in the quantities of energy harvested, typically measured at the
level of milliwatts; U.S. Pat. No. 6,858,970 B2 is representative
of the art of VEH. At the milliwatt level, the total quantity of
energy harvested in any reasonable time is a vanishing percentage
of the energy available in the VS. The principal application of
currently known VEH is to power micro-sized sensors and computers,
with an understanding that the rate of energy extraction from the
VS is too small to have an affect on the vibration field in the
structure. The design of the MEMS energy harvester is consistent
with these applications.
[0010] Rescaling the mass of an available VEH can result in a
comparable rescaling of the quantity of energy harvested, but only
if the more massive device initiates an energy transfer within the
VS, drawing energy from locations remote from that of the device,
to the device. Without this energy transfer, only the energy in the
immediate neighborhood of the device, a percentage of the total
energy available that approaches an infinitesimal, is available of
harvesting. Significantly, the initiation of the energy transport
process requires certain quantifiable design criteria, not known to
the present art, be met. The invention includes the identification
of these criteria, expressed by a limited number of device
parameters, and a design framework for determining the values that
result in an effective device.
[0011] Mechanical devices for damping the vibrations of a structure
are designed to either change the dynamics of the structure,
thereby precluding the introduction of energy, or to dissipate the
energy as heat. Damping devices based on a large number of
oscillators encapsulated in a housing element are in the latter
category. Typically, the mass of an efficient VD is a large enough
percentage of that of the VS to negatively impact design criteria
for the VS not related to the VD. Pub No: US 2009/0078519 A1
describes a class of VD that accomplishes the damping by a much
less massive device than heretofore known. The design according to
this known device results in the rapid transfer of a significant
percentage of the energy drawn into the device, to the internal
oscillators wherein it "remains trapped indefinitely," using the
terminology in the reference. The indefinite trapping of energy in
the oscillators largely eliminates the need for internal "energy
dissipaters" that add mass to the device. The design requires that
the oscillator resonances densely fill a frequency band according
to a formularization, i.e. a prescribed dependence of the values of
the resonant frequencies relative to one another. The dense filling
requirement is expressed by a minimum number of oscillators, which
is also determined by a formularization. Both formularizations are
essential to this known device.
[0012] While the present invention is presented as a vibration
energy harvester, the observation that a significant quantity of
the available energy is removed from the VS suggests it also has a
role as a VD. In this role, the invention is distinguished from the
class described in Pub No: US 2009/0078519 A1 by the active removal
of energy transferred to and entrapped within the internal
oscillators, by converting it to electricity. Thus, the "near
irreversibility" of the vibration damper described in Pub No: US
2009/0078519 A1 is made "absolute" in the case of the present
invention, by the removal of the energy transferred to the interior
oscillators, as electricity. This active removal of energy
fundamentally changes the device design and fabrication, by
eliminating the need for the formularization that determines
precise values for the oscillator resonances relative to each
other. Eliminating this formulation eliminates a major complexity
in the fabrication of the present device.
SUMMARY OF THE INVENTION
[0013] The present invention is a device, described as a
"meta-material vibration energy harvester" (MMVEH), which when
attached to a VS has the capacity to harvest a "reasonable"
percentage of the energy in a spatially diffuse vibration field
contained therein, by drawing energy from throughout the VS, to and
into the device, wherein it is converted from mechanical to
electrical. Further according to the invention are its efficacy for
a virtually inexhaustible range of VS, for virtually all vibration
fields, and for a broad range of attachments of the MMVEH and the
VS. The criterion for "reasonable," in describing the percentage of
energy harvested, is that a small multiplicity of MMVEH; less than
a modest multiple of 10, say; distributed across the VS has the
capacity of harvesting a significant percentage of the total energy
in the vibration field.
[0014] For purpose of this invention, a meta-material is a manmade
material comprised of a large number of mini-structures
encapsulated within a naturally occurring material, also described
as a housing element, which has properties that are not found in
naturally occurring materials. The encapsulation of the internal
mini-structures is understood to preclude both their direct
observation, by mechanical means, and their interaction with an
external environment. The interaction obtains indirectly through
their interaction with the housing material that, in turn,
interacts with the external environment. Their presence is also
observed indirectly, in the changes these cause in the interaction
of the housing material and the external environment. The
usefulness of the concept of meta-materials is the possibility of
designing a manmade material to behave differently from any
material found in nature.
[0015] The invention is a "device" comprising a meta-material that
extracts mechanical energy from a VS via a housing element and
outputs electrical energy via said housing element, for which the
designed behavior is an enhanced "energetics," as measured by the
quantity of mechanical energy extracted and the percentage of the
inputted mechanical energy that is outputted as electrical energy.
The enhanced energetics is occasioned by the operation of the
device over time, whereby the mechanical energy drawn into the
housing element is rapidly transferred to the internal oscillators,
where the energy accumulates and is ultimately converted and
outputted via the housing element, as an electric current. The
mechanical behavior of the invention, which does not obtain for a
housing element comprising any material found in nature but no
internal oscillators, is the device acting both as a one-way
mechanical valve, allowing the energy to enter but not exit, and a
mechanical battery, whereby the accumulating energy is stored.
[0016] In one manifestation of the invention, the housing element
is rigid and the internal structures are single degree-of-freedom
oscillators combined with mechanical/electrical converters. FIG. 1
shows a schematic of the mechanical elements of a more particular
manifestation of the invention for which housing element.sup.20 has
a single degree-of-freedom, represented by a translation
coordinate, and each of the internal oscillators also have a single
degree-of-freedom, and, therefore are represented, mathematically,
as sprung masses.sup.21. The forces shown acting on the internal
oscillator masses represent the converter elements. The negative
work accomplished by these forces in slowing the oscillator masses
quantifies the energy converted from mechanical to electrical.
[0017] For the manifestation in which the housing element is rigid,
the designation of the mechanical aspect as a "meta-mass" may be
more descriptive than "meta-material." The term meta-material is
retained since the invention contemplates other manifestations for
which the housing element is deformable and for which the internal
oscillators have multiple degrees-of-freedom. Also included among
these other manifestations are those for which the internal
oscillators are not mechanical; the internal structures can be
fabricated as mechanical/electrical converters joined with
oscillating electrical circuits.
[0018] The design of a specific MMVEH is expressed in the values of
the multiplicity of physical parameters that describe,
mechanically, the housing element; the internal oscillators; and,
the mechanical effects of the mechanical/electrical converters.
Given the behavior of the housing element depends on the external
environment, i.e. the VS and the vibration field contained therein,
as well as the internal oscillators, an effective MMVEH design
depends on the geometry and composition of the VS and the
description of the vibration field contained therein. There is no
universal design that can be put forth as describing the invention.
The specifics of the design will differ for VS comprised of
structural elements having one, two, and three dimensions and
systems comprised of such elements, and for vibration fields having
differing measures of spatial coherence. Further according to the
invention is that a multiplicity of MMVEH, each of which is
effective for a moderate band of frequencies, forms a system
capable of harvesting the energy in a vibration field that has
broad spectral content. Still further, according to the invention
is a virtually inexhaustible range of ways for attaching the MMVEH
and VS, including both linear and nonlinear attachments, as
occasioned for example via a elastic element that behaves linearly
or nonlinearly. A nonlinear attachment would result in
across-frequency transfer of energy to accompany its transfer from
the VS to the housing element, which can be exploited in choosing
the internal mechanical/electrical energy conversion.
[0019] The spatial coherence of the energy containing vibration
field warrants attention since the invention is intended for a
fully coherent field, an example of which is the low frequency
vibration of a beam caused by a spatially localized force; a fully
incoherent field, an example of which is a higher frequency
vibration in a track caused by a passing train; and, fields that
are partially coherent, an example may be a moderate frequency
vibration in a bridge component caused by automobile traffic. It is
known in the art that appropriate mathematical models that govern
the evolutions of vibration fields depend on the spatial coherence
of the field, with models based on the classical vibration theories
appropriate for fully coherent fields whereas models based on
statistical theories, e.g., "statistical energy analysis" (SEA),
are appropriate for fully incoherent fields. The invention
contemplates application for all vibration fields, requiring only
that the specifics of the MMVEH design accommodate the spatial
coherence of the field.
[0020] The range of harvesting scenarios for which the invention is
applicable presents a challenge to its detailed description,
beginning in the next section. In closing this summary description,
it is noteworthy to observe that available VEH neither exploit nor
require a design that results in enhanced energetics. The quantity
of energy harvested by available VEH, measured relative to that in
the VS, is so small as to make irrelevant, issues of the rates of
transfer of energy between the VS and the VEH.
BRIEF DESCRIPTION OF THE DRAWINGS
[0021] FIG. 1 shows an illustration of a meta-material mechanical
energy converter (MMMEEC), comprising a housing element
encapsulating a large number of internal oscillators to each of
which is connected a mechanical/electrical converter element,
represented by a force;
[0022] FIG. 2 shows a graphical representation of a generic
mathematical model governing the motion of the rigid housing
element of a mechanical meta-material;
[0023] FIG. 3 shows a graphical representation of a mathematical
model governing the motion of the rigid housing element of an
ungrounded mechanical meta-material;
[0024] FIG. 4 shows a graphical representation of a mathematical
model governing the motion of the rigid housing element of a
mechanical meta-material grounded by an elastic element;
[0025] FIG. 5 shows a graphical representation of a mathematical
model governing the response of a dynamical system comprising a
rigid base mass grounded by an elastic element and a dashpot, to
which a multiplicity of masses are each attached by a spring and a
dashpot;
[0026] FIGS. 6a to 6l show the results of simulations for the
motion history of the base mass for the model shown in FIG. 5 for a
range of experiment scenarios;
[0027] FIG. 7 shows a schematic of a simply supported beam to which
a meta-material mechanical energy harvester (MMVEH) is attached at
a "point" location along the beam;
[0028] FIG. 8 shows a schematic of the lowest order vibration mode
for a simply supported beam;
[0029] FIG. 9 shows a graphical representation of a mathematical
model for predicting the n-th modal amplitude history for a
resonant vibrating structure (RVS);
[0030] FIG. 10 shows a graphical representation of a mathematical
model for predicting the n-th modal amplitude history of the RVS
model represented by FIG. 9 when a single MMVEH is directly
attached at some location across the RVS;
[0031] FIG. 11 shows a schematic of a simply supported beam with
multiple MMVEH's attached;
[0032] FIG. 12 shows a graphical representation of a mathematical
model for predicting the n-th model amplitude history for the RVS
model represented by FIG. 9 when multiple MMVEH's are directly
attached at locations across the RVS;
[0033] FIG. 13 shows a schematic of a region of an unboundedly long
beam with a single attached MMVEH;
[0034] FIG. 14 shows a graphical representation of a mathematical
model for predicting the velocity history of the housing element of
a MMVEH directly attached to a non-resonant vibrating structure
(NRVS);
[0035] FIG. 15 shows 3 schematics of the energy flows in a NRVS in
the neighborhood of a directly attached MMVEH, according to
too-light; too-heavy; and ideal MMVEH;
[0036] FIG. 16 shows a schematic of a MMVEH attached to a beam-like
VS via and elastic element;
[0037] FIG. 17 shows a schematic of attaching a MMVEH, as part of a
dynamical system, to a beam-like VS;
[0038] FIG. 18 shows a schematic of attaching a MMVEH to two
locations of a structural frame, using elastic elements;
[0039] FIG. 19 shows a schematic of a deformable meta-material
mechanical/electrical energy converter, for which the housing
element is a rod;
[0040] FIG. 20 shows a generic mathematical model governing the
change in length of the housing element rod illustrated in FIG. 19,
and
[0041] FIG. 21 shows a schematic of a deformable meta-material
mechanical/electrical energy converter, for which the housing
element is a beam, or a plate.
A DETAILED DESCRIPTION OF THE INVENTION
[0042] The novelty of the concept of meta-material devices, as
described in the summary description, and the paucity of studies
reported in the literature, which directly apply to the invention
necessitates a highly technical detailed description. The
description is presented in stages, focusing first on the
energetics of a "mechanical" meta-material (MMM) device that forms
the genesis of the invention, and, second, on the energetics of a
"meta-material mechanical/electrical energy converter" (MMMEEC) in
the context of an experiment scenario that has a direct relation to
energy harvesting. When joined to a VS, which is the source of
input energy to the MMMEEC, the device is a MMVEH. Issues related
to the geometry and composition of the VS and to the coherence of
the vibration field and the role of these in determining an
effective MMVEH design are addressed at this point in the detailed
description.
[0043] Additional sections in the detailed description describe
different type attachments of the MMVEH to the VS; deformable
MMVEH; and, the conversion of mechanical energy to an electrical
current.
The Mechanics of a Meta-Material Device
[0044] The effects of both the external environment and of the
internal oscillators on the motion history of the housing
element.sup.20 are represented by forces, as illustrated in FIG. 2.
The force representing the effects of the external environment has
two components, in general: an "active" component, denoted by F(t),
which is the cause of the housing element motion; and, a "passive,"
or "reactive," component, denoted by F.sub.P(t), which develops in
response to the housing element motion. (An example of an external
reactive force would be that due to an elastic element grounding
the housing element.) With broad generality, the passive component
can be represented by an operator, denoted by K.sub.VS, which maps
the housing element motion history, denoted by x(t), to F.sub.P(t);
one writes a formal "operator" equation, F.sub.P(t)=K.sub.VSx(t).
Given a specific harvesting scenario, the active force history,
F(t), and a representation for the operator, K.sub.VS, can be
determined by pre-calculation. The "net internal force" that
represents the effects of the internal oscillators, denoted by
F.sub.I(t) in the illustration, is passive; it develops in response
to the housing element motion. There are multiple ways one can
represent F.sub.I(t), which are equivalent if not equivalently
convenient; the convenience depends somewhat on the experiment
scenario. The issue of the representation of F.sub.I(t) is
addressed in the context of illustrative experimental
scenarios.
The Energetics of an Ungrounded Mechanical Meta-Material
[0045] Consider an experimental scenario in which the force
representing the external environment has only an active component,
i.e. the housing element is ungrounded, and for which there are no
mechanical/electrical energy converters, i.e. the meta-material is
purely mechanical. The experiment, which does not represent a
realistic harvesting scenario, is a useful prelude for
investigating the energetics of any "mechanical meta-material"
(MMM) device, and for identifying the global design parameters that
result in enhanced energetics. The elimination of the
mechanical/electrical converters results in an "energy conserving"
meta-material; this is necessary for constructing a mathematically
rigorous, analytical framework for quantifying the device
energetics. The experimental scenario is schematically illustrated
in FIG. 3 in which the multiple internal oscillators are
represented by a single, net internal force, F.sub.I(t).
[0046] It is convenient to represent the net internal force
F.sub.I, by an operator, denoted by H*, which maps the external
force, F(t), to an "effective" force, F*(t)=F(t)-F.sub.I(t); i.e.,
F*(t)=H*F (t). The designation of the force F*(t) as "effective"
reflects an understanding that the motion of the housing element of
the meta-material forced by F(t), is the same as is the motion of
the housing element comprised of a naturally occurring material, a
rigid mass in this case, with no internal oscillators, when forced
by it.
[0047] The construction of a spectral, i.e. frequency, space
representation of the H* is a straight-forward exercise for a
practitioner knowledgeable of vibration theory. The result is
written,
H * ( .omega. ) = 1 ( 1 + j N m _ j .omega. j 2 ( .omega. j 2 -
.omega. 2 ) ) = j N ( .omega. j 2 - .omega. 2 ) j N ( .omega. ( j )
2 - .omega. 2 ) , ( 1 ) ##EQU00001##
where .omega. denotes the frequency coordinate; the .omega..sub.j
denote the resonant frequencies of the N internal oscillators; and,
the m.sub.j denote the ratio of masses of the oscillators to the
housing element mass M. The .omega..sup.(j) in the second
expression on the right-hand-side are the N roots of the
denominator polynomial, i.e. are solutions of the equation obtained
on setting the denominator in the first expression equal to zero.
The .omega..sup.(j) are the N, non-zero resonant frequencies of the
ungrounded MMM as a dynamical system; that the MMM is ungrounded
implies that, as a dynamical system, it also has a resonant
frequency equal to 0.
[0048] It is easily concluded from the first expression for the
denominator polynomial that the .omega..sup.(j) and .omega..sub.j
are intertwined, with the N-1, smallest .omega..sup.(j) locating
points on the real frequency line that are each within one of the
N-1 intervals described by the .omega..sub.j. For a scenario for
which the .omega..sub.j "densely" fill a frequency band of width,
.OMEGA., then, these N-1, .omega..sup.(j) also densely fill the
band. Only the largest of the .omega..sup.(j), locates a point
outside the band. Further, referring to the second expression for
the operator representation, each of the interior "poles" of its
spectral space representation can be paired with a corresponding
"zero;" the one nearest to it; with the frequency difference
between each paired pole and zero decreasing for increasing N, and
fixed .OMEGA..
[0049] A knowledgeable practitioner appreciates the operator H* has
a temporal space representation as a convolution described by a
time-series that is the Fourier inverse of its spectral space
representation in Eq. (??). Without loss in either generality or
mathematical rigor, one can write an expression for this
time-series as a sum of four components, a delta function component
that, when convolved with F(t) reproduces F(t), and three
components that, when convolved with each other and with F(t)
determine F.sub.I(t). The latter three components are two that are
separately due to the N-1 interior poles and to the single outlier
pole, and a third that is a convolution of the first two. The
behavior that ultimately results in enhanced energetics is the
contribution from the interior poles, represented by the
time-series,
H I ( t ) = j = 1 N - 1 r _ j .DELTA. j sin ( .omega. ( j ) t ) , (
2 ) ##EQU00002##
where the r.sub.j are the values of "residues," one for each of the
.omega..sup.(j) poles, normalized using the frequency interval
locating the pole, i.e. .DELTA..sub.j. Because of the
normalization, the r.sub.j are all pure numbers that, subject to
the dense filling requirement, have values between 0 and a number
approximately equal to 1.
[0050] Determining the precise values of the r.sub.j requires
detailed descriptions of the sets of .omega..sub.j and m.sub.j
values, and numerical calculations. For understanding the enhanced
energetics of the ungrounded MMM, however, one does not require
precise values of the r.sub.j, all one requires is a conclusion
that the variation across the set will be slow, which is valid
provided the .omega..sub.j are more-or-less evenly distributed
across the frequency band and the m.sub.j are more-or-less equal.
The mild restrictions on the values of the .omega..sup.(j) and the
corresponding r.sub.j significantly constrain the variation in the
time-series, an observation that can be demonstrated by additional
analysis or numerical experimentation. Thus, the time-series admits
of a "universal" description as an unending sequence of pulses with
widths measured in units of .OMEGA..sup.-1, separated by a time
interval measured in units of N.OMEGA..sup.-1. For N substantially
large, then, the pulses are well separated and their effects can be
considered separately. Noteworthy is an observation that a primary
effect of introducing mechanical/electrical energy converters would
be to eliminate the later-arriving pulses, leaving only that which
begins at t=0.
[0051] The first pulse, removed from the later arriving pulses, has
a spectral representation that is described by a dimensionless,
positive real-valued function, r(.omega.), a band-limited function
of frequency, which has a graphical representation obtained by
passing a smooth curve through the set of discrete residue values,
Assuming r(.omega.) is more-or-less symmetrically distributed about
.omega..sub.c, the center frequency of the band spanned by the
oscillators, the first pulse is described by
A.sub.1(t)sin(.omega..sub.ct),
where the amplitude modulation increases from a value of 0 at time
t=0 reaching some maximum value before returning to a value near 0,
all in a time interval of the order of .OMEGA..sup.-1. Moreover,
attached to the end of the pulse is an extended tail during which
its magnitude decreases algebraically with increasing time; the
genesis of the tail is the sharp change in r(.omega.) values near
the ends of the frequency band. The most significant "global"
measure of A.sub.1(t) is its "strength," defined as the integral of
its variation over unbounded time. Normalizing the measure of
strength, it can be described by <
r>.OMEGA..parallel.A.sub.1.parallel., where < r> is the
average of r(.omega.) over the frequency band for which it is
nonzero.
[0052] To appreciate the enhanced energetics of the ungrounded MMM,
consider a specific time-varying force,
F(t)=F.sub.0 sin(.omega..sub.ct).
For this external force history, the effective force history is,
approximately,
F * ( t ) .apprxeq. F 0 ( sin ( .omega. c t ) - 1 2 A 1 ( t ) cos (
.omega. c t ) ) , ##EQU00003##
where, once again, the approximation requires the fractional
bandwidth .OMEGA./.omega..sub.c be substantially small.
[0053] The energetics is quantified by the rate at which the
external force acting on the moving housing element accomplishes
work; this equals the product of F(t) and v(t), the velocity
history of the housing element, obtained on integrating F*(t).
Integrating the first term obtains a contribution to .nu.(t),
(M.omega..sub.c).sup.-1F.sub.0(1-cos(.omega..sub.ct)),
where the harmonically varying component is 90.degree. out-of-phase
with harmonically varying component in the expression for F(t).
Multiplying this contribution by F(t), obtains,
( M .omega. c ) - 1 F 0 2 ( sin ( .omega. c t ) - 1 2 sin ( 2
.omega. c t ) ) , ##EQU00004##
a sum of two terms, both of which vary harmonically in time. This
expression implies there is no net kinetic energy introduced to the
housing element over time; the kinetic energy introduced as
positive work during certain time intervals is removed as negative
work during other time intervals.
[0054] The net internal force, the second term in the expression
for F*(t), results in a second contribution to .nu.(t) that, again
accepting the small fractional bandwidth approximation, equals
(2M.omega..sub.c).sup.-1F.sub.0(.intg..sub.0.sup.tA.sub.1(t')dt')sin(.om-
ega..sub.ct),
which has a harmonic component that is in-phase with the
corresponding variation in F(t) expression. A consequence of this
is the product of this contribution and F(t) contains a term,
(4M.omega..sub.c).sup.-1F.sub.0.sup.2.intg..sub.0.sup.tA.sub.1(t')dt',
which asymptotically approaches a constant,
(4M.omega..sub.c).sup.-1F.sub.0.sup.2
r.OMEGA..parallel.A.sub.1.parallel..
[0055] The last expression describes what is a monotonically
increasing energy inputted to the housing element of the ungrounded
MMM. Significantly, the largest percentage of the inputted energy
does not remain as kinetic energy in the housing element but is
transferred in a time measured in units of .OMEGA..sup.-1, as
vibration energy in the internal oscillators. This is easily
demonstrated by an expression for the work accomplished by the net
internal force acting on the moving housing element.
[0056] Introducing mechanical/electrical energy converters at the
level of the internal oscillators and the necessary circuitry for
collecting the internal electrical currents, changes the nature of
the MMM; it would now be more properly described as a
"meta-material mechanical-to-electrical converter" (MMMEC),
recognizing the mechanical energy inputted to the housing element
is outputted via the housing element as an electric current. Given
the enhanced mechanical energy inputted to the device via the
housing element and its rapid transfer to the encapsulated
oscillators wherein it remains trapped while accumulating, awaiting
conversion to electrical currents, the quantity of energy converted
is substantially larger than would a combination of the housing
element without the encapsulated oscillators, and an
mechanical/electrical converter.
[0057] Removing mechanical energy at the level of the internal
oscillators can be expected to effect the inputting of mechanical
energy to the housing element and the subsequent transfer of this
energy to the internal oscillators, but this effect is secondary to
the effects described by the theoretical analysis. This last
observation can be demonstrated by numerical analysis, presented in
the context of an experiment scenario in which the MMM is grounded,
in a subsequent section.
Lessons from the Theoretical Analysis that Apply to the
Invention
[0058] The conclusions based on the theoretical analysis regarding
the energetics of the meta-material are summarized as follows.
[0059] 1. The transfer of energy from the housing element to the
internal oscillators is measured in units of .OMEGA..sup.-1, where
.OMEGA. is the width of the band of oscillator resonances, and the
time the transferred energy remains trapped in the oscillators is
measured in units of N.OMEGA..sup.-1, where N is the number of
internal oscillators. These conclusions prove to be "universals,"
applicable for all harvesting scenarios and independent of all
other physical measures describing the meta-material; the VS; and,
the connection of the MMVEH to the VS. [0060] 2. There can be a
significant percentage increase in the quantity of energy inputted
to the meta-material by the action of a specified external force,
provided the spectral content of the external force overlaps the
frequency band spanned by the internal oscillators, i.e., that
described by its center frequency, .omega..sub.c, and .OMEGA.. The
increased energy inputted to the meta-material, via the housing
element, is transferred from the housing element to the internal
oscillators as described in the first conclusion. This second
conclusion also proves to be a "universal," applicable for all
harvesting scenarios. [0061] 3. The energy increase is quantified
by a measure of the strength of a net internal force pulse. The
value of the strength measure proves to depend on a multiplicity of
physical parameters that describe the meta-material, primarily the
sum of the masses of the internal oscillators and the mass of the
housing element; the VS; and, the connection of the MMVEH thereto.
That is, the value of the strength measure is not a "universal,"
but depends on the harvesting scenario.
The Energetics of an Elastically Grounded MMM
[0062] Consider an experiment scenario for which the MMM is
grounded by a linearly elastic element.sup.22 of stiffness K, the
mathematical model governing the response of the housing element is
schematically illustrated in FIG. 4. As for the ungrounded MMM, the
internal oscillators are not shown in the illustration, only shown
is the net internal force F.sub.I(t) these cause to act on the
housing element in response to the action of the external force
F(t).
[0063] The grounding of the MMM introduces an additional
characteristic frequency to the geometry and composition of the
dynamical system, the resonant frequency of the housing element
with no internal oscillators, denoted by .omega..sub.0. For the
internal oscillators to have maximum effect on the energetics of
the grounded MMM, the value of .omega..sub.0 must be deep within
the band spanned by the internal oscillator resonances; for
specificity, let .omega..sub.0=.omega..sub.c.
[0064] Neglecting, for a moment, the effects of the
mechanical/electrical energy converters as secondary, the analysis
described for the ungrounded MMM can be slightly generalized for
the grounded MMM. The analysis is expressed in the description of
an operator, H*, that maps an external force, F(t) to an effective
force, F*(t), which now, significantly, acts on the grounded MMM.
Referring to Eq. (??), describing the spectral space representation
of the operator for the case of the ungrounded MMM, the only change
for the grounded MMM are minor changes in the locations of the
poles identified by Eq. (??) and the addition of one zero and pole,
the additional pole described as an "outlier," since its value is
less than the smallest of the resonances of the internal
oscillators.
[0065] Referring to Eq. (??), describing the time-series that is
the contribution of the interior poles to the convolution that
represents H* in temporal space, the principal effects of the
grounding are twofold, a sign change is the residue values for the
.omega..sup.(j) that are smaller than .omega..sub.0, and a
different "rule" for determining the magnitude of the residues in
terms of more primitive physical parameters that describe the
dynamical system.
[0066] Neither change impacts the conclusions that were described
as "universals;" the time-series continues to obtain as a sequence
of pulses with widths measured in units of .OMEGA..sup.-1 separated
by a time interval measured in units of N.OMEGA..sup.-1. The
changes do impact the intra-pulse variations. Regarding the first
pulse, the change in the sign of the residues results in a
variation described by
A.sub.1(t)cos(.omega..sub.0t),
with a 90.degree. phase shift in the harmonic component. The
modulating amplitude pulse, like that for the ungrounded MMM,
begins with a value of 0 at t=0, rising to a maximum and returning
to a value near zero in a time measured in units of .OMEGA..sup.-1,
appended to which is an extended tail within which its magnitude
decreases algebraically with increasing time. The extended tail for
the scenario of a grounded MMM is due to a rapid change in the
value of r(.omega.), from negative to positive, in the neighborhood
of .omega.=.omega..sub.0. The most significant global measure of
A.sub.1(t) is its strength defined as the integral of the variation
over all time. Significantly, the grounding of the MMM impacts the
pulse strength and the rule for determining the pulse strength in
terms of more primitive physical parameters that describe the
dynamical system.
[0067] It proves convenient for discussing the energetics of the
grounded MMM, to convolve the first pulse with the temporal space
representation of the impulse/response function for the grounded
MMM housing element with no internal oscillators, i.e.
(M.omega..sub.0).sup.-1 sin(.omega..sub.0t).
[0068] Accepting the small fractional bandwidth
.OMEGA./.omega..sub.c approximation introduced previously, the
result of the convolution is
-(2M.omega..sub.0).sup.-1(.intg..sub.0.sup.tA.sub.1(t')dt')sin(.omega..s-
ub.0t).
which when combined with the impulse/response function of the
grounded MMM housing element with no internal oscillators, results
in a contribution to an "effective" impulse/response function,
(M.omega..sub.0).sup.-1(1-2.intg..sub.0.sup.tA.sub.1(t')dt')sin(.omega..-
sub.0t).
For times after that of the application of the impulse, t=0, that
are large relative to .OMEGA..sup.-1 but small relative to
N.OMEGA..sup.-1, this reduces to
(M.omega..sub.0).sup.-1(1-2<
r>.OMEGA..parallel.A.sub.1.parallel.)sin(.omega..sub.0t),
where < r>.OMEGA..parallel.A.sub.1.parallel. is the first net
internal force pulse measure.
[0069] The complexity of the grounded MMM as a dynamical system
precludes an analytical formularization of a rule for determining
the strength measure, < r>.OMEGA..parallel.A.sub.1.parallel..
Moreover, should the formularization be accomplished for a MMM with
no mechanical/electrical energy converters and grounded by a simple
spring, it would have little value as a tool for designing an
effective MMVEH. Ultimately, the effects of removing mechanical
energy from a MMM and the design of an effective MMM will require
numerical experimentation. An illustration of the required
experimentation is described in the next section.
A Validation and Extension of the Theoretical Analysis--Numerical
Simulations
[0070] A validation of the theoretical analysis presented for the
grounded MMM and an investigation of factors that cannot be
included therein can be provided by numerical simulations. In lieu
of accomplishing numerical simulations for the purpose of
developing the invention, reference is made to numerical
simulations accomplished by other authors for purposes other than
the development of a vibration energy harvester. These simulations
are reported in the peer reviewed research literature (Vignola et
al; J. Acous. Soc. Am., vol., pp-, 2008). The mathematical model
investigated is schematically illustrated in FIG. 5, taken from the
referenced paper. As seen in FIG. 5) N oscillators.sup.23, each
having a mass.sup.24 m.sub.sj, are shown attached by a
spring.sup.25 with stiffness k.sub.sj and a dashpot.sup.26 with
strength .nu..sub.s j, to a base mass.sup.27 that is grounded by a
spring.sup.28 with stiffness K.sub.s and a dashpot.sup.29 with
strength .nu..sub.s. As indicated previously, for a rigid housing
element, the locations of the internal oscillators are irrelevant
to the motion history of the element. The inclusion of the dashpots
in the dynamical system for which the numerical simulations are
accomplished allows for an investigation of the assertion that the
removal of energy from the system, both at the level of the base
mass and the attached oscillators, has only a secondary effect on
the transfer of energy between the base mass and the
oscillators.
[0071] The reported simulations are summarized by the graphs in
FIG. 6. For all simulations, N is 50; the ratio of the width of the
frequency band spanned by the attached oscillator resonances to the
center frequency of the band is 1/8; the center frequency band
equals the resonant frequency of the grounded base mass in
isolation; and, the strengths of the dashpots are equal. Varied
across the set of simulations presented is the ratio of the sum of
the internal oscillator masses, .SIGMA..sub.jm.sub.sj to that of
the base mass M.sub.s. The variation across the sets of oscillator
masses and the resonant frequencies were chosen by a
formularization that is not germane for the purposes here, except
for an understanding that the masses are about equal and the
distribution of the frequencies across the frequency band is
substantially uniform.
[0072] The graphs in FIG. 6 presenting the results of the
simulations show the velocity histories of the base mass motion
after the action of an impulsive force, and the amplitudes and
phases of the spectral representations of the velocity histories,
for different values of .SIGMA..sub.jm.sub.sj/M.sub.s. Referring to
the time histories, the time coordinate is normalized using
.omega..sub.c, such that for a fractional bandwidth equaling 1/8,
the width of the first pulse is nominally equal to normalized t=8,
and with N=50, the arrival of the second pulse obtains are
normalized t=400, the ending time from the graphs.
[0073] Starting with the top most three graphs: For these the mass
ratio, .SIGMA.m.sub.sj)/M.sub.s=10.sup.-4, a value chosen to be so
small as to result in the internal oscillators having no effect on
the behavior of the housing mass. This is reflected in a housing
mass velocity history having a time harmonic variation at the
resonant frequency of the housing element in isolation, modulated
by an exponentially decaying envelope, appropriate for the strength
of the viscous damper grounding the mass element. The strength of
the viscous damper is such that virtually all of the mechanical
energy is removed from the system before what would be the arrival
time of the second force pulse, t=400. The second and third of the
topmost three graphs show the amplitude and phase spectra defined
on the time series; the three graphs are what one would expect.
[0074] Jumping down to the case for which the mass ratio,
.SIGMA..sub.jm.sub.sj/M.sub.s.apprxeq.0.005, it is clearly seen
that the velocity of the housing element returns to zero in a
normalized time approximately equal to 8, corresponding to
.OMEGA..sup.-1, remaining approximately equal to zero thereafter.
For this case, which is close to optimum, the sum of the masses of
the internal oscillators is approximately 0.5% of the magnitude of
the base mass, a mass ratio that is surprisingly small for the
internal oscillators to have such a profound effect on the base
mass motion. Referring to the amplitude spectrum defined on the
housing mass velocity, this is seen to be relatively fiat over the
frequency band. By association, it is for mass ratios approximately
equal to 0.005 that the strength of the first net internal force
pulse approximately equals 1.
[0075] For mass ratios smaller and larger than 0.005, one again
sees that the velocity of the housing mass does not return to a
value near zero for t.gtoreq.8, approximately. These results, and
others, are explained by the theoretical analysis. The net internal
time-varying force for an impulsive external force of unit
magnitude acting equals the time-space representation of the
operator that maps any external time-varying force acting to the
corresponding net internal force history. The work accomplished by
this force in slowing the mass element gives quantitative measure
to the energy transferred from the mass element to the internal
oscillators. For mass ratios less than 0.005, the "strength" of the
first force pulse is somewhat less than 1, the strength necessary
to transfer the kinetic energy introduced to the mass element by
the impulse force. For mass ratios somewhat greater than 0.005, the
strength of the first pulse is somewhat greater than 1; the first
force acts to remove the total kinetic energy introduced to the
housing element "before" its conclusion. Significantly, the force
pulse continues to act after all the energy has been transferred to
the internal oscillators; the internal force develops to accomplish
motion compatibility at the attachments of the internal springs to
the mass element. This continuing action results in further mass
element motion; the work accomplished by the continuing action of
the force pulse on the moving mass element gives measure to a
secondary energy transfer, by which energy is from the internal
oscillators to the mass element. "Optimum" energy transfer and
entrapment requires a design that results in transfer of energy
from the housing element as the first pulse concludes; for the
other conditions that describe the numerical simulations, this
obtains when the total mass of the internal oscillators
approximately equals 0.5% of M.sub.s.
[0076] Among observations that are consistent with the theoretical
analysis are the appearance to two peaks in the amplitude spectrum
defined on the mass element velocity history. The peaks appear at
the two ends of the frequency band spanned by the uncoupled
oscillator resonances, for mass ratios slightly larger than 0.005,
which become more pronounced and move further from the two ends
with increasing mass ratio values. The genesis of this behavior in
the outlier poles; the simulations show the contributions of these
can be neglected for the smaller mass ratios for which optimum
behavior obtains.
[0077] Not explained by the analytic expression for F.sub.I(t) that
was obtained for an energy conserving system is the observed
behaviors for mass ratios larger than 10.sup.-4 but smaller than
0.005. In this regime, there remains energy in the grounded housing
element after the time t=.OMEGA..sup.-1=8, with the remaining
energy appearing to dissipate at a rate that is faster than that
due to the viscous damper attached to the housing element.
Reference is made to the mass ratio regime between 10.sup.-4, say,
and 0.005, as the "effective dissipation" regime. The explanation
for this regime is the removal of energy from the internal
oscillators impacts the motions at the connections of the internal
springs and the mass element, thereby extending the time of action
of the net internal force pulse. The effect represents an
additional "physics," in coupling the effects of removing
mechanical energy as electricity and the transfer of energy from
the mass element to the internal oscillators. This additional
physics, while of secondary importance to that which applies for
the case of optimum energy transfer, can be exploited in relaxing
somewhat the strict requirement on the total mass of the internal
oscillators as a ratio of the housing element mass.
Lessons from the Numerical Simulations that Apply to the
Invention
[0078] The lessons that are based on the numerical simulations are
summarized as follows. [0079] 1. The conclusion of the theoretical
analysis that for a broad range of experiment scenarios, that a
percentage of kinetic energy introduced to the housing element of a
MMM by the action of an impulsive force is transferred to the
internal oscillators over a time measured in units of
.OMEGA..sup.-1, where it remains trapped for a time measured in
units of M.OMEGA..sup.-1, is validated. [0080] 2. The possibility
of a MMM design, which results in virtually all of the kinetic
energy introduced to the housing element transferred to the
internal oscillators is demonstrated. [0081] 3. The primary design
parameters for this transfer to obtain are the ratio of the total
mass of the internal oscillators to that of the housing element,
and the fractional width of the band spanned by the resonances of
the internal oscillators, .OMEGA./.omega..sub.c. [0082] 4. For a
fractional bandwidth equaling 1/8, the optimum total mass of the
internal oscillators relative to mass of the housing element is
approximately, 0.005, a ratio that is surprising small. [0083] 5.
Removing mechanical energy from the grounded MMM at the level of
the internal oscillators, by converting it to an electrical
current, has no effect on the time for inputting mechanical energy
to the housing element and only minor effect on the transfer time
of energy from housing element to the internal oscillators. The
internal conversion of energy to electrical currents, the sum of
which is outputted via the housing element, changes the nature of
the device, now a MMMEEC. [0084] 6. Removing mechanical energy from
the grounded MMMEEC at the level of the internal oscillators, can
have a beneficial effect in broadening the range of values of the
primary design parameters for which virtually all energy inputted
to the housing element is transferred to the internal oscillators.
Exploiting this potentially beneficial effect can result in an
slight increase in the time required to transfer energy to the
internal oscillators.
The Energetics of a MMVEH
[0085] For a MMMEEC to operate as a MMVEH, it must be attached to a
VS, which both determines the mechanical energy inputted to the
housing element, drawn from the broadly distributed vibration field
contained in the VS, and, figures prominently in determining the
motion history of the housing element in response to the inputted
energy. The ultra-broad range of harvesting scenarios precludes a
definitive description of the design of an effective MMMVEH for the
all scenarios. This said, one can distinguish between two broad
classes, or regimes, of harvesting scenarios, one termed
"resonant," and one termed "non-resonant," with the remaining
harvesting scenarios understood as intermediate these two regimes,
as extremes. For the "resonant" harvesting regime, the VS response
is "global," in the sense that global physical measures defined on
the entirety of the VS determines its behavior at the attachment
location. For the "non resonant" harvesting regime, the VS response
is "local," in the sense that the behavior of the VS at the
attachment location is determined by physical measures that are
local to the attachment.
[0086] The simply-supported beam.sup.30 shown in FIG. 7 with a
MMVEH.sup.31 directly attached at a location along its length can
illustrate both scenarios, each distinguished by the description of
the beam forcing, the source of the vibration field contained
therein.
The Resonant Harvesting Regime
[0087] A necessary condition for a harvesting scenario to be
resonant, in the sense indicated, is for the external forcing of
the VS to act "coherently" for a sufficient time to engage the
entire structure in determining its behavior across the local
region at which a MMVEH is attached. It is well known to
practitioners with knowledge of vibration theory that prediction
models for VS that respond globally are conveniently formulated in
terms of the normal "modes" of the VS. The modes are global
response measures, which can be "synthesized" to form a
representation of any VS response measure, including its local
behavior at the attachment region. As an illustration, FIG. 8 shows
the lowest vibration mode shape.sup.32 for the simply supported
beam.
[0088] Each mode represents a separate degree-of-freedom, with each
modal "coordinate" evolving independently of the others, when there
is no attached MMVEH. Thus, the temporal history of each modal
coordinate is governed by a mathematical model represented by a
grounded mass, illustrated in FIG. 9. The integer, n, identifies a
particular mode; M.sub.n, the modal "mass,".sup.33 is a global
measure of the density distribution throughout the VS; K.sub.n, the
modal "stiffness,".sup.34 is related to a resonant frequency of the
VS, .omega..sub.n= {square root over (K.sub.n/M.sub.n)}; and, the
dashpot measure.sup.35 .nu..sub.n quantifies the rate at which
energy is irreversibly lost to the mode. The modal force Q.sub.n(t)
is a "projection" of the external forcing F(t) that is the source
of vibration energy in the VS.
[0089] Attaching a MMVEH to the VS couples the modes, as it
transfers energy between the VS and the MMVEH; this greatly
complicates the response prediction problem, in general. If,
however, one chooses the frequency band of the internal oscillators
to straddle one of the resonant frequencies, .omega..sub.n, and
accepts that .OMEGA., the width of the band of internal
oscillators, is substantially narrow, measured relative to the
frequency difference separating resonances, the coupling of the
modal coordinates can be ignored, to lowest order. The type of
attachment of the MMVEH to the VS affects the behaviors of both the
MMVEH and the VS. For specificity, one can assume the housing
element is directly attached to a surface of the VS; a brief
section describing other attachments is provided below. For the
directly attached MMVEH, then, the mathematical model for
estimating the housing element motion history is illustrated in
FIG. 10, which can be identified with FIG. 5, by replacing the mass
M.sub.s in FIG. 5 with the sum of a factor, .alpha..sub.n, times
the housing element mass M and the modal mass, M.sub.n, i.e.
(.alpha..sub.nM+M.sub.n), where the factor .alpha..sub.n depends on
the MMVEH attachment location, and the mode. Also distinguishing
the two figures is the absence of the attached oscillators in FIG.
10 with these represented by the net internal force F.sub.I(t) that
the oscillators cause to act on the housing element in response to
Q.sub.n(t). The conclusions demonstrated, both analytically and by
numerical simulations, for the grounded MMM are applicable for the
MMVEH operating in the resonant harvesting regime.
[0090] An observation, readily demonstrated for the simply
supported beam illustration of FIG. 7 and valid in general, is the
modal mass approximately equals the total mass of the VS, a mass
that for a large VS can be too large for a practical device; 1% of
a large mass can, for practical reasons, be too large. A second
observation, again readily demonstrated for the illustrative simply
supported beam and valid in general, can be exploited to mitigate
this practical problem. The observation is that each modal
coordinate represents a "component" vibration field that is
spatially coherent over the extent of the VS. This perfect spatial
coherence together with the high temporal coherence of the
component vibration field allows the use of multiple MMVEH.sup.31
distributed over the extent of the VS, e.g. along the length of the
simply supported beam as illustrated in FIG. 11, all of which will
act in unison. The mathematical model governing the behavior of
this dynamical system is illustrated in FIG. 12, a reproduction of
FIG. 10, now with a multiplicity of MMVEH.sup.31 attached to the
modal mass.sup.33.
[0091] An implication of the MMVEH acting in unison is that 1% of
the total mass can be distributed, albeit not uniformly, among the
multiple MMVEH. A different implication is that for a sub optimum
MMVEH, the energy not immediately transferred to the internal
oscillators is returned to the VS as energy coherent not only
across MMVEH but coherent with the energy in the vibration field.
The invention envisions the use of multiple MMVEH for this
purpose.
The Non-Resonant Harvesting Regime
[0092] The non-resonant harvesting regime includes scenarios for
which the external forcing of the VS is spatially local and acts
for a time that is short when compared to the travel time of an
acoustic disturbance across the VS. It also includes spatially and
temporally extended forcings for which the spatial/temporal
coherence is substantially short. For non-resonant harvesting
scenarios, the interaction of VS and a MMVEH, which is local in
space, is also local in time. Consequently, the passive effects of
the VS in determining the motion history of the housing element are
as though the VS were unbounded in at least one dimension. Assuming
the MMVEH.sup.33 attached to the simply supported beam illustrated
in FIG. 11 is located at a sufficient distance from the end
supports, the passive effects of the beam in determining the
behavior of the MMVEH.sup.31 would be as though the beam.sup.30 is
unbounded in both directions, illustrated in FIG. 13. In
"resisting" the motion of the housing element of the MMVEH, then,
the beam, as representative of a generic VS, acts to remove
mechanical energy, irreversibly. An appropriate mathematical model
governing the motion of the MMVEH housing element.sup.20 is
illustrated in FIG. 14, wherein the grounding dashpot.sup.36
represents the action of the beam in resisting the motion of a
combination of the housing element mass.sup.20 M and the
appropriate local mass.sup.37 of the VS, denoted by M.sub.L.
[0093] The relative magnitudes of the total mass of the MMVEH,
M+.SIGMA..sub.jm.sub.j, to M.sub.L is a critical factor in
determining the transport of energy in the VS, in the neighborhood
of the MMVEH. For a MMVEH mass that is too light relative to
M.sub.L, the magnitude of the internal force that the MMVEH exerts
at its attachment to the VS is too small to accomplish much work.
Consequently, little energy is transferred from the VS to the
MMVEH. Referring to the energy transport in the VS, the fact that
little energy is transferred to the device results in little impact
on the transport in the neighborhood of the VS, near the MMVEH.
This situation is illustrated in the uppermost of FIG. 15, in which
the energy transport near a MMVEH directly attached to a
beam-like.sup.30 VS is represented by arrows. As illustrated in the
uppermost of the figures, a too-light MMVEH has little impact on
the energy transport past the devices, in either direction. This
case applies for a MEMS, VEH. At the other extreme of a MMVEH that
is too heavy relative to M.sub.L, the device acts to impose a
geometric, workless, constraint for which, once again, little
energy transfer to the MMVEH. Referring to the energy transport in
the VS, the imposition of a geometric constraint results in a
reversal in the directions of the energy transport at the
boundaries of the neighborhood of the directly attached MMVEH. This
case is illustrated in the bottommost of FIG. 15 in which the
arrows representing the energy transport are shown reflected at the
boundaries VS. Little energy is transferred, and harvested, because
the MMVEH changes the energy transport in the VS resulting in
little energy its immediate neighborhood. For a scenario for which
the MMVEH mass is neither too light nor too heavy, compared to
M.sub.L, substantial energy transfer to and into the MMVEH via the
housing element obtains. If by design, the energy transfer to the
housing element is subsequently transferred to the internal
oscillators rapidly enough to preclude an immediate return
transfer, instead accumulating in the oscillators until being
converted to electricity, virtually complete harvesting obtains.
This of the MMVEH is transferred to the internal oscillators, for
harvesting, no reverse transfer of energy to the VS will obtain.
This case is illustrated in the middle figure of FIG. 15. The
invention is a MMVEH that seeks to exploit this Goldilocks
range.
Attaching the MMVEH to the VS
[0094] The manner of attaching the MMVEH to the VS is an issue for
effective vibration harvesting. For the harvesting scenarios used
to illustrate the workings of the invention, only direct
attachments were illustrated. The invention contemplates any manner
of attachment. Illustrated in FIG. 16 are cases in which a single
MMVEH.sup.31 is attached to a simply supported beam 30 using a
spring.sup.36; linear, nonlinear and non-elastic springs are
contemplated. In FIG. 16a, the only grounding of the MMVEH.sup.31
is through the VS; in FIG. 16b, the MMVEH.sup.31 is also directly
grounded using a second spring.sup.37. A further case is
illustrated in FIG. 17, in which the MMVEH is part of a two
degree-of-freedom dynamical system, represented by connecting the
MMVEH.sup.31 to a mass element.sup.38 using a spring.sup.36 and
connecting this mass element to the simply supported beam using a
second spring.sup.37.
[0095] For VS that are more geometrically complicated than a beam
element, more geometrically complicated types of attachment are
contemplated. As illustration, FIG. 18 shows a structural
frame.sup.36 as VS, to which a MMVEH.sup.31 are attached at two
locations by spring elements.sup.36.
Deformable MMMEEC'S and MMVEH'S
[0096] For the manifestations of the MMVEH presented above as
illustrative, the housing element is rigid. The invention
contemplates other manifestations for which the meta-material
devices are deformable. In this regard, three deformable
meta-material devices are easily envisioned; those for which the
housing elements are rods; beams; and, plates. FIG. 19 is a
schematic showing a rod.sup.39 encapsulating internal oscillators
represented by sprung masses.sup.21 and energy converter elements
represented by forces acting on the internal masses. The effects of
an external environment, the VS and the vibration field contained
therein, on the meta-rod device are represented the time-vary
forces, F(t) and F.sub.P(t), acting on the end sections of the rod.
FIG. 21 is a schematic showing both a beam.sup.42 and a
plate.sup.43, encapsulating internal oscillators represented by
sprung masses.sup.21 and energy converter elements represented by
forces acting on the internal masses. For a plate, FIG. 21 shows
only a side view. The effects of the external environment on the
meta-beam (meta-plate) are time varying force distribution fields
acting across the face of the meta-beam (meta-plate), a
one-dimensional field for the beam and a two-dimensional field for
the case of a plate.
[0097] As a prelude to describing the energetics of deformable
meta-materials and the design of deformable MMMEEC and MMVEH, it is
convenient to compare and contrast a meta-material with a composite
material, where the latter can be describe as a manmade material
mixtures comprising a large number of mini-sized inclusions of one
material distributed in a matrix of a second material. A
fiber-reinforced plastic is an example of a composite material.
Using this, a meta-material can be said to be a composite material
for which the inclusions are, in the context of the invention,
oscillators joined with energy converter elements. This
understanding is convenient since the behavior of a deformable
meta-material has certain aspects in common with a deformable
composite, and other aspects that are unique to a meta-material.
The unique aspects, again in the context of the invention, are the
enhanced energetics occasioned by the capacity of the internal
oscillators to resonate and thereby can store and accumulate
substantial amounts of energy.
[0098] The effects of distributing inclusions in a rigid matrix is
limited to changing its mass density; the effects of distributing
oscillators in a rigid housing element are such as to suggest the
invention of a MMVEH. The highly technical detailed description of
the invention is occasioned by the absence of a simple, reasonably
complete theory for predicting the behaviors of rigid MMVEH
attached to the broad range of VS for which the invention is
applicable. The absence of an encompassing theory necessitated the
use of illustrative experimental scenarios to demonstrate the
enhanced energetics that is the bases of the invention. The absence
of an encompassing theory also necessitated the division of
harvesting scenarios, in limited classes for which greatly
simplified prediction models could be formulated, thereby
demonstrating the invention could be reduced to practice.
[0099] The simplified prediction models are expected to have a
further role in accomplishing specific MMVEH designs, once the
geometry and composition of a VS and the vibration field contained
therein are specified. The role is to guide the design, perhaps to
accomplish a preliminary design. It is further expected, however,
that completing a specific design for the broad range of VS and
vibration fields for which the invention is intended will require
numerical simulation, using sophisticated finite element computer
software.
[0100] The effects of distributing inclusions in a deformable
matrix are significant, the recognition of which was the occasion
of the formulation of an encompassing theory for predicting the
behaviors of composite materials. Effective modulus theories for
predicting the large-length-scale behaviors of composite material
rods, plates, and beams allow the easy incorporation of these as
elements of VS. An encompassing theory for predicting the behaviors
of meta-material is not available. One might speculate on the
possibility that a suitably encompassing theory will be
accomplished with further mathematical research. Absent an
encompassing theory necessitates a greater reliance on numerical
simulation. This absence is riot germane, however, to the claim of
an invention of a device comprising a deformable meta-material,
provided it is clear that the invention is useful and can be
reduced to practice.
[0101] The consequences of the housing element deformability on the
energetics of a MMMEEC, and hence a MMVEH, is an issue. As for a
device for which the housing element is rigid, the energetics can
be described in phases, the first phase being the inputting of
mechanical energy to the housing element of the device. For a rigid
housing element, the energy inputted is in the form of kinetic
energy and, assuming the grounding of the housing element is energy
conserving, obtains "instantly." By contrast, for a housing element
that is deformable, the energy inputted to the housing element is
in the form of large-length-scale vibrations and obtains, even for
a grounding of the housing element that is energy conserving, over
time. Assuming the time for inputting the energy is rapid enough,
as to be completed before significant motion of the internal
oscillators obtains, the effects of the internal oscillators on
this phase of the energetics are the same as for a composite
material. The conclusion is that while the internal oscillators do
effect the inputting of energy into the large-length-scale
vibrations of the housing element, the effect is properly modeled
as an effective change in the material of which the housing element
is comprised.
[0102] The second phase of the energetics is the transfer of the
energy from the housing element to the internal oscillators. For
the device for which the housing element is rigid, the energy
transfer obtains from the housing element as an entity to the
totality of internal oscillators. For the device for which the
housing element is deformable, the inputted energy is distributed
non-uniformly across the housing element. The energy transfer to
the internal oscillators can be expected to reflect this
non-uniformity, with the transfer incorporating a localness such
that the energy in local regions is transferred to internal
oscillators located in the corresponding local regions. To the
extent that the localness is as complete as suggested by the above
sentence, this second stage energy transfer is easily quantified.
The quantification differs from that for the rigid housing element
only in replacing the housing element mass with a housing element
mass density, and the sum of the internal oscillator masses with a
local sum of the internal oscillator masses. The description of the
energy transfer would the same as for the rigid housing, provided
the non-uniformity in the inputted energy distribution does not
require for its resolution in local regions, regions that are so
small that the number of internal oscillators contained therein is
too small. It is intuitive that there are cases for which the
energetics is as described. There is little doubt that the behavior
of the invention will be as claimed, the closer one is to the cases
for which the energetics is as described.
[0103] The absence of an encompassing theory for predicting the
behavior of meta-material device with a deformable housing element
does not preclude constructing a simplified model for special
cases, the deformable rod meta-material schematically illustrated
in FIG. 19. This case is simple because the interaction of the
device with the external environment is limited to equal and
opposite forces acting at the end sections. The deformable rod is
itself a structural element that is resonant in the sense described
previously. This suggests the model illustrated in FIG. 20 as
governing the "change in the length" of the rod. The mass.sup.40 in
FIG. 20 has a value of the order of the total mass of the rod,
denoted by M.sub.r, which is grounded by a spring.sup.41 with a
stiffness, denoted by K.sub.r that equals
M.sub.r.omega..sub.0.sup.2, where .omega..sub.0 is a resonant
frequency of an effective rod. For definiteness, .omega..sub.0 can
be identified with the smallest, non-zero, resonant frequency of
the rod with the ends free to move. The illustration requires
further description. The displacement of the mass element in the
figure measures not a physical displacement but a modal coordinate,
which represents a deformation mode. For .omega..sub.0 representing
the lowest, non-zero resonant frequency for rod with the ends free
to move, the deformation mode shape, expressed in the deflections
of all section of the rod, has no motion at the rod center section
with maximum motions at the end sections. As already indicated the
displacement of the mass element in the figure also measures the
change in length of the end sections.
[0104] The energetics for device illustrated in FIG. 19, for which
the mathematical model in FIG. 20 applies, can be investigated for
a scenario in which there is no element grounding the rod. For this
case, the spring shown grounding the mass element represents the
stiffness of the rod element to deformation. One can compare the
mathematical model represented by FIG. 20 when F.sub.P is set to
zero, and the mathematical model represented by FIG. 5 when the
attached oscillators shown therein is replaced by a net internal
force the oscillators cause to act on the housing element. The two
mathematical models are identical, even though the physical
variables represented are different. That the two mathematical
models are identical implies the energetics of the two physical
systems represented is identical. This further implies the
simulated results shown in FIG. 6, including the complete transfer
of energy from the base mass (or deformable housing element) for
the force F(t) is rapidly and completely transferred to the
internal oscillators to be dissipated by internal dampers (or
converted to electricity) is identical. The physical manifestations
of the two systems, which include the distribution of the internal
oscillators across the length of the deformable rod, are different;
the energetics of the two systems is identical.
[0105] Constructing a model for the behavior of a meta-material for
which the housing element is a beam, or plate, and which interacts
with the external environment by a force field acting across the
face of the rod, or plate, would be vastly more complicated. Faced
with this task, one would best turn to numerical simulation.
Converting Mechanical Energy to an Electric Current
[0106] The description of the invention accepts, and makes claims
for, any manner of converting mechanical energy to an electric
current at the level of the internal oscillators. The genesis of
one claim made for the invention is the possibility of an MMVEH
design that explicitly incorporates a nonlinear attachment to a VS,
which results is a transfer of energy across frequency to accompany
the transfer of energy from the VS to the MMVEH. The genesis of
another claim for the invention is the accumulation, in time, of
the mechanical energy transferred to the internal oscillators. This
allows the design of a nonlinear mechanical/electrical energy
converter, one that allows the mechanical energy to accumulate to a
design level, before rapidly converting it to an electric current
pulse.
* * * * *