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Surajit SEN

Granular Shock Absorber




Granular chain shock absorber organizes random mechanical energy for conversion to electric power.

[ Related : * LAGIEWKA : Impact Absorber *  /  PALMER : D3O Energy Absorber * PEREZ : Seaweed Shock Absorber * ]










December 18, 2006

In Granular System, Tiniest Grains Absorb Shocks 'Like A Sponge'

A simulation by Surajit Sen exposes the complexity of shock propagation: red and yellow areas show the brunt of it, with sound waves impacting green grains further down the chain.



A University at Buffalo theoretical physicist who published research in 2001 demonstrating that it someday may be possible to build bridges, buildings and other structures that are nearly blast-proof, now has published results based on computer simulations showing how a shock-absorption system might be constructed to accomplish that goal.

Published in October in Physical Review Letters, the research is relevant not only to questions of shock-absorption in these structures, but also to life-saving improvements in tanks and aircraft carriers, as well as bullet-proof vests and other protective clothing for soldiers, law enforcement officers and even outdoor enthusiasts.

The simulations are of critical importance because they allow researchers and manufacturers to see how a potential system might work without having to painstakingly construct the systems and spend $40,000 to conduct a single blast in a test facility.

In earlier UB research by the same scientists, granular systems composed of individual spheres of gradually reduced size -- a "tapered" chain in a casing -- proved to be capable of efficiently absorbing well over 80 percent of input energy.

The main findings of the current research are that it is possible to retain the scalability of the system, reduce its size by a factor of five and make it far more capable of absorbing shock.

The key to achieving the results, according to Surajit Sen, Ph.D., UB professor of physics and co-author of both the current work and the 2001 publication, was the use of interstitial grains of the right sizes to control energy propagation through the chain.

"It turns out that the shock pulse is more easily managed when tiny interstitial grains are placed between the many progressively shrinking spheres or grains that make up the tapered chain," he said.

In the most recent paper, the UB physicists reported that this "decorated, tapered chain" system is capable of absorbing more than 50 percent of the shock that could not be absorbed by previous systems they had simulated.

These greater shock absorption capabilities were attributed to the use of tiny, interstitial grains or

particles of only about a millimeter that were placed in between each sphere, the "decorated" part of the chain; it turned out that the smaller these grains were, the more shock absorption they could achieve.

"These tiny grains were able to accomplish a huge trick," said Sen, co-author of the paper with Robert Doney, doctoral candidate in the Department of Physics in the UB College of Arts and Sciences. "They trap energy as it flows from the larger to the smaller grain, slowing it down. As it slows down, the smaller grain then essentially rattles back and forth between its two bigger neighbors, dissipating much of the energy as heat and sound."

Because the granular shock-absorbing system is strongly nonlinear, he said, the system allows directed energy transfer and the smaller grains undergo rapid rattling, which helps to efficiently distribute and dissipate the energy.

The simulations are significant because they have modeled shock pulses traveling at speeds approaching those encountered in combat situations, Sen said.

"These were simulations of pretty large impact shock pulses, traveling at several hundred meters per second," he explained, "and when we have such large impacts, the grains themselves now behave like sponges, absorbing the energy."

The simulations showed that in some of the larger impacts, the system would remain effective, but that significant and irreversible deformation would occur.

Sen explained that the system is proving to be very scalable, so that it could be designed to handle almost any typical shock.

According to the UB scientists, their earlier predictions about the shock-absorbing capabilities of these "tapered chain shock absorbers" were experimentally confirmed in publications in Granular Matter (2004) by independent researchers at the Colorado School of Mines in collaboration with a group at the NASA Glenn Research Center, as well as in Physical Review E (2006) by researchers at the University of Santiago in Chile and the Superior Institute of Mechanics (SUPMECA) in Paris.

The UB research is funded by the U.S. Army Research Office.





http://www.physics.buffalo.edu/sen/cv_sen_1_6_12.pdf

Surajit Sen and Sourish Chakravarti,

  “Device and concept for novel shock absorbing structures,”



http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.97.155502
Phys. Rev. Lett. 97, 155502
11 October 2006
DOI: http://dx.doi.org/10.1103

Decorated, Tapered, and Highly Nonlinear Granular Chain

Robert Doney and Surajit Sen
Abstract

It has been seen that inertial mismatches in 1D granular chains lead to remarkable energy absorption which increases with the number of spheres, N, and tapering, q. Short chains, however, are limited in that regard, and we therefore present one solution which greatly improves performance for any size chain. These strongly nonlinear and scalable systems feature surprisingly complicated dynamics and are inadequately represented by a hard-sphere approximation. Additionally, such systems have shock absorption capacities that vary as a function of position along the chain. In this Letter, we present results in the form of normalized kinetic energy diagrams to illustrate the impressive mitigation capability of both original and improved tapered chains.





GRANULAR MECHANICAL DEVICE TO DYNAMICALLY EXTRACT DISSIPATED ENERGY AND METHOD
WO2013036971

The present invention may be embodied as a device for harvesting energy comprising a housing with an input element at an input end. A granular alignment is contained within the housing, the granular alignment having a first end grain in mechanical communication with the input element. The device comprises an electrical converter in mechanical communication with a grain of the granular alignment and configured to receive mechanical energy from the alignment. The electrical converter is configured to convert mechanical energy into electrical energy. The invention may be embodied as a method for harvesting energy, wherein mechanical energy is received and transmitted through a granular alignment. The transmitted energy is harvested by, for example, converting the mechanical energy into electrical energy.

Background of the Invention

[0003] Significant amount of energy is continually wasted around us. Some examples of wasted energy are: (i) the energy of ocean waves as they crash on the shore, especially on rocky shore lines; (ii) the energy released at the base of a waterfall; (iii) the energy of oncoming air that is dissipated in the front of a fast moving car, truck, train, or airplane; and (iv) the energy associated with vibrations of train tracks as trains pass on them, etc. Such energy is often released non-uniformly in time and is lost to the environment.

[0004] Typical techniques for conversion of mechanical to electrical energy either use fully resonance-based mechanisms to couple with, and extract energy from, external sources. Other common techniques use turbine or rotor blade based structures. For small scale energy extraction processes, piezoelectric materials are also used. Turbine or rotor blade based technologies can be expensive to install and maintain and are not necessarily highly efficient. Almost all common technologies also have significant bandwidth limitations.

[0005] There is a need for a broadband device for conversion of significant amounts of mechanical to electrical energy that is currently being wasted across a broad spectrum of applications. It is important for these systems to be easy to install and use. Further, the device should be efficient, for example, even an efficiency of 10%, meaning 10% of the available energy is extracted as electrical energy, is reasonable considering presently available technologies. Brief Summary of the Invention

[0006] In an embodiment of the present invention, a device capable of extracting energy from a variety of common energy sources utilizes one or more granular chains in order to organize mechanical energy into a more effective form for generating electrical energy.

[0007] Wave travel is governed by linear and nonlinear principles. Traditionally, wave travel in common media is viewed as strongly linear— the waves gradually disperse and their energy is lost to the medium by distribution along the available degrees of freedom of the system (e.g., a wave caused by dropping a pebble in a pond). Any nonlinear effects are generally weak and are typically ignored. However, in some cases, the nonlinear interactions in a medium are significantly stronger than the harmonic interactions, hence the system is thought of as "strongly nonlinear." In a well-known example, John Scott Russell observed a wave created by the bow of a barge continue down a canal after the barge had stopped. The wave, since termed a "solitary wave," travelled along the canal without dissipating for almost two miles when Russell lost sight of it.

[0008] Granular chains (also referred to as granular alignments) are one-dimensional arrays of spheres in "Hertzian" contact. Such one-dimensional arrangements have been shown to transmit mechanical impulses along the array as solitary waves— nonlinearly. The present invention utilizes granular chains to organize random mechanical energy into more ordered energy in the form of a wave train of solitary waves. In this way, an electrical generator can more efficiently convert the mechanical energy into electrical energy.

Description of the Drawings

[0009] For a fuller understanding of the nature and objects of the invention, reference should be made to the following detailed description taken in conjunction with the accompanying drawings, in which:

Figure 1 is a diagrammatical sketch showing a portion of a device according to an embodiment of the present invention; Figure 2 is a series of graphs showing simulation data of a device according to the present invention over a time sequence; from top to bottom, the sequence of graphs show a perturbation starting out at the left end and moving toward right, and over time, forming a wave train of solitary waves (time (t) is shown for each graph);

Figure 3 is a diagram of a device according to another embodiment of the present invention, having an electric converter comprising a piezoelectric material;

Figure 4 is diagram of a device according to another embodiment of the present invention, having an electrical converter comprising a magnet and solenoid;

Figure 5 is a diagram of a device according to another embodiment of the present invention; and

Figure 6 is a flowchart depicting a method according to another embodiment of the present invention.



Description of the Invention


[0010] The present invention may be embodied as a device 10 for harvesting mechanical energy which comprises a plurality of spheres 14 arranged along a single longitudinal axis 50 (a linear, or one dimensional, array) and a housing 12 configured to contain the spheres 15 of the plurality of spheres 14 and maintain the linear arrangement of the spheres 15. See, e.g., Figure 3. In this manner, the linear arrangement of spheres has a first end sphere 24 and a second end sphere 25. Each sphere 15 of the plurality of spheres 14 may also be referred to herein as a "grain" 15, and each linear arrangement of grains 15 (spheres 15) may be referred to as a "granular alignment" 14.

[0011] Each sphere 15 of the plurality of spheres 14 may be sized to a particular application, further described below. The spheres 15 can have diameters ranging from nano- scale (fractions of millimeters) to several centimeters. The diameters may be outside of this range. In certain embodiments, the spheres 15 may have diameters of 0.1 mm or less, between 0.1 mm and 1 mm, or more than 1 mm. For example, if one is to design a device that will take energy out of the wind power dissipated in front of a moving vehicle on a highway at 55 miles per hour, it may be best to keep the device sufficiently small such that its mass is low and the dissipative effects are modest. For instance, one can consider a granular alignment of 20 grains (spheres), each of which is of 1 mm in diameter. However, if the device is designed to extract mechanical energy released in an airport runway, then the device may need further downsizing to grains that are, for example, 0.5 mm in diameter. In this way, the grains can respond to the small amplitude vibrations effectively. The power density afforded by the device is high, for example for the automobile example above, the power density can be as high as lOOW/kg (when multiple devices are arranged into a system as further described below). The plurality of spheres 14 may comprise any number of spheres 15. For example, the plurality of spheres 14 may comprise 5, 6, 7, 8, 9, or 10 spheres 15 or more.

[0012] The housing 12 may have an elongated shape, and a cross-section of the housing 12, across the longitudinal axis 50, may have any shape. The cross-section may be shaped as a circle, in which case the housing 12 is generally cylindrical. The cross-section may be shaped as a rectangle, a triangle, or otherwise. The housing 12 has an input end 16 where mechanical forces (i.e., mechanical energy) are applied. This may be referred to as an impulse generation assembly. The input end 16 of the housing 12 comprises an input element 18, which may be made of the same material as the bulk of the housing 12 or a different material. The input element 18 of the housing 12 is configured to transmit incident mechanical energy from an exterior side 20 of the input element 18 to an interior side 22, wherein the first end sphere 24 of the granular alignment 14 is in mechanical communication with the interior side 22. A receiving end of the granular alignment, in particular the first end sphere 24 of the granular alignment 14, may be in direct contact with interior side 22 of the input element 18.

[0013] The housing 12 also has a generation end 28. The granular alignment 14 is disposed along the longitudinal axis 50 between the input end 16 and the generation end 28 of the housing 12. An electrical converter 30 is disposed at the generation end 28 of the housing 12 (within the housing 12, exterior to the housing 12, or both). The electrical converter 30 converts mechanical energy (e.g., movement, pressure, etc.) to electrical energy. Such electrical converters 30 are known in the art and include induction generators, capacitive generators, piezoelectric generators, and the like. [0014] In the case of a piezoelectric device (an example of which is depicted in

Fig. 3), a piezoelectric component 32 may be in contact with the second end sphere 25 (generation end) of the granular chain 14 such that pressure of the solitary waves transmitted from the grains 15 to the piezoelectric component 32 causes electrical energy to be created by the piezoelectric components 32.

[0015] In another example of a device 11, depicted in Fig. 4, a magnetic component

34 (e.g., a cylinder formed from a permanent magnet, other shapes, etc.) is in mechanical communication with the second end grain 25 on the generating end of the alignment 14. The magnetic component 34 is at least partially disposed within a solenoid 36 (i.e., a wire coil). In this way, mechanical energy transmitted from the grains 15 to the magnetic component 34 causes the magnetic component to move. As such, the motion of the magnetic component 34 is linked to the dynamics of the granular alignment 14. The incident mechanical vibrations are converted by the chain 14 to more organized energy pulses of fixed spatial width and of a time scale that is controlled jointly by the external input characteristics and the system properties. In this way, the mechanical energy, as function of time, is converted to a voltage, as a function of time.

[0016] In another example, the last grain 15 itself is a strong permanent magnet. An electric coil is disposed around the magnet; the coil is attached to a circuit. The moving permanent magnet will create a flowing current that can be harnessed for practical use. In another embodiment, the end grain 15 comprises a piezoelectric material.

[0017] In other embodiments, the device 60 may comprise more than one granular alignment 64 (see, e.g., Fig. 1). As such, the housing 62 is configured to contain the more than one alignment 64. The alignments 64 may be parallel to one another, or may be disposed in various orientations. The housing 62 may be configured with one input element 68 or more than one input element 68 (e.g., one input element 68 for each granular alignment 64). Each plurality of spheres 64 may have an electrical converter 70.

[0018] When the first end sphere 24 of the granular alignment 14 receives mechanical energy transmitted through the input element 18 of the housing 12, the mechanical energy is then transmitted from sphere 15 to sphere 15, from one end of the granular alignment 14 (the receiving end) to the opposing end (the generating end). Where the received mechanical energy is random, the discrete nature of the spheres 15 of the granular chain, and the Hertzian principles governing the interface between spheres 15, automatically orders the energy into a wave train of solitary waves. See, e.g., Figure 2. Each solitary wave carries approximately 56% of the energy as kinetic energy and approximately 44% of the energy stored as potential energy. While it may be beneficial to maximize the order of the energy, it may not be completely organized into an ordered set of solitary waves.

[0019] The length of the chain, in terms of number of grains, can be varied according to the application. Generally, the length is sufficiently long such that the energy is ordered into a more usable form. In some examples, a small system may typically have between 5 and 10 spherical grains of the same diameter. The speed of solitary wave train formation and the length of the solitary wave train depend upon the magnitude and the time span across which the signal is accepted into the device. The nature of the randomness may also affect the time taken to form a wave train. [0020] The housing and the granular chain may be constructed from any appropriate materials selected for a particular application. For certain applications, it may be best to select the materials in order to minimize the weight of the device. For example, the housing may be constructed from wood, and the spheres of the granular chain may be aluminum.

[0021] The present invention may also be embodied as a system for harvesting energy. The system may comprise a housing configured to hold a plurality of granular alignments. See, e.g., Fig. 1. In this way, a larger area of a body (such as the hull of a ship, or the body of a car, etc.) may be used to harvest energy.

[0022] In one embodiment, the housing of a device is a wooden slab-like structure. In this example, the housing comprises an array of bore holes, each containing a granular alignment. The housing has an input end that can respond to an external driving force and transmit it to the receiving ends of the plurality of granular chains. The input end may be a membrane. The granular alignments, each typically with some 5-10 mono-sized grains, are placed in the bore. At the generating end of the granular alignments are long cylindrical magnets which move in synchrony with the lowest grain of the chains. These magnets are disposed within tightly wound coils. As the magnets move, the flux changes in the coil stimulate a voltage fluctuation.

[0023] The present invention may be embodied as a device 80 for harvesting energy comprising a housing 82 having an input end 83 and a generation end. At least one sphere 84 is contained within the housing 82 and configured to receive mechanical energy. The housing 82 may comprise an input element 86 at the input end 83 for receiving mechanical energy on an exterior side 87 of the input element 86. In such an embodiment (having an input element 86), the at least one sphere 84 is in mechanical communication with the input element 86. The device 80 further comprises an electrical converter 88 in mechanical communication with the at least one sphere 84 and configured to receive mechanical energy transmitted through the at least one sphere 84. The at least one sphere 84 may comprise a plurality of spheres as previously described.

[0024] The electrical converter 88 may comprise a piezoelectric material. As such, when mechanical energy impinges on the electrical converter 88, at least some of the mechanical energy is converted into electrical energy. The electrical converter 88 may be a sphere, similar to the at least one sphere 84, wherein a piezoelectric material has been inserted into the composition of the sphere (i.e., inside one or more grains). Embodiments of the device 80 may comprise at least one additional sphere 90. The device 80 may comprise one or more additional electrical generators 92. For example, the device 80 may have an alignment with a sphere 84, followed by one or more piezoelectric electrical converters 88, 92 interspersed with one or more additional spheres 90. Electrical generators 88, 92 may have the same number or different numbers of spheres between them. The piezoelectric material can be placed between any one or two or more grains.

[0025] The time-dependent behavior and frequency response of devices and systems according to the present invention can be controlled by adjusting the system size, the geometry and the material properties of the grains, and the grain shapes.

[0026] The present invention may be embodied as a method 100 for extracting mechanical energy (see, e.g., Fig. 6). Such mechanical energy may otherwise be wasted (e.g., dissipated to nature). The method 100 comprises the step of receiving 103 mechanical energy at a first end of a linear arrangement of spheres (i.e., a granular alignment). The mechanical energy is transmitted 106 through the granular alignment. As such, with the spheres aligned linearly— in a one-dimensional array— the energy is passed through each sphere from an input end of the granular alignment to a generation end of the alignment. As further described below, the transmission 106 of the energy will organize the energy. The method 100 comprises the step of harvesting 109 the organized mechanical energy at a second end of the plurality of spheres. Harvesting 109 the organized mechanical energy may be done by converting 112 the mechanical energy into electrical energy. The energy may be converted 112 using an electrical converter as previously described.

[0027] Receiving 103 the mechanical energy may further comprise the sub-step of receiving 115 the mechanical energy at an input element. The input element is disposed at an input end of the plurality of spheres. In some embodiments, the received energy is transmitted 118 from the input element through one or more springs. As such, the received mechanical energy is dispersed, or spread out, over time. For example, a short impulse force input at the input element and transmitted through springs may result in multiple oscillations of the lower magnitude. The energy transmitted 118, and dispersed, through the one or more springs is transmitted 121 to the granular alignment.

[0028] The function of the device of the present invention is further described below with respect to an exemplary embodiment.

[0029] EXEMPLARY EMBODIMENT

[0030] In an exemplary embodiment, intended to be non-limiting, the present invention concerns a simple tile- or cube-shaped unit. The unit receives incident mechanical energy at an end and converts the incident energy into electrical energy. The incident energy can be at multiple, various frequencies, can be long- and short-lived impulses or be energy that is randomly incident from a variety of commonly available energy sources. The system may operate without any input power (other than the incident energy) and converts the extracted mechanical energy to electrical energy using a certain arrangement of small elastic grains and mechanical to electrical energy conversion devices. We refer to this system henceforth as a Nonlinear Energy Harvesting ("NEH") tile. In another related realization it is possible to convert oncoming unprocessed mechanical energy into unprocessed electrical energy using appropriately designed microphones or equivalent and then convert the electrical signal using appropriate circuitry to non-dispersive form with some input electrical power. The resultant signal is useful energy— for example, suitable for loading up to the electrical grid.

[0031] Propagation of mechanical waves through a physical system can be a highly complex process. Traditionally, the propagation of mechanical energy through a medium is described in terms of how the energy disperses into the medium. However, there are media through which propagating energy may not significantly disperse at all. These latter systems are examples of strongly nonlinear systems. Non-dispersive energy propagation is typically in terms of propagating lumps of energy that do not significantly change their shape and their energy content as they travel through space and time. These energy lumps are called solitary waves. The NEH tile system uses solitary waves and solitary-wave-like structures to efficiently convert incident broadband mechanical energy to non-dispersive energy lumps. These non-dispersive energy lumps propagate through the tile and a significant part of this energy (between about 10% and nearly 40% or more) is converted to electrical energy. The electrical output obtained can be rectified for consumer needs, and the extracted energy can be plugged into the electrical grid or stored in a battery, as needed.

[0032] Basic experimental and simulational research on how a short-lived impulse is converted to a solitary wave by a single alignment or an array of elastic grains is known in the art. The conversion of a steady, but long-lived, signal into a train of solitary waves is also known. The basic science can be summarized as follows. Consider an alignment of elastic grains (e.g., metallic spheres, ball bearings, etc.) which are all in contact and placed between end walls. When these spheres or grains get compressed against each other, they repel. The repulsive potential is called Hertz law and is described as

V = ad<?>, (1) where a is a force constant that depends on the materials and the shapes of the grains and d is called the overlap between the grains and is a measure of the distance by which the grains, that are initially just touching one another, come closer to one another. For spherical grains n = 5/2. When an impulse is incident on one end of the chain, the system accepts that energy and propagates it as a solitary wave, which is a non-dispersive energy lump. When a long lived impulse is incident on a granular chain, the system makes a solitary wave train as we discuss below. [0033] The present invention applies the above mentioned basic science in a useful device and method. The NEH tile uses a certain multilayered structure to receive all forms of energies and then converts the various inputs using appropriate initial and boundary conditions by taking advantage of the solitary wave and solitary wave train like structures. These objects transform the oncoming energy into non-dispersive forms. This energy can then be subsequently converted to electrical energy.

[0034] An embodiment of the present invention is a device that comprises three independent layers that are designed to work in conjunction with one another. The properties of the layers are described below.

[0035] Layer 1 is a surface layer that can be as simple as a thin but strong layer of a polymeric material or as structurally complex as a layer comprising an array of highly sensitive and strong springs. These springs or related objects would respond to any external perturbation by breaking it down into pulses. The detailed properties of Layer 1 are linked to the specific properties of the external perturbation associated with a given source of energy. For long lived perturbations, Layer 1 will break down the oncoming pulse to a series of shorter pulses by appropriate rapid "flapping" of the surface in response to the oncoming energy.

[0036] Layer 2 is an active layer associated with the broad-band conversion of mechanical energy to non-dispersive energy lumps. This increases the efficiency of device of the present invention, allowing such devices to be largely independent of the maximum amount of energy extracted at different frequency windows. The incident energy on Layer 1 is transmitted to Layer 2. Layer 2 is a block with a dense array of vertical bores. Each vertical bore is filled by an alignment of elastic grains. The grain sizes vary between a few millimeters to a few centimeters or so where the size used relates to the nature of the application at hand. The material of the grain and the block are chosen to optimize the passage of the oncoming energy through the granular alignments. Each alignment comprises 5 to 20 or so grains of the same or of different materials. The shapes of the grains can also be varied to optimize the energy transmission amount and speed (and hence the device's frequency response) as needed. The primary role of Layer 2 is to convert the input energy from Layer 1 into non-dispersive energy lumps and pass it on to Layer 2 for conversion to electrical energy.

[0037] In most realizations, Layer 3 comprises a matrix with vertical bores which are aligned with the structure of Layer 2 in appropriate ways depending upon the nature of the application. The bores have permanent magnetic cylinders that generate an electrical current through the coils via Lenz's law as each impulse is conveyed through a coil. Strong permanent magnets such as neodymium cylinders are an example of a suitable structure, although others will be apparent. Where appropriate, the electrical current generated in each coil may be rectified and added to the currents generated in other coils, before being connected to an electrical grid or a battery. Strong springs are attached at the base of each bore to insure the permanent magnets bounce back and forth as pressure pulses come in.

[0038] Besides using electromagnetic induction based systems, the NEH tile system can also be constructed using piezoelectric crystals that are appropriately embedded inside the granular chains. Such crystals, when wired through an appropriately designed circuit system, generate electricity every time a pressure pulse passes through them. Piezo systems generate small amounts of current but are very convenient to use for a variety of low power applications.

[0039] When a long-lived impulse is incident on a granular chain, the system will convert that input impulse into a solitary wave train. Each solitary wave travels at a speed that is proportional to its energy content. So a solitary wave train progressively spreads out after it forms. Below we describe the typical properties of a solitary wave train that forms out of a random signal.

[0040] We consider a granular chain where all the grains are nominally identical spheres. The extremal distance between two adjacent grains each of mass M and in contact when solitary wave number i passes through a grain-grain contact can be described by: <
<img class="EMIRef" id="122318103-imgf000014_0001" />
>= A(i - 1)<2> + B, A < 0, B > 0, (2) where t denotes time. The quantity i runs from 1 to P, where P refers to the total number of solitary waves in the solitary wave train that forms in the granular chain. In the above equation, i = 1 refers to the leading solitary wave in the train. This wave has the largest amplitude and hence carries the largest energy. As such, < r <aax>{t<'>) >> 0 and this fixes the number of solitary waves in the train. The coefficients A and B are given below,

34.77

<?> = - -<2> (?^) <'> (n-<20133> (F)<0>-<3401>, (3a)

<B> = fe)<"0'6530> (7T-<0378> (F)<0>-<6745>, (3b) where a, T, F refer to the force constant in the interaction potential (see Eq. 1), time period of the extended perturbation and the average/typical amplitude of the oncoming force, respectively. In the above n = 5/2.

[0041] To summarize the exemplary embodiment, Layer 1 receives the oncoming energy and breaks it down into an impulse or a series of impulses. Long lived pulses are automatically broken down by Layers 1 and 2 into a sequence of solitary waves of decreasing energy content. This latter property can be well-characterized as discussed above and may be realizable in terms of appropriate circuits to construct electrical equivalents of Layers 1 and 2. In one realization of Layer 2, piezoelectric crystals can be embedded within the chains to directly generate electrical current. However, piezoelectric crystals make it possible to generate small amounts of current. To generate significant amounts of electricity, it may be possible to use Layer 3 which is an array of strong permanent magnetic cylinders inside densely wound coils. Electromagnetic induction is used to generate electricity in this approach. This approach is also the most efficient way to generate electrical current.

[0042] Although the present invention has been described with respect to one or more particular embodiments, it will be understood that these embodiments are intended to be exemplary and that other embodiments of the present invention may be made without departing from the spirit and scope of the present invention.
    



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