Seismic Wave Damping System

Abstract

A seismic wave damping system, and a corresponding method, includes elements, embedded within a host medium, the elements defining a seismic damping structure, and the elements being arranged to form a border of a protection zone. The seismic damping structure is configured to attenuate power of a seismic wave, traveling from a distal medium to the host medium, that is incident at the protection zone. The seismic damping structure is characterized by a resonance frequency. The system further includes an anti-resonance damping structure positioned within the protection zone and configured to dampen a residual wave propagating within the protection zone at the resonance frequency. Embodiment systems offer synergistic advantages because resonance frequencies of seismic wave damping structures may be predicted by calculation and an anti-resonance damping structure may be built to attenuate waves of primarily only specific resonance frequencies supported by the seismic wave damping structure.

Description

RELATED APPLICATION

This application claims the benefit of U.S. Provisional Application No. 62/769,517, filed on Nov. 19, 2018. The entire teachings of the above application are incorporated herein by reference.

GOVERNMENT SUPPORT

This invention was made with Government support under Contract No. FA8721-05-C-0002 awarded by the U.S. Air Force. The Government has certain rights in the invention.

BACKGROUND

Each year, on average, a major magnitude-8 earthquake strikes somewhere in the world. In addition, 10,000 earthquake-related deaths occur annually, where collapsing buildings claim the most lives, by far. Moreover, industry activity, such as oil extraction and wastewater reinjection, are suspected to cause earthquake swarms that threaten high-value oil pipeline networks, U.S. oil storage reserves, and civilian homes. Earthquake engineering building structural designs and materials have evolved over many years to attempt to minimize the destructive effects of seismic surface waves. However, even under the best engineering practices, significant damage and numbers of fatalities can still occur.

In particular, damage caused by earthquakes to critical structures, such as nuclear power plants, regional hospitals, military installations, airport runways, pipelines, dams, and other infrastructure facilities, exacerbates an earthquake disaster and adds tremendous cost and time of recovery. Even low-energy earthquakes resulting from human activity can cause significant damage. For example, wastewater reinjection practices used by the oil industry resulted in over 900 earthquakes in 2014-2015 in the state of Oklahoma, with a recent 2016 earthquake of magnitude 5.8. These continual earthquakes, although many may be small, can threaten extremely high-value above- and below-ground pipelines that control oil supply, storage, and transport in the U.S. This can present major economic and environmental concerns.

Certain structures have been proposed to protect buildings or other areas from the effects of seismic waves. In one way or another, however, all of those structures previously proposed are inadequate, in that they can only protect against a certain amount of seismic energy and still allow passage, or support propagation, of a certain amount of seismic energy. Previously proposed structures, therefore, are inadequate in that they allow certain types or amounts of seismic waves to enter an area that is intended to be protected.

SUMMARY

Earthquake engineering building practices apply primarily to new construction to decouple seismic energy traveling in the ground between the ground and building foundation, whereas existing high-value structures are typically unlikely to be retrofitted because this is cost prohibitive and because there is typically difficulty in accessing the structures. Historically, there were no free standing subsurface structures used to protect existing high value assets from incoming hazardous earthquake waves. More recently, some research groups have investigated investigating the possibility of using boreholes in front of an area to protect from seismic waves. These efforts have been primarily computer simulations, with one group conducting a small scaled field test. This field test involved using a surface seismic source, which is not representative of a location for an earthquake, because earthquake sources are at depth. This test examined the ability of a few holes to block the seismic energy in its near field. These attempts have been very limited in their usefulness and are not representative of earthquakes, their geometries, raypaths, hypocenters, or seismic wavelengths or amplitudes.

Structures described herein can overcome these challenges by providing broadband redirection and attenuation of ground motion amplitudes caused by earthquakes. Structures can provide for this by implementing an engineered, subsurface, seismic barrier (elastic wave attenuation structure), for example. In some structures, a form of a metamaterial is created. A metamaterial is a material engineered to have a property that is not found in nature. Metamaterials are made from assemblies of multiple elements fashioned from materials not found in the media in which they are embedded. In the case of the earth, boreholes and trenches would be considered metamaterials since they are air-filled or specialized, viscous, attenuating, fluid-filled.

As disclosed herein, a seismic barrier (or metamaterial) can include borehole array complexes or trench complexes that reflect, refract, absorb, divert, or otherwise impede destructive seismic surface waves from a designated “protection zone.” Seismic surface waves against which disclosed wave damping structures can protect include Rayleigh waves (ground-roll), shear waves, Love waves, and compressional waves.

Further, disclosed wave damping structures can overcome the limitations of using a vertical borehole structure only in front of an intended protection zone. Use of a vertical borehole structure only in front of an intended protection zone (between a seismic wave incoming toward the protection zone and the protection zone itself) is not effective enough, since most seismic waves will diffract around a vertical borehole structure vertically and still strike the protection area with considerable force. However, with respect to structures described herein, boreholes or trenches (example “elements” as used herein) formed at an angle with respect to the vertical, with lateral offset into the earth and toward a zone and surface structure to be protected, can better divert seismic waves farther away from the intended protection zone than straight, deep boreholes. In addition, seismic wave damping structures described herein and incorporating such borehole or trench elements have been demonstrated, through numerical modeling and bench scale measurements, to provide broadband seismic wave amplitude reduction.

By using angles for holes that point down below a structure to be protected, seismic wave power can be effectively diverted far underneath the structure. Angled holes forming groupings or tapered structures can be particularly helpful due to the vertical depths that surface waves can reach, which can be hundreds of meters or greater. Moreover, by using angled holes on multiple sides of a structure, an aperture between protective holes can be made small, effectively blocking most seismic energy from diffraction toward the protected zone, thereby significantly limiting any need for deep boring, which can be cost-prohibitive. A structural arrangement formed of at least two angled borehole elements with opposing orientations with respect to a vertical, thus forming a tapered aperture, can be referred to herein as a “muffler.” Such structures are described hereinafter in greater detail with respect to the drawings.

Furthermore, retrofitting a building area with embodiment wave attenuation structures can be done with much flexibility, because disclosed structures can be implemented farther from a structure to be protected, at least at the Earth's surface. Further, certain periodic groupings of boreholes, such as sawtooth-shaped groupings or other geometric groupings, may be employed and can increase a range of seismic wavelengths against which a disclosed structure can be made effective. Such layout geometries can advantageously be configured to cause reflected self-interference of a traveling seismic wave, thus reducing the waves's effective ground motion amplitude. Still further, structures described herein can be used in protecting areas of the ocean or ocean front from the destructive effects of sea waves, such as tsunamis.

In one structure described herein, a seismic wave damping structure includes a structural arrangement of at least two elements, each element defining an inner volume and containing therein a medium resistant to passage of an anticipated seismic wave having a wavelength at least one order of magnitude greater than a cross-sectional dimension of the inner volume of the elements. The structural arrangement can taper from an upper aperture to a lower aperture, the structural arrangement defining a protection zone at the upper aperture, the upper aperture being larger than the lower aperture. The structural arrangement can be configured to attenuate power from the anticipated seismic wave within the protection zone relative to power from the anticipated seismic wave external to the protection zone.

The structural arrangement can be further configured to attenuate power from a Rayleigh wave, or from at least one of a compressional, shear, or Love elastic wave.

The at least two elements can be boreholes in earth, and the upper aperture can be closer to a surface of the earth than the lower aperture. As an alternative, the at least two elements can be trenches in earth, while the upper aperture can still be closer to the surface of the earth than the lower aperture.

The anticipated seismic wave can be a seismic wave in earth, and the medium resistant to passage of the anticipated seismic wave can be air or at least one of a gas, water, or viscous fluid. The anticipated seismic wave can be a water wave, and the medium resistant to passage of the water wave can include a solid material. The upper aperture can be closer to an upper surface of the water in the absence of the anticipated water wave. As an alternative, the upper aperture can be in air, and the lower aperture can be in earth or water in absence of the anticipated water wave.

A depth of the lower aperture in earth or water can be on the order of 100 meters. Each element can further include a structural lining between the inner volume and an exterior of the element. A width of the upper aperture can be on the order of 0.5 km. Particular preferred dimensions for particular upper apertures can be predicted using expressions given hereinafter.

Each of the at least two elements can include a plurality of discrete sub-elements, each of the sub-elements defining a respective sub-element inner volume and containing therein the medium resistant to passage of the seismic wave. Cross-sections of respective discrete sub-elements corresponding to at least one of the elements can be located at points collectively defining a hexagon.

The damping structure can also include a plurality of structural arrangements defining a superstructure, and the protection zone can encompass, at least partially, upper apertures of respective arrangements of the plurality of structural arrangements. The superstructure can be configured to attenuate power from the elastic wave within the protection zone relative to power from the seismic wave external to the protection zone. The protection zone can extend, in length, from one of the at least two elements to the other at the upper aperture. The protection zone can have a width, measured perpendicular to the length, of approximately 5%, 10%, 25%, 50%, 75%, or 100% of the length. The protection zone can be defined by a region, bounded at least partially by the at least two elements, within which the structural arrangement is configured to attenuate power from the seismic wave by at least 10 dB in power within the protection zone relative to power from the seismic wave external to the protection zone. Larger reductions in power have also been demonstrated by the authors using numerical simulations and scaled measurements.

The damping structure can also include an incident grouping of elements situated at a border of the protection zone expected to receive the seismic wave, as well as a transmission grouping of elements situated at a border of the protection zone opposite the incident grouping. The structural arrangement of at least two elements can include one element of the incident grouping and one element of the transmission grouping. Each element of the incident and transmission groupings of elements can have upper and lower ends thereof, and each element of the incident and transmission groupings of elements can define an inner volume and contain therein the medium resistant to passage of the anticipated seismic wave. The incident and transmission groupings can form a superstructure.

Upper ends or lower ends of respective elements of the incident or transmission grouping may be situated along an element row. Upper ends or lower ends of respective elements of the incident or transmission grouping may further be situated along a plurality of substantially parallel rows to form an element array. Upper ends or lower ends of respective elements of the incident or transmission grouping may be situated to form a substantially periodic pattern. The substantially periodic pattern may be a substantially sawtooth pattern, or the pattern may be configured to cause constructive or destructive interference of diffracted portions of the anticipated seismic wave diffracted from respective elements.

The damping structure can also include an electro-mechanical generator configured to generate or store electrical power using mechanical power from the anticipated seismic wave.

In another disclosed seismic wave damping structure the structure may include a structural grouping of elements, each element of the structural grouping defining an inner volume and containing therein a medium resistant to passage of an anticipated elastic wave having a wavelength at least one order of magnitude greater than a cross-sectional dimension of the inner volume of the elements. Each element of the structural grouping may have an upper end and a lower end thereof defining a first line from the upper end to the lower end. The first line can form an acute angle with a second line defining a direction of travel of the anticipated elastic wave toward a protection zone. The structural grouping of elements may be configured to attenuate power from the seismic wave within the protection zone relative to power from the seismic wave external to the protection zone.

The grouping of elements can be further configured to attenuate power from a Rayleigh elastic wave, or from at least one of a compression, shear, or Love seismic wave. Each element may be a borehole in earth or a trench in earth, with the upper end closer to a surface of the earth than the lower end.

At least one of the elements can include a plurality of discrete sub-elements substantially parallel to each other, each of the sub-elements defining a respective sub-element inner volume and containing therein the medium resistant to passage of the elastic wave. Cross-sections of respective discrete sub-elements may be located at points in a cross-sectional plane collectively defining a hexagon.

Upper ends or lower ends of respective elements can be situated along an element row. Furthermore, upper ends or lower ends of respective elements can be situated along a plurality of substantially parallel rows to form an element array. Upper ends or lower ends of respective elements can be situated to form a substantially periodic pattern, and the substantially periodic pattern may be a substantially sawtooth pattern. The substantially periodic pattern may also be configured to cause constructive or destructive interference of diffracted portions of the anticipated seismic wave diffracted from respective elements.

The grouping of elements can be an incident grouping of elements situated at a border of the protection zone at which the anticipated seismic wave is expected to be incident. The damping structure can further include a transmission grouping of elements situated at an opposite border of the protection zone opposite the incident grouping. Each element of the transmission grouping can define an inner volume and contain therein a medium resistant to passage of the anticipated seismic wave having a wavelength at least one order of magnitude greater than a cross-sectional dimension of the inner volume of the element. Each element of the transmission grouping may have an upper end and a lower end thereof defining a first line from the upper end to the lower end, the first line forming an obtuse angle with a second line defining a direction of travel of an attenuated anticipated seismic wave away from the protection zone. The transmission grouping of elements can be configured to attenuate power from the elastic wave within the protection zone relative to power from the elastic (e.g., seismic) wave external to the protection zone and transmitted through or around the incident grouping of elements.

A separation of upper ends of elements of the incident grouping from upper ends of respective elements of the transmission grouping can be on the order of 0.5 km. The protection zone can be further defined by a region, bounded at least partially by the incident and transmission groupings, within which the incident and transmission groupings are configured to attenuate power from the seismic wave by at least 10 dB within the protection zone relative to power from the seismic wave external to the protection zone.

Also disclosed is an elastic wave damping structure that can include first means for damping an anticipated seismic wave and second means for damping an anticipated seismic wave, wherein a combination of the first means and the second means forms an upper aperture and a lower aperture. The upper aperture can taper to lower aperture, the combination defining a protection zone at the upper aperture. The structural arrangement can be configured to attenuate power from the anticipated seismic wave within the protection zone relative to power from the anticipated seismic wave external to the protection zone.

Still another disclosed seismic wave damping structure can include first means configured to attenuate power from an anticipated seismic wave and at least one second means configured to attenuate power from the seismic wave. Each of the first and second means can define a first line from an upper end of the means to a lower end of the means, the first line forming an acute angle with a second line defining a direction of travel of the anticipated seismic wave toward a protection zone.

One limitation of the seismic wave damping structures described above is that they do not block or attenuate all power from an anticipated seismic wave from entering a protection zone. Some seismic power may still enter the protection zone, particularly higher frequencies. Specifically, a seismic wave damping structure is described in the examples above and in other parts of this specification can be very effective at attenuating power from an anticipated seismic wave in lower frequency ranges. Nonetheless, the seismic wave damping structures described above may permit propagation of frequencies, particularly frequencies in a higher frequency range, within the protection zone.

Advantageously, as described herein, the seismic wave damping structures described above may have resonant frequencies that can be predicted based on parameters of the structure and based on properties of the protection zone, particularly earth, soil, ground, etc. within the protection zone, in which the seismic wave damping structure is embedded. As further described hereinafter, the disclosed seismic wave damping structures may be advantageously combined with anti-resonance damping structures described hereinafter according to various embodiments. These combinations may form a seismic wave damping system that is extremely effective at preventing seismic waves from damaging protected structures in a protection zone. While some of the damping structures described herein have previously been known, they have not been used in the sense or combination described in this specification, namely as anti-resonance damping structures. As described hereinafter, when used as anti-resonance damping structures, they may be configured to address, specifically, resonance frequencies supported by the disclosed seismic damping structures, as described above.

When used in combined systems as noted above, and as described further hereinafter, seismic wave damping structures and anti-resonance damping structures have particular synergistic effects when used in combination with each other. On one hand, anti-resonance damping structures described herein may increase effectiveness of seismic wave damping structures described herein in damping residual seismic waves that propagate within the protection zone, which may be allowed to pass the seismic wave damping structures. On the other hand, synergistically, based on properties of a host medium in which the seismic damping structure is embedded, resonance frequencies supported by a given seismic wave damping structure may be specifically predicted based on one or more properties of the host medium and on one or more physical properties of the seismic damping structure. A corresponding anti-resonance damping structure may, therefore, be configured to dampen specifically a residual wave propagating within the protection zone at the resonance frequency. A synergy of this arrangement is that the anti-resonance damping structure need not be configured to address all possible seismic frequencies, potentially requiring more complex and extensive engineering. Instead, the anti-resonance damping structure may be built and configured to dampen, specifically, only one or more resonance frequencies supported by the seismic damping structure, where these resonance frequencies may be specifically predicted based on properties of the host medium and seismic damping structure. Accordingly, according to embodiments described hereafter, particular synergies may be obtained, which have not been contemplated or described before.

In one particular embodiment disclosed herein, a seismic wave damping system includes elements, embedded within a host medium. The elements define a seismic damping structure and are arranged to form a border of a protection zone. The seismic wave damping structure (also referred to herein as “seismic damping structure”) is configured to attenuate power of a seismic wave, traveling from a distal medium outside of the protection zone to the host medium in which the seismic damping structure is embedded. Thus, the seismic wave is incident at the seismic damping structure, at the protection zone. The seismic wave damping structure is characterized by one or more resonance frequencies that may be predetermined (i.e., known by prediction or modeling or calculation).

The resonance frequency may be a function of a depth of the elements embedded in the host medium and of a property (i.e., a physical property) of the host medium.

The anti-resonance damping structure may be configured to dampen the residual wave by being mechanically tuned to the resonance frequency. In other words, the anti-resonance damping structure may be built according to dimensions and specifications that will allow it to specifically dampen the resonance frequency, or the anti-resonance damping structure may be built, and then calibrated in such a manner that it absorbs or scatters preferentially at the resonance frequency, a harmonic of the resonance frequency, or a subharmonic of the resonance frequency. The anti-resonance damping structure may be configured to dampen the residual wave by being mechanically tuned to a harmonic of the resonance frequency or to a subharmonic of the resonance frequency. The anti-resonance damping structure may include two or more anti-resonance damping structures that are configured to dampen the residual wave by being mechanically tuned to two or more respective frequencies selected from the group consisting of (i) the resonance frequency, (ii) harmonics of the resonance frequency, and (iii) sub harmonics of the resonance frequency.

The anti-resonance damping structure may include one or more Helmholtz resonators positioned on or within the host medium within the protection zone. The one or more Helmholtz resonators may be filled with a gas, such as air, or with water or a viscous fluid. The anti-resonance damping structure may be an array of cylinders or other shaped elements, such as meta-concrete cylinders, buried within the host medium within the protection zone.

The anti-resonance damping structure may be a seismic wave absorbing structure configured to dampen the residual wave by absorption. The seismic wave absorbing structure may be a mass-in-mass lattice. The host medium may be earth, and the anti-resonance damping structure may include an array of trees that are not placed in order naturally, but are rather planted in the earth in a specific configuration that includes being spaced periodically within the protection zone. The anti-resonance damping structure may include an array of scattering components that are positioned periodically on or within the host medium within the protection zone. The anti-resonance damping structure may include one or more towers positioned on the host medium within the protection zone. The one or more towers may be one or more flexible, steel-girded towers. The one or more towers may have heights, extending vertically from a surface of the host medium, such as a ground surface, between a few meters and hundreds of meters. In various examples, heights may be on the order of 300 m, on the order of 200 m or, on the order of 100 m, on the order of 50 m, on the order of 25 m, or on the order of 10 m, for example. The one or more towers may have heights, extending vertically from a surface of the host medium, of 100 m or less, such as between about 10 m and about 100 m. Heights of the one or more towers may be specifically configured such that the towers dampen the residual wave at the resonance frequency by dampening one or more harmonics or subharmonics of the resonance frequency. The one or more towers may have, each, a cross-sectional dimension, such as a diameter, on the order of 1 m, on the order of 5 m, on the order of 10 m, or on the order of 15 m, for example.

In another embodiment, a method of constructing a seismic wave protection zone includes embedding elements within a host medium, thus defining a seismic wave damping structure characterized by a resonance frequency and forming a border of a protection zone. The seismic wave damping structure is built or otherwise configured to attenuate power of a seismic wave traveling from a distal medium to the host medium, where the seismic wave may be anticipated to be incident at the protection zone. The method also includes positioning an anti-resonance damping structure within the protection zone and configuring the anti-resonance damping structure to dampen a residual wave propagating within the protection zone at the resonance frequency.

The method may further include positioning the anti-resonance damping structure within the protection zone and configuring the anti-resonance damping structure to dampen a residual wave by building or tuning the anti-resonance damping structure to dampen the resonance frequency, where the resonance frequency is predicted based on one or more properties of the host medium and one or more dimensions or other properties of the elements embedded within the host medium forming the seismic wave damping structure. Configuring the anti-resonance damping structure to dampen the residual wave propagating within the protection zone at the resonance frequency may include configuring the anti-resonance damping structure based on one or more properties of the host medium, such as a wave propagation velocity, and one or more properties of the elements, such as length (depth) in the host medium, such as depth in earth.

In a further embodiment, a method of seismic wave damping includes converting an incident seismic wave propagating in a distal medium outside a protection zone into a residual seismic wave propagating within the protection zone at one or more resonant frequencies. The method further includes dampening the residual wave within the protection zone via anti-resonance damping. The method may optionally include use or incorporation of any of the methods; elements; seismic wave damping structures, superstructures, or arrangements; and anti-resonance damping structures summarized hereinabove pertaining to other embodiments or further described hereinafter in relation to other embodiments.

BRIEF DESCRIPTION OF THE DRAWINGS

The patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawings will be provided by the Office upon request and payment of the necessary fee.

The foregoing will be apparent from the following more particular description of example embodiments of the invention, as illustrated in the accompanying drawings in which like reference characters refer to the same parts throughout the different views. The drawings are not necessarily to scale, emphasis instead being placed upon illustrating embodiments of the present invention.

FIG. 1A is a cross-sectional side view of an embodiment elastic wave damping structure that includes a downward-tapered, V-shaped structural arrangement with two elements.

FIG. 1B is a diagram of the embodiment of FIG. 1A without the above-ground building structure and with additional below-ground details as compared to FIG. 1A.

FIG. 2A is a top-view illustration of the structural arrangement of FIG. 1B.

FIG. 2B is a top-view illustration of a linear structural grouping of elements forming an embodiment elastic wave damping structure.

FIG. 2C is a top-view illustration of an embodiment elastic wave damping structure including an arrangement of two elements formed of pluralities of discrete sub-elements in hexagonal clusters.

FIG. 2D is a top-view illustration two structural groupings of elements, namely an incident structural grouping to 304a of elements that are arranged into substantially parallel element rows 226 on the incident side of the protection zone.

FIG. 2E is a top-view illustration of an embodiment wave damping structure including a superstructure with two of the structural arrangements illustrated in FIG. 1B.

FIG. 2F is a top-view illustration of a superstructure includes trench elements instead of borehole elements.

FIG. 3 is a cross-sectional side-view diagram showing how embodiments can be advantageously used to protect structures in the sea, such as an oil platform.

FIG. 4A is a cross-sectional, side-view illustration of incident and transmission structural element groupings, respectively, of elements on either side of a protection zone.

FIG. 4B is a cross-sectional, side-view illustration of an incident structural grouping of elements that can be employed to protect against incident water waves, such as tsunami waves.

FIG. 5 is a top-view illustration of an arrangement of elements forming an incident structural grouping of the elements in a sawtooth pattern.

FIG. 6 is a top-view illustration of an embodiment wave damping structure with a substantially sawtooth, substantially periodic pattern of elements that form an incident structural grouping.

FIG. 7 is a top-view illustration of an additional incident structural grouping of elements showing interference between reflected wavelets and showing constructive interference used to generate electrical power.

FIG. 8A is a top-view illustration of a protection zone formed between modeled arrays of borehole elements.

FIG. 8B is a three-dimensional (3D) illustration of the protection zone and borehole elements shown in FIG. 8A.

FIG. 8C is a 3D view of modeled trench elements in a 3D finite element meshing.

FIG. 9A shows model equations used in finite element analysis of embodiments.

FIG. 9B is a 3D representation of the finite element analysis.

FIG. 9C is a graph showing a source time function for rupture velocity of an earthquake measured and used as a source time function for the finite element analysis.

FIG. 9D is a graph showing power reduction calculated in a protection zone with and without embodiment wave damping structures.

FIG. 9E is collection of areal and depth view seismic wave snapshots calculated with and without embodiment seismic wave damping structures.

FIG. 10 is a table showing various high-value assets that may be protected, as examples, using embodiment elastic wave damping structures, with expected upper aperture and lateral extent sizes and numbers of boreholes for corresponding structures.

FIG. 11 is a series of diagrams and a graph showing the comparative effects of seismic cloaking on seismic wave propagation for a single, vertical barrier element compared with an embodiment angled structural element, as calculated using a finite difference 2D model of the barrier structures.

FIGS. 12A-12B show a diagram and equations illustrating semi-analytical simulation of acoustic propagation through a “tapered muffler” geometry.

FIGS. 13A and 13B are graphs showing transmission loss as a function of frequency for various acoustic “muffler” parameters, as calculated using the analytical tools shown in FIGS. 12A-12B.

FIG. 14 is a collage of photographs showing apparatus used to study, on a model scale, non-naturally occurring, man-made structures such as borehole arrays and trenches embedded in elastic media analogous to rock and compact soil using a machined, table-top scaled physical model.

FIGS. 15A-15C show measured results that illustrate the effects of a model V-trench machined in Delrin® block table-top experimental configuration. In particular, FIG. 15A and FIG. 15B show accelerometer time series traces measured in a center line across a homogeneous solid Delrin® block relative to the transducer source location, and FIG. 15C shows the model trench barrier structure in schematic form, along with accelerometer locations corresponding to the traces in FIGS. 15A-15B.

FIG. 16 shows earthquake magnitude reduction expected due to subsurface barrier structures, based on extrapolation from the measured and modelled results.

FIG. 17A is a schematic diagram illustrating an embodiment seismic wave damping system in its environment of use.

FIG. 17B is a schematic diagram illustrating, more particularly, the seismic wave damping system of FIG. 17A, without it's contextual environment of use.

FIG. 18A is a perspective, colored or shaded illustration of a V-shaped seismic muffler (seismic damping structure) used in connection with the anti-resonance damping structure of FIGS. 17A-17B to form an embodiment seismic wave damping system.

FIG. 18B is a color-coded or shaded, side-view graphical drawing illustrating particle velocity, in unit length per second, over a cross-sectional area showing the seismic damping structure of FIG. 18A.

FIG. 19A is a cross-sectional diagram illustrating geometry and terms terminology for an example conical shaped muffler seismic damping structure.

FIG. 19B is a diagram illustrating for different muffler geometries and their respective dimensions.

FIG. 19C is a graph illustrating elastic wave transmission loss as a function of wavelength corresponding to seismic frequencies spanning from 0.1-10 Hz for the muffler geometries illustrated in FIG. 19B.

FIG. 19D is a graph illustrating P and S wave transmission loss behavior for the four muffler examples illustrated in FIG. 19B as a function of seismic frequency.

FIG. 20A illustrates four different muffler geometries, wherein the inlet diameter is 0.1 km and the muffler wall slope is constant for all cases except the last case, which shows vertical a vertical wall muffler for comparison.

FIG. 20B is a graph illustrating wave transmission loss as a function of seismic wavelength corresponding to seismic frequencies from 0.1 Hz to 10 Hz.

FIG. 20C is a graph illustrating P and S wave transmission loss for the four muffler examples illustrated in FIG. 20A as a function of seismic wave incident frequency.

FIG. 21A is a diagram illustrating cross-sectional muffler geometry for a shallow sloping muffler model (a) having a wall slope and for a vertical wall muffler model (d).

FIG. 21B is a cross-sectional illustration of the example structures in FIG. 21A, along with color-coded or shaded illustration of numerically calculated damping characteristics.

FIG. 21C is a graph illustrating transmission loss as a function of seismic frequency for the two example structures illustrated in FIG. 21A.

FIG. 22A illustrates a time source a source time function from the Hector Mine 1999 earthquake.

FIG. 22B is a graph showing amplitude as a function of frequency for the Hector Mine earthquake.

FIG. 23A is a color-coded or shaded, cross-sectional illustration of a seismic wave field in presence of an embodiment seismic wave damping system that includes a Helmholtz resonator array anti-seismic damping structure as part of the system.

FIG. 23B is a graph showing the source function for the Hector Mine earthquake, injected by simulation into the model represented in the graph of FIG. 23A.

FIG. 23C illustrates calculations by which damping of a Helmholtz resonator array may be determined.

FIG. 23D is a schematic diagram illustrating the Helmholtz resonator array, anti-resonance damping structure graphically illustrated in FIG. 23A.

FIG. 23E is a graph illustrating seismic amplitude as a function of frequency, as reduced by a seismic wave damping structure alone, and as reduced by the seismic wave damping structure in combination with the Helmholtz resonator array example anti-resonance damping structure of FIG. 23A.

FIG. 24A is a color-coded or shaded, vertical view cross-sectional graph of a seismic wave damping structure embodiment including a tower or tree array example anti-resonance damping structure in presence of a shear wave of the Hector Mine earthquake.

FIG. 24B is a graph showing the source function for the Hector Mine earthquake, which was injected into the model illustrated in FIG. 24A.

FIG. 24C shows an equation that can be used to calculate damping frequency for the array of resonators (towers or trees, e.g.) illustrated in FIG. 24A.

FIG. 24D is a more detailed illustration of the array of resonators illustrated in FIG. 24A.

FIG. 24E is a graph illustrating damping, particularly seismic amplitude decrease, obtained by using the combination of seismic wave damping structure and anti-resonance damping structure of the embodiment system illustrated in FIG. 24A.

FIG. 25A is a cross-sectional illustration of a meta-concrete array example of buried cylinders used as an anti-resonance damping structure as part of an embodiment system, illustrated in presence of the Hector Mine earthquake source function used in the model.

FIG. 25B is a graph illustrating the Hector Mine earthquake source function injected into the model illustrated in FIG. 25A.

FIG. 25C is a cross-sectional view of the meta-concrete array example anti-resonance damping structure illustrated in FIG. 25A.

FIG. 25D is an equation illustrating how resonance frequency of the medic concrete array of FIG. 25A may be calculated.

FIG. 25E is a graph illustrating seismic amplitude reduction damping that may be obtained using the meta-concrete array example seismic anti-resonance damping structure as part of an embodiment seismic wave damping system illustrated in FIG. 25A.

FIG. 26 is a flow diagram illustrating an embodiment procedure for seismic wave damping.

DETAILED DESCRIPTION

A description of example embodiments of the invention follows.

Seismic Wave Damping Structures

FIG. 1A is a cross-sectional side view of an embodiment elastic wave damping structure that includes a downward-tapered, V-shaped structural arrangement with two elements 30a and 30b. The structural arrangement defines a protection zone 40 between the elements 30a and 30b, where a building 20 that is desired to be protected is located above the protection zone. The damping structure is configured to protect the building 20 from incident surface waves 60 that initially impinge on the element 30a as a result of an earthquake 50. A small amount of seismic wave power still enters the protection zone 40, such that there are attenuated surface waves 60′ under the building 20. Some wave energy also travels underneath and around the elements 30a and 30b to become transmitted surface waves 60″.

FIG. 1B is a cross-sectional side view of an embodiment elastic wave damping structure that includes a structural arrangement 100 with two elements 102a and 102b. Each of the elements 102a and 102b defines an inner volume 116 that contains therein a medium resistant to passage of an anticipated incident elastic wave 114. The wave 114 has a wavelength at least one order of magnitude greater than a cross-sectional dimension of the inner volume of the elements, such as the diameter d of the element 102a. The element 102a may be referred to as having an upper end 108 and a lower end 110. Likewise, while not marked in FIG. 1B, the element 102b has upper and lower ends similar to those of element 102a.

The structural arrangement 100 tapers from an upper aperture 104 to a lower aperture 106. The structural arrangement 100 defines a protection zone, also referred to herein as a “structural protection zone” 112, at the upper aperture 104. The structural arrangement 100 is configured to attenuate power from the anticipated, incident elastic wave 114 within the protection zone 112, relative to power from the anticipated elastic wave external to the protection zone. Thus, outside of the protection zone 112, the incident elastic wave 114 has a given power, which can be referred to as seismic power, where the elastic wave 114 propagates in earth 118. However, due to the attenuation of power within the protection zone 112, which is caused by the structural arrangement 100, a component 114′ of the elastic wave 114, which is transmitted into the protection zone 112, has an attenuated seismic power relative to the incident elastic wave 114 that is incident at the structural arrangement 100. In particular, the elastic wave 114 is incident at the element 102a of the structural arrangement 100.

The upper ends of the elements 102a and 102b are located at greater (more positive) Z values along the Z axis that is illustrated in FIG. 1B, while the lower ends 110 are located at smaller positive (more negative), Z values than the upper ends 108.

The structural arrangement 100 can be configured to attenuate power from a Rayleigh wave, or from at least one of a compressional, share, or love elastic wave. Thus, where the elastic weight 114 is a seismic wave, for example, the structural arrangement 100 is, advantageously, effective for attenuating surface seismic waves, as well as seismic waves of other types.

The elements 102a-b can be boreholes in earth, and, as described hereinabove, the upper aperture 104 can be closer to a surface 120 of the earth 118 than the lower aperture 110. However, in other embodiments, the two elements 102a and 102b can be trenches in the earth 118, as described hereinafter in connection with FIG. 2F. For either boreholes or trenches, or other configurations of the elements 102a-b, the upper aperture 104 can be closer to a surface 120 of the earth 118 than the lower aperture 110, as illustrated in FIG. 1B. The lower aperture 110 is at a depth 119 in the earth below the surface 120. In many embodiments, the length of the upper aperture, as also illustrated in FIG. 2A and described below, can be on the order of 0.5 km, for example.

Furthermore, continuing to refer to FIG. 1B, in cases where the anticipated incident elastic wave 114 is a seismic wave in the earth, the medium contained in the inner volume 116 and resistant to passage of the wave 114 can be air, for example. However, in other embodiments, the medium contained in the inner volume 116 can be at least one of a gas, water, or viscous fluid. In all of these cases of air, gas, water, or viscous fluid, for example, the medium does not transmit seismic power nearly as well as the earth 118. Hence, the medium resists passage of the wave 114, and seismic power can be deflected, reflected, diffracted, or otherwise dissipated before it enters the protection zone 112 as the attenuated wave 114 with attenuated seismic power. Moreover, given the tapered shape of the structural arrangement 100, which can also be considered to be angles of the elements 102a and 102b with respect to the z-axis, seismic power can be directed downward, away from any structures or objects desired to be protected within the structural protection zone 112.

The limited-size, lower aperture 106 helps to prevent leakage of seismic power around the element 102a, at which the seismic wave 114 is incident, and up to the protection zone. Such leakage presents some limitation on the effectiveness of the element 102a as a seismic shield; however, it should also be recognized that, in other embodiments, such as that illustrated in FIGS. 4B and 5-7, a grouping of elements, such as the element 102a, may still be advantageously used on an incident side of the structural protection zone 112 in order to limit seismic power reaching the protection zone. Thus, while the element 102b on the side of the protection zone opposite the incident side causes the structural arrangement 100 to be much more effective, embodiments can nevertheless employ an array of elements 102b on the incident side of the protection zone, where the elastic wave 114 is expected to be incident, with some helpful attenuation and deflection of seismic power reaching the protection zone.

The structural arrangement 100 illustrates a key feature of many embodiments, the ability to be effective in broadband shielding against a wide range of seismic wavelengths. In particular, broadband wavelengths longer than the lower aperture 106 are most effectively blocked by the structural arrangement 100. While higher frequency, shorter wavelength seismic waves with wavelengths shorter than the lower aperture 106 are not as effectively shielded by the arrangement 100, the majority of seismic power is typically present at the very low frequencies and longer wavelengths that are less able to enter through the lower aperture.

FIG. 2A is a top-view of the structural arrangement 100 illustrated in FIG. 1B, viewing the X-Y plane, down along a line of sight following the Z-axis illustrated in FIG. 1B. As illustrated in FIG. 2A, a length 222 of the upper aperture 104 can be considered to extend the entire length between the upper end of the element 102a and the upper end of the element 102b. This is because, between the elements 102a and 102b, there is some degree of attenuation of the incident wave 114 at all points, even though the attenuation can vary. Although the cross-sectional profiles of the elements 102a and 102b are round, it should be understand that cross-sectional profiles of elements in other embodiments may have any shape, such as oval, rectangular, or any other shape. However, if drilling is used to form the elements, it is likely most convenient to bore out elements with circular cross sections.

The protection zone 112 can also have a width 224 that can extend, measured perpendicular to the length 222, approximately 5%, 10%, 25%, 50%, 75%, or 100% of the length 222, for example the region defined as the protection zone 112 can also be defined in terms of a particular degree of attenuation of the seismic waves 114 that can be achieved. For example, the protection zone 112 can be defined by a region, bounded by at least partially by the elements 102a and 102b, within which the structural arrangement 100 is configured to attenuate power from the elastic wave 114 by at least 10 dB. This attenuation can be within the protection zone 112, relative to power from the elastic wave 114 external to the protection zone 112. Furthermore, other criteria can be used to select the protection zone, such as the region within which a 3 dB attenuation of seismic power is obtained, or within which more than another given value, such as more than 30 dB of attenuation is obtained, for example.

FIG. 2B illustrates that a structural arrangement can have more than just two elements, such as the elements 102a and 102b in FIG. 1B and FIG. 2A. In particular, four elements 102a are oriented along a line (element row) 226, in the direction of the y-axis, between the incident wave 114 and the protection zone 112. The four elements 102a form a structural grouping 232 of elements. Because these elements are on a side of the production zone 112 that is expected to receive the incident seismic wave 114, the structural grouping 232 can be referred to as an “incident” structural grouping of elements.

FIG. 2C is a top view of two elements 202a and 202b. The element 202a is formed of a plurality of discrete sub-elements 208a. The discrete set of elements 208a can be smaller than the element 102a in FIG. 1B, or they may be of the same size as element 102a. The element 202a forms a cluster with the sub-elements situated to increase structural strength. In particular, in FIG. 2C, the sub-elements 228a are located such that cross-sections of the respective sub-elements 208a corresponding to the element 202a are located at points collectively defining a hexagon 230. Similarly, the element 202b is formed of discrete sub-elements 208b located at points defining the hexagon 230. The cross sections illustrated in FIG. 2C are located in a cross-sectional plane that is the X-Y plane shown in FIG. 2C. However, another example cross-sectional plane in which the elements can be located at points forming a hexagon is a plane perpendicular to the elements 202a, which do not extend downward along the Z axis.

As is understood in the art of mechanical engineering, the hexagon formation for structural elements can be particularly strong. However, it should be understood that discrete sub-elements can be arranged in other orientations, such as in groups of four, in pentagon or other polygon shapes, or in other arrangements, for example.

FIGS. 2D-2F illustrate other embodiment arrangements of elements. In particular, the arrangements illustrated in the top-view illustrations of FIGS. 2D-2F, in which there are a plurality of structural arrangements, constitute other example orientation patterns.

FIG. 2D, in particular, illustrates two structural groupings of elements, namely an incident structural grouping to 234a of elements that are arranged into substantially parallel element rows 226 on the incident side of the protection zone 112, which is the side at which the anticipated seismic wave 114 is anticipated to arrive toward the protection zone. Similarly, at the opposite side of the protection zone 112, the superstructure 240a includes two additional, substantially parallel element rows 226 of elements 102b that include a transmission structural grouping 234b of elements, at the side of the protection zone 112 that receives seismic power transmitted (leaked) through the protection zone 112. The elements 102a on the incident side, in the incident structural grouping 234a, and corresponding elements on the transmission side, in the transmission structural grouping 234b, form respective, structural arrangements similar to the arrangement 100 shown in FIG. 1B.

The element rows of the respective groupings that are closest to the protection zone 112 may be considered to form respective structure arrangements, while the remaining, outer rows can be considered to form respective wider structural arrangements. However, alternatively, an inner row from one grouping may be considered to correspond to an outer row from the opposite grouping, and vice versa, such that all structural arrangements have similar aperture sizes.

As can also be seen in FIG. 2D, the protection zone 112 encompasses, at least partially, upper apertures of respective structural arrangements. It should be noted that, while the elements 102a and 102b each form regular, periodic arrays of elements, in other embodiments, the elements 102a and 102b can each have other formations, and do not need to be situated in element rows. Furthermore, it should be understood that, in other superstructures within the scope of the present disclosure, there can be unequal numbers of elements in an incident structural grouping and a transmission structural grouping. In particular, respective incident-side elements and transmission-side elements can still form upper apertures of one or more respective, structural arrangements, similar to that shown in FIG. 1B.

FIG. 2E is a top-view illustration of a superstructure 240b that includes two structural arrangements. In particular, one structural arrangement is formed by a first of the elements 102a and a first of the elements 102b, while a second structural arrangement similar to that shown in FIG. 1B is formed by a second of each of the groupings 102a and 102b of elements.

FIG. 2F is a top-view illustration of a superstructure 240c that includes trench elements 203a and 203b. In particular, the top view in FIG. 2F shows that the trenches 203a and 203b are not boreholes, but instead are substantially rectangular in their cross-sectional profiles. Each of the trench elements 203a-b defines an inner volume in the earth, filled with a medium that resists transmission of the seismic wave 114 toward the protection zone 112. Furthermore, one pair of the trench elements 203a-b forms a structural arrangement similar to the arrangement 100 in FIG. 1B, while a second pair of the elements 203a and 203b forms a second structural arrangement. The trench elements 203a and 203b form two separate structural arrangements, with respective upper and lower apertures, similar to the upper and lower apertures, 104 and 106, respectively, in FIG. 1B, for example.

FIG. 3 is a cross-sectional side-view diagram showing how embodiments can be advantageously used to protect structures in the sea, such as a sea platform structure 336. The platform 336 stands above a surface 338 of the sea, while legs of the platform are anchored into the earth 118 below the seafloor 337. It is desirable to protect the sea platform 336, which can include an oil rig or other structure in the sea, by forming a protection zone 312 around the platform.

In FIG. 3, the protection zone 312 is formed at the upper aperture of elements 302a and 302b. The elements 302a-b are anchored into the earth 118 at their respective lower ends 110, while their upper ends 108 extend above the sea surface 338. In other embodiments, the lower ends 110 of the elements 302a-b are not anchored into the earth, but instead, the elements 302a-b are suspended mechanically, such that they extend only a certain distance below the surface 338 of the sea. In either case, the elements 302a-b redirect power from an incident elastic, water wave that would otherwise be expected to be incident at the sea platform 336 in its full strength.

The elements 302a-b each form an inner volume containing therein a medium resistant to passage of the wave 314. For the case of water waves, this material can be a solid, for example. In this way, the elements 302a-b may be formed of wood, metal, or another structure. The elements 302a-b can be similar to pylons, for example.

It should be understood that embodiment elastic wave damping structures that are used to protect against water waves are not limited to a single structural arrangement, such as the one shown in FIG. 3. In other embodiments, any one of the groupings of arrangements or discrete sub-elements that are illustrated in FIGS. 2B-2F may be used to protect against water waves, for example. Furthermore, in some embodiment wave damping structures, only a plurality of elements 302a on the incident side of the protection zone are used, such that they form an incident structural grouping. Furthermore, as described hereinafter in connection with FIG. 4B, for example, structural groupings of elements can be used to protect against damaging water waves expected to be incident on land, such as tsunami waves.

FIG. 4A is a cross-sectional, side-view illustration of incident and transmission structural groupings 434a and 434b, respectively, of elements on either side of a protection zone 112. The incident structural grouping 434 includes three elements 102a on the incident side of the protection zone 112. Each of the elements 102a has the upper end 108 and the lower end 110, similar to the elements described in connection with FIG. 1B. Each element 102a defines a line 438a extending from the upper end 108 to the lower end 110. This line forms an acute angle 440a (less than 90°) with a line 436 that defines a travel direction of the wave 114 toward the protection zone.

A grouping of two or more of the elements 102a, with the acute angle for 440 a, can form an elastic wave damping structure having an incident structural grouping of elements. As described hereinabove, the acute angle 440a serves to redirect power from the seismic wave 114 around (below) the elements 102a. Where a depth 119 of the elements 102a (illustrated in FIG. 1B) is sufficiently large, and where the elements extend sufficiently below and toward the protection zone 112, the elements can attenuate power from the wave 114 that reaches the protection zone. Preferred dimensions and orientation of the elements 102a-b can be understood, in part, by reference to an acoustic “muffler” described hereinafter in connection with FIG. 12.

The elements 102a also include an optional structural lining 439 between the inner volume of the element and the exterior of the element (the earth 118). Structural linings may be helpful when used in certain types of earth where there is danger of the structural elements collapsing, or where there is danger of the structural elements filling with water in an undesirable manner, for example. Such structural linings can include PVC, other types of plastic, metal jackets, or any other suitable type of lining material known in the art of civil and mechanical engineering, for example.

While an incident structural grouping, such as the grouping 434a, can provide some protection, it may be useful to include the transmission structural grouping of elements 434b, which is optional. The grouping 434b includes elements 102b, with a line 438b extending from the upper end thereof to the lower end thereof forming an obtuse angle 440b with the line 436 defining the direction of travel of the elastic wave. In this way, an upper aperture and a lower aperture are formed, with the protection zone 112 at the upper aperture. The lower aperture is smaller than the upper aperture, thus preventing leakage of seismic power up into the protection zone 112. It should be understood that, while the incident structural grouping 434a is oriented with successive elements oriented along the X direction, arrays of elements oriented along the Y direction, such as those illustrated in FIGS. 2B and 2D-F provide further advantages, including wider protection zones.

While a superstructure is not specifically annotated in FIG. 4A, it should be understood that the incident grouping 434a and transmission grouping 434b together can be considered to form a superstructure 435. Such a superstructure is particularly well suited to preventing power from incident waves from entering the protection zone 112. Combinations of incident and transmission groupings of elements can significantly reduce a depth to which borehole or other elements need to be drilled in order to reduce seismic power by a given amount. This has significant cost advantages, as increasing bore depths are significantly expensive. In many embodiments, significant attenuation, such as 34 dB of attenuation, is expected, even where depths of lower apertures in the earth, or below the surface of water, are only on the order of 100 m, for example. The incident structural grouping 434a and optional transmission grouping 434b together form a superstructure 435.

FIG. 4B is a cross-sectional side-view illustration of an incident structural grouping 437 of elements 402 that can be employed to protect against incident elastic water waves 314, such as a tsunami wave. The elements 402 are anchored into the earth 118 near a shore of the sea expected to receive an incident wave. The elements 402 can be solid, similar to the elements 302a and 302b illustrated in FIG. 3, for example.

FIGS. 5-7 are top-view illustrations of additional incident structural groupings of elements with different configurations and functions. It should be understood that optional transmission-side structural groupings of elements can have similar configurations and may form part of other embodiment elastic wave damping structures, even though transmission groupings are not illustrated in FIGS. 5-7.

FIG. 5 is a top-view illustration of an arrangement of elements 102a forming an incident structural grouping 534 of the elements in a sawtooth pattern 542. The incident structural grouping 534 is arranged between the incoming, expected incident seismic wave 114 and the protection zone 112. As described hereinafter in connection with FIG. 9E, for example, substantially sawtooth-type groupings of elements can be situated on both the incident side of the protection zone at which the anticipated seismic wave is expected to the incident, and also on the transmission side of the protection zone, the side of the protection zone opposite the incident side. Thus, with sawtooth-type grouping of elements on both sides of the protection zone, elements from respective groupings can form one or more structural arrangements similar to the arrangement 100 in FIG. 1B. Therefore, superstructures including sawtooth-type groupings, just as superstructures including element rows or element array-type groupings, can also have the significant advantage of providing relatively broadband protection against incident seismic waves.

FIG. 6 is a top-view illustration of a substantially sawtooth periodic pattern 642 of elements 102a that form an incident structural grouping 634 of elements.

The substantially periodic, substantially sawtooth pattern 642 has the additional advantage of including more than a single row of elements in order to provide additional attenuation. Furthermore, the multiple-sawtooth pattern can extend a greater width along the y-axis, for example, thus providing a wider protection zone 112.

FIG. 7 is a top-view illustration of an additional incident structural grouping of elements 102a that illustrates interference of reflected wavelets from the elements of a grouping. In particular, the seismic wave 114 impinges on the incident grouping of elements 102a with an incident seismic wavefront 746. As is understood in the art of wave mechanics, the respective elements 102a will reflect some of the seismic power incident on them in the form of seismic wavelets 748. The reflected seismic wavelet 748 from the respective elements 102a will interfere with each other, either constructively or destructively in various positions that depend on separation of the elements 102a, as well as the wavelength of the incident seismic wave.

The reflected seismic wavelength wavelets 748 also can interfere constructively or destructively with the incident seismic wave 114, creating zones of the incident region in which seismic power is potentially greater than that which is incident, and also regions in which incident and reflected waves destructively interfere with each other to diminish significantly the intensity of seismic waves that are present in a given position. This effect can be exploited to protect certain positions on the incident side of the protection zone 112 having elements that are desired to be protected.

Interference effects can also be exploited advantageously to generate electrical power electro-mechanically. In particular, as illustrated in FIG. 7, an electromechanical power generator 744 is located at a position of expected constructive interference between the incident and reflected waves, such that the magnified mechanical power from the waves is converted into electrical power in order to provide power during a power outage due to an earthquake causing the seismic wave, for example.

Finite element modeling has been employed to predict attenuation of waves of various embodiments seismic wave damping structures. FIGS. 8A-8C, 9A-9B, and 9C-9D illustrate some of these methods and results.

FIG. 8A is a top-view illustration of a protection zone 112 formed between model borehole elements 802a and 802b. The model borehole elements 802a are arranged in element rows and include an incident grouping of elements 834a. Similarly, the modeled elements 802b are arranged in rows on the opposite side of the protection zone from the incident grouping to form a transmission grouping 834b. As also illustrated in FIG. 8A, a 3D section of earth with a array of the borehole elements 802a therein is an example of a “metamaterial” as used herein.

FIG. 8B is a three-dimensional (3D) illustration of what is shown in FIG. 8A. In particular, it is noted that the incident grouping 834a of elements 802a, together with the transmission grouping of elements 802b, form a superstructure 840, with the protection zone 112 there between. Individual locations in the 3D meshing represent points used for the finite element analysis.

FIG. 8C is a 3D view of trench elements 803a and 803b in a 3D finite element meshing. The modeled trench element 803a is similar to one or more of the trench elements 203a illustrated in FIG. 2F. Likewise, the modeled trench element 803b is similar to one or more of the trench elements 203b illustrated in FIG. 2F.

The finite element analysis performed on the analytical models illustrated in FIGS. 8A-8C indicate that the borehole based seismic wave damping structure of FIGS. 8A-8B, and the trench-based seismic wave damping structure illustrated in FIG. 8C reduce direct seismic wave power reaching the protection zone by more than 40 dB. It is noted that a diffractive component upwelling through the V-shaped structure has been calculated to be 22 dB lower than the peak seismic wave power observed for the same location in the protection zone without the implementation of the damping structures.

FIG. 9B is a 3D representation of the finite element analysis, with the direction of the incident wave 114 shown.

FIG. 9C is a graph showing a source time function for rupture velocity of an earthquake measured in California in 1991. This source time function was used as input to the finite element analysis. The seismic event is modeled for the response of the source function estimated for the Hector Mine earthquake in 1991 (Magnitude 7.1-USGS). Typical seismic frequencies are less than 1 Hz with minimal power above 1 Hz.

FIG. 9D is a graph showing power reduction calculated in a foundation 912, both with and without the borehole cloak (substantially sawtooth periodic pattern superstructure illustrated in FIG. 9E formed of elements 903a in a substantially sawtooth 942a grouping and elements 903b in a substantially sawtooth 942b grouping).

FIG. 9E is collection of areal and depth view seismic wave snapshots calculated with and without embodiment seismic wave damping (cloaking) structures, providing a finite difference model of the effects on seismic wave propagation from seismic cloaking. In particular, the top row shows an areal view of seismic wave snapshots, with and without cloaking structures. The bottom row shows a depth view of seismic wave snapshots, with and without cloaking structures.

FIG. 9E illustrates that using a single vertical borehole array or trench may significantly reduce the surface wave power reaching a protected region. However, seismic power is able to be directed by diffraction around the barrier. Using a V-shaped muffler (sawtooth) design is, therefore, much more effective in blocking surface waves in the 3D extent.

FIG. 10 is a table showing various high-value assets that may be protected, as examples, using embodiment elastic wave damping structures. FIG. 10 also shows an expected upper aperture widths (see, e.g., upper aperture 804 in FIG. 8B) and lateral extents (see, e.g., lateral extent 805 in FIG. 8A) of a damping structure for each example asset, with a corresponding, example number of boreholes that is expected to be effective in significantly attenuating seismic power within a protection zone, such as the protection zone 112 in FIGS. 8A-8B, including the asset.

Superstructures as described herein, and as exemplified by the superstructure 840 in FIG. 8B, can divert, attenuate, and create destructive interference of hazardous seismic waves that would otherwise reach a high valued asset such as the assets listed in FIG. 10. Example superstructures, also referred to herein a cloaking arrays, may be 50-200 meters wide (in upper aperture), and hundreds of meters to a few kilometers in lateral extent depending on protected asset size and risk. An example, single borehole diameter can be 1 meter for the structures listed in FIG. 10, where 400 boreholes can populate an example 100 square meter region. For most applications, borehole depths can be an estimated 150 meters or less. These depths can be particularly relevant where surface seismic waves are of greatest concern.

FIG. 11 is a series of diagrams 1160-1162, and a graph 1163, showing the comparative effects of seismic cloaking generally on seismic wave propagation for a single, vertical barrier element compared with an embodiment structural element, as calculated using a finite difference 2D model of the barrier structures. The seismic event is modeled for the response of the source function estimated for the Hector Mine earthquake in 1991 (Mag. 7.1-USGS), as illustrated in FIG. 9C. Typical seismic frequencies are less than 1 Hz, with minimal power above 1 Hz.

In particular, the diagram 1160 shows an areal view of seismic wave snapshots without cloaking, while the diagram 1161 shows the effect of a single, frontal vertical borehole 1164. Further, the diagram 1162 is a depth view snapshot of the effect of cloaking using a structural arrangement including two angled borehole elements 102a, 102b as described in relation to FIG. 1B.

As illustrated in the comparison, using a single vertical borehole array or trench may significantly reduce the direct surface wave energy reaching a protected region. However, energy is still able to diffract around the barrier and enter the protected region. Using a V-shaped muffler (structural arrangement) formed of elements 102a and 102b is much more effective in blocking surface waves in the 3D extent. The graph 1163 of seismic power as a function of time reaching the protection zone further bears out this fact. A curve 1164 corresponds to graph 1160, with no boreholes and the highest power; a curve 1165 corresponds to graph 1161 with one vertical borehole and somewhat diminished power reaching the protection zone. However, a curve 1166 corresponds to the angled boreholes case of graph 1162, with greatly diminished power entering the protection zone between the angular boreholes.

FIGS. 12A- 12 B are a diagram and equations illustrating semi-analytical simulation of acoustic propagation through a “tapered muffler” 1200 geometry. The tapered muffler 1200 and associated equations can be found in Easwaran and Munjal, J. Sound and Vibration 152 (1992), 73-93 and can be used as part of understanding the efficacy of embodiment structural arrangements, such as the arrangement 100 illustrated in FIG. 1B, in proving a barrier against seismic waves.

The muffler 1200 includes a tapered funnel with a subwavelength opening (entrance; lower aperture) 1206 with respect to a wavelength of an acoustic radiation source 1214. The mathematical expressions shown in FIG. 12B provide a simple, analytical, transfer matrix approach to evaluating energy attenuation as a function of the entrance 1206, an exit (upper aperture) 1204, a depth 1219, and related angle dimensions.

FIGS. 13A and 13B are graphs showing transmission loss as a function of frequency for various acoustic “muffler” parameters assuming a wave speed of 1500 meters per second (m/s). The calculations were performed using the analytical tools shown in FIGS. 12A-12B in order to at least begin to understand optimization of parameters for structural arrangements such as the arrangement 100 in FIG. 1B.

In general, the calculations for mufflers suggest that small entrance, shallow depth and steep angle can all help to maximize attenuation of acoustic waves incident at the entrance aperture. Specific trends observed in calculations such as those shown in FIGS. 13A-13B include: (i) Increasing the depth at constant angle increases attenuation at the lower frequencies, but reduces attenuation at the higher frequency due to multiple nodes; (ii) Increasing the depth at constant top & bottom aperture dimensions (i.e., while reducing the angle) reduces attenuation across the entire frequency range; and (iii) Increasing the bottom aperture opening reduces attenuation across the entire frequency range. These trends can be applied advantageously to design of structural apertures for seismic wave attenuation in embodiment elastic wave damping structures in order to maximize seismic wave attenuation.

FIG. 14 is a collage of photographs showing apparatus used to study, on a model scale, non-naturally occurring, man-made structures such as borehole arrays and trenches embedded in elastic media analogous to rock and compact soil using a machined table-top scaled physical model. The effort focused on examining the effects of borehole arrays and trenches (metamaterials) on seismic wave propagation, diversion, scattering, and attenuation through spatial measurements from controlled seismic sources. The time series measurements were then compared and analyzed with computer model simulations.

The solid model was composed of Delrin® plastic 1464 with a P-wave speed of 1700 m/s, S-Wave speed of 855 m/s, and a density of 1.41 g/cm3. Delrin® blocks were machined to contain boreholes in prescribed patterns or trenches defining a V-shaped muffler and compared with homogeneous solid blocks. The Delrin® blocks contained boreholes in prescribed patterns or trenches defining a V-shaped muffler.

The model muffler was aimed at significantly reducing the elastic wave power reaching a ‘protected keep-out’ zone from a controlled elastic wave source. Each borehole had a diameter of 3 mm and was separated 3 mm apart from neighboring boreholes, forming a single line, where the line extended the entire length of the block. A near and far borehole line V-shaped pattern was formed where the near and far borehole line spacing is 3 inches apart on the Delrin® surface, the boreholes are sloped with a 5 inch length (4 inch vertical depth), and provided an aperture opening at the V-borehole barrier structure base of 0.5 inches.

Similarly, a V-trench barrier structure was machined in a separate Delrin® block where the 3 mm diameter boreholes were in contact, forming continuous hollow walls on both sides of the barrier structure. A Modal-Shop® variable transducer 1462 was used to vertically load on the Delrin® block surface to prescribed loading functions. A 10 kHz Ricker waveform was used to act as the seismic input function to generate the elastic wave propagation in the Delrin® blocks. PCB® model 352C33 accelerometers 1460 were used to measure the temporal and spatial vibration distributions observed on the Delrin® surface. An IOTECH® wavebook 516E was used to record each time series trace using a synchronized 70 kHz sample rate per channel.

FIGS. 15A-15C show measured results that illustrate the effects of a model V-trench machined in Delrin® block table-top experimental configuration. FIG. 15A shows four accelerometer time series traces measured in a center line across a homogeneous solid Delrin® block relative to the transducer source location. Receivers 1, 3, 4, and 7 were 1, 3, 4, and 7 inches from the source, respectively. Particle velocities were computed by integrating the measured accelerations and then applying a high-pass filter to remove low frequency drift. A single 10 kHz Ricker vertical load burst was recorded as it traveled from its source. Each trace records a similar time series, showing a direct surface wave arrival (circled outline) followed by later reflection arrival interference from the Delrin® block side and bottom boundaries. The first break of the direct arrivals shows a wave speed of 1693 m/s. Spherical spreading and Delrin® attenuation losses are not compensated in the measurement plots. At the observed wave speed, the P-wavelength was estimated at 17 mm.

FIG. 15B shows four accelerometer time series traces measured in the center line across the Delrin® block that contains a V-trench barrier structure perpendicularly oriented to the direction of elastic wave propagation relative to the transducer source location. Receivers 1, 3, 4, and 7 were 1, 3, 4, and 7 inches from the source, respectively, where receivers 3 and 4 were between the near and far trench walls.

FIG. 15C shows the trench barrier structure in schematic form. The times series traces show that the direct surface wave is reflected off the near trench wall, where very little direct wave is observed inside the keep out zone between the near and far trench walls. The reflected arrival interference from the bottom surface is observed at the surface between the trench walls. In this geometry, elastic waves were able to leak through the aperture at the trench bottom and travel to the surface. These amplitudes, however, are lower than those of the peak surface wave that would be observed in the same locations if the barrier cloaking structure were not present.

FIG. 16 shows earthquake magnitude reduction expected due to subsurface barrier structures, based on extrapolation from the measured and modelled results. In this simple analysis, the power drop observed in the measurement and model studies are presented in terms of Mw reduction. The V-trench structure shows that a magnitude 7.0 earthquake energy intensity can be reduced to 5.4-5.0 for the peak power of the direct destructive surface wave. The leakage through the aperture is measured to show a modest reduction. However, when modeling the earth, where the boundaries are infinite, diffraction leakage through the aperture is small, and the structure would be expected provide a significant reduction in wave energy. Notably, modeled and measured Delrin® block waveforms and amplitudes agreed within 3 dB.

Seismic Wave Damping Systems and Corresponding Methods

In addition to the seismic wave damping structures described hereinabove, embodiments may include seismic wave damping systems that provide significant benefits beyond the benefits of the structures alone. Seismic wave damping structures, together with anti-resonance damping structures described herein, have particular synergistic effects when used in combination with each other.

On one hand, seismic wave damping structures described above can provide significant protection against damaging seismic power, particularly against the most damaging, lower-frequency range of seismic waves. On the other hand, separately, various alternative types of earthquake protection that may be generally less-effective against lower-frequency waves have been proposed, including Helmholtz resonators, tree or tower arrays, meta-concrete arrays, etc. A problem of the various proposed alternative types of protection is that they may particularly effective only at particular frequencies for which they are designed. Thus, these alternatives may not provide protection for a given expected earthquake, whose specific seismic frequencies may not be known in advance. Further, while various proposed alternative types of protection may be built in the same location to address different frequencies, engineering in this manner is redundant and expensive.

A problem of the seismic wave damping structures described above is that they may be characterized by one or more resonant frequencies that may propagate well within a protection zone defined by the structures. The present inventors have recognized that this problems of both the seismic wave damping structures and the various proposed alternative types of protection may become special advantages in a system that incorporates both a seismic wave damping structure, as described hereinabove, and also one or more of the various proposed alternative types of earthquake protection, which are referred to in the context of the embodiment systems of the present application as “anti-resonance damping structures” and by other terms describing specific examples thereof

An advantage of the seismic wave damping structures described hereinabove is that their resonant frequencies may be predicted in advance. Then, with that knowledge, it is not only helpful, but also especially advantageous to configure an anti-resonance damping structure to address the particular resonant frequencies supported by the seismic wave damping structures. Thus, when used together, the two structures act synergistically to increase protection against seismic waves in significant manner, beyond the benefits that could be expected from either seismic wave damping structures or anti-resonant damping structures acting individually.

Further, when combined as described herein, the benefits are beyond those that may be predicted simply from a hypothetical combination of any two given earthquake protection structures. As noted above, both the seismic wave damping structures and the anti-resonance damping structures have disadvantages when used separately, namely that higher-frequency resonances are allowed to propagate, and that only protection against particular frequencies is generally provided, respectively. However, the inventors have realized that when used in combination, these very disadvantages each become advantageous. Particularly, there is a way described to predict resonance frequencies that may be supported by a seismic wave damping structure, and instead of attempting to engineer an alternative earthquake protection structure to protect against primary (incident) seismic waves, an alternative structure (“anti-resonance damping structure,” as used herein) may be engineered to protect against specific resonant frequencies predicted for the seismic wave damping structure.

Based on properties of a host medium in which the seismic damping structure is embedded, resonance frequencies supported by a given seismic wave damping structure may be specifically predicted based on one or more properties of the host medium and on one or more physical properties of the seismic damping structure. A corresponding anti-resonance damping structure may, therefore, be configured to dampen specifically a residual wave propagating within the protection zone at the resonance frequency. A synergy of this arrangement is that the anti-resonance damping structure need not be configured to address all possible seismic frequencies, potentially requiring more complex and extensive engineering. Instead, the anti-resonance damping structure may be built and configured to dampen, specifically, only one or more resonance frequencies supported by the seismic damping structure, where these resonance frequencies may be specifically predicted based on properties of the host medium and seismic damping structure. Accordingly, according to embodiments described hereafter, particular synergies may be obtained, which have not been recognized previously.

As noted hereinabove in the Summary section, a certain amount of seismic wave power will be able to enter a disclosed seismic damping structure. In FIG. 1A, this seismic wave power is referred to as “attenuated surface waves 60′,” as opposed to the incident surface waves 60 that are initially incident at the elements 30a-b of the seismic damping structure. Hereinafter, this residual seismic wave power propagating within the protection zone may be referred to as “residual waves,” which may be residual surface waves or other types residual seismic wave power that enter the protection zone. The residual wave power that propagates within a protection zone, such as the protection zone 40 illustrated in for FIG. 1A, will propagate most strongly at seismic wave frequencies that are resonant in the seismic damping structure.

FIG. 17A is similar to FIG. 1A in some respects, but also illustrates a seismic wave damping system, in its environment of use, the benefits of which extend beyond those described above for seismic damping structures alone. FIG. 17A also includes terminology specific to seismic damping systems disclosed herein, as further described hereinafter. In FIG. 17A, in addition to the elements 30a and 30b, which constitute a seismic damping structure, there is illustrated an anti-resonance damping structure 1770 that dampens residual wave 1760′ that propagate within the protection zone at resonance frequencies at a resonance frequency defined by the seismic damping structure. As seismic waves 1760 travel from the earthquake center 50 in a distal medium 1719, the seismic wave 1760 are incident at the element 30a, which forms a border of the protection zone 40 together with the element 30b. The anti-resonance damping structure 1770 is positioned within the protection zone 40 and is configured to dampen the residual wave 1760 from 1760′ propagating within the protection zone at the resonance frequency. Example resonance frequencies are illustrated hereinafter in connection with FIG. 24E, for example.

Example anti-resonance damping structures 1770 may include a Helmholtz resonator or array of Helmholtz resonators, as illustrated in FIG. 23A and FIG. 23D, an array of trees planted in the protection zone, or an array of towers placed on the protection zone, as illustrated in FIGS. 24A and 24D, and array of buried cylinders, such as the meta-concrete array illustrated in FIGS. 25A and 25C, among other structures that are described herein or are otherwise or are known to those of skill in the art.

FIG. 17B is a schematic diagram illustrating the elements 30a and 30b, which form a seismic damping structure 1700, together with the anti-resonance damping structure 1770, as illustrated in FIG. 17A. Together, the structure 1770 and the seismic damping structure 1700 form an embodiment seismic wave damping system 1772. It should be understood that seismic damping structures (also referred to herein by “seismic wave damping structures,” “elastic wave damping structures,” “superstructures,” “arrangements” of elements, and similar terms) that are described throughout the application may be used in an embodiment seismic wave damping systems. These structures include structural arrangements, such as the arrangement 100 in FIG. 1B, superstructures, such as the superstructure 240a illustrated in FIG. 2D, incident and transmission groupings of elements as described in connection with FIGS. 6 and 8A-8C, for example, and any other structures described herein. For each of these structures, in view of the disclosure herein, a person of skill will understand how to calculate resonance frequencies from a property of the host medium and from one or more properties of the elements embedded within the host medium, such as depth or length thereof. A depth 119 of an element 102b, for example, is illustrated in FIG. 1B. A further property of an element may include a length thereof, such as a length from the upper and 108 to the lower and 110 of the element 102a illustrated in FIG. 1B.

The host medium 1718 shown in FIG. 17A can include earth, such as the earth 118 illustrated in FIG. 4A. Furthermore, any natural or man-made host medium, such as rock, sand, gravel, etc. or concrete, bricks, or other building materials may form part of the host medium 1718. Furthermore, it should be understood that the distal medium 1719 may be of the seven of the same material as the host medium 1718, forming one continuous medium. The distal medium and host medium are delineated only by the seismic damping structure, comprising the elements 30a and 30b in FIGS. 17A-17B, which form the boundary of the protection zone.

It should be further understood that the anti-resonance damping structure 1770 may be configured to dampen the residual wave 1760′ at the resonance frequency by being tuned to dampen, and being mechanically tuned to a harmonic or subharmonic of the resonance frequency. Furthermore, in some embodiments, anti-resonance damping structures may be configured to dampen a residual wave by being mechanically tuned to the resonance frequency, one or more harmonics of the resonance frequency, one or more sub harmonics of the resonance frequency, or any combination thereof. An example harmonic of a resonance frequency of 5 Hz is 10 Hz, for example. An example subharmonic of a resonance frequency 5 Hz is a frequency of 2.5 Hz, for example.

FIG. 18A illustrates general “seismic muffler” geometry for disclosed seismic damping structures described hereinabove. In the example of FIG. 18A, a borehole array muffler, also referred to herein as a seismic damping structure 1800, is configured to dampen seismic waves and form a protection zone around a structure asset 20, in this case a nuclear power plant, to be protected. Also illustrated in FIG. 18A is the anti-resonance damping structure 1770 of FIG. 17A, which can further enhance seismic protection in the system formed by the combination of the structure 1800 and the anti-resonance damping structure 1770 by damping resonance frequencies defined by the seismic damping structure 1800. The structure 1800 is a V-shaped seismic muffler that diverse, dissipates, and reduces ground motion from hazardous seismic waves prior to reaching the critical infrastructure 20. Seismic wave types caused by earthquakes can include surface and body waves. Surface seismic waves, such as Rayleigh and Love waves, commonly cause significant damage and destruction to buildings and structures.

These resonance frequencies characterizing an example V-shaped muffler (seismic wave damping structure) may be determined by numerical modelling, as will be understood by one of skill in the art in view of this disclosure. Alternatively, the resonance frequencies may be calculated more easily as follows. We separate the description of the total transmission loss (Eqn. (1)) into the entrance loss TL_ent (Kinsler et al., 2000) and the loss through the conical muffler TL_con (Easwaran and Munjal, 1992). In terms of seismology, the entrance loss is the frequency dependent reflected wave power that does not couple into the muffler inlet. The transmission loss through the muffler as the coupled wave travels within the muffler to its outlet (exit) is attenuation.

TL=TL_con+TL_ent   (1)

The entrance loss is caused by the acoustic impedance of the open pipe with the subwavelength opening, as described by the lumped element theory (Kinsler et al., 2000).

$\begin{array}{cc}TL_{\mathrm{ent}}=-10\log \left(1-{\frac{0.25{\left(k_0b\right)}^2+0.6i\left(k_0b\right)-1}{0.25{\left(k_0b\right)}^2+0.6i\left(k_0b\right)+1}}^2\right)& \left(2\right)\end{array}$

Furthermore, the transmission loss through the conical muffler can be expressed in terms of a transfer matrix relating the pressure and velocity at muffler inlet (entrance) (p1, v1) to those at the muffler outlet (exit) (p2, v2).

$\begin{array}{cc}\left[\begin{array}{c}{p}_1\\ v_1\end{array}\right]=\left[\begin{array}{cc}{T}_{11}& T_{12}\\ T_{21}& T_{22}\end{array}\right]\left[\begin{array}{c}{p}_2\\ v_2\end{array}\right].& \left(3\right)\end{array}$

Using the above transfer matrix, we can describe the TL_con term in Eqn. (1) for a conical muffler as follows (Easwaran and Munjal, 1992):

$\begin{array}{cc}TL_{\mathrm{con}}=20\log \left(0.5\frac{b}{t}\times T_{11}+\frac{4{\mathrm{vT}}_{12}}{\pi t^2}+T_{21}\frac{4v}{\pi b^2}+{\left(\frac{t}{b}\right)}^2T_{22}\right),& \left(4\right)\end{array}$

in which the terms in the Eqn. (4) are in turn defined below:

$\begin{array}{cc}{T}_{11}=\frac{z_2}{z_1}\cos \left(k_0l\right)-\frac{\sin \left(k_0l\right)}{k_0z_1},& \left(5\right)\\ T_{12}=i\frac{4{\mathrm{vz}}_1}{\pi b^2z_2}\sin \left(k_0l\right),& \left(6\right)\\ T_{21}=i\frac{\pi b^2\sin \left(k_0l\right)}{4v}\left(\frac{z_2}{z_1}+\frac{1}{{\left(k_0z_1\right)}^2}\right)-i\frac{\pi b^2\cos \left(k_0l\right)}{4{\mathrm{vk}}_0z_1^2},& \left(7\right)\\ T_{22}=\frac{z_1}{z_2}\cos \left(k_0l\right)+\frac{\sin \left(k_0l\right)}{k_0z_2},& \left(8\right)\\ z_1=\frac{\mathrm{bl}}{t-b},& \left(9\right)\\ z_2=z_1+l,& \left(10\right)\\ k_0=\frac{2\pi f}{v}.& \left(11\right)\end{array}$

Term definitions for the muffler characteristics specified in Eqns. (1)-(11) are described in Table 1.

TABLE 1
Term Definitions for Muffler Characteristics
in Equations (1) to (11)
Equation Term      Definition
TL                 Total transmission loss through muffler structure
TLent              Transmission loss across muffler inlet
TLcon              Transmission loss through the conical muffler
Pi(1, 2)           Pressure at muffler inlet (entrance) and outlet (exit)
vi(1, 2)           Elastic wave velocity at the muffler inlet (entrance)
                   and outlet (exit)
k0                 Wavevector in free space, defined in equation (11)
B                  Muffler inlet (entrance) diameter
T11, T12, T21, T22 Transfer matrix elements, defined in equations (5-8)
T                  Muffler outlet (exit) diameter
L                  Muffler length
zi(1, 2)           Geometrical muffler quantities, defined in
                   equations (9-10)
v                  Elastic wavespeed in medium
F                  Elastic wave frequency

The above formalism of the conical muffler depicted in Eqns. (1)-(11) was used to estimate elastic wave transmission and attenuation performance for various muffler geometries and sizes scaled to nominal areas for example infrastructure protection vulnerable to damage from earthquakes, treating seismic waves as example elastic waves. Muffler (seismic wave damping structure) performance results calculated based on Eqns. (1)-(11) are depicted in FIGS. 19A-21C. From the performance results, resonance frequencies characterizing seismic wave damping structures may be determined, as will be illustrated in examples described hereinafter.

In particular, the one or more resonance frequencies of a seismic wave damping structure may be determined as a function of depth of the elements forming the muffler (L, as used in Table 1 and illustrated in FIG. 19B). The one or more resonance frequencies may further be determined as a function a physical property of the host medium such as wavespeed in the host medium, which is v in Table. 1. A particular manner in which the equations and parameters noted above lead to a determination of resonance frequencies will become further apparent to one of skill in the art in view of the examples provided hereinafter.

In summary, in order to achieve the maximum attenuation of seismic motion for a broadband frequency response, the muffler parameters should include small inlet diameters, shallow depths with shorter lengths, and steep muffler wall angle (shallow slope). In these calculations, a 25-30 dB transmission loss through the muffler was possible for a shallow, steep angle structure at 1 Hz of seismic frequency, with greater reductions at frequencies less than 1 Hz. Seismic frequencies from 0.1 to 1 Hz are common for many large magnitude earthquakes. This frequency band appears to be well suited for the muffler structure. The predicted transmission loss at these frequencies indicates that the impact of larger earthquakes can be reduced to a less destructive level by the employment of conical mufflers.

Additional details regarding these calculations may be obtained in “Seismic Muffler Protection of Critical Infrastructure from Earthquakes,” Robert W. Haupt, et al., Bulletin of the Seismological Society of America, Vol. 108, No. 6, pp. 3625-3644, Dec. 2018, which is hereby incorporated herein by reference in its entirety.

FIG. 18B is a color-coded or shaded, side-view graphical drawing that illustrates particle velocity, in unit length per second, over a cross-sectional area showing the seismic damping structure 1800 in cross section.

FIGS. 19A-19D are cross-sectional diagrams and graphs, respectively, illustrating the effect of a muffler inlet diameter and wall slope on seismic wave transmission loss behavior through conical shaped mufflers (seismic damping structures for different wall slope inlet diameters. FIG. 19A specifically is a cross-sectional diagram of a seismic damping structure illustrating muffler geometry and parameters situated in a homogenous, isotropic, solid host medium. FIG. 19A illustrates the same basic geometry shown in FIG. 12A, wherein the terms entrance (lower aperture), exit (upper aperature), and length, among other terms, are defined.

FIG. 19B is a diagram illustrating four different muffler geometries and their respective dimensions. These structural geometries are contrasted where the outlet diameter is 0.5 km and the muffler length (also referred to herein as muffler “depth” or as a “depth” of elements forming a seismic wave damping structure is 0.25 km for all cases a-d. The inlet diameter is varied from 0.1 km, to 0.25 km, to 0.35 km, to 0.5 km.

FIG. 19C is a graph illustrating elastic wave transmission loss as a function of seismic wavelength corresponding to seismic frequencies spanning from 0.1-10 Hz. These calculations treat seismic waves as elastic, which is a helpful approximation. This is shown for all four cases a-d corresponding to FIG. 19B.

FIG. 19D is a graph illustrating P and S wave transmission loss (spectral ratio of muffler outlet and inlet) behavior for the four muffler examples of FIG. 19B as a function of seismic frequency. The solid lines in FIG. 19B represent P-wave transmission losses with a P wave velocity of 1550 m/s, while the dotted lines represent S wave losses with an S wave velocity of 17th of 700 m/s.

FIGS. 20A-20C illustrate the effect of muffler length and output diameter for various damping structure seismic damping structure geometries. In particular, seismic wave transmission losses through proportional conical mufflers for different muffler length and output and outlet diameters are illustrated, where wall slope and inlet diameter are constant. FIG. 20A illustrates four different muffler geometries that are contrasted, where the inlet diameter is 0.1 km and the muffler wall slope is constant for all cases. The outlet diameters are 0.1 km, 0.2 km, 0.4 km, and 0.6 km, where the corresponding muffler length are 0.18 km, 0.55 km, and 0.9 km, respectively. A vertical wall muffler is shown for comparison purposes.

FIG. 20B is a graph showing elastic wave (approximate for a seismic wave) transmission loss as a function of seismic wavelength corresponding to seismic frequencies spanning 0.1-10 Hz.

FIG. 20C is a graph illustrating P and S wave transmission loss (spectral ratio of muffler outlet and inlet) behavior for the four muffler examples illustrated in FIG. 20A as a function of seismic wave incident frequency. Solid lines represent P-wave transmission losses with a P velocity of 1500 m/s, while dotted lines represent the S wave losses with an S velocity of 700 m/s.

FIGS. 21A-21C illustrate a comparison of the analytical calculations to 2D numerical finite difference models for two muffler examples, namely a shallow sloping muffler model (a) having a wall slope and a vertical wall muffler model (d) (also shown in FIG. 19B). FIG. 21A illustrates cross-sectional muffler geometry for these examples. FIG. 21B is a cross-sectional illustration of the example structures (a) and (d). FIG. 21C shows calculated P and S wave transmission loss (spectral ratio of muffler outlet and inlet) behavior for the two muffler examples of FIG. 21A as a function of seismic frequency. Solid lines represent P-wave transmission losses with a P-velocity of 1500 m/s, while dotted lines represent S-wave losses with an S-velocity of 700 m/s. The symbol markings correspond to the numerical model computations for single frequency—continuous wave (CW) seismic tones.

FIG. 22A illustrates a source time function from the Hector Mine 1999 earthquake (Mag. 7.1-USGS), which was estimated from the slip velocity distribution. The 2D vertical-depth view and 2D aerial-plan view simulations described hereinafter use a source time function proportional to the normalized slip velocity illustrated in FIG. 22A. FIG. 22B is a graph showing a frequency distribution for the Hector Mine earthquake. The source time function exhibits its peak amplitude at 0.3 Hz with minimal amplitudes above 3 Hz.

FIG. 23A is a color-coded or shaded vertical depth view of the seismic wave field after nine seconds for the up-going shear waves, where a protection zone is formed by elements 30a-30b, combined with a Helmholtz resonator array 2370, which together form a seismic wave damping system 2372.

FIG. 23B is a graph showing the source function for the Hector Mine earthquake, which was injected by simulation at the bottom boundary shown in the graph, over time, into the horizontal component as a source as a line source.

FIG. 23C is an equation showing a transmission factor T for a finite number N of resonators. In this manner, a net effect on residual wave damping for a Helmholtz resonator array may be calculated.

FIG. 23D is a schematic diagram illustrating the Helmholtz resonator array, anti-resonance damping structure 2370 that is pictorially illustrated in FIG. 23A.

The following are two examples for how to use the equations in FIG. 23C in order to configure a Helmholtz resonator array to address a particular resonance frequency of a seismic damping structure. When the Helmholtz resonator array, or another anti-resonance damping structure within the scope of embodiments, is built or otherwise configured to address a particular seismic wave frequency, as used herein, it is considered to be configured to dampen residual waves having the frequency by being “mechanically tuned” to the resonance frequency. “Mechanical tuning” in some embodiments may be accomplished by virtue of construction with particular mechanical dimensions, properties, or parameters. In other embodiments, “mechanical tuning” may be accomplished in whole or in part by adjustments that follow actual construction of the anti-resonance damping structure.

In some embodiments, anti-resonance damping of residual waves in a protection zone may be accomplished by mechanically tuning an anti-resonance damping structure to a harmonic or subharmonic of the resonance frequency. Furthermore, as will be understood in view of this description, a seismic wave damping structure may be characterized by two or more resonance frequencies. Accordingly, two or more resonance frequencies for residual seismic waves may be targeted for attenuation by mechanically tuning one or more anti-resonance damping structures to two or more respective frequencies, whether resonance frequencies, harmonics or subharmonics thereof, or any combination of these.

In the case of a seismic damping structure having a resonance frequency at 5 Hz, the Helmholtz resonator array anti-resonance damping structure may be built with cylinder width of 1.7 m, cylinder height of 1.3 m, neck width of 0.1 m, and neck height of 0.2 m. As a second example, for a resonance frequency of a seismic damping structure at 10 Hz, a cylinder width for the Helmholtz resonator array may be 1.7 m, with cylinder height 1.3 m, neck width of 0.24 m, and neck height of 0.2 m. Additional details regarding parameters for Helmholtz resonators may be found, for example, in Wang et al., J. Appl. Phys. 103 (2008) 064907, which is hereby incorporated herein by reference in its entirety.

FIG. 23E is a graph illustrating the benefits of using the Helmholtz resonator array (anti-resonance damping structure) 2370 in the seismic wave damping system 2372. A curve 2373 shows seismic amplitude as a function of frequency where there is no seismic wave damping structure provided to create a protection zone. Correspondingly, where only a seismic wave damping structure is used without an anti-resonance damping structure, a curve 2376 shows a greatly reduced seismic amplitude as a function of frequency. However, as will be noted, the curve 2376 shows several resonances 2374 (also referred to herein as “resonance frequencies” or “resonant frequencies”), that are present in the spectrum 2376. By contrast with the curve 2376 and the curve 2373, a dashed curve 2378 shows the effect of a full seismic wave damping system the full seismic wave damping system 2372 including the Helmholtz resonator array 2370. As illustrated, the resonances 2374 are dramatically damped at the higher frequencies associated with the seismic wave damping structure alone. This illustrates the significant damping benefit that can be obtained by embodiments herein when a system includes an anti-resonance damping structure that is configured specifically to dampen the one or more resonance frequencies of the seismic wave damping structure.

FIG. 24A is a color-coded or shaded, vertical depth view graphs of a seismic wave field after nine seconds for the up going shear wave, where the source function for the Hector Mine earthquake, as illustrated in FIG. 24B, was injected at the bottom boundary over time into the horizontal component as a line source. A representative above ground tower or tree anti-resonance damping structure 2470 is illustrated, which, together with the seismic wave damping structure formed by the combination of elements 30a and 30b, forms a seismic wave damping system 2472.

FIG. 24C illustrates an equation that can be used to calculate a frequency that can be damped by the array 2470 of resonators (in FIG. 24A, towers or trees, for example) illustrated in FIG. 24A.

FIG. 24D is a more detailed illustration of the array of resonators 2470, which have a lattice constant L and height H above the ground. L may have various values in different embodiments, depending on the number of resonators desired, the resonant frequency to be addressed, the diameter or other cross-sectional dimension of the elements, and other factors. At the right of FIG. 24D is shown a cross-sectional view of a single one of the resonators 2469, which can be a tower or tree, for example. The resonator 2469 has a cross-sectional area A and a diameter D. The diameter D, for a cylindrical tower or tree, is an example of a cross-sectional dimension. In general, embodiments may include cross-sectional dimensions, such as diameters, on the order or 1 m, on the order of 5 m, on the order of 10 m, for example. Using the information of FIGS. 24C-24D, a person of skill can readily calculate a resonance frequency that may be addressed by a given array of tower towers or trees, and a person of skill may configure the tower or tree array 2470 to dampen a residual wave propagating within a protection zone at one or more given resonance frequencies. It will be understood that while the equation in FIG. 24C is for the infinite array of resonators 2470, similar results may be obtained for a large array, or calculated, or numerically obtained via numerical modeling to address particular resonance frequencies of associated seismic damping structures. An array of trees may be an array that does not naturally occur, such as an array of trees that are periodically spaced, for example.

Heights H of various embodiments, extending vertically from the surface of the host medium, may be between a few meters and hundreds of meters, such as 100 m or less, for example. A particular advantage of embodiment systems is that an anti-resonance damping structure need not be built to attempt to counteract the direct influence of incident seismic waves, the most destructive frequencies of which are usually at the lower end of the seismic frequency range, such as between 0.1 Hz and 3 Hz, as an example. Instead, in embodiment systems, anti-resonance damping structures are configured to dampen seismic waves propagating at the resonance frequency or frequencies of the respective seismic damping structures. The seismic damping structures are particularly effective at damping lower seismic frequencies, while the resonance frequencies tend to be higher, such as above 3 Hz, above 5 Hz, or above 10 Hz, for example. For damping at higher frequencies, towers, trees, or other structures may generally be smaller, shorter, and more feasible and inexpensive to construct.

For exemplary wood towers, example parameters can include v_p=2200 m/s, p_r=450 kg/m3, vs=1200 m/s, and A=0.071 m². Example ground host medium parameters can include v_p=900 m/s, p_g=1200 kg/m³, and v_s=500 m/s, with r=v_s/v_p, L=50, μ_g=1×10⁸ for a shear modulus of soil, and E_r=1×10⁹ for Young's modulus of the wood. With these parameters, in particular examples, where an example height of the resonators is 50 m, the resonant frequency addressed will be 5 Hz, while in a second example, with a height of 25 m, the resonant frequency addressed will be 10 Hz. Furthermore, assuming that a resonant frequency is 5 Hz for example, a height of 25 m may still be used, thus tuning the array of resonators 2472 to dampen the harmonic frequency 10 Hz, which will still dampen a residual wave propagating within the protection zone at 5 Hz. Further information regarding these calculations may be found, for example, in Colombi et al. (2016), “A seismic meta-material: The resonance meta-wedge,” Sci. Rep. 6, 27717, which is hereby incorporated herein by reference in its entirety.

FIG. 24E is a graph showing seismic amplitude as a function of frequency in connection with the infinite array of resonators 2470 illustrated in FIGS. 24A and 24D. The curve 2373 shows a case where no seismic wave protection is provided. The curve 2376, as in FIG. 23E, illustrates a case where only the seismic wave damping structure formed by the elements 30a and 30b is provided, wherein the structure is characterized by resonances the resonances 2374. However, where the muffler (seismic wave damping structure) is combined with the array of tower resonance absorbers 2470, the resonances 2374 are significantly damped.

FIG. 25A is a color-coded or shaded, vertical depth view graph showing seismic power in the form of particle velocity as a function of depth and cross range for a seismic wave damping system 2572. In this case, similarly, the Hector Mine earthquake source function, as illustrated in FIG. 25B, was injected at the bottom border over time into the horizontal component as a line source the system 2572 includes the seismic wave damping structure formed by the elements 30a-30b and also a meta-concrete array 2582. The meta-concrete array 2582 is an example of buried cylinders, and geometry of the buried cylinders, particularly because the meta-concrete array 2582, is illustrated further in FIG. 25C.

FIG. 25C is a cross-sectional view of the meta-concrete array 2582 that is illustrated in FIG. 25A. A particular cylinder 2580 of the array 2582 is also shown in greater detail.

FIG. 25D illustrates how resonance frequency may be obtained in terms of any lost an elastic modulus E_S of a soft coating of the cylinders 2580 and in terms of a core size R1 of a heavy core in the cylindrical elements 2580. This equation permits the person of ordinary skill in the art to configure an array of buried cylinders to dampen resonance frequencies predicted for a particular seismic wave damping structure. In addition, further information regarding these example structures, which may be termed “meta-concrete arrays,” may be found in “Meta-concrete: designed aggregates to enhance dynamic performance,” Journal of the Mechanics and Physics of Solids 65 (2014) 69-81, which is hereby incorporated herein by reference in its entirety.

FIG. 25E is a graph illustrating seismic amplitude as a function of frequency that can be reduced using the meta-concrete array 2582 as part of the seismic wave damping system 2572 illustrated in FIG. 25A. The curves 2373 and 2376 that were previously described are shown, together with the route of the resonance the resonance frequencies 2374. A curve 2578 (dashed) illustrates that in the case of the meta-concrete array anti-resonance damping structure, similar benefits may be obtained by specifically addressing the resonance frequencies 2374 of the seismic wave damping structure by appropriate configuration of the anti-resonance damping structure meta-concrete array 2582.

FIG. 26 is a flow diagram illustrating a procedure 2600 seismic wave for constructing a seismic wave protection zone, according to an embodiment herein. At 2684, elements are embedded within a host medium, thus defining a seismic wave damping structure that is characterized by a resonance frequency, and also thereby forming a border of a protection zone. The seismic wave damping structure is particularly configured to attenuate power of a seismic wave traveling from a distal medium to the host medium and which is incident at the protection zone formed by the elements.

At 2686, an anti-resonance damping structure is positioned within the protection zone and is configured to dampen a residual wave propagating within the protection zone at the resonance frequency. As described hereinabove, configuring the anti-resonance damping structure to dampen the residual wave propagating within the protection zone at the resonance frequency may include determining the resonance frequency as a function of a depth of the elements in the host medium and of a physical property of the host medium.

It should be understood that the procedure 2600, in other embodiments, may be modified to use or take advantage of any embodiment structure or system described hereinabove.

In a further embodiment not directly illustrated in the drawings, but which will be clearly understood by those skilled in the art in view of the other illustrations in the drawings and description herein, a procedure for seismic wave damping includes converting an incident seismic wave propagating in a distal medium outside a protection zone into a residual seismic wave propagating within the protection zone at one or more resonant frequencies. The procedure further includes dampening the residual wave within the protection zone via anti-resonance damping. The method may optionally include use or incorporation of any of the methods; elements; seismic wave damping structures, superstructures, or arrangements; and anti-resonance damping structures summarized hereinabove pertaining to other embodiments or further described hereinafter in relation to other embodiments. For example, the distal medium may be earth, and also the protection zone may earth or another building foundation or medium. The conversion may occur via the incident seismic waves of any of the types described hereinabove being incident at a seismic wave damping structure, superstructure, arrangement, or grouping formed by elements such as borehole elements embedded in the earth. A casing (i.e., liner) may be inserted into each borehole to maintain its shape and structure. Dampening the residual wave within the protection zone via anti-resonance damping may include use of any of the anti-resonance damping within the scope of the claims listed hereinafter or within the scope of the embodiments otherwise described hereinabove.

The teachings of all patents, published applications and references cited herein are incorporated by reference in their entirety.

While this invention has been particularly shown and described with references to example embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the scope of the invention encompassed by the appended claims.

Claims

1. A seismic wave damping system comprising:

elements, embedded within a host medium, defining a seismic damping structure, the elements arranged to form a border of a protection zone, the seismic damping structure configured to attenuate power of a seismic wave, traveling from a distal medium to the host medium, that is incident at the protection zone, the seismic damping structure characterized by a resonance frequency; and

an anti-resonance damping structure positioned within the protection zone and configured to dampen a residual wave propagating within the protection zone at the resonance frequency.

2. The seismic wave damping system of claim 1, wherein the resonance frequency is a function of a depth of the elements in the host medium and of a physical property of the host medium.

3. The seismic wave damping system of claim 1, wherein the anti-resonance damping structure is configured to dampen the residual wave by being mechanically tuned to the resonance frequency.

4. The seismic wave damping system of claim 1, wherein the anti-resonance damping structure is configured to dampen the residual wave by being mechanically tuned to a harmonic of the resonance frequency.

5. The seismic wave damping system of claim 1, wherein the anti-resonance damping structure is configured to dampen the residual wave by being mechanically tuned to a subharmonic of the resonance frequency.

6. The seismic wave damping system of claim 1, wherein the anti-resonance damping structure includes two or more anti-resonance damping structures configured to dampen the residual wave by being mechanically tuned to two or more respective frequencies selected from the group consisting of (i) the resonance frequency, (ii) harmonics of the resonance frequency, and (iii) subharmonics of the resonance frequency.

7. The seismic wave damping system of claim 1, wherein the anti-resonance damping structure includes one or more Helmholtz resonators positioned on or within the host medium within the protection zone.

8. The seismic wave damping system of claim 1, wherein the anti-resonance damping structure is an array of cylinders within the host medium within the protection zone.

9. The seismic wave damping system of claim 1, wherein the anti-resonance damping structure is a seismic wave absorbing structure configured to dampen the residual wave by absorption.

10. The seismic wave damping system of claim 9, wherein the seismic wave absorbing structure is a mass-in-mass lattice.

11. The seismic wave damping system of claim 1, wherein the host medium is earth, and wherein the anti-resonance damping structure includes an array of trees planted in the earth within the protection zone and spaced periodically.

12. The seismic wave damping system of claim 1, wherein the anti-resonance damping structure includes an array of scattering components positioned periodically.

13. The seismic wave damping system of claim 1, wherein the anti-resonance damping structure includes one or more towers positioned on the host medium within the protection zone.

14. The seismic wave damping system of claim 13, wherein the one or more towers are one or more flexible, steel-girded towers.

15. The seismic wave damping system of claim 13, wherein the one or more towers have heights, extending vertically from a surface of the host medium, between a few meters and hundreds of meters.

16. The seismic wave damping system of claim 13, wherein the one or more towers have heights, extending vertically from a surface of the host medium, of 100 m or less.

17. The seismic wave damping system of claim 13, wherein each of the one or more towers has a cross-sectional dimension on the order of 1 m or on the order of 10 m.

18. A method of constructing a seismic wave protection zone, the method comprising:

embedding elements within a host medium, thus defining a seismic wave damping structure characterized by a resonance frequency and forming a border of a protection zone, wherein the seismic wave damping structure is configured to attenuate power of a seismic wave traveling from a distal medium to the host medium and incident at the protection zone; and

positioning an anti-resonance damping structure within the protection zone and configuring the anti-resonance damping structure to dampen a residual wave propagating within the protection zone at the resonance frequency.

19. The method of claim 18, wherein configuring the anti-resonance damping structure to dampen the residual wave propagating within the protection zone at the resonance frequency includes configuring the anti-resonance damping structure based on one or more properties of the host medium and one or more properties of the elements.

20. A method of seismic wave damping, the method comprising:

converting an incident seismic wave propagating in a distal medium outside a protection zone into a residual seismic wave propagating within the protection zone at one or more resonant frequencies; and

dampening the residual wave within the protection zone via anti-resonance damping.