BLUNT FORCE PROTECTION SYSTEM

FIELD AND BACKGROUND OF THE INVENTION

The present invention, in some embodiments thereof, relates to blunt force protection and, more particularly, but not exclusively, to a wearable blunt force protection system.

High energy impacts may be experienced in vehicle collisions, airplane and train crashes, underwater shocks applied to ships and sea-based oil platforms, highway barrier collisions, abrupt, high level forces applied to airplane landing gear components or between components of any vehicles or machinery, and other physical interactions and may result in extensive damage to the applicable equipment (e.g., the vehicle, vessel, airplane, roadside barrier, etc.), and injury to passengers and personnel. During such interactions, shock energy may either be reflected towards the source of the energy, accumulated by the receiver, transmitted through the receiver to surroundings, or some combination thereof. Accumulated energy may be dissipated, stored and retrieved, and/or converted to another form of energy.

Traditional techniques for protecting objects from being damaged by incident energy pulses are sometimes applied to or integrated into materials used for protecting the equipment. Such materials include elastically deformable components, e.g., coil springs, foam, sand, gels, rubber or other elastomeric materials, or shock absorbing pads with polyurethane or other similar materials, and supporting welds or other metal anchors.

Much effort and ingenuity has been directed over the past decades into protecting the heads of human beings, in the growing recognition of possibly long-term neurological deficits that can be caused by concussions or other brain traumas (traumatic brain injuries). The problem extends to a wide variety of activities, including recreational activities such as sport and recreation activities (football, hockey, lacrosse, baseball, motorcycle riding, bicycle riding, horseback riding, parachuting, skydiving, skateboarding, kiteboarding, skating, rollerblading, surfing, surfboarding, skiing, snowboarding, wingsuit flying), military operations (paratrooper jumps, effects of explosions from improvised explosive devices being one particular area of concern, and blunt force impacts and ballistic impacts), and occupations such as construction work. U.S. Published Application No. 20150272255 discloses a helmet having a shell, and a pad within the shell. The pad includes a shock-absorbing material having a carrier liquid and solid particles with pores that are lyophobic with respect to the liquid.

U.S. Published Application No. 20120204329 discloses a helmet including an outside layer that serves as first contact with a source of kinetic energy, and a layer that at least partially liquefy to yield a liquid, when a threshold shear yield is met.

SUMMARY OF THE INVENTION

According to some embodiments of the invention the present invention there is provided a blunt force protection system. The system comprises an outer layer, a liner layer and intermediate layer comprising liquid extending continuously within a volume defined between the outer layer and the liner layer.

According to an aspect of some embodiments of the present invention there is provided a method of manufacturing a blunt force protection system. The blunt force protection system may include but not limited to a helmet. The method comprises forming an outer layer and a liner layer, and introducing a liquid into a volume defined between the outer layer and the liner layer to extend continuously within the volume.

According to some embodiments, the method includes determining the characteristics of the outer layer, liner layer, liquid or any combination thereof based on the expected forces (such as blunt force) parameters to be applied on the force protection system (such as blunt force protection system). According to some embodiments, the characteristics may include, physical properties (such as but not limited to, elasticity, rigidity, flexibility, tensile stress, layer thickness, density, homogeneity/heterogeneity, etc. or any combination thereof) and/or chemical properties. According to some embodiments, the expected forces (such as blunt force) parameters may include force frequency (in the time domain) and/or spatial domain.

According to some embodiments of the invention the outer layer is softer than the liner layer.

According to some embodiments of the invention a bending rigidity si of the outer layer and a bending rigidity s ₁₆ of the liner layer satisfy Si/s ₁₆ < 0.3. According to some embodiments of the invention an effective viscosity μ of the liquid, a thickness ho of the intermediate layer and a bending rigidity si of the outer layer, satisfy 12μ1ΐο ³ /8ι < 0.0008t _e , where t _e is a time parameter from about 10 ^"4 s to about 2 s. According to some embodiments of the invention an effective viscosity μ of the liquid in units of Pa-s, a density pi of the liquid in units of kg/m ³ , and a thickness ho of the intermediate layer in units of meters, satisfy ρι1ΐο ² /μ < 0.3t _e , where t _e is a time parameter from about 10 ^"4 s to about 2 s.

According to some embodiments of the invention an effective viscosity μ of the liquid in units of Pa-s, a thickness ho of the intermediate layer in units of meters, and a bending rigidity si of the outer layer in units of Pa-m ³ satisfy 12μ/(1ΐο ³ 8ι) < 750te// _e ⁶ , where t _e is a time parameter from about 10 ^"4 s to about 2 s, and wherein l _e is a length parameter from about 0.01 m to about 0.15 m.

According to some embodiments of the invention an effective viscosity□ of the liquid in units of Pa-s, a thickness hO of the intermediate layer in units of meters, and a bending rigidity si of the outer layer in units of Pa-m ³ satisfy 1ΐο ³ /(144μ ² 8ι) < 31 _e ⁶ /(t _e ² f _e ³ ), where t _e is a time parameter from about 10 ^"4 s to about 2 s, wherein l _e is a length parameter from about 0.01 m to about 0.15 m, and wherein f _e is a force parameter from about 2000N to about 20,000N. According to some embodiments of the invention an effective viscosity μ of the liquid is from about 10 ^"4 Pa-s to about 500 Pa-s.

According to some embodiments of the invention a thickness of the intermediate layer is from about 0.01 cm to about 1 cm.

According to some embodiments of the invention a thickness of the outer layer is from about 0.5 cm to about 4 cm.

According to some embodiments of the invention a density of the liquid is from about 900 to about 1400 kg/m ³ .

According to some embodiments of the invention the outer layer is elastic.

According to some embodiments of the invention a Young modulus of the outer layer is at least 3000 Pa. According to some embodiments of the invention a Young modulus of the outer layer is from about 3000 Pa to about 500 GPa.

According to some embodiments of the invention the outer layer is a woven fabric. According to some embodiments of the invention the outer layer is a non- woven fabric.

According to some embodiments of the invention the outer layer comprises a material selected from the group consisting of expended polystyrene, aluminum, Rubber, Stainless steel, polycarbonate and polyaramid. According to some embodiments of the invention the liquid comprises a material selected from the group consisting of silicon oil, glycerol, and water.

According to some embodiments of the invention the liquid is a multi- component solution.

According to some embodiments of the invention the system is wearable. According to some embodiments of the invention the system is a reinforcing element of a wearable system.

According to some embodiments of the invention the system is, or serves as a component of, a helmet.

According to some embodiments of the invention the system is, or serves as a component of, a garment.

According to some embodiments of the invention the system is, or serves as a component of, a shoe.

According to some embodiments of the invention the system is, or serves as a component of, a shoe. According to some embodiments of the invention the system is, or serves as a component of, a glove.

According to some embodiments of the invention the system is, or serves as a component of, a limb protector. According to some embodiments of the invention the system is, or serves as a component of, a knee protector, an elbow protector, a heel protector or a shoulder protector.

According to some embodiments of the invention the system is, or serves as a component of, an electronic device.

According to some embodiments of the invention the system is, or serves as a component of, a packaging.

According to an aspect of some embodiments of the present invention there is provided a method of dissipating kinetic energy caused by a blunt force acting on an object. The method comprises placing the system as delineated above and optionally and preferably as further detailed below onto a surface of the object receiving the force.

Unless otherwise defined, all technical and/or scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which the invention pertains. Although methods and materials similar or equivalent to those described herein can be used in the practice or testing of embodiments of the invention, exemplary methods and/or materials are described below. In case of conflict, the patent specification, including definitions, will control. In addition, the materials, methods, and examples are illustrative only and are not intended to be necessarily limiting.

Implementation of the method and/or system of embodiments of the invention can involve performing or completing selected tasks manually, automatically, or a combination thereof. Moreover, according to actual instrumentation and equipment of embodiments of the method and/or system of the invention, several selected tasks could be implemented by hardware, by software or by firmware or by a combination thereof using an operating system. For example, hardware for performing selected tasks according to embodiments of the invention could be implemented as a chip or a circuit. As software, selected tasks according to embodiments of the invention could be implemented as a plurality of software instructions being executed by a computer using any suitable operating system. In an exemplary embodiment of the invention, one or more tasks according to exemplary embodiments of method and/or system as described herein are performed by a data processor, such as a computing platform for executing a plurality of instructions. Optionally, the data processor includes a volatile memory for storing instructions and/or data and/or a non- volatile storage, for example, a magnetic hard-disk and/or removable media, for storing instructions and/or data. Optionally, a network connection is provided as well. A display and/or a user input device such as a keyboard or mouse are optionally provided as well.

BRIEF DESCRIPTION OF SEVERAL VIEWS 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 drawing(s) will be provided by the Office upon request and payment of the necessary fee.

Some embodiments of the invention are herein described, by way of example only, with reference to the accompanying drawings. With specific reference now to the drawings in detail, it is stressed that the particulars shown are by way of example and for purposes of illustrative discussion of embodiments of the invention. In this regard, the description taken with the drawings makes apparent to those skilled in the art how embodiments of the invention may be practiced.

In the drawings:

FIG. 1A-FIG. 1C are schematic illustrations of a blunt force protection system, according to some embodiments of the present invention;

FIG. 2A and FIG. 2B are schematic illustrations of the blunt force protection system in embodiments of the invention in which the system is a helmet system;

FIG. 3 is a schematic illustration of a configuration and coordinate system used in a theoretical and experimental study performed according to some embodiments of the present invention;

FIG. 4A and FIG. 4B show similarity shape function vs. a signal velocity coefficient, as obtained in a theoretical and experimental study performed according to some embodiments of the present invention; FIG. 5A and FIG. 5B show dynamics during and after application of a temporally uniform external force, as obtained in a theoretical and experimental study performed according to some embodiments of the present invention;

FIG. 6A shows a liquid pressure divided by external pressure, as obtained in a theoretical and experimental study performed according to some embodiments of the present invention;

FIG. 6B shows a ratio of liquid pressure at a center of an impact to an external pressure, at a moment of maximal external pressure, as obtained in a theoretical and experimental study performed according to some embodiments of the present invention; FIG. 7 is a schematic illustration of an experimental setup used in an experimental study performed according to some embodiments of the present invention;

FIG. 8 shows mean value of four experimental measurements and theoretical predictions vs. a radial coordinate r, as obtained in a theoretical and experimental study performed according to some embodiments of the present invention; FIG. 9 is a schematic illustration of an experimental setup used in an experimental study performed, according to some embodiments of the present invention;

FIG. 10A, FIG. 10B, FIG. IOC and FIG. 10D show dynamics of a configuration with a lower rigid surface (i.e., Z ₂ → ±∞). FIG. 10A shows a pressure ratio magnitude IP/Pel, FIG. 10B shows a phase between the liquid pressure and the external pressure fig. IOC shows a normalized magnitude of deformation \Di/P _e \ ,

and (d) is the phase of between the upper sheet deformation and the external pressure ZDi— ZPe- The solid curve, defined by a ² 5Zi/3, indicates maximal magnitude of deformation and liquid pressure for constant a ² . The dashed curve indicates maximal magnitude of liquid pressure for constant Zl this curve coincides with the solid curve in FIGIOC and FIG. 10D.

FIG. 11A, FIG. 11B, FIG. 11C, FIG. 11D, FIG. HE and FIG. 11F show dynamics of a configuration consisting of two sheets with identical impedance, Z\ = Z ₂ . FIG. 11A shows

Gray colored area indicates that the value of the variable exceeds the maximal value of the color bar and is singular at Z\ ~ 0. The smooth line denotes values of Z\ yielding extrema for predefined or.

FIG. 12A, FIG. 12B, FIG. 12C, FIG. 12D, FIG. 12E and FIG. 12F show magnitude ratio and phase between the liquid pressure to external pressure as a function of a ² , and Zl . FIG.12A, FIG. 12C, and FIG. 12E show \P/P _e \. FIGs. 12B, 12D, and 12F show LP- LP _e . FIG. 12A and FIG. 12B present Z ₂ = -6, FIG.12C and FIG. 12D present Z ₂ = -3, and FIG. 12E and FIG. 12F present Z ₂ =-l. The smooth (eq. 3.26a) and dotted lines (eq. 3.26b) present extremum of deflection with regard to Zl (for set values of α , Z ₂ ) and with regard to or (for set values of Zl, Z2), respectively;

FIG. 13A, FIG. 13B, FIG. 13C, FIG. 13D, FIG. 13E and FIG. 13F show scaled average deformation magnitude \D\APe\ (FIG. 13A, FIG. 13C and FIG. 13E), and scaled relative deformation \D APe\ (FIG. 13B, FIG. 13D, and FIG. 13F), as a function of a ² , and ¾. FIG. 13A and FIG. 13B present Z ₂ = -6; panels FIG. 13C and FIG. 13D present Z2 = -3, and FIG. 13E and FIG. 13F present Z ₂ = -1. The smooth and dotted lines present extremum of deflection with regard to Zl (for set values of a ² , Z2) and with regard to a ² (for set values of Zl, Z2), respectively;

FIG. 14A, FIG. 14B, FIG. 14C and FIG. 14D are illustrations of the additional resonance frequency emerging due to parallel motion of fluid. FIG. 14A presents the normalized displacement of the upper sheet for equal sheet impedance Zi = Z _2; FIG. 14B presents the fluid pressure amplitude of identical sheets Zi=Z ₂ ; FIG. 14C presents the normalized displacement of the upper sheet for a configuration of Z2 »Z1 ; FIG. 14D presents the fluid pressure amplitude for a configuration of Z ₂ »Zi;

FIG. 15A, FIG. 15B and FIG. 15C show frequency response of elastic Hele- Shaw cells for three physical configurations in which the bottom surface is rigid (FIG. 15A), in the bottom surface is mass-less and compliant (FIG. 15B); and the bottom surface mass is finite and compliant (FIG. 15C). Vertical lines depicted by the letters 'a', 'b', and 'c' correspond to resonance frequencies of the upper sheet, the lower substrate, and a combine reference configuration with a constraint of constant gap between the upper sheet and the lower surface. DESCRIPTION OF SPECIFIC EMBODIMENTS OF THE INVENTION

The present invention, in some embodiments thereof, relates to blunt force protection and, more particularly, but not exclusively, to a wearable blunt force protection system.

Before explaining at least one embodiment of the invention in detail, it is to be understood that the invention is not necessarily limited in its application to the details of construction and the arrangement of the components and/or methods set forth in the following description and/or illustrated in the drawings and/or the Examples. The invention is capable of other embodiments or of being practiced or carried out in various ways.

Referring now to the drawings, FIG. 1A- FIG. 1C are schematic illustrations of a blunt force protection system 10, according to some embodiments of the present invention, preferably for reducing or eliminating a damage caused by a blunt force 11. System 10 is preferably used as a wearable system, but other applications of system 10 are also contemplated. For example, system 10 can be, or serve as a component of, an electronic device (e.g. , a display device, particularly, but not necessarily a mobile device having a display, such as, but not limited to, a mobile phone or a tablet), a packaging, or any blunt force vulnerable object. When system 10 is wearable, it can be a reinforcing element of a wearable system, or it can be the wearable system itself. Wearable systems contemplated according to some embodiments of the present invention include, without limitation, a helmet, a garment (e.g. , body protection article), a shoe, a glove, a limb protector, a knee protector, an elbow protector, a heel protector, a shoulder protector, and the like. When system 10 is a helmet, it can be the basis for helmets for a variety of uses, and can include any of a variety of configurations appropriate for use, and can include any of a variety of additional features, such as face masks, eye protection, built-in communication devices, built-in display, built-in cameras, ventilation holes, retention devices, such as chin straps, and the like. System 10 preferably comprises an outer layer 12 and liquid 14 extending continuously within a volume defined between outer layer 12 and a surface 20 of the object 22 to be protected. In any of the embodiments described herein, system 10 preferably, but not necessarily, comprises also a liner layer 16. In these embodiments, system 10 has an intermediate layer 18 which comprises the liquid 14 extending continuously within a volume defined between outer layer 12 and liner layer 16. Layers 12 and 16 are illustrated in FIG. 1A and FIG. IB as planar. However, this need not necessarily be the case, since, for some applications, the layers are preferably non-planar or curved. Thus, layers 12 and 16 (in embodiments in which layer 16 is employed), can have any shape. Typically, the layers are shape-wise compatible with the surface of the object 22. A representative example of a curved shape for layers 12 and 16 is illustrated in FIG. 1C. The illustration in FIG. 1C can correspond, for example, to an embodiment in which system 10 is a helmet protecting a head 22 of an individual from blunt force trauma.

In a search for the problem of dissipating blunt forces, the present inventors examined the dynamics of a liquid film enclosed between two layers. As demonstrated in the Examples section that follows, it was found that the presence of a viscous film between the outer layer 12 and the liner layer 16 or surface 20 of object 22 successfully re-distributes localized external forces. The existence of liquid 14 re-distributes the force both in the time domain and in the spatial domain, such that the volume of the region experiencing the impact caused by the force is widened and the duration of the impact caused by the force is lengthened. Such spatiotemporal re-distribution of the force is advantageous in systems in which it is desired to dissipate kinetic energy caused by a blunt force acting on an object.

System 10 preferably protects object 22 by reducing both direct missile trauma and secondary kinetic effects. System 10 receives contact of an external object (not shown) that transfers kinetic energy to layer 12. Viscous-elastic transient dynamics transfer the kinetic energy laterally with respect to surface 20 of object 22 and also delays the propagation of the energy, thus providing a spatiotemporal re-distribution of the force.

The outer layer 12 of system 10 is optionally, but not necessarily, softer than the liner layer 16. Alternatively, layer 12 can be harder than layer 16. The latter embodiments are particularly useful when surface 20 of object 22 is made of a hard material, in which case it is not necessary for layer 16 to be hard since the hardness is provided by surface 20 itself. In some embodiments of the present invention both Layer 12 and liner layer 16 are softer than surface 20 of object 22. . Preferably, the outer layer 12 is elastic. For example, the Young modulus of outer layer 12 can be at least 3000 Pa. Typically, but not exclusively, the Young modulus of outer layer 12 is from about 3000 Pa to about 500 GPa, more preferably from about 1 MPa to about 500 GPa, more preferably from about 100 MPa to about 500 GPa, more preferably from about 1 GPa to about 250 GPa, more preferably from about 1 GPa to about 150 GPa.

As used herein "Young modulus" refers to the ratio between the tensile stress and the extensional strain of the respective material, at a linear segment of the stress- strain curve, or at a linearized segment of the stress-strain curve.

The thickness of intermediate layer 18 is preferably less than 1 cm, more preferably from about 0.01 cm to about 1 cm. The thickness of outer layer 12 is preferably from about 0.5 cm to about 4 cm, more preferably from about 0.5 cm to about 3 cm. Liquid 14 can generally be any viscous liquid. Preferably, the effective viscosity μ of liquid 14 is from about 10 ^"4 Pa-s to about 500 Pa-s, more preferably from about 10 ^{" 2} Pa-s to about 500 Pa-s, more preferably from about 1 Pa-s to about 500 Pa-s, more preferably from about 1 Pa-s to about 300 Pa-s. The density of liquid 14 is preferably from about 900 to about 1400 kg/m ³ . As used herein "effective viscosity" refers to the outcome of a viscosity measuring experiment performed on a Newtonian liquid, or, in case the liquid is a non- Newtonian, to the outcome of a viscosity measuring experiment performed on the non- Newtonian liquid but calculated as the liquid were a Newtonian liquid.

Unless otherwise indicated, any value of a temperature-dependent physical quantity recited herein (such as, but not limited to, density and viscosity), corresponds to the value of the respective physical quantity at a temperature of 25°C.

In any of the embodiments of the present invention, the effective viscosity μ of liquid 14, the thickness ho of intermediate layer 18, and the bending rigidity si of outer layer 12, preferably satisfy the relation: where Xi is a dimensionless parameter, and t _e is a time parameter. Typically, Xi=0.0008, more preferably Xi=0.0006, more preferably Xi=0.0004, more preferably Xi=0.0002, more preferably Xi =0.0001.

As used herein "bending rigidity" of a layer is defined as s=Eb ³ /12(l-v ² ), where E is the Young modulus of the layer, b is the thickness of the layer, and v is the Poisson ratio of the layer.

In any of the embodiments of the present invention, the effective viscosity μ and density pi of liquid 14, and the thickness ho of intermediate layer 18, preferably satisfy:

where X ₂ is a dimensionless parameter, t _e is the aforementioned time parameter, and where μ is expressed in units of Pa-s, pi is expressed in units of kg/m ³ , and ho is expressed in units of meters. Typically, X ₂ =0.3 , more preferably X ₂ =0.2, more preferably X ₂ =0.1, more preferably X ₂ =0.05, more preferably X ₂ =0. 01.

In any of the embodiments of the present invention, the effective viscosity μ of liquid 14, the thickness ho of layer 18, and a bending rigidity si of outer layer 12 preferably satisfy:

where X ₃ is a dimensionless parameter, t _e is the aforementioned time parameter, where l _e is a length parameter, and where μ is expressed in units of Pa-s, ho is expressed in units of meters, and si is expressed in units of Pa-m ³ . Typically, X ₃ =750, more preferably X ₃ =500, more preferably X ₃ =250, more preferably X ₃ =150, more preferably X ₃ =75.

In any of the embodiments of the present invention, the effective viscosity μ of liquid 14, the thickness ho of layer 18, and the bending rigidity si of outer layer 12, preferably satisfy:

where X ₃ is a dimensionless parameter, t _e is the aforementioned time parameter, l _e is the aforementioned length parameter, and f _e is a force parameter. Typically, X ₄ =3 , more preferably X ₄ =2, more preferably X ₄ =l, more preferably X ₄ =0.5, more preferably X ₄ =0.1.

As used herein, a "time parameter" is a parameter having a dimension of time (for example seconds), a "length parameter" is a parameter having a dimension of length (for example centimeters), and a "force parameter" is a parameter having the dimension of a force (for example newtons).

The time, length and force parameters described herein can optionally be viewed as parameters that characterize the force to be spatiotemporally re-distributed by system 10. For example, t _e can represent the duration of the force, once applied, l _e can represent the diameter of the area over which the force is applied, once applied, and f _e can represent the magnitude of the force, once applied. However, while the time, length and force parameters may be interpreted as such force characteristics, in some embodiments of the present invention they serve as threshold parameters for defining the relations between the various geometrical and physical characteristics of layer 12 and liquid 14 (as formulated in the left-hand- sides of the above inequalities). Thus, it is not necessary to apply a force that is characterized by these parameters in order to determine whether or not layer 12 and liquid 14 satisfy the respective inequality.

The time parameter t _e is typically from about 10 ^"4 s to about 2 s, more preferably from about 10 ^"4 s to about 1 s, more preferably from about 10 ^"4 s to about 0.1 s. The length parameter l _e is typically from about 1 cm to about 15 cm, more preferably from about 1 cm to about 10 cm, more preferably from about 1 cm to about 5 cm, more preferably from about 1 cm to about 4 cm, more preferably from about 1 cm to about 3 cm.

The force parameter f _e is typically from about 2000N to about 20,000N, more preferably from about 3000N to about 20,000N, more preferably from about 5000N to about 20,000N.

In embodiments of the invention in which layer 16 is harder than layer 12, the bending rigidity si of outer layer 12 and the bending rigidity si ₆ of liner layer 16 and the preferably satisfy the relation: where X5 is a dimensionless parameter. Typically, X ₅ =0.3, more preferably X ₅ =0.2, more preferably X ₅ =0.1, more preferably X ₅ =0.05, more preferably Xs=0.001.

Notice that the Xi (i=l, 2, 5) parameters are different from the t _e , l _e and / _e parameters, since Xi are all dimensionless parameters, while t _e , l _e and f _e are all dimensional parameters.

In a preferred embodiment of the invention, the effective viscosity μ and density pi of liquid 14, the thickness ho of intermediate layer 18, and the bending rigidity si of outer layer 12, satisfy at least two of EQs. 1-4, above, more preferably at least three of EQs. 1-4, above, more preferably each of EQs. 1-4, above. In some embodiments of the present invention layer 12 is a solid structure, in some embodiments of the present invention layer 12 is a woven fabric, and in some embodiments of the present invention layer 12 is a no n- woven fabric. Layer 12 can, for example, be a composite composed of a discrete reinforcement and a continuous binder, such as a polymer, or can be any other material such as Kevlar™ or steel that can spread kinetic energy. A discrete reinforcement can be composed of particles that can have differing sizes, shapes and can be different materials such as ceramic or glass. The reinforcement preferably has particles with a size greater than one micron. Representative examples of material suitable for forming layer 12 including, without limitation, expanded polystyrene, expanded polypropylene, aluminum, Rubber, Stainless steel, polycarbonate and polyaramid (e.g., Kevlar™).

Layer 16 can be made of any material, including, without limitation, any of the materials described above with respect to layer 12.

Representative liquids that can be included in, or serve as, liquid 14 include, without limitation, silicon oil, glycerol, and water. In some embodiments of the present invention, liquid 14 is a multi-component solution, and in some embodiments of the present invention, liquid 14 is a multi-component suspension.

As used herein, "multi-component solution" refers to a liquid having at least one solvent and at least one additional substance dissolved therein.

As used herein, "multi-component suspension" refers to a heterogenous liquid containing at least a dispersed phase (e.g., solid particles), and a dispersed liquid medium. FIG. 2A and FIG. 2B are schematic illustrations of system 10 in embodiments of the invention in which system 10 is a helmet system. Although helmet system 10 is depicted as a motorcycle helmet, a helmet incorporating features of at least some embodiments of the present invention may be implemented as other types of helmets, including, without limitation, a bicycle helmet, an industrial safety helmet, a recreation activity helmet (e.g. , football helmet, a hockey helmet, a lacrosse helmet, a baseball helmet, a motorcycle riding helmet, a bicycle riding helmet, a horseback riding helmet, a parachuting helmet, a skydiving helmet, a skateboarding helmet, a kiteboarding helmet, a skating helmet, a rollerblading helmet, a surfing helmet, a surfboarding helmet, a skiing helmet, a snowboarding helmet, a wingsuit flying helmet, etc.), a military helmet, or any other helmet, without departing from the scope of the present embodiments.

Apart for layers 12 and 18, and preferably also 16, described above, helmet system 10 optionally and preferably comprises a face shield 104 and a chin bar 106. Chin bar 106 can optionally and preferably have a plurality of ventilation intakes 108 which are optionally adjustable to allow a controlled amount of air to enter the helmet for the purpose of reducing fogging of face shield 104 in humid weather and/or for ventilation of the head 22 of the user. Helmet system 10 can further comprise one or more of air intakes 110. Optionally and preferably, the rear part of layer 12 is shaped to include a rear spoiler 123, so as to reduce lift and/or wind buffeting of the helmet and associated noise at high speed. In some embodiments of the present invention, helmet system 10 comprises a forward-facing camera 114, and in some embodiments of the present invention helmet system 10 also comprises a rear facing camera 160, preferably having a field-of-view of at least 150° or at least 170°. In the illustrative embodiment, rear- facing camera 160 is mounted within rear spoiler 164 of layer 12 thereby utilizing what would otherwise be wasted space.

Preferably, but not necessarily, helmet system 10 comprises a display device 124 mounted to the rear surface 126 of chin bar 106. Display device 124 can be in the form of a virtual image display consisting of a liquid crystal or LED display. Display device 124 can be attached to chin bar 106 by means of a hinge element 128. In some embodiments of the present invention display device 124 produces a virtual image 144 which optionally and preferably appears to the rider to be behind the chin bar 106 at optical infinity. Because the virtual image 144 appears to be behind (i.e., passing through) the chin bar 106, virtual image 144 appears in an area that is already obscured from the rider's field of view and therefore does not interfere with or reduce the rider's field of view. Optionally, virtual image 144 can be positioned at the top edge of chin bar 106.

When helmet system 10 comprises forward-facing camera 114, display device 124 optionally and preferably receives image data to be displayed from forward-facing camera 114, thereby allowing helmet system 10 to provide an augmented reality experience to the rider. When helmet system 10 comprises rear-facing camera 160, display device 124 optionally and preferably receives image data to be displayed from rear-facing camera 160, thereby providing the rider with a rear view image, obviating the necessity of the rider to turn his/her head in order to view oncoming traffic. Optionally, rear facing camera 160 may be gyroscopically stabilized. Alternatively or additionally, display device 124 can also receive image data from an external source such as a portable mobile device, e.g. , a smartphone, a tablet, a phabet, a smartwatch or a GPS (not shown), being in wireless or wired communication with display device 124.

An additional chin spoiler 130, preferably formed of a soft material may optionally and preferably be attached to chin bar 106 to further reduce wind noise and lift at high speeds. Optionally, chin spoiler 130 may include a microphone (not shown) and/or additional electronics (not shown) for operating display device 124.

As used herein the term "about" refers to ± 10 %.

The word "exemplary" is used herein to mean "serving as an example, instance or illustration. "Any embodiment described as "exemplary" is not necessarily to be construed as preferred or advantageous over other embodiments and/or to exclude the incorporation of features from other embodiments.

The word "optionally" is used herein to mean "is provided in some embodiments and not provided in other embodiments. "Any particular embodiment of the invention may include a plurality of "optional" features unless such features conflict. The terms "comprises", "comprising", "includes", "including", "having" and their conjugates mean "including but not limited to".

The term "consisting of means "including and limited to".

The term "consisting essentially of means that the composition, method or structure may include additional ingredients, steps and/or parts, but only if the additional ingredients, steps and/or parts do not materially alter the basic and novel characteristics of the claimed composition, method or structure.

As used herein, the singular form "a", "an" and "the" include plural references unless the context clearly dictates otherwise. For example, the term "a compound" or "at least one compound" may include a plurality of compounds, including mixtures thereof.

Throughout this application, various embodiments of this invention may be presented in a range format. It should be understood that the description in range format is merely for convenience and brevity and should not be construed as an inflexible limitation on the scope of the invention. Accordingly, the description of a range should be considered to have specifically disclosed all the possible subranges as well as individual numerical values within that range. For example, description of a range such as from 1 to 6 should be considered to have specifically disclosed subranges such as from 1 to 3, from 1 to 4, from 1 to 5, from 2 to 4, from 2 to 6, from 3 to 6 etc., as well as individual numbers within that range, for example, 1, 2, 3, 4, 5, and 6. This applies regardless of the breadth of the range.

Whenever a numerical range is indicated herein, it is meant to include any cited numeral (fractional or integral) within the indicated range. The phrases "ranging/ranges between" a first indicate number and a second indicate number and "ranging/ranges from" a first indicate number "to" a second indicate number are used herein interchangeably and are meant to include the first and second indicated numbers and all the fractional and integral numerals therebetween.

It is appreciated that certain features of the invention, which are, for clarity, described in the context of separate embodiments, may also be provided in combination in a single embodiment. Conversely, various features of the invention, which are, for brevity, described in the context of a single embodiment, may also be provided separately or in any suitable subcombination or as suitable in any other described embodiment of the invention. Certain features described in the context of various embodiments are not to be considered essential features of those embodiments, unless the embodiment is inoperative without those elements.

Various embodiments and aspects of the present invention as delineated hereinabove and as claimed in the claims section below find experimental support in the following examples.

EXAMPLES

Reference is now made to the following examples, which together with the above descriptions illustrate some embodiments of the invention in a non-limiting fashion. Example 1- Theoretical Considerations

In the present Example, the transient dynamics of a viscous liquid contained in a narrow gap between a rigid surface and a parallel elastic plate is studied. The elastic plate is deformed due to an externally applied time-varying pressure-field. The flow- field is modeled via the lubrication approximation and the plate deformation by the Kirchhoff-Love plate theory. A self- similarity solution is obtained for the case of an external point force acting on the elastic plate. The pressure and deformation field during and after the application of the external force are derived and presented by closed form expressions. A distributed external pressure, spatially uniform and linearly increasing with time, acting on the elastic plate is examined over a finite region and during a finite time period, similar to the viscous-elastic interaction time-scale. The interaction between elasticity and viscosity is shown to reduce by order of magnitude the pressure within the Hele-Shaw cell compared with the externally applied pressure.

Problem Formulation

Transient creeping flow is studied in the narrow gap between a rigid surface and a parallel elastic plate due to time- varying external pressure acting on the elastic plate. In this model, the lower object/plate is considered more rigid than the upper plate. The configuration and coordinate system are defined in FIG. 3. Referring to FIG. 3, a lower plate 200 is more rigid than an upper plate 201 and a viscous liquid 202 is depicted therebetween. Plates 200 and 201 are parallel to the x-y plane. is the external

pressure field and h is the gap between the rigid surface and the elastic plate.

Hereafter, underlined or bold quantities denote vectors, asterisk superscripts denote characteristic values, and Capital letters denote normalized variables. The subscript _L denotes the vector component perpendicular to the x-y plane and no subscript denotes the two-dimensional vector components parallel to the x-y plane. The liquid pressure is denoted p, the liquid velocity is the gap between

the surfaces is h(t)=ho+d(t), where ho is the gap before application of force onto the upper plate, and d(f) is the deformation of the plate with respect to its centerline, the liquid viscosity is denoted μ, the liquid density is denoted pi, the deformation of the plate is denoted d, the plate bending resistance is denoted s, the plate thickness is denoted b, the plate density is denoted p _s , and the external pressure field is denoted p _e . The characteristic length-scale in the x-y plane is denoted / ^* , the characteristic liquid pressure is denoted p ^* , the initial liquid film height is denoted ho, the characteristic deformation is denoted d ^* , the characteristic speed in the x-y plane is denoted u and the characteristic speed in the z direction is denoted the characteristic stress resultant

acting perpendicular to z direction is denoted n .

A set of small parameters is defined as follows:

corresponding to assumptions of shallow geometry, small ratio of transverse plate deformations to viscous film height, small Womersley number, negligible solid inertia and negligible membrane effects, respectively. Under the assumptions given in EQs. 1. la-e, the Hele-Shaw cell's upper elastic plate dynamics are governed by the Kirchhoff- Love equation:

where order of magnitude analysis yields p =sd II . The boundary condition is selected such that the deformation vanishes sufficiently far from the location of the external force

The Newtonian, incompressible fluid located within the elastic cell is governed by the continuity and momentum equations:

with the boundary condition that pressure is uniform sufficiently far from the location of the external pressure

as well as no-slip and no-penetration at the solid-liquid interfaces

where the term represents in-plane velocities due to angular speed.

Normalized variables are defined as follows

corresponding to the normalized pressure P, coordinates (X, Xi), fluid velocity (U, U±), and solid deformation D.

Substituting EQs. 1.8a-d into EQs. 1.5 and 1.4 yields a lubrication approximation, in leading order,

and where order of magnitude yields

From the above equations, the leading order boundary conditions is given by

Substituting EQ. 1.9 into EQ. 1.10, integrating over Z, provides

The governing equation in terms of deflection D is, therefore:

with boundary condition Alternatively, the governing equation can be presented in terms of the pressure:

with boundary condition

EQs. 1.13 and 1.14 are the linearized, inhomogeneous form of a sixth order thin film equation [see, for example, Flitton et al., 2004, European Journal of Applied Mathematics, 15, 06, 713].

Combining results from order of magnitude analysis, the characteristic viscous- elastic time t ^* is obtained, as well as alternative expressions of plate deformation d ^* and in-plane fluid velocity u ,

Since an infinite configuration is examined, there is no inherent length- scale / to the problem and only a relation between t ^* and I ^* may be obtained. However, t ^* and I ^* may be defined by the external actuation p _e . For an external pressure field with characteristic pressure p ^* , characteristic length l _e and characteristic time

or t*=t _e can be set (but not necessarily both). If setting viscous- elastic dynamics may be neglected and one obtains For

the case in which setting the characteristic parameters may be

computed from EQ. 1.15. However, if setting the scaling is

inconsistent and the viscous-elastic dynamics takes place on a length-scale much greater than l _e . In this case the characteristic time scale is preferably defined by the external actuation instead of its length-scale. The characteristic length-scale of the viscous-

elastic interaction can then be estimated from

In addition, substituting EQ. 1.15 into EQ. 1.1 allows defining the range of validity of the assumptions, yielding an equivalent definition of the Womersley number and negligible plate's inertia requirement

Utilizing order of magnitude analysis of the plate's in-plane force equilibrium the Inventors of the present invention obtain that the requirement for

negligible membrane effects simplifies to Therefore, the requirements of shallow geometry

are sufficient in order to neglect membrane effects.

Below, external pressures modeled are examined as the Dirac delta function in time. The Dirac delta function represents an external pressure field which satisfies of interest can be chosen arbitrarily, where,

effectively, only dynamics after the end of the external application of pressure are examined. Due to the self-similar nature of the problem, the condition of order of magnitude smaller length-scale Ul ^* «l may be translated into additional condition on the time-scale. Thus, any external pressure distributed over a length-scale of l _e , may be modeled as the Dirac delta function for sufficiently large The validity

of the assumptions EQ. 1.1 during the pressure application period are examined in this Example with regard to t _e and le, representing dynamics during the pressure application, not the arbitrarily chosen t ^* and I ^* time- and length-scale. Green's functions and self-similarity

The Green's function of EQs. 1.13 and 1.14 is:

where are the location and time of the delta function, respectively. EQ. 1.16

represents the solution for the evolution of the pressure for external pressure,

Similarly, EQ. 1.16

represents the solution for the evolution equation of the deformation, EQ. 1.13, for

noted that external pressures of the form do not create deformation

of the plate.

EQ. 1.16 can be interpreted as the inverse Fourier transform, where the argument of transformation is Furthermore, radial symmetry of the argument

allows representation of EQ. 1.16 by the inverse Hankel transform:

Expressing the Bessel function in a series form, and integrating each element according to

where Γ(γ) is the Gamma function, yields:

where

The series is optionally and preferably decomposed into three separate

thus yielding a closed-form expression in term of generalized hyper-geometric functions:

While the function presented in EQ. 1.22 can be used by convolution to obtain a general solution, more insight may be obtained from a solution for the case of

This may be achieved without convolution by applying the

Laplacian operator in terms of X on the equation defining the Green's function:

Thus, the deformation- field due to a unit impulse is

where

A similar approach can be used to obtain the pressure-field due to a unit impulse:

where

G _p , Gd thus provide direct insight regarding the response to external forces and can be used to convolve P _e directly for general solutions, similarly to a regular Green's function. It is noted that the characteristic liquid pressure where j _e is the

magnitude of impulse.

Without loss of generality, From η, the radial speed of the signal propagation is obtained as r| _re fT ^"5/6 , where r| _re f is a reference state.

Utilizing Eq. 1.9 and 1.10 together with the radial G _u and transverse G _w fluid speed are obtained: respectively.

Substituting into the plate's constitutive equations the bending moment

are obtained.

FIGs 4A and FIG. 4B show the similarity shape function vs. the signal velocity coefficient, η, defined in EQ. 1.19. FIG. 4A shows (solid, dashed and dotted lines, respectively), and FIG. 4B shows (solid, dashed and dotted

lines, respectively). are similar decaying oscillating functions of η.

However, a difference in the decay rate in time exists due to the different powers of T multiplying Ψ, where the slowest time decay is of the deformation, scaling as T ^2/3 .

Impact mitigation and response dynamics

The functions in EQs. 1.19, 1.24 and 1.26 are used to examine the fluid pressure- field and plate deformation-field created during and after application of spatially and temporally distributed external forces. For a temporally uniform external force applied over a finite time interval, defined as

where Θ is the Heaviside function and T _e is the instant of release, the pressure-field is obtained froni EQ. 1.19 as

Substituting EQ. 1.29 into EQ. 1.12 and integrating with respect to yields the

deformation of the elastic plate.

Reference is now made to FIG. 5A and FIG. 5B which show dynamics during and after application of a temporally uniform external force applied a

FIG. 5A presents the fluid pressure-field at two instants durin and

after the application of external force of the form of EQ. 1.28, where

During the application the pressure in the impact locus decays with time and

acts to resist the temporally constant external force. After the application period

the pressure at the locus instantaneously changes sign, now working to resist the plate's relaxation and increases with time. The inset in FIG. 5A focuses on pressure at the locus of application of the external force, given by

From EQ. 1.30, the rate of decay during application is of the same order of magnitude as T ¹¹³ , and a discontinuity in pressure occurs at the instant of release T=T of the external force.

FIG. 5B presents the deformation-field created in the elastic plate. While the deformation near the impact locus is negative, there is a region of significant positive deformation adjacent to the minima at the locus. An infinite number of maxima points of the deformations are expected from EQ. 1.24. However, all other maxima points are small in magnitude compared with the deformation near the locus. The maxima of this primary positive deformation region propagate radially both during and after the application of the external pressure field. The inset in FIG. 5B focuses on deformation at the locus, given by

From EQ. 1.31, the deformation during force application is shown to increase as a power of and have a discontinuity in the rate of deformation at the moment of

release of the external pressure The solution is therefore particularly useful for

time-scales greater than the inertial plate time-scale where l _e is the finite

actuation length-scale. It is noted that the characteristic liquid pressure is defined as where l _e is the magnitude of the external force.

The relation between the externally applied pressure field and the fluidic pressure field will now be explored in order to examine impact mitigation properties of such configurations. To this end, finite external pressures distributed both spatially and temporally are examined. Consider firstly the case of a suddenly applied external pressure, uniform in both space and time with a time period T _e ,i, spatial radius L _e and a constant total impulse of 1, given by

The pressure ratio between the externally applied pressure and the fluidic pressure at the center X=0 can be estimated for as

FIG. 6A shows the liquid pressure at X=0 divided by the external pressure during application period vs. r|Le, according to EQ. 1.33. The solid line denotes the ratio of pressures as a result of an external pressure rapidly applied at T=0 and constant throughout the application period. The dashed line denotes the ratio of pressures as a result of an external pressure linearly increasing with time. Both external pressures are distributed evenly on a disk of radius L _e . The inset shows a schematic illustration of the evolution of the external pressures in time for both cases.

Three distinct periods are evident: (i) an initial period of the impac where the fluidic pressure closely follows the external pressure, (ii) the interval,

shows small oscillations of the pressure ratio going from mitigation to amplification and vice versa, and (iii) the period where mitigation occurs and grows with time.

Consider secondly, time-varying external pressures with rise time of order of magnitude of the viscous-elastic time-scale. In the present Example, an external pressure field, evenly distributed on a disk of radius L _e , linearly increasing in magnitude with respect to time until and then decreases linearly until vanishing at is modeled.

The total impulse is 1, and is thus given by

The ratio of pressures at the center for is:

where is the Meijer G-function.

FIG. 6B shows the ratio of the liquid pressure at the center of impact to the external pressure, P _e , ₂ , at the moment of maximal external pressure, T = T _e ,i, according to EQ. 1.35. External radii L _e = 0.1, 0.3, 0.6, 1.2, 2.4, 4.8 correspond to lines as depicted in the Figure. The inset shows schematically the evolution of the external pressure in time. As shown, for any width of external pressure L _e , mitigation may be achieved if the application time is sufficiently long

Specifically, for the case of L _e =0.1, mitigation of more than 90 % is achieved for external pressures applied over the period of or longer.

Experimental Verification Experiments were conducted to illustrate and verify some of the theoretical results presented above.

The Experimental setup is illustrated in FIG. 7. The experimental setup includes an elastic plate 701 of Polyurethane rubber (Econ®-60 Urethane rubber) floating over a viscous liquid 702 of thin silicon-oil film (Xiameter® PMX-200 Silicone Fluid) which is positioned over a rigid surface 700. The center of plate 701 was deformed at a constant velocity of 10 mm/s due to application of a spherical indenter 703 with radius of 5mm. Indenter 703 was connected to a linear actuator (Thorlabs™ DRV013) 704 and a load cell 705 measuring the force applied on the elastic plate. The radial deformation profile created during force application was sampled by a laser profilometer (MicroEpsilon™ ScanControl 2650-100) 706.

Relevant physical properties of the configuration are elastic plate thickness b=5.5 mm, elastic plate diameter 238 mm, plate bending resistance s=0.01 Pa-m ³ , plate material density p _s =954 Kg/m ³ , liquid viscosity μ=60 Pa-s, liquid density pi=987 Kg/m ³ and initial gap ho=6 mm. FIG. 8 shows the mean value of four experimental measurements (hollow square, filled circle and hollow circle, corresponding to t=0.4s, t=0.6s and t=0.8s, respectively) and theoretical predictions (lines corresponding to t=0.4s, t=0.6s and t=0.8s, as indicated by arrows) vs. the radial coordinate r. The inset in FIG. 8 shows the mean value of force measurements f _e (N), applied by the actuator on the center of the plate vs. time t(s). Error bars indicate one standard deviation. The theoretical deformation is obtained by convolution of the external force measurements with EQ. 1.24 (see also EQ. A.9, below). The radial location of the minimal radius measured by the laser profilometer was estimated by correlation to the analytic solution as r=20mm. No other fitting parameters were used and good agreement between the analytical results and experimental data is evident.

Example 2 - Exemplified Systems

Following are several exemplified systems, according to some embodiments of the present invention.

In some embodiments of the present invention layer 12 is made of expended polystyrene and liquid 14 is a silicon oil, with the following geometrical and physical properties:

. For a length, force and time parameters of

which can correspond to external force applied over an area enclosed by

a circle of diameter, reaching its maximal value of 9000N after t _e =5ms, the

inequalities EQs. 1-4 are calculated as follows:

In some embodiments of the present invention layer 12 is made of aluminum as and liquid 14 is a silicon oil, with the following geometrical and physical properties:

For a length, force and time parameters of

and which can correspond to external force applied over an area enclosed by a circle diameter, reaching its maximal value of 9000N after t _e =5ms, the inequalities EQs. 1-4 are calculated as follows:

Additional exemplified systems are listed in Table 1, below. In Table 1, Ei is expressed in Pa, μ is expressed in Pa-s,f _e is expressed in newtons, l _e is expressed in cm, t _e is expressed in seconds, pi is expressed in kg/m ³ , bi is expressed in cm, and ho is expressed in cm.

Table 1

Appendix A - Results in dimensional form

Presented here are some of the equations and results in dimensional form. These include:

the Green's equation

the governing equation for deformation, EQ. 1.13

governing equation for pressure, EQ. 1.14

the Green's function for EQ. A.1

the Green's function for EQ. A.2

and the Green's function for EQ. A.3

where η for EQs. A.2-A.6 is

The deformation and pressure distribution as a result of a specific external pressure, p _e =p _e (x,y,f), are obtained by the convolutions

Example 3 - Frequency response and resonance of elastic Hela-Shaw cells

In the present Example, the steady-state oscillations of an elastic Hele-Shaw cell excited by traveling pressure waves over its upper surface is studied. The fluid within the cell is bounded by two asymmetric elastic sheets which are connected to a rigid surface via distributed springs. The fluid is modeled by the unsteady lubrication approximation and the sheets are modeled by the linearized plate theory. Modal analysis yields the frequency response of the configuration as a function of three parameters: the fluidic Womersley number and the ratio of solid stress to viscous pressure for each of the sheets. These ratios, analogous to the Capillary number, combine the effects of fluid viscosity and the sheets inertia, bending and tension. The resonance frequencies of the configuration include the resonance frequency of the upper sheet, the resonance frequency of both sheets, and a new resonance frequency related to the interaction between the fluidic motion parallel to the elastic solids and the relative elastic displacements. Near the resonance frequency of the upper sheet, the fluid pressure is identical in amplitude and phase to the external excitation. For configurations where both sheets are near resonance, small changes in frequency yield significant modification of the fluidic pressure. The amplitude ratio of the fluidic pressure to the external pressure is presented vs. frequency for several characteristic solid and fluid properties, yielding a bandpass filter behavior. The results presented here suggest fluid embedded structures may be utilized as protective surfaces and mechanical filters. \Problem Formulation

The steady- state oscillations of two parallel elastic sheets containing a thin liquid film was examined. The fluid flow and solid displacements were excited by an external pressure wave with prescribed frequency and wavelength, which may be readily generalized to an arbitrary external forcing. The elastic sheets were modeled by the linear plate theory and include bending, tension, and inertial effects.

The configuration and the Cartesian coordinate system are defined in

FIG. 9. Referring to FIG. 9, both elastic sheets 1 and 2 are parallel at rest to the x-y plane. P _e (x, y, t) is the external propagating pressure wave. Optionally, a gap between elastic sheets 1 and 2 includes viscous film 3. 2ho represents height of the gap between sheets 1 and 2 at rest, di and d ₂ are the displacements of plates/sheets 1 and 2 in the z- direction, respectively. Elastic spring array 4 may which are connect to elastic sheets 1 and 2 to a rigid surface (not shown). Further, 11 subscript denotes two-dimensional vectors in the x- y plane. The x-y plane is parallel and of equal distance to both plates at rest. The subscripts 1 and 2 denote the upper and lower plates, respectively, and the gapbetweentheplates atrestis 2ho- The fluidic pressure isp, fluid velocity is

fluidic density is p, fluidic viscosity is μ, elastic sheet bending stiffness is s _n , sheet tension is t _n , sheet thickness is b _n , sheet mass-per-area is m _n and sheet displacements are

dn) (where n = 1, 2).

The dynamics of the Newtonian, incompressible fluid is governed by the Navier- Stokes equation:

and the continuity equation

supplemented by no-slip and no-penetration at the fluid-solid interfaces

consistent with the Kirchhoff hypothesis of linear displacement with regard to

the z- direction. The sheet deflections are governed by the linearized Kirchhoff-Love sheet theory

where ¾, i ₂ are considered uniform and isotropic. (The boundary conditions (4) may be reduced to a free surface description by setting where γ is

surface tension)

The external propagating pressure wave is the real part of the function

where pe is the amplitude of the wave, is the wave vector, the

wavenumber, and ω is the angular frequency For the purpose of separating

the flow problem from the bulk deformation of the structure, hereafter the deformation of the sheets is denoted by the average and relative

deformation, w is defined as

where dd /6t represents fluid speed due to the mean motion of both sheets and w is thus fluid speed due to the relative motion of the sheets.

Next scaling and order-of-magnitude analysis were performed. Hereafter, asterisk superscripts denote characteristic values and Capital letters denote normalized variables. The characteristic plane fluid velocity is , the characteristic z-direction

fluid velocity is , the characteristic fluid pressure is the characteristic mean

deformation i and the characteristic relative deformation isd *. The following small parameters were defined:

corresponding to requirements of slender configuration and small relative deformation to fluid gap height. Normalized coordinates and time T are defined

normalized mean sheet deflections

normalized fluid velocity

and normalized fluid pressure P and external pressure P _e

Substituting EQ. 3.8a/b/c/d into EQ. 3.1-EQ. 3.2 yields the leading order momentum equations:

and continuity equation where is me Reynolds number and is the Womersley number.

Order of magnitude analysis of and

Since the linear time-derivative inertial term in the LHS of

(EQ. 3.1) scales with a while the non-linear convective terms

w) scale with the linearized Navier-Stokes momentum equations is

obtained without any further assumptions. In addition, from a restriction on the

phase velocity of the wave is obtained

Substituting yields the leading

order boundary conditions:

EQ. 3.10d

In are necessary conditions for applying linear plate

theory and is a requirement of negligible effect of longitudinal

displacements of the sheets on fluid velocity.

Hereafter the focus is o Thus, the order of

magnitude of the sheet displacements ar and the fluidic pressure due to transverse acceleration is negligible Leading order

governing equations, boundary conditions and order of magnitude for the case of dominant effect of transverse acceleration on fluidic pressure are presented in Appendix B.

PHASE AND AMPLITUDE OF TRAVELING- WAVE SOLUTIONS

To study steady state oscillations traveling- wave solutions of frequency a and wave vector k equal to the external excitation pressure wave are examined. Without loss of generality, the focus was on two-dimensional configurations where the wave vector is parallel to the x-direction. Thus seeking solutions of the form:

By substituting EQ. 3.11 into EQ. 3.9a/b/c-EQ. 3.10a/b/c/d, the governing equations (EQ. 3.9a,b, EQ. 3.9c) were simplified

as well as the boundary conditions EQ. 3.10a/b/c/d to where the dimensionless parameters are defined by (n = 1, 2)

which may be interpreted as the ratio between the inertial and elastic stress within the solid and the traction applied by the fluid due to the viscous squeeze flow.

analogous to the Capillary number for thin films, C _a .) The limits of

correspond to resonance of elastic sheets 1 and 2, respectively. Negative values of

are associated with dominant elastic bending and tension effects whereas positive values are associated with dominant inertial effects. Initially EQ. 3.12 was solved together with EQ. 13 to obtain the longitudinal fluid velocity

EQ. 3.15 was substituted into EQ. 3.12, integrate with respect to Z, and apply EQ. 3.13a to obtain the transverse liquid velocity

Substituting into EQ. 3.13b yields the relative sheet deflection

EQ. 3. 17 was substituted into EQ. 3.13 to obtain the mean sheet deformation

Finally, EQ. 3.13 was subtracted and EQ. 3.17 was substituted to obtain the liquid pressure

Thus obtaining representing the steady-state dynamics by a complex

amplitude which is a function of the Womersley numbe and the sheets impedance

Solutions 3.15-3.18 in their dimensional form may be found in appendix C, as well as the three- dimensional response dynamics for an arbitrary external pressure field.

For the limit of negligible fluidic inertial effects may be further

simplified. In this limit, the liquid pressure is

the mean and relative deformations of the sheets are

and the fluid longitudinal and transverse speeds are

MAPS AND EXTREMA LINES OF FLUID PRESSURE AND ELASTIC DISPLACEMENTS

FIG. 10- FIG. 13 map the amplitude and phase of liquid pressure and sheet deflection for various configurations. In all figures smooth lines represent values of Zl yielding extrema of the amplitude of the examined parameter for set values of (a2, Z2). Similarly, dotted lines represent values of a2 yielding extrema points for set values of Extrema of the fluidic pressure are readily obtained from EQ. 3.15-3.19, where for simplicity extrema of IP (a2, Zl, Z2)/Pel is seek by examining IPe/P 12, defined by

By differentiation of equation 3.23 values of yielding extrema of EQ. 3.23 for

predefined were obtained

Similarly, values of Z ₂ yielding extrema of EQ. 3.23 for predefined are

Combining EQ. 3.26b into 3.26a yields the extrema Z _\ = Z ₂ = 0 for a predefined value is singular and is determined by the limit of the ratio

Expressions for the extrema of the amplitude of the average and

relative D~ '(Zl, Z2, a2) solid displacements are presented in Appendix D. Rigid lower surface, Z2→+∞

FIG. 10A- FIG. 10D present EQ. 3.17 and EQ. 3.19 for the limits of Z2→ ±∞, corresponding to a fixed lower surface D2 ~0. FIG. 10A shows the magnitude ratio of liquid pressure to external pressure FIG. 10B shows the relative phase of liquid pressure FIG. IOC shows the magnitude of deformation normalized by the

external pressure \D\/Pe\, and panel FIG. 10D shows the relative phase of deformation

FIG. 10A and FIG. IOC present maxima of the liquid pressure as well as

sheet deformation along a smooth black line defined by

(or in dimensional form representing linearization of the extrema condition of EQ. 3.26a with regard to a ² around Fluid

inertia thus reduces the resonance frequency of the configuration, which emanates from increasing the mass accelerated during oscillations. On the line the fluidic

pressure decreases with a ² , the amplitude of deformation increases with a ² , and the sheet velocity is syn- chronized with the external pressur Thus for a given deformation amplitude, external oscillating pressure waves apply the maximal external work on the sheet, and thus maxi- mal dissipation, at the line (27). The sheet deformation reaches a global maximum for , corresponding to the

resonance frequency of the upper sheet and negligible fluid inertia parallel to the sheets. At the resonance frequency of the upper plate (on the line Z _\ - 0), the fluidic Womersley number does not affect the liquid pressure which is equal to the external

pressure in both magnitude and phase. In addition, the liquid pressure amplitude is equal to the external pressure on the line on which however

Equal impedance of the lower and upper sheets,

FIG. 11 A- FIG. 11 F show dynamics for configurations where Z dimensional terms FIG. 11A

presents

shows FIG. HE shows \ D2\/ \ P _e \ , and FIG. 11F shows For

predefined values of a ² and ratio of sheet impedance yielding

extrema of fluidic pressure are defined by

which can be approximated by (presented by smooth lines in

figure 11) in the limit of a ² → 0. The limit of yields various values

of pressure and displacements depending on the ratio of For lim(z, ,z ₂ )→(0 _, 0) Z _\ / Zi = — 1, resonance dynamics are accompanied by significant increase in the magnitude of the fluidic pressure. In contrast, for lim(z \ the fluidic

pressure equals half of the external pressure excitation.

FIG.11A presents the extremum line of EQ. 3.28 which is accompanied by a maximum of the relative displacement This maximum line is similar to the

modified resonance of the upper plate presented in figure 10A. FIG. 11C and FIG. 1 IE present the solid resonances alon 0 and an additional fluidic maximum of the upper plate near t he line eq. 3.28. FIG. 1 ID and FIG. 1 IF yield a sharp phase difference of D _\ and D ₂ between Both D _\ and D ₂ are near anti

phase for small negative Z _\ and in-phase for small positive Z _\ . In contrast, the phase of the fluidic pressure (FIG. 1 IB) presents gradual change near resonance. FIG. 11 A, FIG. 11C and FIG. 1 IE show that the resonance of the solid deflection for Z _\ - Zi→ 0, yields similar oscillations of the upper and lower sheets. These oscillations are accompanied by synchronous liquid pressure with half the magnitude of the external pressure ( | P Thus, both sheets are applied with identical

excitations at resonance.

Asymmetric elastic sheets

The effect of modifying the properties of the lower surface on the frequency response and extrema for configurations wher for the range

< 10 and 0 < a < 10 was examined. FIGs. 12A- 12F present scaled fluid pressure amplitude IP/Pel (left column) and relative phase (right column). FIGs. 12A

and 12B correspond to Z2 = -6, FIGs. 12C and 12D correspond to and FIGs.

12E and 12F correspond to The smooth lines (EQ. 3. 26a) represent values of

yielding extrema of the liquid pressure amplitude for set values o The dotted

lines (EQ. 3.26c) present, similarly, values of for set values o

FIG. 12A presents maxima and saddle points along (EQ. 3.26c) where the maximum associated with not on this case the fluid

inertia decreases the resonance frequency, similarly to the cases presented in FIGs. 10 and 11. However, as Z ₂ increases additional maximum emerge FIG. 12C and coalesce

(see FIG. 12E) to a single line nea in which the magnitude of the

derivative of fluid pressure amplitude and phase with regard to Zl sharply increases. This new extremum emanates from a combined fluid-elastic interaction increasing the fluidic pressure (see EQ.3.19) and not a modified solid resonance. As becomes smaller, configurations with positive values o become increasingly synchronized with the external pressure and the liquid pressure, whereas configurations with negative approach the inverse phase of The fluidic pressure amplitude is equal to the

external amplitude on two curves, one of which is the upper sheet resonance frequencies line Z _x = 0. FIGs. 13A- 13F focus on elastic deflection and presents the magnitude of average deformation (left column), and the relative deformation \D \APe\ (right

column) as a function o for identical parameter range as in figure 12A-F.

The smooth and dotted lines represent extremum of deflection with regard to Z _x (for set values of a , Z2) and with regard to a (for set values of Zl, Z2), respectively, see Appendix D. The patterns presented for both the scaled average deflection \D\APe\ and scaled relative deflection \D \APe\ closely follow that of the fluidic pressure. The average deflection \D\APe\ does not involve viscous flow and is thus undamped, in contrast with

\D \APe\. Hence, while far from resonance the magnitude of are similar, near resonance frequencies (e.g. near the lin

-1), the ratio increases indefinitely and eventually invalidates the model assumption of

Referring to configurations where the average and relative displacements are of similar order of magnitudes In the limit of (presented in

Appendix B), although liquid pressure is created by elastic displacements, it is not significant in determining the displacement dynamics.

The response of an elastic Hele-Shaw cell was presented in terms of the parameters (a, Z\, Z£), which combine effects of elasticity, viscosity, fluid arjd solid inertia, as well as the frequency and wavelength of the excitation. While FIG. 10- FIG. 13 describe a wide range of parameters, the effect of changing excitation frequency or wavelength of a specific configuration requires following curved line ( g

interpolation between panels (as in FIG. 12 and FIG. 13). The emergence of an additional fluid-elastic resonance frequency

In the limit of negligible fluidic effects the dynamics of the upper surface will approach the dynamics of an isolated sheet. Alternatively, in the limit of a highly viscous fluid the configuration will be similar to two elastic sheets with a constraint of constant gap (see Appendix E). Thus, multiple elastic resonance frequencies defined by Z _\ = 0, Z ₂ = 0 and Zi + Z ₂ = 0 are expected to appear. However, from FIG. 10- FIG. 13 an additional response frequency is evident, which involves the interaction between motion of fluid parallel to the elastic sheets and elastic displacements and external actuation perpendicular to the sheets. This fluid-elastic resonance is presented in FIGs 14A- FIG. 14C for two illustrative configurations. FIG.14A and FIG. 14B examine the normalized displacement of the upper sheet (FIG. 14A) and fluid pressure amplitude (FIG. 14B) of identical sheet (defined by

2 ^{3 3 3}

and FIG. 14D similarly present upper sheet

normalized displacement (FIG. 14C) and fluid pressure amplitude (FIG. 14D) for configuration where

All deformations are scaled by d* = 3 ·

I0 ^~ ^m. For both configurations presented in FIGs 14A and FIG. 14B and in FIG.14C and FIG. 14D, at lower frequencies corresponding to IZll» l, the amplitude of fluid pressure decreases with ω and displacement of the top sheet is identical to that of an isolated elastic sheet. For the opposite limit of large co, corresponding to IZll,IZ2l« l, the elastic Hele-Shaw cell oscillates as two elastic sheets with a constraint of constant gap (see Appendix E). For FIG.14A and FIG. 14B, in accordance with the results presented in FIG. 11, the identical properties of the bottom and top sheets yield a single solid resonance frequency ω~14.9ΚΗζ accompanied by \p/pe\= 0.5. In addition to this elastic resonance frequency, a clear additional extremum frequency is evident at co~5.05 KHz, near the extremum of the fluidic pressure at ω ~ 5.05 KHz. A similar deformation extremum is presented in FIG. 14C and FIG. 14D for ω ~ 1.11 KHz, near the pressure extremum of ω ~8.09 KHz, in which it is the dominant resonance frequency due to the rigid lower elastic sheet eliminating other solid resonances.

Elastic Hele-Shaw Cell as a Mechanical Filter

FIG. 15A- FIG. 15C present the magnitude and phase of liquid pressure vs. excitation frequency ω for wavelengths of / = 0.06, 0.08 and 0. lm, denoted by the solid, dashed, and dotted lines, respectively. Vertical lines denote resonance frequency of the reference solid configurations of the upper sheet (line a), bottom surface (line b) and two sheets with a constraint of constant gap (line c). Referring to FIG. 15 A- FIG. 15C, the properties of the upper sheet and fluid layer are defined by bending resistance s\= O. SPam 3 , sheet thickness b\= lcm, sheet density pi = 954Kg/m 3 , liquid viscosity μ = 60Pa-s, liquid gap height 2h0 = 5mm, and liquid density pi = 150Kg/m (characteristic to rubber and silicon oil). In panel (a) the bottom surface is rigid. In panel (b) the bottom surface is an elastic spring array with sk = YlGPa/m. Referring to FIG. 15C, the bottom surface is an elastic spring array with sk = YlGPa/m and mass-per-area of m2 =

25Kg/m ² . In all of the configurations, sufficiently small frequencies yield negligible liquid pressure amplitude. An intermediate range of frequencies, which include the resonance frequency of the elastic sheet, leads to liquid pressure with amplitude and phase identical to the external pressure. In the configuration of FIG. 15A large frequencies lead to decay of the liquid pressure due to growing dominance of the elastic sheet inertia. Thus, for Z2→ ±∞, the response of the system is similar to a bandpass filter and the fluidic pressure cannot exceed the external excitation. In configuration (b), similar behavior is observed with an additional peak of the fluidic pressure near the elastic resonance frequency defined by EQs. 3.26a-3.26c At greater

frequencies both the liquid pressure and phase decay to zero. The configuration of FIG. 15C involves an additional resonance frequency of the lower surface, yielding a minima of the pressure near the combined reference resonance

Appendix B: Leading order equations fo

For dynamics characterized b order of magnitude yields that the convection terms in equation 3.1 scales as and may

not be neglected. In addition, the sheets mean acceleratio yields significant pressures gradients in the transverse direction Thus, the leading order

governing equations (3.9) are of the form

and leading order boundary conditions (EQ. 3.10) are

Equations (B 1)-(B2) yield that in the leading order dynamics for

the liquid pressure is created by elastic displacements, but is not significant in determining displacement dynamics of steady- state solutions.

Appendix C: Results in dimensional form

The steady-state oscillating solutions in dimensional form is presented herein. The dimensional liquid pressure (EQ. 3.19) is

the dimensional longitudinal (EQ. 3.15) and transverse (EQ. 3.16) liquid velocities are

The dimensional relative deformation (EQ. 3.17) is

The three-dimensional response dynamics for an arbitrary external pressure field is therefore given in dimensional form by

Where and y denote the x, y direction velocity

components and s calculated from the inverse transformation

EQ. C7

Equations C6 and C7 require the dimensional form since the normalization of the two- dimensional problem was dependent o

Appendix C: Extrema points o

Extrema points of the magnitude of were obtained. The solution

consists of the liquid pressure multiplied by

Thus, extrema points with respect to Zi, Z ₂ (where the remaining variables are kept constant) will yield the same expressions as for the liquid pressure - EQs. 3.26a and 3.26b, respectively. Therefore obtaining extremum points with respect to a ² . The problem was simplified by investigating

Differentiating with respect to or and equating to zero yields

Therefore obtaining extremum points of the magnitude of ■ The equation

defining the extremum point of with respect to is the same as the liquid pressure and

relative deformation and is defined by EQ. 3.26a. Next an expression for the extrema point of D for or was obtained. The problem was simplified by investigating

By differentiating D3 with respect to and equating to zero yields an inexplicit

relationship

as a quadratic equation in terms of Z\ and obtain a functional relationship between Z\

Appendix E: Comparison between dynamics of two sheets connected by a stiff spring array to an elastic Hele-Shaw cell

The phase and amplitude of the steady state oscillations for a reference configuration consisting of two elastic sheets connected by a spring array with coefficient s{l was calculated. The response of two elastic sheets with a constraint of constant gap or an isolated upper sheet is obtained directly from limits of the spring array coefficient.

The governing equation of the upper elastic sheet is

and the governing equation of the lower elastic sheet is

where 5^2 is the spring stiffness connecting the sheets. The wave form was substituted

for all variables, and obtain the upper sheet deformation

and the lower sheet deformation

wher

resonance is obtained when

isolating ¾ yields

In case the spring is much stiffer than the sheets the resonance will occur at z\=-z2- The case of which the spring is much softer than zl yields an isolated upper sheet.

Although the invention has been described in conjunction with specific embodiments thereof, it is evident that many alternatives, modifications and variations will be apparent to those skilled in the art. Accordingly, it is intended to embrace all such alternatives, modifications and variations that fall within the spirit and broad scope of the appended claims.

All publications, patents and patent applications mentioned in this specification are herein incorporated in their entirety by reference into the specification, to the same extent as if each individual publication, patent or patent application was specifically and individually indicated to be incorporated herein by reference. In addition, citation or identification of any reference in this application shall not be construed as an admission that such reference is available as prior art to the present invention. To the extent that section headings are used, they should not be construed as necessarily limiting.