EP4676831A1

EP4676831A1 - Flapping wing and application thereof

Info

Publication number: EP4676831A1
Application number: EP24766620.9A
Authority: EP
Inventors: Bat-El PINCHASIK; Gal RIBAK
Original assignee: Ramot at Tel Aviv University Ltd
Current assignee: Ramot at Tel Aviv University Ltd
Priority date: 2023-03-05
Filing date: 2024-03-05
Publication date: 2026-01-14
Also published as: WO2024184882A1; EP4676831A4

Abstract

A flapping wing for an aerial vehicle comprises a support structure having an arrangement of struts, and a membrane coupled to the support structure, wherein a deformation of the support structure during an up- stroke phase of a flapping cycle of the wing is a mirror of a deformation of the support structure during a down-stroke phase of the flapping cycle of the wing.

Description

FLAPPING WING AND APPLICATION THEREOF

RELATED APPLICATIONS

This application claims the benefit of priority of U.S. Provisional Patent Application Nos. 63/450,029 filed on March 5, 2023, and 63/467,359 filed on May 18, 2023, the contents of which are all incorporated by reference as if fully set forth herein in their entirety.

FIELD AND BACKGROUND OF THE INVENTION

The present invention, in some embodiments thereof, relates to aerodynamics and, more particularly, but not exclusively, to a flapping wing and application thereof. Some embodiments of the present invention relate method of fabricating a flapping wing.

The field of unmanned, powered vehicles operating in air has seen accelerated growth since around 2009. Such vehicles exist across the full range of scales, from tiny toy drones to large cargo carriers. Most of the known miniature unmanned aerial vehicle (UAV) fly using rotary wings with the number of rotors varying between one and many (multi-rotor UAVs). Most popular are quadcopters with four rotating wings. However, other types of UAV are also known. These include the fixed-wing type and the flapping wing type.

The flapping wing UAV is a kind of minute vehicle that imitates birds or insect flight, and has grown up in the nineties of 20th century. Insect wings are morphologically diverse, supporting a variety of flight styles. They comprise a flexible membrane and stiffer wing veins, with the latter differing in number, thickness, cross-section shape and spatial arrangement among species. The thin wings undergo elastic wing deformations during each flapping cycle. The geometry of the wing veins determines the pattern and magnitude of these deformations and affect the aerodynamics of flapping flight.

SUMMARY OF THE INVENTION

According to an aspect of some embodiments of the present invention there is provided a flapping wing for an aerial vehicle. The wing comprises a support structure having an arrangement of struts, and a compliant membrane coupled to the support structure, wherein during hovering of the vehicle a deformation of the support structure during an up-stroke phase of a flapping cycle of the wing is a mirror of a deformation of the support structure during a down-stroke phase of the flapping cycle of the wing.

According to an aspect of some embodiments of the present invention there is provided a flapping wing for aerial vehicle. The wing comprises a support structure having an arrangement of struts, and a membrane coupled to the support structure, wherein during forward flight of the vehicle an up-stroke and a down-stroke phase of the flapping cycle an in-plane and an out-of-plane deformations of the support structure increases the aerodynamic force generated by the wing, such that the increased aerodynamic force is vertical during the down-stroke phase and horizontal during the up-stroke phase.

According to an aspect of some embodiments of the present invention there is provided a flapping wing for aerial vehicle. The wing comprises a support structure having an arrangement of struts, and a membrane coupled to the support structure, wherein a deformation of the support structure during flapping of the wing is such that a sectional lift-to-drag ratio is generally uniform along the wing, wherein for any section of the wing, a lift-to-drag ratio of the section is no less than 80% or no less than 90% or no less than 95% of the maximal possible lift-to-drag ratio among all possible angle-of-attacks and curvatures of the section, irrespectively of the load on the wing.

According to an aspect of some embodiments of the present invention there is provided a flapping wing for aerial vehicle. The wing comprises a support structure having an arrangement of struts, and a membrane coupled to the support structure, wherein a deformation of the support structure during flapping of the wing is such that for any section of the wing and a for any lift value of the section, a drag value of the section is smallest among a set of drag values corresponding to the lift value.

According to some embodiments of the invention at least one of the struts is tapered along the strut.

According to some embodiments of the invention at least one of the struts has a circular cross-section.

According to some embodiments of the invention at least one of said struts has a U-shaped cross-section.

According to some embodiments of the invention at least one of the struts has an oval crosssection with a generally horizontal major axis.

According to some embodiments of the invention at least one of the struts has an oval crosssection with a generally vertical major axis.

According to some embodiments of the invention the struts are joined at one end but spaced apart at an opposite end.

According to some embodiments of the invention the support structure comprises at least one elongated stringer member connecting two of the struts.

According to some embodiments of the invention the support structure is printed. According to some embodiments of the invention at least one of the struts, more preferably each of the struts, is characterized by a Young’s modulus of from about 1000 to about 3000 MPa.

According to some embodiments of the invention at least one of the struts is characterized by a flexural modulus of from about 1000 to about 3000 MPa.

According to an aspect of some embodiments of the present invention there is provided an aerial vehicle. The flying vehicle comprises a vehicle body, at least two controllable flapping wings pivotally coupled to the vehicle body, and at least one motor for providing a flapping motion to the wings. Each of the wings is as delineated above and optionally and preferably as further detailed below.

According to an aspect of some embodiments of the present invention there is provided aerial vehicle. The flying vehicle comprises a fixed wing flying vehicle having a pair of fixed wing, and at least one flapping wing mounted on a distal end of each fixed wing of the pair. The flapping wing is as delineated above and optionally and preferably as further detailed below.

According to some embodiments of the invention the aerial vehicle comprises a passive pitching mechanism connected to a root of each flapping wing and configured to vary a pitching angle of the flapping wing during a flapping cycle of the flapping wing.

According to an aspect of some embodiments of the present invention there is provided an aerial vehicle. The flying vehicle comprises a vehicle body, at least two controllable flapping wings pivotally coupled to the vehicle body, at least one motor for providing a flapping motion to the wings, and a passive pitching mechanism connected to a root of each flapping wing and configured to vary a pitching angle of the flapping wing during a flapping cycle of the flapping wing.

According to an aspect of some embodiments of the present invention there is provided aerial vehicle. The flying vehicle comprises a fixed wing flying vehicle having a pair of fixed wing, at least one flapping wing mounted on a distal end of each fixed wing of the pair, and a passive pitching mechanism connected to a root of each of the at least one flapping wing and configured to vary a pitching angle of the flapping wing during a flapping cycle of the flapping wing.

According to some embodiments of the invention the variation of the pitching angle is generally harmonic. According to some embodiments of the invention the passive pitching mechanism comprises an elastic element configured to exhibit elastic vibration about a vertical axis during the flapping cycle.

According to some embodiments of the invention the variation of the pitching angle is described by a generally piecewise linear function of a time within the flapping cycle. According to some embodiments of the invention the variation of the pitching angle is generally trapezoidal. According to some embodiments of the invention the passive pitching mechanism comprises a bearing configured to allow the flapping wing to rotate about a horizontal axis during the flapping cycle.

According to an aspect of some embodiments of the present invention there is provided a kit. The kit comprises an aerial vehicle body connectable to two or more flapping wings, and having one or more motors for providing a flapping motion to the wings. In some embodiments, the aerial vehicle of the kit is connectable to the wings by means of a passive pitching mechanism configured to vary a pitching angle of the flapping wing during a flapping cycle of the flapping wing, as delineated above and optionally and preferably as further detailed below. Separately from the vehicle body, the kit comprises a plurality of sets of flapping wings pivotally connectable to the vehicle body. Each set of wings is adapted to provide a different flight mode.

According to some embodiments of the invention the sets comprise at least one set adapted to provide a hovering flight mode.

According to some embodiments of the invention the sets comprise at least one set adapted to provide a forward flight mode.

According to some embodiments of the invention the sets comprise at least one set having wings characterized by wing loading which is higher than a wing loading of any wing in other sets.

According to some embodiments of the invention the sets comprise at least one set having wings characterized by lift-to-drag ratio which is higher than a lift-to-drag ratio of any wing in other sets.

According to some embodiments of the present invention the aerial vehicle is an unmanned aerial vehicle.

According to an aspect of some embodiments of the present invention there is provided a method of fabricating a support structure for a flapping wing. The method comprises receiving CT scan data of veins within a wing of an insect; generating a CAD model based on the scan data; and, based on the CAD model, operating a three-dimensional printer to print a support structure having an arrangement of struts, corresponding to at least a portion of the veins.

According to some embodiments of the invention the method comprises modifying the CAD model prior to the printing.

According to some embodiments of the invention the method comprises printing a sacrificial structure wherein the struts are embedded in the sacrificial structure, removing the sacrificial structure to expose the struts, and placing the exposed struts in a template substrate patterned with grooves compatible with the struts so as to stabilize a shape of the struts in the template substrate.

According to some embodiments of the invention the sacrificial structure is removed by heating.

According to some embodiments of the invention the sacrificial structure is removed by immersing the sacrificial structure in liquid. According to some embodiments of the invention the liquid comprises oil.

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 THE SEVERAL VIEWS OF THE DRAWING(S)

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. 1 is a schematic illustration of an exemplary arrangement of struts based on veins of a beetle wing obtained in experiments performed according to some embodiments of the present invention by CT scan;

FIGs. 2A-Z are schematic illustrations of structural modification, which can be employed according to some embodiments of the present invention;

FIGs. 3A-C are schematic illustrations of various profiles of struts which can be employed according to some embodiments of the present invention;

FIGs. 4A-F show a 3D printing scheme of bioinspired beetle wings, as employed in experiments performed according to some embodiments of the present invention;

FIGs. 5A-G show static bending tests on 3D printed wings with the same wing mass and 2D geometry but differences in cross section of the wing-struts, as obtained in experiments performed according to some embodiments of the present invention;

FIGs. 6A-F show drag force measurements of 3D printed wings, as obtained in experiments performed according to some embodiments of the present invention;

FIGs. 7A-N show structural variations in bioinspired 3D-printed wings, as obtained in experiments performed according to some embodiments of the present invention;

FIG. 8A is a schematic illustration of a flapping wing according to some embodiments of the present invention;

FIG. 8B is a schematic illustration of an aerial vehicle which includes a flapping wing, according to some embodiments of the present invention;

FIG. 9 is a schematic illustration of a kit which includes a plurality of sets of flapping wings connectable to a body of an aerial vehicle according to some embodiments of the present invention; and

FIG. 10 is a flowchart diagram of a method suitable for fabricating a support structure for a flapping wing according to various exemplary embodiments of the present invention.

FIGs. 11A-D show synthesis of a 3D-printed wing, according to some embodiments of the present invention.

FIGs. 12A-H show mechanical response of an individual strut with uniform and tapered structures, as obtained in experiments performed according to some embodiments of the present invention. FIGs. 13A-G show mechanical response of a wing according to a cross-section geometry, as obtained in experiments performed according to some embodiments of the present invention.

FIGs. 14A-D show results of experiments directed to study elastic deformations of 3D- printed wings and the wings of the rose chafer beetle, as obtained in experiments performed according to some embodiments of the present invention.

FIGs. 15A-E show aerodynamic performance of 3D-printed wings as obtained in experiments performed according to some embodiments of the present invention.

FIGs. 16A and 16B show lift production efficiency as a function of the angle-of-attack (AoA) and strut cross-section shape, as obtained in experiments performed according to some embodiments of the present invention.

FIGs. 17A and 17B show a Rose chafer beetle (FIG. 17A), a pitching of the Rose chafer beetle’s wings during the flapping cycle (FIG. 17B). The small sketch of the beetle in FIG. 17B depicts the pitching motion of the wings.

FIGs. 17C-H show flapping mechanisms (FIGs. 17C-D and 17F-G) and corresponding qualitative profiles of resulting changes in pitching angles during a flapping cycle (FIGs. 17E, and 17H), obtained in experiments performed according to some embodiments of the present invention.

FIGs. 18A-F are schematic illustrations detailing connections between a flapping system, pitch mechanisms, and wings which were used in experiments performed according to some embodiments of the present invention.

FIGs. 19A-F show a wing fabrication process employed in experiments performed according to some embodiments of the present invention.

FIGs. 20A-F are images showing flapping devices assembled in experiments performed according to some embodiments of the present invention.

FIG. 21 is a schematic illustration of experimental setup used according to some embodiments of the present invention for testing the aerodynamic performance of the devices.

FIGs. 22A and 22B are images captured using the experimental setup shown in FIG. 21.

FIGs. 23A-C show lift forces measurement results obtained in in experiments performed according to some embodiments of the present invention.

FIG. 24 shows power-to-lift ratios results obtained in experiments performed according to some embodiments of the present invention.

FIG. 25 is a schematic illustration formulating a change in pitching angle of a spring mechanism according to some embodiments of the present invention.

FIG. 26 is a schematic illustration of a leaf spring used in experiments performed according to some embodiments of the present invention. FIGs. 27A-C show results of simulations directed to investigate a deflection of a leaf spring, as obtained in experiments performed according to some embodiments of the present invention.

FIGs. 28A-D are images of maximal pitching angles obtained in experiments performed according to some embodiments of the present invention.

DESCRIPTION OF SPECIFIC EMBODIMENTS OF THE INVENTION

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.

FIG. 8A is a schematic illustration of a flapping wing 104, according to some embodiments of the present invention. Flapping wing 104 can be connected to any areal vehicle having a flapping actuator, optionally and preferably an electrically driven flapping actuator, such as, but not limited to, a motor. The flapping wing of the present embodiments is particularly useful for use in an unmanned aerial vehicle, particularly a miniature unmanned aerial vehicle.

A representative example of an areal vehicle suitable for some embodiments of the invention is illustrated in FIG. 8B, and is described below.

The flapping wing 104 of the present embodiments is formed of a support structure 106 having an arrangement of struts 108, and a membrane 110 coupled to the support structure 106. In some embodiments of the invention support structure 106 is fabricated by additive manufacturing, such as, but not limited to, three-dimensional printing.

The basic operation of any additive manufacturing system consists of slicing a three- dimensional computer model into thin cross sections, translating the result into two-dimensional position data and feeding the data to control equipment which manufacture a three-dimensional structure in a layerwise manner.

Additive manufacturing entails many different approaches to the method of fabrication, including three-dimensional printing, laminated object manufacturing, fused deposition modeling and others.

In three-dimensional printing processes, for example, a building material is dispensed from a printing head having a set of nozzles to form layers of deposited material on a supporting structure. Depending on the building material, the layers may then be cured or solidified using a suitable device. The building material may include modeling material, which forms the object, and support material, which supports the object as it is being built.

Support structure 106 can be made of any suitable material, preferably a hardened material formed by solidifying a building material deposited by three-dimensional printing. Suitable materials include, but are not limited to, a photopolymer resin, polyurethane, methyl methacrylate- acrylonitrile-butadiene- styrene copolymer, resorbable materials such as polymer-ceramic composites, etc. Examples of commercially available materials are: DSM Somos® series of materials 7100, 8100, 9100, 9420, 10100, 11100, 12110, 14120 and 15100 from DSM Somos; ABSplus-P430, ABSi, ABS-ESD7, ABS-M30, ABS-M30i, PC-ABS, PC ISO, PC, ULTEM 9085, PPSF and PPSU materials from Stratasys; Accura Plastic, DuraForm, CastForm, Easerform and VisiJet line of materials from 3-Systems; the PA line of materials, PrimeCast and PrimePart materials and Alumide and CarbonMide from EOS GmbH. The VisiJet line of materials from 3- Systems may include Visijet Flex, Visijet Tough, Visijet Clear, Visijet HiTemp, Visijet e-stone, Visijet Black, Visijet Jewel, Visijet FTI, etc. Examples of other materials may include Objet materials, such as Objet Fullcure, Objet Veroclear, Objet Digital Materials, Objet Duruswhite, Objet Tangoblack, Objet Tangoplus, Objet Tangoblackplus, etc. Another example of materials may include materials from the Renshape 5000 and 7800 series.

The membrane 110 can be made from a flexible material such as a polymer film, such as, for example but not limitation, ABS or Nylon. The membrane 110 can be attached to one or more of the struts 108 of support structure 106 using an adhesive, such as, but not limited to, epoxy glue, polyamide glue, phenolic resin glue, polyolefin glue, cellulose gum, butadiene-styrene rubber, saturated polyester glue, polyurethane adhesive, isocyanate glue, polyvinyl chloride glue, and polyimides glue.

Typically, but not necessarily, the thickness of membrane 110 is at least 10pm, or at least 20pm. Embodiments in which the thickness of membrane 110 is less than 10pm are also contemplated.

In some embodiments of the present invention at least some of the struts 108 are joined at one end 114 of the struts but are spaced apart at an opposite end 118 of the struts.

In some embodiments of the present invention support structure 106 comprises at least one elongated stringer member 120 connecting two of struts 108. Elongated stringer member 120 can be made of the same material as struts 108 but is optionally and preferably, but not necessarily, has a cross-section which is different in size or shape from the cross section of the struts it connects. FIGs. 8 A and 8B illustrate one elongated stringer member but any number of such stringer members can be utilized, including zero.

Struts 108 are preferably flexible so that during the flapping cycle the support structure 106 deforms. Preferably, struts 108 are made of a material having a Young’s modulus of from about 1000 to about 3000 MPa. In some embodiments of the present invention struts 108 are made of a material having a flexural modulus of from about 1000 to about 3000 MPa. A flexural modulus is meant the ratio of stress to strain during flexural deformation, which slope can be determined from the slope of a stress-strain curve produced by a flexural test such as the ASTM D790. Flexural modulus may be determined, for example, according to ASTM D-790-04.

At least one of struts 108, more preferably each of struts 108 has a tapering geometry. Preferably, the struts are tapered such that they are thicker at their base and thinner at their tip. The Inventors found that tapering geometry provides a structural advantage. Considering that the bending moment, due to load applied at the free tip, is maximal at the leading edge, the tapering provides a higher safety margin against failure of the struts at their base.

In various exemplary embodiments of the invention a deformation of support structure 106 during an up-stroke phase of a flapping cycle of wing 104 is a mirror of a deformation of support structure 106 during a down- stroke phase of the flapping cycle of the wing. These embodiments are particularly useful when it is desired to allow the areal vehicle employing the sing to hover. Representative examples of arrangements of struts suitable for these embodiments are described in Table 1 and FIGs. 2A-D of the Examples section that follows.

In various exemplary embodiments of the invention there are different chordwise bending in the up stroke and down stroke of wing 104. For example, the arrangement of struts 108 can be selected such that an in-plane deformation of support structure 106 increases the aerodynamic force generated by wing 104 on surrounding air during both the up-stroke and the down-stroke phases of the flapping cycle, except that the increased aerodynamic force is vertical during the downstroke phase and horizontal during the up-stroke phase. These embodiments are particularly useful when it is desired to allow fast forward flight for the vehicle employing the wing. Representative examples of arrangements of struts suitable for these embodiments are described in Table 2 and FIGs. 2E-N of the Examples section that follows.

In various exemplary embodiments of the invention the deformation of support structure 106 during flapping of wing 104 is such that a sectional lift-to-drag ratio is generally uniform along the wing. The inventors found that such uniformity improves the sectional lift through tuning local chordwise curvature. These embodiments are particularly useful when the vehicle employing wing 104 is configured for carrying a load at the expense of flying at a reduced speed, typically between a zero speed (the aforementioned hovering mode) and high speed (the aforementioned fast forward flight mode). Typically, for any section of the wing, a lift-to-drag ratio of the section is no less than 80% or no less than 90% or no less than 95% of the maximal possible lift-to-drag ratio among all possible angle-of-attacks and curvatures of the section, irrespectively of the load on the wing. Representative examples of arrangements of struts suitable for these embodiments are described in Table 3 and FIGs. 2O-U of the Examples section that follows.

In various exemplary embodiments of the invention the deformation of support structure 106 during the flapping of wing 104 is such that for any section of wing 104 and a for any lift value L of that section, a drag value D of the section is smallest among a set {Di} of drag values corresponding to the lift value L. The set {Di} includes a plurality of drag values that can be induced by a given lift L for various geometries of the wing and various angles of attack. In embodiments in which a drag value of the respective section is minimal for all wing sections, the lift-to-drag ratio L/D is maximal, thereby reducing the power consumption of the vehicle employing wing 104 and increasing the endurance of the flight. Representative examples of arrangements of struts suitable for these embodiments are described in Table 4 and FIGs. 2V-Z of the Examples section that follows.

The profile of struts 108 of structure 106 can have any shape. For example, one or more of struts 108 can have a circular or an oval cross-section. Also contemplated are embodiments in which at least one of the struts of structure 106 has a circular cross-section, and at least one of the struts of structure 106 has an oval cross-section. Further contemplated are embodiments in which one or more of the struts has at least one segment with a circular cross-section and at least one segment with an oval cross-section. When a particular strut or a segment thereof has an oval crosssection, the major axis of the oval can be horizontal or vertical, or it can be tilted, where the terms "horizontal," "vertical," and "tilted," are defined with respect to the gravity when the membrane generally engages a plane that is perpendicular to gravity. A representative example of oval crosssection with a horizontal major axis 302 relative to gravity g is illustrated in FIG. 3A, and a representative example of oval cross-section with a vertical major axis 304 relative to gravity g is illustrated in FIG. 3B. The skilled person would know, based on the details shown in FIGs. 3 A and 3B how to configure a strut with a tilted major axis. Other profiles for the struts are also contemplated. For example, one or more of struts can have a U-shaped cross-section, as illustrated in FIG. 3C.

FIG. 8B is a schematic illustration of an aerial vehicle 100 according to some embodiments of the present invention. Vehicle 100 is optionally and preferably an unmanned aerial vehicle, more preferably a miniature unmanned aerial vehicle. Vehicle 100 comprises a body 102 having a longitudinal axis 101, a flapping wing 103, and a flapping system 113 that connects the wing 103 to the body 102, and is configured to impart at least a flapping motion to wing 103. The flapping motion generated by system 113 can be horizontal, vertical or combination thereof. Preferably, the flapping motion is generally horizontal so as to mimic the flapping motion of beetles. The connection of wing 103 to flapping system 113 can in some embodiments of the present invention be by a joint 116, such as, but not limited to, a flexible joint. Wing(s) 103 is/or connected at the sides 121a, 121b, of body 102, and longitudinal axis 101 is typically defined by the head and tail ends 123a and 123b of body 102. The forward flight direction of vehicle 100 is typically along the longitudinal axis 101. A lateral axis 105 of vehicle 100 is an axis that is perpendicular to axis 101 and passes through the sides 121a, 121b (e.g., at the points at which the wings are connected to body 102). It is appreciated that both axes 101 and 105 are imaginary lines.

In some embodiments of the present invention wing 103 is enacted by wing 104 described above with reference to FIG. 8A. However, this need not necessarily be the case, since, for some applications, it may not be necessary for the flapping wing 103 to be of the same type as wing 104. Thus, the present embodiments contemplate any type of flapping wing for device 100. Representative examples of flapping wings suitable for use as wing 103 according to some embodiments of the present invention include, without limitation, the flapping wings discussed in References [17] and [28, the contents of which are hereby incorporated by reference.

Body 102 of vehicle 100 can take any form suitable for use as a body of a flapping wing aerial vehicle, including, without limitation, UAV, and an insect-like robot. In some embodiments of the present invention flapping system 113 is connected, for example, by joint 116, directly to body 102, and in some embodiments of the present invention vehicle 100 is a fixed wing flying vehicle having a pair of fixed wings 122, in which case flapping system 113 (e.g., wing 104) is mounted, e.g., by means of flexible joint 116, on a distal end 124 of the fixed wing(s) 122. While FIG. 8B illustrates a case in which flapping system 113 is mounted on a fixed wing 122 of vehicle 100 it is to be understood that it is not necessary for vehicle 100 to include a fixed wing, in which case the flapping system 103 is connected, as stated, to the body 102 of vehicle 100. When vehicle does not include a fixed wing, the flapping wing 103 (e.g., wing 104) serves for providing both lift and locomotion, and when vehicle 100 includes a fixed wing 122, the fixed wing 122 serves for providing lift, and flapping wing 103 (e.g., wing 104) serves for providing locomotion and optionally and preferably, but not necessarily, also lift.

Preferably, flapping system 113 is connected, e.g., by joint 116, to structure 106 at end 114 at which the struts of the wing are joined. Also contemplated are embodiments in which one or more of the struts of the wing is joined to another strut at a location other then at joint 116. Flapping system 113 can be of any type that creates reciprocal linear motion along a direction perpendicular to the plane engaged by body 102 of vehicle 100. Representative examples of such types systems include, without limitation, a crank and a slider, a cylinder and piston, a cam and a follower, and a solenoid and a plunger, but other types of reciprocal motion systems are also contemplated. A representative example of a system that can be employed as flapping system 113 according to some embodiments of the present invention is illustrated in FIG. 18F. In this embodiment, flapping system 113 comprises a crank 82 having a crankshaft 84 and a crank arm 86 connected at its proximal end 88 to crankshaft 84 away from a rotation axis 90 of crankshaft 84. The crank arm 86 translates the rotation of the crankshaft 84 to a linear reciprocal motion at the distal end 92 of crank arm 86. A toothed slider 94 is connected to crank arm 86 at distal end 92 to reciprocally move therewith. Slider 94 engages a pair of pinions 96, each arranged to rotate about an axis 98. The joint 116 of each wing (the wing is not shown in FIG. 18F) is connected to one of the pinions 96. Since the linear motion of slider is reciprocal, the rotation of each of the pinions 96 alternates between a clockwise and a counterclockwise rotation, creating the upstroke and downstroke parts of the flapping motion.

Flapping system 113 can is driven by a motor 112, which may be controlled to vary the speed and optionally the direction of the flapping motion, so as to allow controlling the vehicle's flight characteristics. For example, motor 112 can be arranged to rotate crankshaft 84 of crank 82. Motor 112 is typically mounted on body 102. Motor 112 can be connected to flapping system 113 by means of a gear transmission 206 (not shown, see FIG. 18D). In experiments performed by the Inventors a transmission ratio of 1:6 was used, but any other transmission ratio can be used.

When joint 116 is flexible, its flexibility enhances the flapping motion of wing 103 (e.g., wing 104). The joint 116 is optionally and preferably designed to provide a range of motion, preferably up and down, but optionally also forward and backward, and side to side. Alternatively, wing 103 can be rigidly connected to flapping system 113, in which case the flexibility of the flapping motion is provided only by the structure and materials of wing 103.

In some embodiments of the present invention vehicle 100 comprises a pitching mechanism 117 that is connected to a root (e.g., end 114) of each flapping wing 103 (e.g., wing 104) and that is configured to vary a pitching angle of wing 103 during its flapping cycle. Pitching mechanism 117 can be connected between the root of wing 103 and joint 116.

Herein "pitching angle" refers to the angle of rotation around an axis parallel to lateral axis 105 of vehicle 100.

Pitching mechanism 117 is preferably a passive pitching mechanism. As used herein "passing pitching mechanism" refers to a pitching mechanism that varies a pitching angle solely due to the inertia of the wing's flapping motion, in the absence of any motor that actuate a pitching motion.

A passive pitching mechanism is preferred from the standpoint of cost and simplicity of manufacturing. The Inventors found that use of passing pitching mechanism is efficient, and that it can significantly amplify the lift.

The variation of the pitching angle is preferably gradual during the flapping cycle of wing 103. The gradual variation can be generally linear (e.g., within about 10% deviation from linearity) or nonlinear (e.g., with more than 10% deviation from linearity) as a function of the time within the flapping cycle. A representative example of a nonlinear variation is a harmonic (e.g., sinusoidal) variation. FIG. 17E is a graph that schematically illustrate a nonlinear variation in embodiments in which the variation is harmonic. The graph is constructed in a manner that the pitching angle is defined as a positive angle during the downstroke part of the flapping cycle and a negative angle during the upstroke part of the flapping cycle. When the variation is generally linear, it is optionally and preferably according to a generally piecewise linear function.

A generally piecewise linear function is a mathematical function that is defined by multiple generally linear segments (e.g., within about 10% deviation from linearity). FIG. 17H is a graph that schematically illustrate a generally linear variation in embodiments in which the variation is described by a piecewise linear function. As in the case of FIG. 17E, the graph in FUG. 17H is constructed in a manner that the pitching angle is defined as a positive angle during the downstroke part of the flapping cycle and a negative angle during the upstroke part of the flapping cycle. In the schematic illustration of FIG. 17H, the piecewise linear function is trapezoidal, but other piecewise linear shapes are also contemplated.

In some embodiments of the present invention mechanism 117 comprises an elastic element 202 (not shown in FIG. 8B, see FIGs. 17D, 18A-C) configured to exhibit elastic vibration about a vertical axis (perpendicular to both axes 101 and 105) during the flapping cycle of wing 103. The elastic element can be in the form of an elastic beam (e.g., a metallic plate) that act as a leaf spring. Preferably, the elastic element is connected between the root of the and the joint 116. This embodiment is illustrated in FIGs. FIGs. 17D, 18A, 18B.

The elastic element vibrates due to the inertial and aerodynamic loads applied on the wings. As the elastic element bends out of the vertical plane the wing's root is slightly displaced and rotated by the deflection angle of the elastic element (see angle 0 in FIG. 17D). The rotation changes the pitch of the wing relative to the horizontal plane (defined by axes 101 and 105), thus reducing the AoA of the wing from its value at rest (90 ⁰ when system 113 is not operative). At stroke reversals, the wings switch direction and so does the bending direction of the elastic element, changing the AoA for lift generation at both half strokes of the flapping cycle. The deflection angle 0 depends on the magnitude of the loads, and on the length, thickness and elastic modulus of the elastic element.

Elastic element 202 in mechanism 117 is particularly useful when it is desired to have a nonlinear, more preferably a harmonic (e.g., sinusoidal), variation of the pitch angle during the flapping cycle.

In some embodiments of the present invention mechanism 117 comprises a bearing 204 (not shown, see FIGs. 17G, 18B, 18C) configured to allow the flapping wing 103 to rotate about the lateral axis 105 (which is horizontal when the vehicle is horizontal) during the flapping cycle. Barring 204 can be of any type known in the art, such as, but not limited to, a ball bearing, a roller bearing, a needle bearing, and the like. Thus, bearing 204 in mechanism 117 is particularly useful when it is desired to have a generally piecewise linear variation (e.g., trapezoidal) of the pitch angle during the flapping cycle. A trapezoidal variation can be achieved by configuring the barring to a predetermine (less that ±180°) angular range of rotation. For example, barring can be preferably configured to rotate at pitching angles of ±30°, or ±40°, or ±45°, or ±50°. The barring is rotated during the flapping cycle due to the inertia of the wing and the drag. As the wing changes its flapping direction at stroke reversals, the pitching angle of the wing changes until the wing reaches a hard stop at the end of the rotation range. Aerodynamic loads centered on the wing posterior to the rotation axis can ensure that this pitching angle remains constant until the next stroke reversal.

The Inventors found that mechanism 117 substantially increases the lift force applied to vehicle 100. In experiments, performed according to some embodiments of the present invention, and described in the Examples section that follows (see Example 4), it was found that the lift is increased in proportion to the cubic power of the flapping frequency when mechanism 117 comprises elastic element 202, to the square power of the flapping frequency when mechanism 117 comprises bearing 204. Use of elastic element 202 in mechanism 117 is preferred from the standpoint of elastic energy recycling at stroke reversals, and having pitch angles which adapt themselves to the load, thus improving the lift, particularly at higher flapping frequencies. Use of bearing 204 in mechanism 117 is preferred from the standpoint of mechanical stability and pitching predictability, particularly at lower flapping frequencies.

Typically, vehicle 100 comprises more than one flapping wing (two are shown in FIG. 8B, but any other number of wings can be used). Preferably, the number of wings is even. The wings can be operated in synchrony to provide lift and propulsion, or they can be operated independently to achieve different flight characteristics. The wings may be identical in shape and size, or be different as desired. For example, wings at opposite sides of the body 102 may be mirror images of one another.

Vehicle 100 can be controlled using a variety of means, including remote control or onboard computer systems (not shown). In some embodiments, vehicle 100 can include sensors to detect and respond to changes in the environment, such as wind or obstacles, and may also includes other sensors for surveillance purposes (e.g., a light sensor, an imager, a sound sensor or the like). For example, vehicle 100 may include a gyroscope or accelerometer to detect changes in its orientation, or a proximity sensor to detect nearby objects.

The ability of the flapping wing of the present embodiments to be manufactured easily for a specific flight mode (e.g., a wing specifically designed for hovering, or a wing specifically designed for fast forward flight, or a wing specifically designed for high wing loading, or a wing specifically designed for endurance), makes the wing suitable for being mounted on vehicles designed for such a specific flight mode. Alternatively, two or more set of wings can be manufactured in advance, each set for a specific flight mode. In this case the same vehicle can be used for many types of flight modes, where for each flight mode a different set of flapping wings is mounted on the vehicle. Thus, with reference to FIG. 9, according to some embodiments of the present invention there is provided a kit 130, which comprises an aerial vehicle body (e.g., body 102) connectable to at least two flapping wings (e.g., wings 104), and having at least one motor (e.g. , motor 112) for providing a flapping motion to wings 104. The kit can also include, separately from vehicle body 102, a plurality of sets 132 of flapping wings 104 pivotally connectable to vehicle body 102 (for example, by means of joints 116) wherein each set of is adapted to provide a different flight mode.

In some embodiments of the present invention sets 132 comprise at least one set adapted to provide a hovering flight mode, as further detailed hereinabove, in some embodiments of the present invention sets 132 comprise at least one set adapted to provide a forward flight mode, as further detailed hereinabove, in some embodiments of the present invention sets 132 comprise at least one set having wings characterized by improved wing loading as further detailed hereinabove (e.g., wing loading higher than a wing loading of any wing in other sets), and in some embodiments of the present invention sets 132 comprise at least one set having wings characterized by a high lift-to-drag ratio as further detailed hereinabove (e.g., lift-to-drag ratio which is higher than a lift- to-drag ratio of any wing in other sets).

FIG. 10 is a flowchart diagram of a method suitable for fabricating a support structure for a flapping wing according to various exemplary embodiments of the present invention. It is to be understood that, unless otherwise defined, the operations described hereinbelow can be executed either contemporaneously or sequentially in many combinations or orders of execution. Specifically, the ordering of the flowchart diagrams is not to be considered as limiting. For example, two or more operations, appearing in the following description or in the flowchart diagrams in a particular order, can be executed in a different order (e.g., a reverse order) or substantially contemporaneously. Additionally, several operations described below are optional and may not be executed.

The method begins at 400 and continues to 401 at which CT scan data of veins within a wing of an insect is received. The CT scan data can be obtained from a CT scanner. Generally, an X-ray source generates X-ray radiation which penetrates through the wing of the insect and is detected on the other side using a group of detecting elements to thereby acquire projected data. The X-ray source and the detecting elements rotate around the wing, and the scan data is generated from the projected data by reconstructing X-ray absorption distribution. The scan data can be, for example, in DICOM format, which is a standard format used in medical imaging.

The method proceeds to 402 at which a CAD model is generated based on the scan data. The CAD model preferably includes an arrangement of struts (e.g., struts 108) forming a support structure (e.g., support structure 106), wherein each strut has a shape of one of the veins in the CT scan data. This can be done using any computer software product known in the art. The software can automatically segment the CT scan data to separate the wing the surrounding tissue. The segmentation can be by one or more techniques such as, but not limited to, thresholding, region growing, and the like. The software can then generate a surface mesh from the segmented data and convert it into the CAD model. The surface mesh can be generated, for example, by marching cubes, level sets, or any other mesh generation algorithms known in the art. A representative example of a computer software product suitable for the present embodiments includes Solidworks®, AutoCAD®, Inventor®, CATIA®, Onshape®, and the like.

The software can optionally and preferably provide tools for editing and refining the CAD model, such as smoothing, simplification, and feature recognition. The software can also provide tools for measurement and analysis of the CAD model, such as volume, surface area, and cross- sectional area. In some embodiments of the present invention the method proceeds to 403 at which the CAD model is modified, preferably using the software tools. The modification can be done for the purpose of varying the number, arrangement, shape, and/or size of the struts so as to adapt the support structure for a specific flight mode as further detailed hereinabove. The modification can be done for the purpose of adding one or more elongated stringer members connecting two of the struts as further detailed hereinabove. The method proceeds to 405 at which a three-dimensional printer is operated to print a support structure having an arrangement of struts, corresponding to at least a portion of the veins, based on the CAD model. The printer can be of any type, including, without limitation, jet printing, stereolithography, digital light processing, and the like. Typically the printer includes or is associated with a processor that process the graphic elements of the surface mesh and transforms the graphic elements to a grid of voxels that define the internal shape of the struts, and that are arranged as a plurality of slices, each comprising a plurality of voxels describing a layer of the support structure. The slices are then used by the printer to build the support structure in layers as known in the art of three-dimensional printing.

In some embodiments of the present invention operation 405 is preceded by 404 at which the three-dimensional printer is operated to print a sacrificial structure wherein the struts are embedded in the sacrificial structure. The sacrificial structure can be made of any solidifiable substance that can be thereafter separated from the struts. A representative example of such a substance is wax.

At 406 the sacrificial structure (in embodiments in which 404 is employed) is removed to expose the struts. This can be done, for example, by heating and/or immersing it in liquid such as, but not limited to, oil, that dissolves the solidifiable substance of the sacrificial structure. The Inventors found that the removal of the sacrificial structure may render the veins soft and deformable. Thus, in some embodiments of the present invention the method proceeds to 407 at which the exposed struts are placed in a template substrate that is patterned with grooves, where the grooves are compatible with the struts, so as to stabilize a shape of the struts in the template substrate. The template substrate can be printed in advance by the same or other printer based on the modified CAD model. The stabilization can be ensured by allowing the struts to cool and/or dry within the template structure.

The method ends at 408.

As used herein the term “about” refers to ± 10 %

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 Exemplified Support Structures

This Example considers four flight modes, and demonstrates several optional support structure configurations for each flight mode. A first flight mode is hovering. It this mode, the wing deformation in the up-stroke and down- stroke optionally and preferably mirror each other and wing compliance attenuates asymmetries between contralateral wings.

A second flight mode is a fast forward flight. In this mode, the wing deforms differently during the up-stroke and down-stroke to provide most of the vertical force during the down-stroke and forward thrust during the up-stroke. The passive deformations vary with the stroke plane of the flapping wings and the flight speed.

A third flight mode is a high wing loading mode. In this mode, the flight speed is lower than the fast forward flight mode but higher than the hovering flight mode. The wings of the present embodiments deform chord- wise to provide the maximal sectional lift over the entire wing span.

A fourth flight mode is an endurance flight mode. The wing deforms optionally and preferably to maximize the lift-to-drag ratio of the wing in each section and in both the up- stroke and down-stroke. The Inventors found that this reduces power consumption and allows staying more time in air on the same energy source.

Each of the above modes can be ensured by judicious selection of the struts of the support structure. Representative structural parameters that can be selected for ensuring a flight mode include, without limitation, (a) cross-section, (b) arrangement, and (c) initial camber and twist of the wing. Each of these structural parameters can optionally and preferably be a modification of structural feature of veins of an insect's wing, as demonstrated in Example 2, below.

In (a), the cross-section area of the struts can be kept the same or approximately the same as the vein of an insect's wing, so as not to change the relative mass distribution over the wing area. Yet, the shape of the cross-section is varied, for example, from circular to oval or from circular to U-shaped, or from oval to U-shaped. Both oval and U-shaped cross-sections can have their major axis parallel to the wing plane, or may form any angle with the wing plane (e.g., the major axis can be perpendicular to wing plane). The changes in the second moment of cross-section area vary the flexural stiffness of the struts to forces applied on the wing hence also the distribution of structural stress. By such manipulation of individual struts, the passive morphing of the wing when flapping are controlled.

In (b), modification is applied to the number, position and size of small inter-veins that link larger radial wing veins that provide most of the structural support. This is achieved by selecting one or more elongated stringers to connect between struts. The location of the elongated stringers along the strut can be selected to impart a desired local stiffness of the trailing edge of the wing, and also adjust the ability of the wing to undergo twist. The twist is assumed to vary as a function of the distance between the tip of the elongated stringers and the wing base (hinge), with higher twist for elongated stringers that are closer to the base of the wing.

In (c), an initial camber is optionally and preferably applied to control the asymmetry in the passive deformation of the wing between the up-stroke and down-stroke maneuvers. For example, wings with X% camber with a maximum camber at Y% of the wing chord (measured from the leading edge) can be used, where X can be from about 5 to about 15, or more, and Y can be from about 15 to about 45 or more. Also contemplated is a gradient of increased maximal wing camber towards the wing tip. The gradient and proximity of the maximum camber to the leading edge is inspired by insect wings. The Inventors found that this can affect both the lift and the mechanical properties of the wings. With respect to the latter, the camber increases the rigidity closer to the leading edge, but results in asymmetry in wing twist between the up-stroke and down-stroke maneuver.

The Inventors found that a cambered wing has a higher flexural rigidity when forces are applied on it from its concave side. Such an asymmetry changes the wing from a wing suitable for low speed flight and hovering mode to a wing suitable for fast forward flight.

FIG. 1 illustrates an exemplary arrangement of struts based on a beetle wing (obtained by CT scan, see Example 2), and Tables 1-4, below, provide exemplified and non-limiting structural modification of this arrangement, for four different flight modes: hovering (Table 1), fast forward flight (Table 2), high wing loading (Table 3), and endurance (Table 4).

The struts in FIG. 1 are enumerated as strut 1 through strut 11, and are referred to in Tables 1-4 below using this enumeration.

Table 1 Table 2

Table 3

Table 4

Example 2

Experimental

This Example relies on the natural morphology of the wings of the rose chafer beetle (Protetia cupred). This includes the veins arrangement and structural gradients. The cross-section of the natural structure was modified in order to induce asymmetry in flexural response. Several constructing materials of the struts and membrane were used, with varying mechanical properties, in order to increase the mechanical stability of the printed wings. The information provided in this Example allows fabricating customized wings for robotic flapping flyers.

Materials and methods

Computed tomography (uCT)

A pCT scan of the Protaetia cuprea beetle wing was obtained and a 3D model was constructed by Solidworks®.

3D printing of wings

The model was 3D printed using Multijet 2500 Plus, 3D Systems®, 3D Systems Inc., USA. The material used was VisiJet-M2R CL (Young’s modulus: 1500-2000 MPa, flexural modulus: 1700-2200 MPa). The printed samples were then cleaned using paraffin oil to remove the support material and then sonicated in a sonication bath with DI water and soap for 5 min at 40 °C. The samples were then left for drying for 24 hours. For the membrane, a thin film of nylon sheet was placed gently on a flat surface. The printed struts were glued to the nylon sheet with dual-phase epoxy glue and let dry. The contour of the wing was cut using a sharp knife. For mounting the 3D printed wing onto the rotating motor system, connectors were designed and 3D printed.

Lift force measurements

A revolving wing apparatus was used to measure the vertical aerodynamic force generated by the printed wing models. A mirror image of each wing prototype was 3D printed and the wing pair was attached horizontally to the vertical shaft of a 3V DC electrical motor. A potentiometer on the power source (LE305D, Lion) was used to adjust the voltage to gain the same rotation rate of 715 RPM for all the prototypes. The wingtip speed at this rotation rate was 7 ms'¹ and the Reynolds number using that speed and the mean chord (2.7 cm) was 12,600. The apparatus was mounted on a force transducer (3N, accuracy ± 0.015%, Zimech LB9H) connected to an A2D card and sampled at 100 Hz. The sampled force data was filtered using a low pass filter and averaged over time. The actual RPM of the wings was measured with a high-speed camera (Fastcam SA3_120K, Photron) and all force measurements were normalized by the square of wingtip speed. A measurement of the vertical force without the wing prototypes but with the motor spinning gave the baseline for the vertical force measurement. Since all wings prototypes had the same planform area and shape, the normalized force measurement is comparable. Torque measurements

Torque was measured once indirectly by reading the voltage and current delivered by the power source and calculating the power required to rotate the wings and motor at the given speed as current times voltage. A second, direct, measurement was made on a subsample of the wing prototypes by measuring the counter torque acting on the revolving wing apparatus using a torque meter. The torque measured on the system without the wing prototypes was used to subtract the counter torque generated of the spinning motor. Torque measurements were normalized by dividing them by the cubic power of the actual wingtip speed. The geometric angle-of-attack (AoA) of the wing pair was adjusted manually each time and measured with the high-speed camera. For each wing prototype, force and torque were measured at a range of AoA of 0-60 degrees.

Results and Discussion

FIGs. 4A-F show 3D printing scheme of bioinspired beetle wings. FIG. 4A shows an optical image of the beetle wing. Scale bar corresponds to 4 mm. FIG. 4B shows a pCT scan of the whole wing and FIG. 4C shows a pCT scan of the veins, isolated from the membrane. FIG. 4D shown a 3D model of the beetle wing based on the pCT scan (blue). FIGs. 4E and 4F show optical images of the printed wing: before (FIG. 4E) and after (FIG. 4F) incorporation of the membrane. The cross-section of the printed struts are round, vertical oval and horizontal oval as sketched at the top. Scale bar corresponds to 15 mm.

FIGs. 5A-G show static bending tests on 3D printed wings with the same wing mass and 2D geometry but differences in cross section of the wing-struts. FIG. 5A shows the measuring setup. The printed wing (enlarged in the insert in the bottom left corner) is clamped as a cantilevered plate and pressed with a needle connected to a force transducer that is mounted on a micromanipulator. The force-displacement relationship was measured for displacement of 0.25-1.5 mm at increments of 0.25 mm. FIGs. 5B-F show the four points of force application over the printed wing’s trailing edge. FIG. 5G shows differences in the “beam” flexural-stiffness of the two printed wing models. The flatter wing veins had lower flexural rigidity.

FIGs. 6A-F show drag force measurements of the 3D printed wings.

FIGs. 7A-N show structural variations in bioinspired 3D-printed wings. FIG. 7A shows four locations along the struts at which cross-sectional images were captured. The locations are denoted as lines A— A, B— B, C— C, D— D. FIGs. 7B-M are optical images of the three different cross section shapes: round (FIGs. 7B, 7E, 7H, and 7K), oval horizontal (FIGs. 7C, 7F, 71, and 7L), and oval vertical (FIGs. 7D, 7G, 7J, and 7M), with cuts along line A— A (FIGs. 7K-M), line B— B (FIGs. 7E-G), line C -- C (FIGs. 7H-J) and line D— D (FIGs. 7B-D). Scale bar (see FIG. 7K) corresponds to 200 pm. FIG. 7N is similar to FIG. 7A, except that the strut's diameter remains constant along the strut.

Example 3

Improved Lift and Lift Production Efficiency

This Example describes 3D-printed wings based on the rose chafer beetle wings. By modulating the wing structure the Inventors designed 12 different wing models that differ in the shape of the veins’ cross-section, tapering geometry, and membrane thickness. The mechanical and aerodynamic properties of these models were compared, to establish guidelines linking wing form to function. This Example shows that (i) the geometry of the veins' cross-section allows engineering in-plane and out-of-plane deformations; (ii) tapering veins improve the wings’ mechanical stability, and (iii) the membrane merges the mechanics of the individual veins into an integrated aerodynamically favorable structure. This Example demonstrates These resulted in 16% higher lift and 27% improvement in lift production efficiency (N/Watts) in a revolving wing setup.

Introduction

The morphology of insect wings is extremely diverse [2]. They comprise a flexible membrane and stiffer wing veins [3], with the latter differing among species in number, thickness, cross-section shape, and spatial arrangement [4]. The thin wings undergo significantly large elastic deformations during each flapping cycle [5], with the shape, arrangement, and mechanical properties of the venal structure determining the pattern and magnitude of these flexural deformations [6,7].

The veins are thicker and their arrangement denser at the wing base. As they radiate towards the wingtip and trailing edge, they taper. This results in span-wise and cord-wise gradients in flexural stiffness [7]. The structural gradients of natural formations, with high aspect-ratio, are used to reduce weight through minimization of the construction materials, while maintaining mechanical stability against collapse or failure [8]. The specifics of each vein’s cross-section, and the arrangement of all the veins combined, fine-tune the mechanical properties of the wing’s structure and determine its elastic deformations [9]. In turn, these deformations affect the aerodynamic performance in insects’ flapping flight [6,10-12], supporting a variety of flight styles [13], such as hovering, flying with heavy loads, maneuvering and fast forward flight [13,14]. Such adjustments in venal arrangements and vein geometry are evolution’s way of adjusting the wing structure to the insect’s flight style.

While such diverse wing morphologies shed light on the different evolutionary paths [11], the relationship between insect wing venal structure and aerodynamic performance is obscured by the combination of complex geometry and non-homogenous material properties. In contrast, physical (artificial) models offer the ability to isolate and manipulate each trait systematically. This enables the establishment of specific guidelines for the design of artificial wings with tailor-made mechanical features [15].

Various methods have been applied to the design and fabrication of artificial miniature wings in recent years in order to enable the creation of tiny flapping drones [28]. These include laser-cutting [15], molding [29], and a few examples of 3D-printing of generic wings [30,31].

This study relates to the natural wing morphology of the rose chafer beetle (Protaetia cuprea) and studies how specific modifications of the vein’s cross-section results in changes in the aerodynamic performance. The rose chafer was chosen due to its superb flight versatility. It can hover, fly forward at high speed, perform sharp maneuvers, and land stably on small perches [34,35]. Like many beetles, it flies with a high wing loading compared to other insects with similar wing size [36]. The capabilities of its flight apparatus are therefore suitable for the development of miniature (e.g. , centimeter-scale) drones that are required to function with a relatively high payload due to the electronic components they carry [37].

By manipulating the beetles’ wing-vein cross-section geometry the Inventors introduced anisotropy into the 3D-printed wings, without altering their area or mass. The Inventors then eliminated the diameter gradients of the veins and changed the membrane thickness to test the deformations and aerodynamics of the resulting wings prototype in a revolving wing set-up.

Results

Wing fabrication

Basic design and 3D-printing

The process of design and fabrication of the beetle-inspired 3D-printed wing of this example is provided in FIGs. 4A-F. An original rose chafer wing (FIG. 4A) was scanned using micro-computed tomography (pCT) (FIG. 4B). The scanned veins were then isolated from the membrane (FIG. 4C) and used to construct a 3D computer-aided-design (CAD) model in Solidworks© that captured the size, position, orientation, and tapering of each strut that mimics the wing's vein (FIG. 4D). Next, the CAD model was up-scaled 3:4 and the strut cross-sections were modified, as explained below, while the position and curvature of the struts were retained at high fidelity to the original beetle wing. The modified CAD models were 3D-printed using a high- resolution 3D printer (Multijet 2500 Plus, 3D Systems Ltd., USA) (FIG. 4E). A membrane was then glued onto the struts (FIG. 4F). The membrane comprised either a thin, 5-7 pm, polyethylene sheet or a thicker, 80 pm, sheet (see Extended Methodology, below). The high aspect-ratio, overhanging structure, of the wing struts is highly challenging in terms of 3D printing. A step-by-step illustration of the necessary adaptations to the 3D-printing and post-processing process is provided in FIGs. 11A-D. The entire venal structure, including the overhanging thin struts, was embedded in wax directly after printing (FIG. 11 A). The wax was removed by immersing the sample in oil at 60°C (see Extended Methodology). The heating renders the struts soft and deformable (FIG. 11B). Their original shape is then restored by placing the structure in a frame and leaving it to dry in air for 10-15 minutes (FIG. 11C). During the drying process, the venal structure stiffens, restoring the original shape of the wing (Fig. 11D).

Structural modifications

The original CAD model was used to modify the structural gradient (tapering) of the struts and their cross-section shape. FIGs. 7A-N present the 3D-printed wings, as further detailed hereinabove. Thus, the shape of the struts’ cross-section was altered in order to induce anisotropy of the structure. The cross-section shape of all the struts was altered to either: round (‘O’); oval with the major radius perpendicular to the plane of the wing (‘V’); or oval with the major radius in the wing plane (‘H’). Both oval cross-sections had a 2.2: 1 ratio between the major and minor radii, but the same area as the round cross-section. Consequently, all three cross-sections had the same volume (and therefore mass) but differed in the second moment of area (MoA) for bending in or out of the wing’s plane: the struts of the O prototype had an equal MoA for bending in-plane or out-of-plane; those of the H prototype had the highest MoA for bending in-plane and lowest MoA for bending out-of-plane; and those of the V prototype had the highest MoA for bending out-of- plane and lowest MoA for bending in-plane. The cross-section geometry of the three printed wing prototypes (O, H, and V), was verified by cutting at four selected points along the struts.

A second set of three prototypes had O, H and V cross-sections but with constant cross- sectional area throughout the struts’ length. The fixed diameter was the mean one for each strut, so that the masses of the uniform and tapering struts were identical.

Mechanical testing

Flexible insect-inspired wings are designed to deform when forces are applied on them. The shape of the deformation is primarily dictated by the stiffness of the venal structure. The individual struts spread radially from the wing base, with their free ends subject to bending as beams. When forces are applied on the structure, the tensile resisting membrane links between the struts, distributing local forces as stresses over the entire structure. The struts’ free ends and the tapering towards these ends contribute to the overall chordwise gradient of flexural stiffness. This results in higher compliance at the proximal trailing edge. This study evaluated how modification in mass distribution (taper) and cross-section shape (Mo A) of the wing struts affect the elastic deformations. Static load was simulated by applying the finite element (FE) method to an individual strut and experimentally measuring the deflection of the entire wing structure, including the membrane.

Static bending tests: simulations and experiments

FIGs. 12A-E show mechanical response of an individual strut with uniform and tapered structures. Shown are CAD sketches of uniform and tapered designs of the wing's support structure (FIG. 12A), where the dotted black lines denote the veins used for the mechanical simulations. FIGs. 12B(i) and 12B(ii) show top view of the stress distribution in the strut under study in in response to a point-like force at the tip using finite element simulations in uniform [FIG. 12B(i)] and tapered [FIG. 12B(ii)] geometries. FIGs. 12C(i) and 12C(ii) are similar to 12B(i) and 12B(ii) except that they show a side view of the stress. FIF. 12D shows a deflection along the strut in response to the applied force at the tip (30 mm), and FIG.12E shows the stress along the strut. FIGs. 12F(i)-(vi) show experimental deflection of the strut under study from for all the 3D-printed wings with uniform [FIGs. 12F(i)-(iii)] and tapered [FIGs. 12F(iv)-(vi)] geometries, for the three prototypes, differing in their cross-section shape. The wing was fixed at the base (marked in red dotted lines in FIGs. 12A). FIGs. 12G and 12H show comparison of the empirical net displacement of the 3D-printed loaded vein to the unloaded initial position of the uniform (FIG. 12G) and tapered (FIG. 12H) structures. Fine colors denote the shape of the prototype cross-section: V - light blue, O - red, H - yellow.

Using the CAD models, the CuA strut was isolated from the venal structure with a round cross-section and simulated the load and displacement along the strut using finite-element analysis (ABAQUS Inc., CA, USA, v. 6.14). A point load of 5 mN was applied at the free tip. The boundary conditions at the fixed base were defined as encastre. One simulation was applied for a tapering strut and another for a strut of the same length and mass but with a uniform cross-section area. From the simulation von Mises stresses and the displacement along the struts were extracted [FIGs. 12B(i)-C(ii)]. The area adjacent to the encastre base was excluded to avoid boundary condition effects. FIG. 12D reveals that the two struts do not vary substantially in the displacement of the free tip but do vary in the distribution of stresses and the curvature of the bending strut. Specifically, tapering leads to a more even distribution of stresses (FIG. 12E) and to a higher deflection angle further away from the support. For a wing strut such as CuA, this leads to stiffening of the leadingedge area, followed by a steeper chordwise deflection towards the trailing edge (FIG. 12D).

On the printed wings, two static bending tests were performed to verify the difference in the mechanical properties of the modified wings. The first test empirically replicated the CuA strut bending in the simulations [FIG. 12F(i)-(vi)] but with a different support at the strut base. The support structures of the wing prototypes (without the membrane) were clamped at the wing base (red dashed line in FIG. 12A) to a rigid horizontal plate, with the wing structure protruding from the plate edge as a cantilever beam. A vertical point-load of about 0.4 mN was applied 3 mm from the free tip of the CuA, using an entomological pin fixed to a force transducer (Lsb200, Futek) mounted on a micromanipulator. A digital camera and image analysis were used to capture and measure the displacement along the struts. Similar to the computer simulations, it was found that the tapered wings had a larger displacement at the free end and a gradually increasing displacement angle towards the tip.

Due to the lower load applied in the empirical measurement, displacements were mostly evident in the H prototypes. The tapered struts had larger displacements at the free end compared to the uniform struts.

The second test focused on the entire wing (including the struts and the membrane), in which the application of point loads on one strut led to distributed stresses throughout the wing area, via the membrane. FIGs. 13A-G shows mechanical response of the wing as obtained during static bending tests that were applied on 3D-printed wings with the same wing mass and 2D geometry but different cross-section shape of the strut. FIG. 13A illustrates the experimental setup. FIG. 12B shows the mean flexural stiffness of the three strut profiles, at four points along the wing trailing edge, depicted in FIGs. 13C-G. Columns in FIG. 13B represent the mean of four repetitions from two wings, and error bars indicate standard deviation. “Thick” and “Thin” refer to the membrane type, and “uniform” and “taper” correspond to strut cross-section.

In the experiment, the wing was clamped at its base (FIG. 13A). The same force-transducer and micromanipulator set-up were used to measure the force needed to displace the tip of the strut by 1 and 2 mm. Measurements were performed at four locations on the wings, where struts terminated at the trailing edge (FIGs. 13C-G denoted as AA, CuA, MP and WT). The forcedisplacement curves were used to estimate the flexural stiffness of the wing at each point. At the CuA point, the measurement was repeated without the wing membrane and the difference between the measurements with and without membrane was used to estimate the contribution of the two membrane types to the overall flexural stiffness of the wing.

The measurements confirmed that the V prototype displayed a higher stiffness for bending out of plane, compared to the O prototype, which was stiffer than the H prototype. The wings with tapering struts were stiffer than the uniform wings at point WT, while the opposite was found at the other three (proximal) trailing edge locations. The entire structure, including the struts and membrane, supported at the base, differed from the single bending beam condition. The thicker membrane increased the stiffness of the wing by roughly two-fold. The two membranes contributed roughly 15 and 0.5 mN-m² to the flexural stiffness of point CuA in the thicker and thinner membrane, respectively.

Aerodynamic performance

Wing deformations

A high-speed video camera (Photron, Fastcam SA3) and a revolving wing set-up [38] were used to observe the passive wing deformations elicited by the aerodynamic loads applied on the revolving wings.

FIGs. 14A-D show results of experiments directed to study elastic deformations of the 3D- printed wings and the wings of the rose chafer beetle. FIG. 14A illustrates the experimental set-up. FIG. 14B shows 3D-printed wings with tapered and uniform cross-sections of the three prototypes (V,0,H). The arrows highlight significant wrinkles at the trailing edge. The lower panel of FIG. 14B depicts the deformation of a Protetia cuprea beetle (Pc) actual wing. FIGs. 14C and 14D show measured projected area of the deformed wings with thin and thick membranes, respectively. Filled and empty symbols correspond to tapered and uniform veins, respectively, and the colors denote the shape of the prototype cross-section: V - blue, O - red, H - yellow.

In the experiments, a pair of wings of each prototype was mounted horizontally in a propeller arrangement and rotated about the vertical axis by an electric DC motor at 13 revolutions per second (wing-tip speed of 7 m-s ¹). The wings were mounted and filmed rotating, once at a geometrical angle-of-attack (AoA) of 30° and again at an Ao A of 40°. Due to the wing mount, its base had a radial distance of 3.5 cm from the axis of rotation.

As a basis for comparison, real rose chafer wings were rotated in the same experimental set-up, but the wing mount was downscaled and the rotation rate was upscaled to achieve similar wing-tip speeds and Rossby number conditions [39] (see Extended Methodology). The high-speed camera was horizontal with the field of view, aligned with the vertical rotation axis and horizontal plane of the revolving wing. This allowed to qualitatively evaluate the wing deformations in both the span- wise and chord-wise views of the revolving wings in the vertical plane.

Wing deformations were visible in the wings with a thinner membrane. FIG. 14B presents the 3D-printed wings with tapered and uniform veins at an AoA of 30°. The V prototype displayed substantial in-plane displacement of the veins towards each other, leading to wrinkling of the membrane between veins at the trailing edge (FIG. 14B, arrows). These wrinkles were not present in the O and H prototypes of the tapering wings but were present in all three prototypes of the uniform wings. The wings with a thicker membrane did not display visible trailing edge wrinkles in any of the prototypes. From the image showing the wingspan perpendicular to the optical axis of the camera, the projected area of the wing on the vertical (camera’s field of view) plane was measured. This was then compared to the expected projection for a rigid wing mounted at the same AoA (FIGs. 14C and 14D). For the thicker membrane, all area measurements were higher than expected for a rigid wing. This was due to the wing passively twisting to assume a higher AoA than that established by the wing base. This was more evident at an AoA of 30° than at 40°, and the measured projected area increased in the following order, H > O > V, matching the stiffness of the prototypes for out- of-plane bending.

For the thinner membrane, all three prototypes of the tapered wings had a projected area that was smaller than expected, due to the tip of the veins buckling towards the suction side of the wing, resulting in reduced effective wing area. This buckling was not present in the uniform wings nor in any wing prototype with the thicker membrane.

Quantification of aerodynamic forces

Next, using force transducers, we measured and compared the aerodynamic forces generated by the revolving wings. The measured lift and torque (about the rotation axis) were used to calculate the lift and drag coefficients of the wings as a function of AoA in the range of 0 °< AoA < 50°. FIGs. 15A-D show aerodynamic performance of the 3D-printed wings. Polar curves showing the lift (Cl) and drag (Cd) coefficients of the revolving wings. Symbol colors denote the shape of the prototype cross-section: V - blue, O - red, H - yellow. Each data point is the mean of three repetitions (three printed wings of the same prototype) at a given AoA. Error bars are equivalent to the standard error of the measurements. Each plot represents a different wing design: wings with tapering (FIG. 15B) and uniform (FIG. 15C) cross-section struts attached to a thin membrane; and wings with tapering (FIG. 15D) and uniform (FIG. 15E) cross-section struts attached to a thick membrane.

FIGs. 16A and 16B show lift production efficiency as a function of AoA and strut crosssection shape in the wings with a thicker membrane and tapering struts. FIG. 16A shows lift efficiency calculated as Cl/P*, where Cl is the lift coefficient and P* is the non-dimensional electrical power consumed by the DC motor (see Extended Methodology). FIG. 16B shows relative lift efficiency of the V (blue) and H (yellow) prototypes with tapering struts and thicker membrane compared to the mean lift efficiency of the O prototype. The mean lift coefficient of the three repetitions of the O prototype, Cf_o , is also shown as a function of AoA by the red circles with a dashed line. In the experiments, the bench top DC power source (LE305D, Lion, Israel) used to power the motor also concurrently measured the current (7) and voltage (V) supplied to the rotating motor and thus gave the electrical power input (P = IV).

At the higher AoA, lift and drag forces were up to -30% higher for the wings with a thicker membrane.

Wings with uniform struts had higher variance in the measurement of drag as a function of AoA. The wings with tapered struts and a thin membrane (FIG. 15B) displayed the lowest forces, with no distinct difference in CL or CD between the three prototypes. As noted above, this resulted from the buckling of the highly compliant trailing edge, reducing the effective area and AoA exposed to the flow lowering both lift and drag. Uniform struts (FIG. 15C) had thicker strut tips and, therefore, experienced less buckling at the trailing edge. However, they displayed distinct wrinkling of the membrane (FIG. 14B) at the trailing edge. Consequently, they demonstrated about 25% higher lift at large AoAs (30° -50°) but also higher and variable drag. In the tapering wings with a thicker membrane (FIG. 15D), lift at the highest AoA (=45°) was higher in the V prototype by 16%, compared to O and H, but without a substantial increase in drag. In the wings with uniform struts (FIG. 15E), although the advantage of V for lift production vanished, the V prototype produced the least drag at low Ao As. Lift efficiency (measured lift /measured electrical power input) was highest at AoA = 15°-20° in all the prototypes (FIG. 16A) but at that AoA, the lift force was less than 37-42% of the lift achieved at AoA =45°. The higher lift/drag ratio of the tapered V prototype at the largest AoA (FIG. 15D) translated into 27% higher lift efficiency relative to the O prototype. Meanwhile, the H prototype generated less lift at a higher power input and consequently, had the lowest lift efficiency, which was 17% lower than the O prototype (FIG. 16B).

Discussion

The shape of the strut's cross-section constituted a criterion for design, enabling alteration of the mechanical response of each strut without altering its mass or length. The flexural stiffness of the three wing prototypes (H, O, V) matched the expected differences in the struts’ MoA, where V was stiffer for out-of-plane bending than O, which was stiffer than H. The differences in displacement in the entire wing (with membrane) were more pronounced at the points farthest from the wing base (MP and WT). For simplicity and consistency, the beam length was defined as the linear distance from the base, which is not the actual length of the curved struts. This measurement allowed to quantify differences in stiffness between the three wing prototypes, due to all three being mounted and measured in the same way.

Wootton et al. [9] demonstrated that the cross-section shape of wing struts varies according to the struts’ different positions on the same wing. This implies that the cross-section shape too, and not just the diameter, is used to tune mechanical properties locally [40]. Modifying the crosssection shape is particularly useful for achieving anisotropy. A strut with the same cross-section area can be made to resist wing bending in-plane more than bending out-of-plane. The V profile, which was the stiffest for out-of-plane bending (FIGs. 13A-G), presented a projection in the vertical plane (FIGs. 14C and 14D) that was closer to 1.0 (rigid wing), compared to in O and H. In the V prototype wings with thicker membrane, this resistance to bending caused higher lift generation at higher AoAs compared to the O and H prototypes. The increased lift was not however coupled with an increase in drag, resulting in a substantial improvement of the lift-to-drag ratio and lift efficiency of the V prototype. While the V profile had the highest stiffness for out-of-plane bending, it had the lowest stiffness for in-plane bending. This resulted in the struts approaching each other when the membrane became taut. This was particularly evident in the thinner membrane, which did not add its own bending resistance. Hence, the cross-section shape is demonstrated by the Inventor to be a useful tool to allow adjustment of the mechanical properties of the venal structure without altering the wing’s mass, area, or shape. When properly tailored to the complex fluidflexible structure interaction, this adjustment of the mechanical properties can be used to improve lift production and the lift-to-drag ratio (FIGs. 15A-E).

This example demonstrates that the tapering nature of the wing struts provides a structural advantage and to shape the chord- wise curvature (camber) of the wing. Compared to uniform struts with the same mass, the tapered struts are thicker at the base and thinner at the tip. Considering that the bending moment, due to load applied at the free tip, is maximal at the leading edge, tapering thus provides a higher safety margin against failure of the struts at their base. This is exemplified in a comparison of the distribution of stresses of the two geometries in the computer simulations (FIG. 12E). Tapering of the radial struts also contributes to the chord-wise and span-wise gradients in structural stiffness over the entire wing. Unlike in the uniform wing, where displacement takes place almost linearly along the strut, the tapered wing has minimal displacement at the base (near the proximal leading edge) and a higher curvature along the strut (FIG. 12D). This increases the wing camber and shifts the maximal curvature away from the leading edge. In addition, buckling of the tapering struts, due to aerodynamic loading, will be limited in this case to the trailing edge rather than the entire wing. FIG. 14B demonstrates that this buckling at the trailing edge tends to add tension to the membrane in this area, preventing it from wrinkling, unlike in the wings with uniform struts. These wrinkles seem to be associated with high drag at low Ao As and an overall lower dependence of the drag on the AoA (FIGs. 15A-D).

The selected membranes had an effect on the produced forces and deformations. The thin, fully compliant, membrane was less efficient in distributing the stresses over the entire wing area. This resulted in wrinkles, which were not observed in the thicker membrane. The thicker membrane, although having high compliance in itself, resulted in a substantial (>2-fold) stiffening of the wing structure. Therefore, rather than deforming locally, resulting in buckling out-of-plane or wrinkling due to in-plane displacements, the struts of the entire wing deformed together. Under the aerodynamic load exerted by the revolving wing set-up, this led to a passive wing twist, which increased the effective AoA.

The combination of tapering struts and a thicker membrane gave the best aerodynamic performance for the revolving insect-inspired wings in terms of maximum lift and maximal lift-to- drag ratio. Although deformations were challenging to detect visually in the wings with thicker membrane, the three prototypes nonetheless differed in their aerodynamic properties (mostly at higher AoA). The 16% and 27% improvement in maximal lift and lift efficiency, respectively, of the V prototype (at AoA =45°), compared to the O prototype, was achieved despite the three prototypes having identical wing planform shape and area and having the same mass and material properties. Hence, the improvement in aerodynamic performance and lift efficiency is solely attributed to the mechanical properties of the flexible wings being adjusted by the struts’ crosssection shape.

The V-shaped cross-section prototype with tapering struts and a thick membrane gave the highest lift and the best lift per unit power input at maximal lift. Combined, these geometrical features contributed to a less compliant structure that tended to twist when rotated rather than to produce wrinkles and buckling at the trailing edge.

This example provided a design and material guidelines for creating flexible, 3D-printed wings. The empirical approach presented herein, using revolving wings, provides a benchmark for studying the connection between geometry, mechanical properties, and aerodynamic performance of light flexible wings. The empirical approach described herein can be used to fabricate miniature man-made flapping UAVs

Extended Methodology

3D-printing

The struts were printed using the polymer VisiJet® Armor M2R-CL (3D Systems Ltd.) which has a Young’s modulus (1.5-2.0 GPa) and a flexural modulus (1.7-2.2 GPa) similar to that of an insect cuticle (Young’s modulus 1-20 GPa, [41]). The printed struts were cleaned using rapeseed oil to remove the support material (wax) residue, and then immersed in a bath filled with water and soap, for 5 min at 60°C. The thin and thick membranes were created from a thin nylon sheet (5-7 pm) (Clean Spot Ltd.) and laminating sheet (Gloss Laminating Pouch Film 80 pm, Polyethylene), respectively. The membrane was first placed gently on a flat surface. The printed struts were then placed on a 3D-printed template and glued to a nylon sheet using spray adhesive (Super 77, 3M Industrial Adhesives and Tapes Division, St. Paul, USA) and left to dry. The contour of the wing was cut using sharp scissors. Ad-hoc connectors were designed and used to mount the 3D-printed wing onto the rotating motor system.

Flexural stiffness estimates

To convert the force-displacement data from the static bending tests to flexural stiffness (£7 ) each measurement was treated as that of a cantilever beam where F is the force [N] measured by the transducer, I is the linear (shortest) distance [m] between the wing base and the point where the force was applied, fl is the displacement [m] at the point of force application, and C is a constant that depends on the specific conditions (support, type of load).

Since the focus in this example was on comparing between the wing prototypes, and all measurements were taken under the same conditions at the same four points on the trailing edge,

Fl³

C was omitted and — was used as the quantification of El .

Aerodynamic performance

A revolving wing apparatus, similar to that used by Usherwood and Ellington [38], was used to rotate the wings about a vertical axis while measuring the vertical force (lift) with a force transducer (Zimech LB9H, 3N, accuracy ± 0.015%) and the torque due to horizontal force (drag) with a torque meter (ME-systems, TD70, 300 mNm). Herein any vertical force is referred to as

"lift" and any horizontal force as "drag", neglecting the flow conditions around these wings. The resulting lift and drag coefficients for a given AoA were calculated based on the blade-element principle [38]

2L

Cl= (EQ. 2) pS₂fF

(EQ. 3) where Cl and Cd are the lift and drag coefficient, respectively, L and Q are the measured lift force and torque, respectively, fl is the angular speed of the revolving wing, p is air density (1.2 kgm’³), and S₂ and S₃ are the second and third moments of wing area [42].

The moments of wing area were found by measuring the span-wise distribution of the wing chord from digital images, as in Strutak et al. [43], while accounting for the offset distance (3.5 cm) between the rotation axis and the wing base.

The angular velocity was set to fl = 81.7 rad s’¹ to give a tangential wing-tip speed of 7 m- s’ ^x. This wing-tip speed is similar to the mean wing-tip speed of a rose chafer flapping its wings (wing length 2 cm) at 100 Hz [34]. The force measurements were repeated at geometrical AoAs varying from 0° - 50°. Both the manually set AoA of the wing prior to rotation and the rotation rate during the experiment were confirmed using a high-speed video camera (Fastcam SA3, Photron).

The electrical power input needed to rotate the motor and wings at the above speed and at a given AoA was measured directly from the power supply. The measured electrical power (P) was non-denationalized as P*= P/pSsQ³. The force per unit power input Cl/P* was defined as the ‘lift efficiency’^44,45

Measurement of wing deformation in the revolving wings

To quantify the passive elastic deformations of the revolving wings a high-speed camera was used the to capture images in the vertical plane when the wing span was perpendicular to the camera’s optical axis. The images included the wing at AoAs of 40° and 30°. Using ImageJ, the area of the wing in the images (the area projected on the vertical plane, Si) was measured and compared to the expected projection for a rigid wing at the same AoA, (Sy). where S_6J is the planform area of the wing (S_6J = 17.280cm²). To observe the deformation of real beetle wings under similar conditions wings from a rose chafer beetle were mounted and rotated at a higher angular velocity to achieve the same wing-tip speed (7 ms'¹). Real beetle wings are 3-fold smaller than the printed wings and so the offset between the rotation axis and wing base was downscaled by 3-fold in order to keep the Rossby number the same [46].

Example 4

Improved Flapping Kinetics

Insects flap their wings through a highly specialized and coordinated musculoskeletal system that allows the wings to rotate about three degrees of freedom. Consequently, the wingtip trajectory is adjustable in three-dimensions, and accompanied with appropriate wing feathering (wing pitch). The complex flapping motion is achieved by thoracic muscles acting on the wing hinge. The wings themselves do not possess muscles, rather, they adjust their shape and orientation by elastically deforming as a result of the loads applied on them during flapping. Previous attempts to develop insect-inspired flapping drones have mostly focused on simplified linear flapping mechanisms, which do not seem to utilize the interaction between the wing flexibility and flapping kinematics to its full potential.

This Example describes biomimetic mechanisms that improve the similarity of the flapping motion to that of insects. One biomimetic mechanism is an elastic beam mechanism that allows the wing root to swing during flapping. Another biomimetic mechanism is a passive wing pitch mechanism that allows the wing to rotate at stroke reversals. This Example describes experiments directed to test these two mechanisms using the 3D printed wings of the present embodiments. The experimental results show a six-fold improvement of aerodynamic performance compared to linear flapping kinetics of the same flexible wings.

Introduction

Insect flight comprises a complex wing motion. Unlike traditional manmade flight, it involves flapping wings mid-flight, [58-66] rather than gliding through the air with the assistance of wind or a thruster. Furthermore, unlike birds or bats, [67,68] insects do not have muscles in their wings. Instead, they rely on the wing’s geometry and flexibility to adjust the wing shape during the flapping cycle [59,69,57]. Insect wings are elastic, comprised of stiff skeletal veins of variable cross-sections that dictate the basic geometry of the wing, and an elastic anisotropic membrane that connects between these veins [62,70-73]. Together, the veins and membranes combine to a light weight elastic structure that is adapted for the maneuverability and high flapping frequency required by small insects [63,64,74]. The wing flapping motion of insects is a cyclic rotational motion about the wing hinge that can be roughly divided to changes in the flapping angle that forms the motion of the wing in a stroke plane, and changes in the wing pitch angle that adjusts the orientation of the wing chord relative to this plane. [61,65,71,75] The two rotations (flapping and pitch) repeat in each flapping cycle but occur out-of-phase from each other. Flapping is actuated by the insect flight muscles, whereas wing pitch may be actively achieved using the action of direct flight muscles [76,77] or passively achieved by the inertia and aerodynamic loads acting on the wing at stroke reversals. [78,79] The change in pitch orientation of the wing and its rate are presumably also affected by the elastic deformations of the veins and the wing’s membrane. [62] This results in the wing structure twisting to achieve a favorable Angle-of-Attack (AoA). [75,80] The elastic wing deformations, for a given flapping motion, are determined by the geometry and mechanical properties of the wing, including spanwise and chordwise gradients in wing compliance [81].

Flapping flight has the potential to outperform fixed or rotary wing designs in miniature drones [82,83]. Inspired by insect flight, several such Flapping Micro Air Vehicles (FMAVs) were developed in the last decades, attaining untethered controlled flight using low weight, flexible membranous wings. They achieved mean lift in the range of up to a few dozen gf (gram-force) at frequencies of up to roughly 50 Hz, using wings with span of 170-210 mm [84-88].

Table 5 summarizes some technical details and performance of some conventional FMAV systems. The table compares the maximal mean lift as well as the flapping frequency and wingspan. The type of the actuating mechanism and the shape of the wings are also compared as they influence the aerodynamic efficiency of the FMAV system. Table 5

The FMAVs developed so far utilize various types of mechanisms, such as planar 4-bar mechanisms [84,85] or spatial bar- linkage mechanisms that allow deviation from a plane [90]. Nevertheless, these mechanisms generate a relatively linear flapping motion that does not fully mimic the wingtip trajectory of insects’ flapping motion. The wings themselves are typically made of a stiff skeletal frame with reduced geometrical complexity compared to real insect wings, and a soft membrane affixed to the simple frame.

To resemble the dynamic twist of structurally complex insect wings during flapping [19, 80, 81] the simplified FMAV wings are secured to the actuator at two points of contact [90, 91, 93]. This forces a rigid wing to pitch or a compliant membrane to twist like a sail, in order to create lift in both the downstroke and upstroke sections of the flapping cycle. This ‘sail effect’ might be similar to the actual elastic wing deformations that appear in the flight of some insect [70,85] . Other FMAV systems have utilized more complicated designs and spatial mechanisms [90,93,94], to allow the wing path to emulate that of an insect or hummingbird. The Inventors found that these solutions overlook opportunities to improve flight performance and minimize energy consumption by passive wing pitch [75,85,87,90,92,93], achieved by tailoring the interaction between the wing’s mechanical properties and the flapping kinematics [83]. The Inventors successfully achieved these adjustments without adding significant weight and/or power demands to the FMAV that, owning to miniaturization, is already operating close to maximal capacity.

This Example describes an improvement of a simple planar mechanism by introducing passive wing flapping mechanisms. The approach is inspired by the rose chafer beetle (Protaetia cuprea, FIG. 17A) that achieves superb flight stability and maneuverability using highly flexible wings. The deformations of these flexible wings contribute to dynamic wing pitch, according to the loads applied on the wings during flight [71]. In this Example, a passive co-rotational wing pitch motion was introduced at the wing hinge. The pitch motion was indirectly actuated by the 1- DOF planar 4-bar mechanism flapping the wing. This passive pitch rotation is realized using a single point of contact between the wing and the flapping mechanism [87]. Two alternative mechanisms have been designed and tested, according to some embodiments of the present invention. Each mechanism imposes a different pitching profile for the wings throughout the flapping cycle. A first mechanism utilizes a leaf spring between the hinge and the wing, and a second mechanism uses a miniature ball-bearing, allowing the wing to rotate about the wing length axis. The effect of these modifications on the aerodynamic performance of the flapper were studied by quantifying the lift generation and power-to-lift ratio of both flapping mechanisms, when connected to the same high-fidelity beetle inspired 3D-printed wings.

Results and Discussion

Flapping mechanisms

FIGs. 17A and 17B show a Rose chafer beetle (FIG. 17A), a pitching of the Rose chafer beetle’s wings during the flapping cycle (FIG. 17B). The small sketch of the beetle in FIG. 17B depicts the pitching motion of the wings. Wing pitch angles during the upstroke have been transformed to negative angles. During the flapping cycle, the wing pitch angle of the Rose chafer beetle changes between 58-138 ⁰ (angle of the wing chord relative to the flapping plane, see FIG. 17B). In order to mimic the pitching of the wings in our robotic flapping system, the Inventors of the present invention developed and implemented two passive flapping mechanisms. For the first mechanism a flexible beam that act as a leaf spring was added between the root of the wing and the wing hinge of the flapping mechanism. This mechanism is referred to below as the "spring mechanism." For the second mechanism a ball bearing was added at the hinge, to allow the wing to rotate freely inside a constrained arc. This mechanism is referred to below as the "rotating mechanism." Each of these two mechanisms provides a different variation of the pitching angle along the flapping cycle.

FIGs. 17C-E show the flapping device of the present embodiments with the spring mechanism (FIG. 17C), a magnified view of the spring mechanism (FIG. 17B) focusing on the leaf spring at the wing hinge, and a schematic representation of the wing pitch kinematics during the flapping cycle (FIG. 17E). Wing pitch angles during the upstroke have been transformed to negative angles.

FIGs. 17F-H show the flapping device of the present embodiments with a rotating mechanism (FIG. 17F), a magnified view of the rotating mechanism (FIG. 17G), and a schematic representation of the wing pitch kinematics during the flapping cycle (FIG. 17H). Wing pitch angles during the upstroke have been transformed to negative angles. A more detailed illustrations of the connection between the mechanisms and the wings according to some embodiments of the present invention is illustrated in FIGs. 18A-C. The top and middle parts of FIG. 18A illustrates the axes of rotation of the wings for two cases of the spring mechanism, and the bottom part of FIG. 18A illustrates the axes of rotation of the wings for the rotation mechanism. FIGs. 18B and 18C illustrate back views (FIG. 18B) and side viewers (FIG. 18C) a of the spring (top) and rotating (bottom) mechanisms.

The leaf spring was implemented as a custom manufactured thin metallic plate (see FIG. 17D) which bends vertically like a beam due to the inertial and aerodynamic loads applied on the wings.

Table 6 lists dimensions and materials properties of the springs that were used in experiments performed according to some embodiments of the present invention.

Table 6

The wing root was fixed to the free end of the vertical bending beam. As the beam bends out of the vertical plane the root is slightly displaced and rotated by the beam’s deflection angle. The rotation changes the pitch of the wing root relative to the horizontal flapping plane, thus reducing the AoA of the wing from its value at rest (90 ⁰ when the mechanism is not flapping). At stroke reversals, the wings switch direction and so does the beam’s bending direction, changing the AoA for lift generation at both half strokes of the flapping cycle. The beam’s deflection angle, relative to the vertical plane (see angle Q in FIG. 17D), depends on the magnitude of the loads, length and thickness of the beam, and the spring’s Young’s modulus.

The wing pitch profile (FIG. 17E) of the spring mechanism is close to harmonic, and appears like a sinusoidal wave, disregarding higher order harmonics. This is mostly due to the elastic behavior of the spring, interacting with the time-varying inertial loads on the wing that peak at stroke reversal. Additionally, it results from the weaker aerodynamic loads on the wing that peak at midstroke [64] and act like a damper. FIG. 25 is a schematic illustration formulating the change in pitching angle of the spring mechanism of the present embodiments. The passive pitching of the wing by angle 9 (relative to the axis normal to the stroke plane), as a result of flapping the wing in harmonic motion, can be modeled to show that the pitching motion is harmonic-like. To simplify the analysis of the dynamics involved, the leaf spring is approximated to be a linear spring with stiffness, k, attached to the wing mass, m, at a spanwise distance R_s from the wing hinge. The air provides a damping term, c. The spring has a deflection of u, perpendicular to the direction of the yaw rotation.

The equation of motion to this problem can be written as:

The damping term is expected to be much smaller than the inertial term and is neglected herein.

Since the wing root is actuated in a sine wave motion, the position of the wing-mass r_tot depends on the rigid body rotation and the deflection of the spring.

Using the Lagrange formulation to derive the equations, one obtains: and the Euler- Lagrange formalism provides the following equation of motion: it + (m² — d²)u = —aR_s (EQ. 8)

This is a case for a non-homogeneous harmonic oscillator equation since the coefficient of u is not a constant. Because a(t) is close to a harmonic function (due to the four bar linkage kinematics) its derivatives is also a harmonic function.

It is noted that when the amplitude of the yaw angular velocity is larger than the natural frequency of the spring, m, then the spring equation stops acting harmonically and becomes exponential in its behavior. Therefore, for harmonic motion to occur the natural frequency of the spring is set to be higher than the amplitude of the yaw angular velocity.

The leaf springs of the mechanism were designed considering the dimensions of the flapping system, the wings and the basic simplified equation of motion described above. FIG. 26 is a schematic illustration of the leaf spring that was used in this example. The second moments of area for the leaf spring shown in FIG. 26 are:

Using the constants listed in Table 6, the deflection of the leaf spring can be simulated. The results of the simulations are provided in FIGs. 27A-C, showing the pitching angle 0 of the wing, in degrees, as a function of the time t, in seconds, for different flapping frequencies (10 Hz, 15 Hz, and 20 Hz, in the present example), and for three values of the length parameter, /, of the leaf spring: 1=6 mm (FIG. 27 A), 1=9.5 mm (FIG. 27B), and 1=13.5 mm (FIG. 27C). The simulations show the harmonic nature of the pitching profile during the flapping cycle. The amplitude grows with the frequency and the length of the spring. The motion of the spring remains periodic with the flapping frequency, but some additional vibrations are occurring as well during the flapping motion. The higher the frequency, the stronger these vibrations are compared to the overall amplitude of the motion. A damping term can be added to the simulation but due to its smaller size it is not expected to change the underlying harmonic nature of the pitch profile. Images of the maximal pitching angle during the flapping cycle in the spring mechanism, are shown in FIGs. 28A-D, for four values of the length parameter, /, of the leaf spring: 1=6 mm (FIG. 28A), 1=9.5 mm (FIG. 28B), 1=11.5 mm (FIG. 28C), and 1=13.5 mm (FIG. 28D). In FIG. 28A, a maximal pitching angle of 12.9° was obtained for flapping frequency of 21.739 Hz, in FIG. 28B, a maximal pitching angle of 16.78° was obtained for flapping frequency of 15.873 Hz, in FIG. 28C, a maximal pitching angle of 21.08° was obtained for flapping frequency of 17.544 Hz, and in FIG. 28D, a maximal pitching angle of 24.61° was obtained for flapping frequency of 15.152 Hz,

In the spring mechanism of the present embodiments the wing pitch axis is anterior to the leading edge of the wing, so that the chordwise distance of this axis from the wing’s center of mass varies with the spring length (see FIG. 18A top and FIG. 18A middle). In this Example, several spring lengths were tested ranging from about 6 mm to about 13.5 mm to adjust the passive wing pitching dynamics. All other spring parameters were the same amongst the tested springs.

The second wing pitch mechanism is based on free, low-friction rotation of the wing root about the wing pitch axis. In this mechanism, the wing pitch rotation axis is on the leading edge (FIG. 18A bottom). Consequently, as the wing changes flapping direction at stroke reversals, the inertia of the wing and drag changes the pitching angle of the wing about the leading edge, until the wing reaches its hard stop at a maximal pitching angle with respect to the stroke plane. In the present Example, the maximal pitching angle was set to be about 45 °. Aerodynamic loads centered on the wing posterior to the rotation axis ensure that this pitching angle remains constant until the next stroke reversal. The resulting pitching profile is a trapezoidal wave, with a relatively quick transition time compared to the longer periods of fixed pitch angles.

The advantage of the spring mechanism is that it allows for elastic energy recycling at stroke reversals. Furthermore, the wing pitch angle of the same spring varies with the load on the wings. The advantage of the rotating mechanism is that it provides stable and predictable pitching of the wing. Flapper design

The flapping robotic system was fabricated by high-resolution 3D printing and is illustrated in FIGs. 18A-C. Magnified views of the flapping system are shown in FIGs. 18D-F.

The flapping system included a planar 4-bar mechanism in the form of a slider-crank. The flapping system was driven by a motor. The motor drives the crank which, in turn, moves a slider back and forth. The slider acts as a rack that drives two pinions, moving the wings in a periodical flapping motion [77]. The motor 112 on the upper frame is illustrated in FIG. 18D, a 1:6 gear transmission 206 connected between the motor and the crank 82 is illustrated in FIGs. 18D and 18E, and the crank 82 and pinions 96 are illustrated in FIG. 18F.

This actuating system is responsible for rotating the wings in a horizontal plane about the vertical (yaw) axis. The second mechanism is the hinges of the wings, which add another rotation (pitch) axis, allowing lift generation at both half strokes of the flapping cycle. This is done by allowing the wings to passively adjust their AoA. In both mechanisms, the change in wing pitch is coupled to the flapping motion by the inertia of the wings and, therefore, it is not considered as another controllable Degree of Freedom (DoF) for the flapping device. This is advantageous from the strand point of cost in mass, size, control, and energy. Thus, the passive wing pitch of the present embodiments is practical, in particular for miniature flapping drones

Wings

The wings were fabricated according to the procedure described in Example 3, above. Briefly, with reference to FIGs. 19A-F, wings of P. Cuprea (FIG. 19A, scale bar 50 mm) were pCT scanned (FIG. 19B). The scans were converted to CAD models (FIG. 19C) and were then 3D printed (FIG. 19D, scale bar 1 cm). A membrane was integrated post printing (FIG. 19E), and was glued to the 3D printed venal structure (FIG. 19F).

For the rotating mechanism, two sizes of wings were tested. A first size, referred to below as "S3," had a wing length of 6 cm and was a 3:1 upscale of the real beetle wing shown in FIG. 19A. The second size, referred to below as "S4," was upscaled 4:1 relative to the real beetle wing shown in FIG. 19A. When up-scaling the wings, the wing veins’ diameters were increased non- linearly at a power of 1.25 to the length dimension, to account for the increased rigidity suggested by the observed wing deformations of real beetles varying in body size [34].

FIGs. 20A-F are images showing the fully assembled flapping devices, where FIGs. 20A, 20C, and 20E show a flapping device including a spring mechanism, and FIGs. 20B, 20D, and 20F show a flapping device including a rotating mechanism. FIGs. 20A and 20B show the device in its entirety, FIGs. 20C and 20D show magnified views of the wings, and FIGs. 20E and 20F show side views of the device. Scale bar correspond to 2 cm. The white arrows in FIGs. 20C and 20D point to the spring and rotation mechanisms, respectively. The device weighed approximately 34 gr and was fully modular, as the parts were 3D-printed and were therefore easily adjustable and replaceable.

Aerodynamic performance

FIG. 21 is a schematic illustration of experimental setup used according to some embodiments of the present invention for testing the aerodynamic performance of the devices. The aerodynamic performance were quantified based on the lift generated and power needed to flap the wings. The flapping frequency was adjusted using an Arduino driver circuit, connected to a stable laboratory (24 V) DC power source. The lift was measured by mounting the device on a force transducer, and measuring the decrease in weight during flapping. The force transducer measures at 100 samples per second and the instantaneous force was averaged during 2 seconds at the middle of each trial. A high-speed camera (2000 fps) was used to measure the flapping frequency and observing the flapping in slow-motion. The electrical power consumed was measured from the mean current and voltage supplied by the power source (measured with an Ampere-meter and a Volt- meter, respectively).

Images captured using the camera shown in FIG. 21, are shown in FIGs. 22A and 22B. FIG. 22A shows an image series of the wing during a flapping cycle using the spring mechanism and FIG. 22B shows an image series of the wing during a flapping cycle using the rotating mechanism. The scale bar corresponds to 2 cm.

Lift forces measurement results are shown in FIGs. 23A-C. Measured lift forces as a function of the flapping frequency are shown in FIG. 23A for the spring mechanism and in FIG. 23B for the rotating mechanism. The horizontal dashed lines denote the maximum attainable lift, with no wing pitch adjustment, within the range of tested frequencies. In FIG. 23A, the curved dashed lines are the least square fitting power functions for the data points (circles) for spring lengths I of 6 mm, 9.5 mm, 11.5 mm, and 13.5 mm. The least square fits for these spring lengths are, respectively, O.OOO8 ^{3 0909}, 0.00 Lf^{3 1222}, 0.001 Lf^{3 1437}, and 0.0013-f^{3 1826}. In FIG. 23B, the curved dashed lines are the least square fitting power functions for the data points (circles) for the S3 and S4 wing sizes. The least square fits for these wing sizes are, respectively, O.OO77-/²²⁵⁰⁸, 0.0338-/²⁰³³⁹. The shaded areas around each fit in FIGs. 23A and 23B denote the predicting interval of the relationship at 95 % confidence level for the respective fit.

As demonstrated in FIGs. 23 A and 23B, the lift generated scales differently with the flapping frequency across mechanisms. Lift is increased in proportion to the cubic and square power of the flapping frequency in the spring and rotating mechanisms, respectively. FIG. 23C shows ratios between the fitted lift value generated by the spring mechanism and the fitted lift value for the rotating wing mechanism with the S3 wing. Ratios above 1.0 show that the spring mechanism is favorable at the respective flapping frequency. As demonstrated, the rotating mechanism improves the ability of the device to generate higher lift at lower flapping frequencies, and the spring mechanism outperforms the rotating mechanism at higher flapping frequencies. This result derives from the system dynamics in two ways. The rotating mechanism enforces a fixed Ao A of 45° at midstroke, while the Ao A in the spring mechanism depends on the spring length and flapping frequency. At low flapping frequencies both the inertial and aerodynamic loads on the wings are weaker. These loads are sufficient to rotate the wing at stroke reversals in the low-friction rotating mechanism, resulting in a fixed AoA, regardless of the flapping frequency. In the spring mechanism, the weak loads need to act against the springs, resulting in a reduced pitching of the wing. The aerodynamic loads that provide damping for the elastic return of the wing to AoA = 90° are lower as well. Consequently, the wing quickly returns to an AoA that is too high for effective lift production. At a flapping frequency of 6 - 15Hz (depending on the spring length) the wing loads become dominant relative to the spring’s resistance, and the AoA becomes low enough for efficient lift production.

The experimental results also shoed that at high flapping frequencies and spring length I = 13.5 mm, the AoA reached 65 ⁰ at stroke reversal, while it was higher at lower frequencies and shorter springs. For insect wings, the lift coefficient reaches a maximum at high AoA of approximately 45°- 65° [99], which is close to the value of wing pitch observed in this Example. The shorter springs likely resulted in AoAs that were above the angle giving the maximum lift coefficient. Note that while the two passive wing pitching mechanisms enforce a geometric AoA at the wing root, the effective AoA for both mechanisms is different from the geometric AoA, and depends also on the elastic deformations of the wings themselves (wing twist and camber). The deformations result from the time-varying inertial and aerodynamic loads acting on the wings, that were likely not identical between the two mechanisms.

Another distinction between the two mechanisms is that the spring mechanism rotates the wing root at stroke reversal but the spring’s elastic resistance provides a constant link to transmit forces and torques from the wings to the mechanism and vice versa. In contrast, the rotation of the wing in the rotating mechanism is through a low friction ball bearing and, therefore, the pitching torque cannot contribute to the lift of the mechanism when the wing rotates.

The pitch rotation of insect wings at stroke reversals can augment the lift forces produced [100]. The timing of this wing rotation relative to stroke reversal affect the augmented force and insects such as fruit flies vary this timing to adjust their aerodynamic power output [74,100,101]. A particular feature of the spring of the device of the present embodiments is that it introduces a time delay and sinusoidal kinematics to the wings passive pitch, introducing an option to adjust the phase shift between the wing elastic deformation and the flapping of the device. Therefore, adding elastic components to the wing hinge is advantageous since it boosts the lift of flapping drones for a given flapping frequency. From the standpoint of mechanical stability, the rotating hinge mechanism is advantages it proved to be more robust and provided higher endurance over time than the spring mechanism.

FIG. 24 shows the power-to-lift ratio of the spring and rotating mechanisms. Specifically, the electric power (Pe) consumption of the motor was divided by the measured lift (L) for each frequency tested, and for both mechanisms. This gives a measure of the added power needed by the system for increasing drone weight (lift). Shown are data points and the curves Pe/L=2.1127L’ 0.6109, _e/ ₌i.456L’⁰-⁴⁶³⁹, for spring lengths of 6 mm and 13.5 mm, respectively, Pe/L=\ .656L’⁰³⁴³⁵, and Pe/L=1.4107L’°-⁴⁷³⁷, for the S3 and S4 wing sizes, respectively.

FIG. 24 demonstrates a diminishing decrease in the Pe/L ratio with increasing lift, in both mechanisms, reaching a plateau at about 0.5 W/gf for wings with 6 mm and 13.5 mm spring lengths, corresponding to the shortest and longest springs, respectively. For the rotating mechanism, the plateau is reached at roughly 0.6 W/gf. These values correspond to frequencies of 15 Hz in the spring mechanism and about 10 Hz in the rotating mechanism. The increase in lift efficiency (Pe/L) at higher lift values is a direct result of Pe increasing with the flapping frequency to the power of 2.0 (R² = 0.936) or 2.1 (R² = 0.974) for the longest and shortest spring, respectively (data not shown), and to the power of 1.44 in the rotating wing (with S3, R² = 0.741 while lift for these mechanisms increases with the flapping frequency to a larger power of about 3 and about 2.25, respectively. Thus, despite the difference in lift productions between the two mechanisms, the power needed to generate a given lift is roughly the same, at least in the higher flapping frequencies, where lift is maximal.

Experimental

Flapping mechanism

The mechanism was designed in Solidworks © and 3D printed using Multijet 2500 Plus, (3D Systems, USA), using mainly VisiJet-M2R CL as the material for most of the parts. For the axles and locking rods, we used brass. The coupler between the motor and the driving gear was made from 6061 Aluminum to allow tapped holes.

All parts were affixed using screws, either to a printed tapped hole or to a nut. Moving parts like the motor coupler to the driving gear were made specifically out of metal to have strong and resilient tapped holes even when moving. The motor is a HobbyKing LD1510A-02-P Micro

BLDC.

3D printing of wings

An established procedure was used an in order to 3D-print and fabricate the wings, inspired by the P. cuprea beetle wing, see Example 3.

Lift force measurements

The forces were measured using Zemic L6B-06 Force Sensor, with a measuring rate of 100 samples per second. The lift is the difference between the weight of the flapping system at rest and the readout of the sensor while flapping. The lift is measured over two seconds after the system reaches a steady state at the desired flapping frequency. The period of force measurements overlapped a video recording by a high-speed camera (Fastcam SA3, Photron, USA) recording at 2000 frames per second. The video gave the exact flapping frequency from the number of video frames in a flapping cycle.

Power-to-Lift ratio measurements

A volt-meter and an Ampere-meter were connected a to the motor to measure the RMS values of the voltage and current that the motor requires to actuate the flapping system. The product of these two measurements is the electronic power of the motor.

By dividing this power and the measured lift for the same measured flapping frequency, the ratio can be calculated.

Conclusions

This Example demonstrates that the advancement of the next generation of flapping drones can be achieved using improved flapping mechanism. The employed flapping mechanism can mimic the flapping of insects, by introducing pitching during the flapping cycle. While controllers are a viable option to do so, this Example demonstrated that incorporating passive mechanisms for wing pitching is efficient. The mechanisms used in this study significantly amplify lift, surpassing a flapping mechanism lacking pitching by a factor of six. This Example showed that a mechanism relying on elastic adjustment of the pitch angle (spring mechanism) is more efficient in increasing lift generation, and that a mechanism which relies on non-elastic collisions (rotating mechanism) is less efficient in producing lift for the same flapping frequency, but is more robust and durable over time.

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.

It is the intent of the applicant(s) that all publications, patents and patent applications referred to in this specification are to be incorporated in their entirety by reference into the specification, as if each individual publication, patent or patent application was specifically and individually noted when referenced that it is 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. In addition, any priority document(s) of this application is/are hereby incorporated herein by reference in its/their entirety.

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Claims

WHAT IS CLAIMED IS:

1. A flapping wing for an aerial vehicle, comprising a support structure having an arrangement of struts, and a membrane coupled to said support structure, wherein a deformation of said support structure during an up-stroke phase of a flapping cycle of the wing is a mirror of a deformation said support structure during a down-stroke phase of said flapping cycle of the wing.

2. A flapping wing for an aerial vehicle, comprising a support structure having an arrangement of struts, and a membrane coupled to said support structure, wherein during an upstroke and a down-stroke phases of a flapping cycle of the wing an in-plane deformation of said support structure increases aerodynamic force generated by the wing on surrounding air, such that said increased aerodynamic force is vertical during said down-stroke phase and horizontal during said up-stroke phase.

3. A flapping wing for an aerial vehicle, comprising a support structure having an arrangement of struts, and a membrane coupled to said support structure, wherein a deformation of said support structure during flapping of the wing is such that a sectional lift-to-drag ratio is generally uniform along the wing, wherein for any section of the wing, a lift-to-drag ratio of said section is no less than 80% of the maximal possible lift-to-drag ratio among all possible angle-of- attacks and curvatures of said section, irrespectively of the load on the wing.

4. A flapping wing for an aerial vehicle, comprising a support structure having an arrangement of struts, and a membrane coupled to said support structure, wherein a deformation of said support structure during flapping of the wing is such that for any section of the wing and a for any lift value of said section, a drag value of said section is smallest among a set of drag values corresponding to said lift value.

5. The flapping wing according to any of claims 1-4, wherein at least one of said struts is tapered along said strut.

6. The flapping wing according to any of claims 1-4, wherein at least one of said struts has a circular cross-section.

7. The flapping wing according to any of claims 1-6, wherein at least one of said struts has an oval cross-section with a generally horizontal major axis.

8. The flapping wing according to any of claims 1-7, wherein at least one of said struts has an oval cross-section with a generally vertical major axis.

9. The flapping wing according to any of claims 1-8, wherein at least one of said struts has a U-shaped cross-section.

10. The flapping wing according to any of claims 1-9, wherein said struts are all joined at one end but spaced apart at an opposite end.

11. The flapping wing according to claim 10, wherein support structure comprises at least one elongated stringer member connecting two of said struts.

12. The flapping wing according to any of claims 1-11, wherein said support structure is printed.

13. The flapping wing according to any of claims 1-12, wherein at least one of said struts is characterized by a Young’s modulus of from about 1000 to about 3000 MPa.

14. The flapping wing according to any of claims 1-13, wherein at least one of said struts is characterized by a flexural modulus of from about 1000 to about 3000 MPa.

15. An aerial vehicle, comprising a vehicle body, at least two controllable flapping wings pivotally coupled to said vehicle body, and at least one motor for providing a flapping motion to said wings, wherein each of said wings is the flapping wing according to any of claims 1-14.

16. An aerial vehicle, comprising a fixed wing flying vehicle having a pair of fixed wing, and at least one flapping wing mounted on a distal end of each fixed wing of said pair, said at least one flapping wing being the flapping wing according to any of claims 1-14.

17. The aerial vehicle according to any of claims 15 and 16, comprising a passive pitching mechanism connected to a root of each flapping wing and configured to vary a pitching angle of said flapping wing during a flapping cycle of said flapping wing.

18. An aerial vehicle, comprising a vehicle body, at least two controllable flapping wings pivotally coupled to said vehicle body, at least one motor for providing a flapping motion to said wings, and a passive pitching mechanism connected to a root of each flapping wing and configured to vary a pitching angle of said flapping wing during a flapping cycle of said flapping wing.

19. A aerial vehicle, comprising a fixed wing flying vehicle having a pair of fixed wing, at least one flapping wing mounted on a distal end of each fixed wing of said pair, and a passive pitching mechanism connected to a root of each said at least one flapping wing and configured to vary a pitching angle of said flapping wing during a flapping cycle of said flapping wing.

20. The aerial vehicle according to any of claims 17-19, wherein said variation of said pitching angle is generally harmonic.

21. The aerial vehicle according to claim 20, wherein said passive pitching mechanism comprises an elastic element configured to exhibit elastic vibration about a vertical axis during said flapping cycle.

22. The aerial vehicle according to any of claims 17-19, wherein said variation of said pitching angle is described by a generally piecewise linear function of a time within said flapping cycle.

23. The aerial vehicle according to claim 22, wherein said passive pitching mechanism comprises a bearing configured to allow said flapping wing to rotate about a horizontal axis during said flapping cycle.

24. A kit, comprising: an aerial vehicle body connectable to at least two flapping wings, and having at least one motor for providing a flapping motion to said wings; and separately from said vehicle body, a plurality of sets of flapping wings pivotally connectable to said vehicle body, wherein each set of said plurality of sets is adapted to provide a different flight mode.

25. The kit according to claim 24, wherein said plurality of sets comprises at least one set adapted to provide a hovering flight mode.

26. The kit according to claim 24, wherein said plurality of sets comprises at least one set adapted to provide a forward flight mode.

27. The kit according to claim 24, wherein said plurality of sets comprises at least one set having wings characterized by wing loading which is higher than a wing loading of any wing in other sets.

28. The kit according to claim 24, wherein said plurality of sets comprises at least one set having wings characterized by lift-to-drag ratio which is higher than a lift-to-drag ratio of any wing in other sets.

29. The kit according to any of claims 24-28, wherein said areal vehicle is connectable to a root of each of said at least two wings via a passive pitching mechanism configured to vary a pitching angle of said flapping wing during a flapping cycle of said flapping wing.

30. The kit according to claim 29, wherein said variation of said pitching angle is generally harmonic.

31. The kit according to claim 30, wherein said passive pitching mechanism comprises an elastic element configured to exhibit elastic vibration about a vertical axis during said flapping cycle.

32. The kit according to claim 29, wherein said variation of said pitching angle is described by a generally piecewise linear function of a time within said flapping cycle.

33. The kit according to claim 32, wherein said passive pitching mechanism comprises a bearing configured to allow said flapping wing to rotate about a horizontal axis during said flapping cycle.

34. A method of fabricating a support structure for a flapping wing, comprising: receiving CT scan data of a veins within a wing of an insect; generating a CAD model based on said scan data; and operating a three-dimensional printer to print a support structure having an arrangement of struts, corresponding to at least a portion of said veins, based on said CAD model.

35. The method according to claim 34, comprising modifying said CAD model prior to said printing.

36. The method according to any of claims 34 and 35, wherein said operating said three- dimensional printer comprises printing a sacrificial structure wherein said struts are embedded in said sacrificial structure, and the method comprises removing said sacrificial structure to expose said struts, and placing said exposed struts in a template substrate patterned with grooves compatible with said struts so as to stabilize a shape of said struts in said template substrate.

37. The method according to claim 36, wherein said removing said sacrificial structure comprises heating.

38. The method according to any of claims 36 and 37, wherein said removing said sacrificial structure comprises immersing said sacrificial structure in liquid.

39. The method according to claim 38, wherein said liquid comprises oil.