Microstructures using carbon fiber composite honeycomb beams

A drive assembly for a wing of a micromechanical flying insect. The drive assembly comprises a honey comb structure. A method for flying a micromechanical flying insect comprising moving a wing with a drive assembly having a stiffness to weight ratio greater than about 16×1010 N/mKg.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention is related in general to micromechanical flying insect (MFI) devices. More specifically, embodiments of the present invention provide a drive assembly for a wing of a MFI.

2. Description of the Background Art

Micro flapping structures such as are used in the MFI are required to produce large displacements at high resonant frequencies, while performing complex kinematic patterns. The MFI wings must be capable of independently going through a wing stroke of 120°, while being able to rotate 90° at a resonant frequency of 150 Hz. To do this, the body of the MFI includes actuators, two wings, each driven by separate thorax structures. The thorax structures consist of actuators, mechanically amplifying four-bar structures, and a differential. Since the work done on the air is proportional to the velocity of the wing squared, an important requirement is a high resonant frequency. Conventional MFI thorax is produced by using stainless steel beams as the structural members and polymer flexures to act as joints. This has drawbacks of having high inertias, thus lowering the resonant frequency, as well as being difficult to construct.

SUMMARY OF EMBODIMENTS OF THE INVENTION

Embodiments of the present invention also provide a drive assembly for a wing of a micromechanical flying insect. The drive assembly includes a honey comb structure, and an actuator including a piezoelectric material and a bonding layer. The actuator comprises a single crystal piezoelectric or an amorphous piezoelectric layer.

Embodiments of the present invention further also provide a method for flying a micromechanical flying insect comprising moving a wing with a drive assembly having a stiffness to weight ratio greater than about 16×1010N/mKg. The drive assembly comprises a honeycomb structure.

These provisions together with the various ancillary provisions and features which will become apparent to those artisans possessing skill in the art as the following description proceeds are attained by devices, assemblies, systems and methods of embodiments of the present invention, various embodiments thereof being shown with reference to the accompanying drawings, by way of example only, wherein:

DETAILED DESCRIPTION OF EMBODIMENTS OF THE INVENTION

Referring now toFIG. 1there is seen a drive assembly, generally illustrated as10, for a wing12of an MFI, generally illustrated as11. The drive assembly10is supported by an airframe, generally illustrated as13inFIG. 4. The drive assembly10includes a four (4) bar linkage assembly, generally illustrated as49inFIG. 1. The links or bars of the structure inFIG. 1are constructed using a manufacturing process involving uncured fibers pre-impregnated with epoxy, a laser micromachining system, and two dimensional computer aided design software. It is desired that these links be machined down to feature sizes of 50 μm while comprising a material of Young's modulus greater than 200 GPa and density of less than 2000 kg/m3.

The air frame13comprises a generally parallelepiped frame (e.g., a generally rectangular frame), generally illustrated as15inFIG. 4, and a pair of depending leg frames17aand17bconnected to the parallelepiped frame15in any suitable manner. The parallelepiped frame15has a pair of side bars19aand19b, and a transverse bar33and a pair of end bars21and29, all connected to the pair of side bars19aand19b. Leg frame17aconnects to side bar19aand includes a base bar23having opposed ends23aand23bto which actuators (identified as “14” below) connect. Similarly, leg frame17bconnects to side bar19band includes a base bar25having opposed ends25aand25bto which actuators connect. A pair of support platforms31aand31bis respectively connected to transverse bar33, end bar29and to transverse bar33, end bar21. Support platforms31aand31beach have a pair of openings31c–31cthrough which an actuator passes, as best shown inFIG. 3B.

The drive assembly10broadly includes actuators14, slider cranks16pivotally coupled to actuators14and to bars18(e.g., part of four bar linkage assembly49). Bars18are pivotally attached to bars20(e.g., also part of four bar linkage assembly49). Bars18are pivoted to lug flexures24at26. Bars20are pivotally coupled to a wing12which is pivotally coupled to lug flexures30at32.

The drive assembly10more specifically includes four (4) actuators14—14—14—14, four (4) slider cranks16—16—16—16, four (4) bar linkage assembly49including bar linkages50—50—50—50, two (2) differential assemblies54—54, and two (2) wings12—12. Each of the support platforms31aand31bsupports a pair of bar linkages50—50. As best shown inFIG. 3C, each bar linkage50includes bar18and bar20. Each bar18is coupled to a lug plate58by a flexure connection60. Each bar20is coupled to bar18by flexure connections62. Each bar20is also coupled to one of the support platforms31aor31b. Lug plates58are coupled to support platforms31aand31bvia flexure connections68. As best shown inFIGS. 3B,3C and5, a slider crank16interconnects an end of actuator14to bar18via flexure connections70and72. Each actuator14is coupled to suitable electronics (not shown) for having a voltage applied to the actuator14and creating a field across the thickness of the actuator14.

To drive the drive assembly10illustrated inFIGS. 1,3A and3B, piezoelectric bending actuators14are used. To achieve the high mechanical power density required for the drive assembly10, the actuators14for various embodiments of the invention may comprise single crystal piezoelectric materials and high modulus carbon fiber based passive layers. Under internal loading, the maximum achievable strain for an amorphous piezoceramic material (e.g. PZT-5H) is approximately 0.3%. For single crystal piezoelectric materials the fracture strain is increased to a value greater than 1% (e.g., PZn-PT). Utilizing the thermal expansion properties of various composite materials for various embodiments of the invention allows for extrinsically increasing the fracture toughness of these actuator materials. A 1% compressive strain bias placed to the piezoceramic layer increases the strain energy density by 300%. Since the piezoelectric coupling coefficients of these single crystal piezoelectric materials are much higher than their amorphous counterparts, the strain energy density of the actuators14may be is increased by a factor of 10.

The actuators14may be constructed by laminating together a piezoelectric layer and an anisotropic passive layer in an ordered fashion and curing them together. The orientations, mechanical, and piezoelectric properties of the constituent materials are of importance for the performance of the actuators14. With a mixture of piezoelectric materials and non-piezoelectric materials (e.g., anisotropic passive constituent layers(s)) within the actuators14, either symmetric extension/contraction or uniform bending will occur when an electric field is applied to the piezoelectric material. Referring now toFIG. 3Ethere is seen an electrical schematic drawing showing the electrical field and poling direction when an electrical circuitry is hooked up to a PZT layer. Extension or contraction occurs when the piezoelectric materials are symmetric about the neutral axis while bending will occur when this symmetry does not exist. The anisotropic passive constituent layers produce a unidirectional composite that is capable of bending-twisting or extension-twisting coupling.

For various embodiments of the present invention, each actuator14comprises as best illustrated inFIG. 12a piezoelectric layer80, and a passive composite elastic layer82coupled to the piezoelectric layer80by a bonding layer84. The bonding material for the bonding layer84may be any suitable bonding material, preferably a matrix epoxy from the composite prepreg. The bonding material for the bonding layer84may be purchased commercially from YLA Inc.

The piezoelectric layer80may comprise a single crystal relaxor-based piezoelectric material (e.g. PZN-PT, PMN-PT) and/or amorphous polycrystalline (PZT) ceramic piezoelectric material. Single crystal piezoelectric materials exhibit a greater piezoelectric coupling coefficient than the PZT materials. However, the PZT materials exhibit a higher fracture toughness which may be more suitable for certain embodiments of the present invention.

The passive composite elastic materials for the passive layer82comprise unidirectional ultra high modulus (UHM, such as Young's modulus greater than about 50 GPa) carbon or graphite fiber, and an epoxy resin (e.g. an uncured epoxy resin). The passive layer82comprises from about 30% by vol. to about 80% by vol. of carbon or graphite fiber, more preferably from about 40% by vol. to about 70% by vol., most preferably from about 50% by vol. to about 60% by vol. of the carbon or graphite fiber. The carbon or graphite fiber may be purchased under the product name M60J from YLA Inc.

The epoxy resin may be any suitable thermosetting resin based on the reactivity of the epoxide group. A suitable epoxy resin is made from epichlorohydrin and aromatic bisphenol A (or aliphatic polyols, such as glycerol having glycidyl ether structures). Another suitable epoxy resin comprises polyolefins oxidized with peracetic acid. A suitable epoxy resin may be catalyzed with any suitable catalyst. For various embodiments of the present invention the passive layer82comprises from about 20% by vol. to about 70% by vol. of the epoxy resin, more preferably from about 30% by vol. to about 60% by vol., most preferably from about 40% by vol. to about 50% by vol., of the epoxy resin.

The passive layer(s)82comprise(s) a Young's modulus in the horizontal or longitudinal direction ranging from about 50 GPa to about 500 GPa; preferably from about 100 GPa to about 400 GPa, most preferably from about 200 GPa to about 300 GPa, and a Young's modulus in the transverse direction ranging from about 1 GPa to about 10 GPa, preferably from about 3 GPa to about 7 GPa, most preferably from about 4 GPa to about 6 GPa. To provide flexibility in the design and construction of the actuators14, the thickness of the bonding and passive layer(s) is preferably as thin as possible. In various embodiments of the invention, the thickness of the passive layer(s)82is less than about 40 μm (e.g. from about 10 μm to less than about 30 μm) more preferably less than about 30 μm (e.g. from about 10 μm to about 20 μm) most preferably less than about 20 μm (e.g. from about 5 μm to about 15 μm).

Depending upon the desired motion of the actuators14, the number and relative orientations of the respective combined layers are determined and each layer is cut using a laser-micromachining stage. Then each layer is assembled together, still in an uncured state, and cured in a vacuum oven and subsequently released.

Referring now toFIGS. 5 and 6, there is seen on the actuators14comprising piezoceramic (PZT) layer82(a dielectric material) and elastic layer80. Applying an electric field to the piezo layer82forms a strain in the piezo layer82. Since the PZT motion is restricted by the elastic layer80, a stress develops. This stress within the actuator14may vary through the cross section of the actuator14; thus, there is an effective moment MOin the beam as illustrated inFIG. 5, causing a deflection. The strain in the PZT layer82is given by the following equation (1):

ɛ1p=1Ep·σ1p-d31⁢Vappt(1)
where: Epis PZT modulus, σ1Pis stress in the fiber direction, t is PZT thickness, d31is the piezoelectric constant and Vappis the applied voltage. More generally, this strain may be given in the following form:

Assuming that the piezoelectric layer82is transversely isotropic (d31=d32), d36is taken to be 0; thus, there is no shearing forces or twisting moments applied by the piezoelectric. Solving equation (2) for the stresses in the PZT layer gives the following:

In equations (1) and (3), the [Qij]Pterms are the material constants of the PZT as given in Table 1 below. Similarly, the stresses in the elastic layer80may be given as follows:

The forces and moments may be given as a function of the ply stresses:

In equation (5), z is the linear variable through the thickness direction, and the term h is the total actuator thickness; thus, to solve for Nijand Mijin accordance with the following equations (6) the integrals need to be split into a summation over all layers of the actuator14:

[Nij]=∑k⁢∫tk-1tk⁢[σij]k⁢⁢ⅆz[Mij]=∑k⁢∫tk-1tk⁢[σij]k⁢z⁢⁢ⅆz(6)
where K is the total number of layers in the actuator14.

Subsequently, the actuator properties may be determined as a function of the ply lay-up using laminate plate theory. First, the relationship between the midplane strains and curvatures and the forces and moments may be given by the following equation (7):

In equation (7) N and M are the external forces and moments acting on the actuator14, and NPand MPare the piezoelectric forces and moments generated within the actuator14. Also, the A, B, and D terms may be determined by the following equation (8):

Assuming that there are no external forces and moments (i.e., that all extension and curvature is a result of the piezoelectric effect), there would be two terms of importance within equation (8): the curvature in the displacement direction, Kxand the twist curvature, Kxy. These two quantities are related to the linear displacement of the tip of the actuator and the output twist angle in accordance with the following equations (9):

δ=12·κx·l2γ=tan-1⁡(κxy·(l2+w2w))(9)
where l is actuator length, w is actuator width, δ is the linear displacement of the tip of the actuator in meters, and γ is the output twist angle in radians. Because of the desired kinematics of the wing12, it is preferred to have γ be as large as possible while keeping δ roughly the same as conventional actuators. Thus, the indicated numerical approach generates γ and δ as a function of the lay-up for a given input voltage. A Matlab script may be employed for searching a confined parameter space and determining the optimum γ and δ for each iteration. The results of this search gave a lay-up of [PZT /θ/0/0/θ], where θ are the ply angles. For this lay-up, γ and δ are given inFIGS. 19 and 20as a function of ply angle, θ.

The graph inFIG. 19illustrates that the maximum output twist angle occurs when θ=63 degrees, which corresponds to γ=12 degrees, and δ=300 μm. For these parameters, a four bar transmission ratio of 10 will produce a preferred output angle of 120 degrees.FIG. 19also illustrates that the twist and displacement are robust to small changes in the ply angles.FIG. 21shows a simulated end-on view of the actuator14through one cycle, where x2is the width direction and x3is the thickness direction.

For various embodiments of the present invention, actuators14comprise an average displacement of approximately 400 μm, and an average twist angle of about 6 degrees. The following Table 1 comprises various PZT design parameters for various embodiments of the present invention.

The drive assembly10uses a slider crank16which relies upon the buckling strength of flexures within the slider crank16to determine the serial stiffness. Because of the non-linear motions of the actuator14and the four bar linkage49; the slider crank16is preferably designed not to buckle. Preferably, the slider crank16increases the serial stiffness by ensuring that regardless of the position of the elements of the four bar linkage49, there will be flexures within the slider crank16that are in tension. This concept has been attempted by using the standard steel beam construction, however it is ineffective since the construction is bulky, requiring too complex a structure to obtain links with the required stiffness. The stiffness of the slider cranks16should be higher than the rest of the thorax links within the four bar linkage49, since the slider crank16will incur the highest forces. The complexity of the structure of the slider crank16could cause alignment errors to arise, leading to kinematics singularities and an increase in the effective parallel stiffness. Thus, if the stiffness of the sections of the slider crank16could remain high, while using a planar slider crank structure to decrease the complexity, the no-buckling slider crank16could be made in a small and efficient form factor.FIGS. 7–10shows various embodiments of the completed no-buckling slider crank16, made with various layers, [90/0]swith the polyester flexure layer in the middle.

For one embodiment of the invention as illustrated inFIGS. 7 and 8, the slider crank16is constructed by layering composite link92between composite links90aand90bsuch as to sandwich at least one thin flexure layer, generally illustrated as94. Preferably, flexure layer94comprises flexures94a,94b,94cand94d. Composite link92comprises a composite body92ahaving a pair of depending legs92b–92b, each leg92bterminating in a beveled surface92c. Composite links90aand90bhave respective openings90cand90dfor receiving depending legs92band92b.

In another embodiment of the invention as illustrated inFIGS. 9 and 10, the slider crank16comprises a composite link100sandwiched between composite links102aand102b. A flexure layer, generally illustrated as104, is conveniently disposed between composite link100and composite links102aand102b. Preferably, flexure layer104comprises flexures104a,104b,104c,104d,104eand104f. The composite link100comprises a composite body101having upper spaced links100aand100bseparated by an opening100c, and spaced lower links100dand100eseparated by an opening100f. The composite link102aincludes an opening102cfor receiving spaced upper links100aand100b. Composite link102aalso includes a tongue102ewhich lodges within opening100cof the composite link100. Similarly, composite link102bincludes an opening102dfor receiving spaced lower links100dand100e. Composite link102balso includes a tongue102fwhich lodges within opening100fof the composite link100.

The composite material for the various composite layers (e.g., links90a,90b,92,100,102aand102b) comprises unidirectional ultra high modulus (UHM, such as Young's modulus greater than about 50 GPa, such as about 200 GPa) carbon or graphite fiber, and an epoxy resin (e.g. an uncured epoxy resin). The composite material may be isotropic with respect to the Young's modulus, and may be constructed using the following lay-up: [0/90/flexure/90/0] where 0 and 90 are the relative angles of the plies and the flexure comprises flexure layer94or flexure layer104.

For various embodiments of the invention, the composite material for the various links of the slider crank16comprises from about 30% by vol. to about 80% by vol. of carbon or graphite fiber, more preferably from about 40% by vol. to about 70% by vol., most preferably from about 50% by vol. to about 60% by vol. of the carbon or graphite fiber. The carbon or graphite fiber may be purchased under the product name M60J from YLA Inc.

The epoxy resin in the composite material for the composite links of the slider crank16may be any suitable thermosetting resin based on the reactivity of the epoxide group. A suitable epoxy resin is made from epichlorohydrin and aromatic bisphenol A (or aliphatic polyols, such as glycerol having glycidyl ether structures). Another suitable epoxy resin comprises polyolefins oxidized with peracetic acid. A suitable epoxy resin may be catalyzed with any suitable catalyst. For various embodiments of the present invention the composite material for the composite links of the slider crank16comprises from about 20% by vol. to about 70% by vol. of the epoxy resin, more preferably from about 30% by vol. to about 60% by vol., most preferably from about 40% by vol. to about 50% by vol., of the epoxy resin.

The composite material for the composite links of the slider crank16comprises a Young's modulus in the horizontal or longitudinal direction ranging from about 50 GPa to about 500 GPa; preferably from about 100 GPa to about 400 GPa, most preferably from about 200 GPa to about 300 GPa, and a Young's modulus in the transverse direction ranging from about 1 GPa to about 10 GPa, preferably from about 3 GPa to about 7 GPa, most preferably from about 4 GPa to about 6 GPa.

The thickness of the combined layers/links of the slider crank16is preferably as small as possible to allow for unocluded pivotation at the joints. In various embodiments of the invention, the thickness of the combined layers/links of the slider crank16is less than about 100 μm (e.g. from about 50 μm to less than about 100 μm), more preferably less than about 50 μm (e.g. from about 25 μm to less than about 50 μm), most preferably less than about 25 μm (e.g. from about 5 μm to less than about 25 μm).

For various embodiments of the present invention, each layer/link of the composite material in the slider crank16is laser micromachined on a single ply basis and placed into a mold along with one of the flexure layers, such as flexure layer94or flexure layer104. All links/layers are subsequently cured together under vacuum and released.

The flexure layer(s) (i.e., flexure layer94or104) is preferably as thin and compliant as possible without plastically deforming in the presence of large tensile loads. The flexure layer(s) comprises any suitable material. For various embodiments of the invention the flexure layer(s) comprises any suitable polymer, preferably polyester. The thickness of the flexure layer(s) is less than about 20 μm (e.g. from about 10 μm to less than about 20 μm), more preferably less than about 10 μm (e.g. from about 5 μm to less than about 10 μm), most preferably less than about 5 μm (e.g. from about 1 μm to less than about 5 μM).

The flexure connections (i.e., flexure connections60,62,64and68) are preferably as thin and compliant as possible without plastically deforming in the presence of large tensile loads. The flexure connections comprise any suitable material. For various embodiments of the invention the flexure connections comprise any suitable polymer, preferably polyester. The thickness of the flexure connections is less than about 20 μm (e.g. from about 10 μm to less than about 20 μm), more preferably less than about 10 μm (e.g. from about 5 μm to less than about 10 μm), most preferably less than about 5 μm (e.g. from about 1 μm to less than about 5 μm).

The polyester for all flexure layer(s) and all flexure connections for various embodiments of the present invention may comprise polyester resin from any of a group of synthetic resins, which are polycondensation products of dicarboxylic acids with dihydroxy alcohols. The polyester resin may comprise ethylenic unsaturation, generally introduce by unsaturated acids (e.g., maleic and fumaric acids). The unsaturated polyesters are typically cross-linked through their double bonds with a compatible monomer, also containing ethylenic unsaturation, and thus become thermosetting.

The actuators14drive the four bar linkage assembly49which includes the bar linkages50—50—50—50having a structure40comprising generally hollow beams as links and flexures (e.g., polymer flexures such as polyester flexures) as joints. As previously indicated, between any actuator14and a bar linkage50is a slider crank16which converts the approximately linear motion of the tip of the actuator14to a partial rotation or pivotation at the base of the bar linkage50. In an embodiment of the present invention, any actuator14includes an electrode, a PZT layer, another electrode, a bonding layer (e.g., a matrix epoxy), and an elastic layer. The electrodes are deposited onto the PZT layer by the manufacturer. More specifically, referencingFIGS. 3D and 3E, for the actuation a voltage from a power source118is applied to the actuator electrode(s)119for creating an electric field120across the thickness of the piezoelectric material82which causes it to contract. Since this contraction is limited by the passive elastic layer80, a moment Mois generated which causes the actuator structure to bend or buckle. This small deflection initially needs to be converted to a pivotation/partial rotational motion, and then amplified. The slider crank16attached on the distal end of the actuator14moves in direction of arrow B towards bar18, causing bar18to pivot or rotate in accordance with arrow C away from actuator14and converting a generally linear motion (i.e., a buckling or bending motion) of the actuator14to a pivotation/partial rotation at the base (i.e., at the base of bar18) of the bar linkage50. The bar link (bars18,20, including lug plates58) lengths are not uniform, thus creating a mechanical amplifier. The input pivotation/partial rotation caused by the slider crank16is amplified by the respective bar linkage50, and the output of the respective bar linkage50is thus a large partial rotational motion as evidence by arrow D inFIG. 3D. Because the buckling, bending motion is a periodic motion (going back and forth) from the periodic application and release of the voltage from the power source118, the various initial movements the bar linkage50(bars18,20, including lug plates58) and slider crank16are reversed in direction, causing a corresponding reversal in the large partial rotation motion (about 120°) as indicated by arrow D. The partial rotation motion and reverse of the partial rotation motion as indicated by arrow D, causes an appended wing12to flap as a wing of a flying insect.

Referring in detail now toFIGS. 13–18, there is seen inFIG. 13a top plan view of a face sheet210(a carbon face sheet) for the bars (e.g., bars18and20), of the bar linkage assembly49.FIG. 14is a top plan view of a honeycomb core comprising a carbon fiber frame212and transverse from carbon fiber ribs214. There is no material between the ribs214and the frame212, and a pair of face sheets210—210would sandwich the frame212and ribs214.FIG. 15is a top plan view of a similar core structure comprising frame220, transverse ribs224, and criss-cross ribs220. The frame220and ribs224and220are molded from a suitable plastic, such as polyurethane. For this embodiment of the invention, a pair of face sheets210—210would sandwich the frame220and ribs224and222.

Referring now toFIG. 16, there is seen a vertical sectional view of two quadrilateral-shaped links/bars as illustrated inFIG. 18and a flexure (i.e., flexure60,62, and64). The three face sheets210are preferably carbon fiber, the flexure is preferably polyester, and the core230may comprise at least one of: the carbon fiber ribs ofFIG. 14, the molded polyurethane ribs ofFIG. 15, or a syntactic foam, such as honeycomb structure/foam ofFIG. 2.

Referring now toFIG. 17there is seen a vertical sectional view of a prior art bar/beam240. The triangular cross section has nothing in the middle and the shell is constructed from stainless steel.FIG. 18is a cross section of the improved bars/links250for the bar linkage49. The top and bottom layers250aand250bare preferably comprise carbon (e.g., carbon fiber), and the sides250cand250dwould be determined by what core material (e.g., carbon, polyurethane, foam, honeycomb, etc) for the frames (e.g., frames212,220, etc) employed.

The links (e.g., bars18and20) within the four bar assembly49have dimensions ranging from about 0.5 mm to about 5 mm for lengths and ranging from about 0.5 mm to about 1 mm for widths. Thus, for any of the previously indicated honeycomb structure, using a cell wall size of an order of magnitude less than the smallest dimension within the four bar linkage assembly49gives about 50 μm as a dimension for one of the smaller beam/bar widths. For fiber-reinforced (e.g., carbon fiber-reinforced) beams/bars with fiber diameters of around 10 μm, this would be approximately one of the limiting size for the links including the associated materials. In producing the links (e.g., bars18and20) of the bar linkage assembly49, uncured layers/materials are preferably employed. Uncured layers/materials have the benefit- of being able to lay-up the laminae for the links and the flexures a polymer for the joints at one time, and cure this laminate without the need of extra adhesive layers. In am embodiment of the present invention, the links (e.g., bars18and20) may possess the lamina parameter listed in the following Table 2:

For the maximum weight savings and for various embodiments of the invention, the beams/links for the four bar assembly49are preferably not solid structures. Preferably, a honeycomb configuration as illustrated inFIGS. 14–16may be employed. An analysis for the beams/links for four bar assembly49is aimed at matching the stiffness of the conventional stainless steel beams such as that illustrated inFIG. 17while minimizing the weight of the beam/links. First, the stiffness of a double supported cantilever beam/link may be determined by the following equation (10):

K=48⁢EIl3(10)
wherein E is the Young's modulus, I is the cross sectional moment of inertia, and l is the beam length.

For conventional beams four bar assembly49and having a hollow triangular cross section as illustrated inFIG. 17, the cross sectional moment of inertia may be determined by the following equation:

The cross sectional moment of inertia of a honeycomb structure for embodiments of the invention may be determined in accordance with the following equation:

Is=112⁢(BH3-bh3)(12)
where B is the outer width, H is the outer height, b is the inner width, and h is the inner height.

FIGS. 17 and 18illustrate the cross sections defined respectively in equation (11) and (12). For conventional parameters of a conventional four bar linkage assembly for a MFI, the link stiffness is 39348 N/m and the mass is 2.34 mg. An objective is to match the conventional stiffness using the honeycomb structure for embodiments of the present invention by optimizing over the geometric parameters shown inFIG. 18. To simplify this, the parameter tbaseis set to 50 μm since up to two 25 μm plies can be cut at once. This is done to simplify and expedite construction. Also, the parameter B is set to 1 mm so to allow the beams/links of the four bar assembly49to fit into the current form factor. Thus, a two-parameter optimization can be done over twalland H, with a restriction that twallcannot be less than the carbon fiber diameter. By matching the stiffness of the beams/links for various embodiments of the present invention with conventional link stiffness and for the given geometric parameters, the mass of the individual beams/links for various embodiments of the present invention ranges from about 0.75 mg to about 2.1 mg, more preferably from about 0.90 mg to about 1.5 mg, most preferably from about 1.0 mg to about 1.3 mg, such as about 1.16 mg, which is roughly half the conventional link mass.

The capabilities of the carbon fiber MFI thorax for the present invention may be compared to the capabilities from a conventional steel version with the same dimensions, and using a similar actuator. The two main dynamic parameters which are affected by the lower inertia are the resonant frequency and the mechanical Q. The Q is required to be low since it determines the ratio between the inertia force and the aerodynamic force. The resonant frequency is desired to be high since an increase in velocity increases the work done by the wing12on the air. For a given stiffness, reducing the mass by a factor of two will give a rise in resonant frequency by a factor of 1.414. The resonant frequency of conventional models is about 120 Hz with a Q of 3.5 while the four bar assembly49of the present invention gives a resonant frequency of 190 Hz with a Q of 2.5.

The beams/links of the four bar assembly49for the drive assembly10preferably comprises a structure40, such as a honey comb structure generally illustrated as42inFIG. 2. A honey comb structure includes any structure that resembles a honeycomb in structure or appearance. A honey comb structure may comprise a cellular structural material or any structure comprising cavities like a honeycomb.

The structure40has a stiffness to weight ratio greater than about 16×1010N/mKg, more preferably greater than about 18×1010N/mKg, most preferably greater than about 20×1010N/mKg including greater than about 24×1010N/mKg, where N is newtons, and “m” is meters, and Kg is kilograms. The structure40(e.g., honey comb structure42) comprises carbon, more preferably a carbon material sold under the product name M60J UHM carbon fiber reinforced epoxy by YLA incorporated.

In an embodiment of the invention the stiffness to weight ratio of the structure40(e.g., the honey comb structure42) ranges from about 16×1010N/mKg to about 50×1010N/mKg, more preferably from about 18×1010N/mKg to about 40×1010N/mKg, most preferably from about 20×1010N/mKg to about 30×1010N/mKg, including from about 24×1010N/mKg to about 28×1010N/mKg (e.g., about 26×1010N/mKg).

The structure40has a longitudinal Young's modulus greater than about 200 GPa, preferably greater than about 250 GPa, most preferably greater than about 300 GPa including greater than about 325 GPa. In an embodiment of the invention the longitudinal Young's modulus of the structure40(e.g., the honey comb structure42) ranges from about 200 GPa to about 600 GPa, more preferably from about 250 GPa to about 500 GPa, most preferably from about 300 GPa to about 400 GPa, including from about 325 GPa to about 375 GPa (e.g., about 350 GPa).

The structure40has a ply thickness greater than about 13 μm, preferably greater than about 18 μm, most preferably greater than about 22 μm including greater than about 24 μm , where “μm” is micrometers. In an embodiment of the invention the ply thickness of the structure40(e.g., the honey comb structure42) ranges from about 13 μm to about 50 μm, more preferably from about 18 μm to about 40 μm, most preferably from about 22 μm to about 30 μm, including from about 24 μm to about 28 μm (e.g., about 25 μm).

The structure40has a density of less than about 2200 kg/m3, preferably less than about 2000 kg/m3, most preferably less than about 1800 kg/m3including less than about 1700 kg/m3, where “kg” is kilograms and “m” is meters. In an embodiment of the invention the density of the structure40(e.g., the honey comb structure42) ranges from about 200 kg/m3to about 2200 kg/m3, more preferably from about 1000 kg/m3to about 2000 kg/m3, most preferably from about 1400 kg/m3to about 1800 kg/m3, including from about 1550 kg/m3to about 1750 kg/m3(e.g., about 1650 kg/m3).