Beam-based nonlinear spring

Abstract

Nonlinear spring. In one embodiment, the spring includes two opposed curved surfaces curving away from one another. A flexible cantilever member is disposed between the two opposed curved surfaces and a mass is attached to a free end of the cantilever member wherein the flexible cantilever member wraps around one of the curved surfaces as the cantilever member deflects to form a nonlinear spring. Energy harvesting devices and a load cell are also disclosed.

Claims

1. A nonlinear load sensitive structure comprising: a. a first rigid structure with at least one curved contact surface; b. a second rigid structure with at least one curved contact surface; and c. at least one curved flexible member disposed between said first and second rigid structures, one end of said at least one curved flexible member attached to said first rigid structure and other end of said at least one curved flexible member attached to said second rigid structure, such that when said first rigid structure is displaced toward said second rigid structure, said at least one curved flexible member bends and makes increasing contact with the contact surfaces of the rigid structures to provide an increasing force as a function of displacement of said rigid structures toward each other.

2. The nonlinear load sensitive structure of claim 1 further comprising an optical or eddy-current sensor configured to measure the displacement between said first and second rigid structures.

3. The nonlinear load sensitive structure of claim 1 further comprising one or more strain gages attached to said at least one of said flexible structures to measure the force required to cause displacement between said first and second rigid structures.

4. A nonlinear load sensitive structure comprising: a. a first rigid structure with at least one concave contact surface and one convex curved contact surface facing each other; b. a second rigid structure with at least one concave contact surface and one convex curved contact surface facing each other; and c. at least one curved flexible member disposed between said first and second rigid structures' contact surfaces, one end of said a least one curved flexible structure attached to said first rigid structure and other end of said at least one curved flexible structure attached to said second rigid structure, such that when said first rigid structure is displaced towards said second rigid structure, said at least one curved flexible member bends and makes increasing contact with the concave or convex contact surfaces of the rigid structures to provide an increasing force as a function of displacement of said rigid structures towards or away from each other, respectively.

5. The nonlinear load sensitive structure of claim 4 further comprising an optical or eddy-current sensor configured to measure the displacement between said first and second rigid structures.

6. The nonlinear load sensitive structure of claim 4 further comprising one or more strain gages attached to said at least one flexible member to measure the force required to cause displacement between first and second rigid structures.

7. The nonlinear load sensitive structure of claim 4 wherein each rigid structure has two each of said concave and convex surfaces and two said flexible members wherein the surfaces and flexible members are approximately circular arcs.

8. The nonlinear load sensitive structure of claim 4 configured as an energy harvesting device where a magnet is attached to said first rigid structure and a wire coil is in proximity to the magnet and attached to said second rigid structure.

9. A nonlinear load cell comprising: a first member further including a first symmetrical member with a left and a right curved surface; a second member displaced parallel to the first member and including a second symmetrical member with a left and a right curved surface, the first and second members having first and second anchor points to receive tensile or compressive forces; and a total of four flexible structures including a left and a right flexible structure projecting out from center regions of both said first and second members, respectively; and left and right connection structures connecting left flexible structure ends to each other and right flexible structure ends to each other, respectively.

10. The nonlinear load cell of claim 6 further comprising a motion sensor between at least one of the connection structure and one of said first or second members.

11. The nonlinear load cell of claim 9 further comprising an optical or electromagnetic linear motion sensor between the first and second members.

Description

BRIEF DESCRIPTION OF THE DRAWING

(1) FIG. 1 is a side view showing an embodiment in an oscillator application, where there is no force applied to the cantilever tip.

(2) FIG. 2 is a side view showing an embodiment in an oscillator application, where a downward force is applied to the cantilever tip.

(3) FIG. 3 is a side view showing a possible feature modification to the surfaces 2a and 2b.

(4) FIG. 4 is a side view showing a possible assembly method of an embodiment in an oscillator application.

(5) FIG. 5 is a side view showing an embodiment as a component in a two degree-of-freedom oscillator application.

(6) FIG. 6 is a side view showing an embodiment as components in a two-degree-of-freedom oscillator application.

(7) FIG. 7 is a front view showing an embodiment in a load cell application, where there is no force applied to the top and bottom.

(8) FIG. 8 is a front view drawing showing an embodiment in a load cell application, where there is a compressive force applied to the top and bottom.

(9) FIG. 9 is a front view showing an embodiment in a load cell application, where there is no force applied to the top and bottom.

(10) FIG. 10 is a front view showing a load cell embodiment when the flexible members of the load cell are naturally curved beams.

(11) FIG. 11 is a cross-sectional view of a nonlinear spring implemented by a cantilever beam that vibrates between two curved surfaces, according to an embodiment of the invention.

(12) FIG. 12a is a cross-sectional view of an embodiment of the invention disclosed herein showing a load cell with rigid connections and deflected in compression.

(13) FIG. 12b is a cross-sectional view of a load cell having rigid connections in an undeflected stage with rotational spring connections physically realized by 270 arcs.

(14) FIG. 13a is a free body diagram of a free cantilever segment in a load cell.

(15) FIG. 13b illustrates a free cantilever section of the free body diagram.

(16) FIG. 13c is a three-quarter ring section of the free body diagram.

(17) FIG. 14 is a schematic illustration of a circular load cell in an embodiment of the invention disclosed herein.

(18) FIG. 15 is a free body diagram and schematic of a bottom right quarter of the circular load cell in compression.

DESCRIPTION OF THE PREFERRED EMBODIMENT

(19) FIG. 1 is a side view of an embodiment of the invention disclosed herein. The embodiment in FIG. 1 has a depth into the page. In this embodiment, a beam 3a is clamped between top surface 1a and bottom surface 1b. In this embodiment, the beam 3a is a cantilever. In other embodiments, the right end of beam 3a may have another boundary condition or be attached to another object. In this embodiment, curves 2a and 2b are flat at their leftmost ends so that they clamp beam 3a. To the right of the flat segment, Curves 2a and 2b have decreasing radius of curvature along their lengths in the right direction.

(20) FIG. 2 shows that cantilever 3a wraps around the bottom surface curve 2b when a downward force is applied to the tip of cantilever 3a. In this embodiment, for increasing forces, an increasing segment of the cantilever contacts the surface, starting at the root. If an upward force is applied to the cantilever tip 3a, then the cantilever would wrap around the top surface curve 2a.

(21) The present invention can be made to be any size and out of a large range of materials. Dimension limitations and applied force limitations are related to the stress in the deflected beams 3a and rotational springs 20 (FIG. 9).

(22) Other embodiments may have different features. Some of these features may be teeth cut along the curves 2a and 2b, as shown on curve 2n in FIG. 3. These teeth may be useful for reducing mechanical damping of the oscillator. Other features may be that curves 2a and 2b have radii of curvature that do not necessarily decrease as the distance from the leftmost end increases.

(23) Further, the concept of a stiffening member can be extended from a one-dimensional beam 3a wrapping around a one-dimensional curved surface 2a or 2b to a two-dimensional or three-dimensional flexure. For example, the two-dimensional flexure may be a conical coil spring or a plate. The three-dimensional flexure may be a shell, for example. For a two-dimensional flexure, the surfaces 2a and 2b may be two-dimensional shapes where the curvature changes as the radius from the origin increases, for example. For a three-dimensional flexure, the surface may be a three-dimensional sphere or ellipsoid, for example.

(24) FIG. 4 shows a possible assembly method of the embodiment. End-mass 4a may be fixed to the cantilever tip. Holes 6 may be used to bolt bottom surface 1b to back plate 10. Slots 5 may be used to bolt surface 1a to back plate 10. Holes 8 may be used to bolt top bar 7 to back plate 10. Bolts 9a-9b may be screwed through holes in the top of top bar 7 so that they push down on surface 1a. This assembly clamps cantilever 3a in between the surfaces 1a and 1b.

(25) FIG. 5 shows an embodiment of the stiffening spring as a component in a two degree-of-freedom oscillator application. Mass 4b is connected by spring 13a to the device outer casing 11a. Mass 4b is connected by linear spring 13a to end-mass 4a. The embodiment in FIG. 5 may be an electromagnetic energy harvester if masses 4a and 4b are magnets. 12a and 12b may be coils rigidly attached to outer casing 11a. Coil 2c may be rigidly attached to mass 4b. Mechanical energy may be dissipated due to the relative motion of coil 12c and mass 4b when 4b is a magnet. Mechanical energy may also be dissipated due to the motion of 4a relative to coil 12a and of 4b relative to 12b when 4a and 4b are magnets.

(26) FIG. 6 shows the embodiment as components in a two-degree-of-freedom oscillator application. 4a is one mass of the oscillator. Surfaces 1c and 1d, which are rigidly attached to one another, are the second mass of the oscillator. Spring 13c connects surface 1c to the outer casing 11b, and spring 13d connects surface 1d to outer casing 11b. Spring 13e connects end mass 4a to outer casing 11b.

(27) The present invention may be used as a spring component in other oscillators and systems as well. Other systems may use any number and configuration of the present invention. In energy applications, the present invention may be used with transducers such as electromagnetic systems, piezoelectric systems, and electrostatic systems among others. Electromagnetic system configurations, for example, may use the masses 4a and 4b as magnets or coils. The piezoelectric system, for example, may use cantilever 3a as the piezoelectric element.

(28) In the embodiment shown in FIG. 7, surfaces 1f and 1e are rigid. Rigid vertical bars 15a connect cantilever 3e to 3h and 3g to 3f. The rigid vertical bar 15a may have holes 18 for purposes such as attaching an optical sensor or eddy current sensor 18a. In the embodiment of FIG. 7, a sufficiently large compressive force applied to 1e and 1f causes beam 3e to begin to partly wrap around curve 2e, beam 3g to begin to wrap around curve 2g, beam 3f to begin to wrap around curve 2m, and beam 3h to begin to wrap around curve 2j. In the embodiment of FIG. 7, a sufficiently large tensile force applied to 1e and 1f causes beam 3e to begin to partly wrap around curve 2f, beam 3g to begin to wrap around curve 2h, beam 3f to begin to wrap around curve 2k, and beam 3h to begin to wrap around curve 2i.

(29) Surfaces 1e and 1f may have holes 16e-16m cut into the roots of surface curves 2e-2m. Holes 16e-16m may be necessary to satisfy manufacturing practices that may not be able to cut a point at the intersections of 3e with 1e, 3g with 1e, 3f with 1f, and 3h with 1f. When surfaces 1e 375 and 1f have holes 16e-16m, insert 17 may be made to fit into all or some of holes 16e-16m. The presence of insert 17 extends the length of curves 2e-2m.

(30) The top surface 1e be may be connected to the object of interest while the bottom surface may be connected to the tabletop. The displacement measurements of this load cell embodiment could be measured by an optical sensor or eddy-current sensor that compares the displacement of the top surface 1e to the bottom surface 1f. The force acting on the load cell could also be determined by measuring the strain on a strain gage 20a located on the flexible member 3 or 20 (see FIG. 9).

(31) For example, if an optical sensor can detect changes as small as 0.1 m, then to achieve 1% accuracy in the force measurement requires a change in displacement per force: dy/dF110.sup.7 m/0.01F. For F=0.01 N [1 gram], it is desirable, then, to have a stiffness of K=dF/dy1000 N/m. For F=1,000 N [100 Kg], it is desirable to have K1e8 N/m.

(32) FIG. 8 shows the deflected load cell embodiment when a compressive force is applied on surfaces 1e and 1f.

(33) FIG. 9 shows the undeflected load cell with rotational springs 20 connecting the tips of cantilevers 3 and vertical bars 15b. Rotational spring 20 may be realized by a 270-degree curved beam. The rotational spring 20 may be useful for reducing the stress in deflected beams 3n, 3p, 3q, and 3r.

(34) FIG. 10 shows an undeflected embodiment of the nonlinear load cell 28. Curved beams 3t wrap around rigid outer surface curves 2n when a compressive force is applied to the rigid surfaces 1g, 1h. Curved beams 3t wrap around the rigid inner surface curves 2p when a tensile force is applied to the rigid surfaces 1g, 1hpossibly via the holes 27. Load cell 28 may be manufactured from one monolithic part with cut-outs 16e-16m to satisfy manufacturing finite cutting constraints. Interlocking bodies 19e and 19f may be used to limit load cell deflection in both tension and compression by contacting each other when a certain deflection between the rigid surfaces 1g and 1h has occurred.

(35) The concept of stiffening members 3e, 3f, 3g and 3h in load cells 24 and 26, as shown FIGS. 7 and 9, can be extended from one-dimensional beams 3 wrapping around one-dimensional curved surfaces 2 to two-dimensional or three-dimensional flexures. For example, the two-dimensional flexures 3 may be a conical coil spring or a plate. The three-dimensional flexure 3 may be a shell, for example. For a two-dimensional flexure, the surfaces 2 may be two-dimensional shapes where the curvature changes as the distance from the origin increases, for example. For a three-dimensional flexure, the surface may be a three-dimensional sphere or ellipsoid, for example.

(36) The present invention can be made to be any size and out of a large range of materials. Dimension limitations and applied force limitations are related to the stress in the deflected beams 3 and rotational springs 20.

(37) Here we briefly summarize the theory for the force, deflection, and stress relationships for designing the vibrating spring and load cells. Designing the spring or load cell maximum stress to remain below a certain value increases its fatigue lifetime. Further details and equation derivations can be found in the journal article J. M. Kluger et al, Robust Energy Harvesting from Walking Vibrations by Means of Nonlinear Cantilever Beams, Journal of Sound and Vibrations (2014).

(38) As shown in FIG. 11, we choose a surface with the curve

(39) $\begin{array}{cc}S={D\left(\frac{z}{L_{\mathrm{Surf}}}\right)}^n,& \left(1\right)\end{array}$

(40) where D is the gap between the surface end and undeflected cantilever, and n is an arbitrary power greater than 2 (a requirement for essential nonlinearity), z is a spatial coordinate measured from the cantilever/surface root, and L.sub.Surf is the surface length. The theory derived below should apply to any surface with a monotonically increasing curvature,

(41) $\frac{{}^{2}S}{z^2}.$
When a sufficiently larger force F is applied to the beam tip, the cantilever begins to wrap around the surface. The contact point z.sub.c is the axial coordinate where the cantilever stops wrapping around the surface and becomes a free beam of length L.sub.Free. To the left of the contact point, we assume that the beam is tangent to (equal to) the surface shape given by eq. (1). For the free beam segment to the right of the contact point, the boundary conditions on the beam are

(42) $\begin{array}{cc}w\left(x=0\right)=S\left(z_c\right),\frac{w\left(0\right)}{x}=\frac{S}{z}{.Math.}_{z=z_c},\frac{{}^{2}w\left(x=L_{\mathrm{Free}}\right)}{x^2}=0,\frac{{}^{3}w\left(x=L_{\mathrm{Free}}\right)}{x^3}=\frac{-F}{\mathrm{EI}},& \left(2\right)\end{array}$

(43) where w is the beam deflection along its free length, F is the force applied to the mass, L.sub.Free is the cantilever segment to the right of the contact point z.sub.c, EI is the cantilever rigidity, S is the surface shape defined in eq. (1), z.sub.c is the contact point between the cantilever and surface for the given force, and x is the spatial coordinate with its origin at z.sub.c. Based on Euler-Bernoulli beam theory and solving

(44) $\frac{{}^{4}w}{x^4}=0,$
the deflection along the free beam length, x, is

(45) $\begin{array}{cc}w\left(x\right)=\frac{1}{\mathrm{EI}}\left(S\left(z_c\right)+\frac{S\left(z_c\right)}{z}.Math.x+\frac{{\mathrm{FL}}_{\mathrm{Free}}}{2}{x}^2-\frac{F}{6}{x}^3\right).& \left(3\right)\end{array}$

(46) Substituting x=L.sub.Free into eq. (3), the beam tip deflection due to the force F is

(47) $\begin{array}{cc}=\frac{{\mathrm{FL}}_{\mathrm{Free}}^3}{3\mathrm{EI}}+\frac{S}{z}{.Math.}_{z=z_c}.Math.L_{\mathrm{Free}}+S\left(z_c\right),& \left(4\right)\end{array}$

(48) We can slightly modify eq. (4) to describe the deflection of the end-mass center .sub.Mass by accounting for the beam tip angle:

(49) $\begin{array}{cc}{}_{\mathrm{Mass}}=+\frac{}{z}\frac{L_{\mathrm{Mass}}}{2},& \left(5\right)\end{array}$

(50) where L.sub.Mass is the length of the undeflected end mass in the z direction. In eq.s (4) and (5), we assume that L.sub.Mass is small and causes a negligible moment on the beam tip. Eq.s (4) and (5) and the following equations may straightforwardly be modified for larger L.sub.Mass and other beam loading conditions.

(51) The location of the contact point z.sub.c along the surface is the point at which the cantilever curvature equals the surface curvature (surface contact condition):

(52) $\begin{array}{cc}\frac{{}^{z}S}{z^2}{.Math.}_{z=z_c}=\frac{{}^{2}w}{z^2}{.Math.}_{z=z_c}=\frac{{}^{2}w}{x^2}{.Math.}_{x=0}.& \left(6\right)\end{array}$

(53) This is the case because the free cantilever curvature decreases along its length (cantilever gets flatter), while the surface curvature is constant (n=2) or increases (n>2) along its length (surface gets rounder). z.sub.c is the point where the surface would no longer prevent the natural curvature of the free cantilever. Alternatively, at z.sub.c, the curvature at the root of a free cantilever of length L.sub.Free subject to tip force F equals the surface curvature to which it is tangent. The boundary condition defined by Eq. (6) is required for static equilibrium because no external moment is applied to the beam at the contact point.

(54) The free beam length is the full beam length minus the beam length in contact with the surface. Assuming a slender beam, the beam length in contact with the surface is approximately equal to the surface arc length from z=0 to z.sub.c. For small deflections, one can assume that L.sub.Free=L.sub.Cantz.sub.c.

(55) Further using the slender Euler-Bernoulli beam theory, the maximum stress magnitude, , in the beam cross-section is

(56) $\begin{array}{cc}=-\mathrm{Ec}\frac{{}^{2}w}{z^2},& \left(7\right)\end{array}$

(57) where E is the beam elastic modulus,

(58) 0$C=\frac{h}{2}$
is half the beam height and

(59) $\frac{{}^{2}w}{z^2}$
is the beam curvature. For the beam segment in contact with the surface,

(60) $\frac{{}^{2}w}{z^2}$
can be found by using w(z)=s(z) and differentiating eq. (1). For the free beam segment,

(61) $\frac{{}^{2}w}{z^2}$
can be found by differentiating eq. (3).

(62) As shown in FIGS. 12a and b, each load cell consists of a 22 symmetrical grid of nonlinear spring elements. Load cell deflection occurs between the top and bottom rigid blocks. The nonlinear springs are physically realized by cantilevers wrapping around the rigid surfaces as they deflect, splitting each cantilever into a cantilever segment in contact with the surface and free cantilever segment. The junction between the cantilever segment in contact and free 485 cantilever segment is the contact point, x.sub.c. The tips of the bottom cantilevers connect to the tips of the top cantilevers by rigid bars, which cannot rotate due to symmetry. The cantilever tips may be rigidly connected to these vertical bars (FIG. 12a) or connected to the vertical bars by moment compliance rings hereafter referred to as rings (the 270 arcs shown in FIG. 12b).

(63) Below, we describe the relationship of the load cell's applied force F, contact point x.sub.c, tip moment M.sub.Tip, and deflection . When F is applied to the load cell and the load cell deflects by , the complete load cell experiences the applied force 2F and deflection 2.

(64) The surface shape follows the curve

(65) $\begin{array}{cc}S={D\left(\frac{x}{L_{\mathrm{Surf}}}\right)}^n,& \left(8\right)\end{array}$

(66) where D is the gap between the surface end and undeflected cantilever, and n is an arbitrary power greater than or equal to 2, x is a spatial coordinate measured from the cantilever/surface root, and L.sub.Surf is the surface length. The theory derived below should apply to any surface with a monotonically increasing curvature,

(67) $\frac{{}^{2}S}{z^2}.$

(68) Referring to FIGS. 13a, b and c, the internal moment along the free cantilever segment as a function of distance x from the full cantilever root is
M.sub.Internal,Cant=F(Lx)+M.sub.Tip,(9)

(69) where F is the applied force on the load cell, L is the length of the full cantilever, and M.sub.Tip is the moment applied at the junction of the cantilever and ring. The internal moment along the ring is
M.sub.Internal,ring=FR(1+sin )+M.sub.Tip,(10)

(70) where R is the radius of the ring and is the angle along the ring.

(71) Using Euler-Bernoulli beam theory and equating the angle of the cantilever tip to the angle of the ring at =3/2, the value of M.sub.Tip as a function of the applied force F and contact point x.sub.c is

(72) $\begin{array}{cc}{M}_{\mathrm{Tip}}=\frac{\left(-3R^2-2R^2+L_{\mathrm{Free}}^2\right)F+2\frac{S\left(x_c\right)}{x}\mathrm{EI}}{3R+2L_{\mathrm{Free}}},& \left(11\right)\end{array}$

(73) where L.sub.FreeLx.sub.c (using small beam deflection approximation) is the free cantilever segment length,

(74) $\frac{S\left(x_c\right)}{x}$
is the slope of the surface at the contact point found by differentiating eq. (8), and EI is the cantilever and ring rigidity.

(75) At the contact point, the cantilever curvature must be continuous because there is not an applied external moment. To the left of the contact point, we assume that the cantilever segment in contact with the surface is tangent to the surface. Then, the contact point relates to the applied force F and tip moment M.sub.Tip by

(76) $\begin{array}{cc}\frac{{}^{2}S}{x^2}{{.Math.}_{x_c}=\frac{M_{\mathrm{Internal},\mathrm{Cant}}}{\mathrm{EI}}.Math.}_{x_c}.fwdarw.\frac{{}^{2}S}{x^2}{.Math.}_{x_c}=\frac{F\left(L-x_c\right)-M_{\mathrm{Tip}}}{\mathrm{EI}}.& \left(12\right)\end{array}$

(77) Eq.s (11) and (12) can be simultaneously solved to relate the applied force F, tip moment M.sub.Tip, and contact point x.sub.c.

(78) Having determined F, M.sub.Tip, and x.sub.c, the deflection of the load cell indicated in FIG. 2 relative to the rigid block is the summation of four components:
=.sub.1+.sub.2+.sub.3+.sub.4.(13)

(79) The first component is the cantilever deflection at the contact point x.sub.c. This deflection component is the vertical location of the surface curve at x.sub.c:
.sub.1=S(x.sub.c)(14)

(80) The second component is the deflection of the free cantilever segment due to the cantilever's slope at the contact point. Since the beam is tangent to the surface at the contact point, its slope equals the surface slope. The free length of the beam rotates by this slope (i.e. small angle) about the contact point, which results in the deflection:

(81) $\begin{array}{cc}{}_2=\frac{S}{x}{.Math.}_{x_c}.Math.L_{\mathrm{Free}},& \left(15\right)\end{array}$

(82) where L.sub.Free=Lx.sub.c is the length of the free cantilever segment, assuming small deflection and small surface curves, S(x.sub.c). The third deflection component is due to the free cantilever segment bending. Using Euler-Bernoulli beam theory, integrating the moment-curvature relation given by eq. (9), and using boundary conditions that the deflection and slope due to bending equal zero at the free cantilever segment root (the contact point, x.sub.c), this deflection component is:

(83) 0$\begin{array}{cc}{}_3=\frac{{\mathrm{FL}}_{\mathrm{Free}}^3}{3\mathrm{EI}}-\frac{M_{\mathrm{Tip}}{L}_{\mathrm{Free}}^2}{2\mathrm{EI}}.& \left(16\right)\end{array}$

(84) The fourth deflection component is due to the ring bending. When an infinitesimal segment of the ring, l=Rd, bends, it rotates the segments of the ring on either side of it by an angle =l with respect to each other, where is the change in the curvature of the beam at the infinitesimal segment due to bending

(85) $\left(=\frac{M_{\mathrm{Internal},\mathrm{Ring}}}{\mathrm{EI}}\right).$
Based on geometry and the small angle approximation, the vertical tip deflection due to this change in angle is the horizontal distance between the infinitesimal segment and the tip, X=R(1+sin ), multiplied by the change in angle, . Integrating this infinitesimal deflection along the curved beam results in the total deflection of the curved beam due to bending:

(86) $\begin{array}{cc}{}_4={}_{l_{\mathrm{Ring}}}.fwdarw.{}_4=\frac{\left(9.Math.+8\right){\mathrm{FR}}^3+\left(6.Math.+4\right)M_{\mathrm{Tip}}{R}^2}{4\mathrm{EI}}.& \left(17\right)\end{array}$

(87) The load cell stiffness is K=dF/d. The entire load cell stiffness is also K=dF/d.

(88) Again using Euler-Bernoulli beam theory, the normal stress in the cantilever segment in contact with the surface can be found by

(89) $\begin{array}{cc}=\frac{Eh}{2},& \left(18\right)\end{array}$

(90) where E is the cantilever elastic modulus, h is the beam height, and

(91) $=\frac{{}^{2}S}{x^2}$
(assuming the cantilever segment in contact with the surface is tangent to the surface). The normal stress in the free cantilever segment and ring can be found by

(92) $\begin{array}{cc}=\frac{M_{\mathrm{Internal}}h}{2I},& \left(19\right)\end{array}$

(93) where M.sub.Internal is the internal moment in the free cantilever segment (eq. (9)) or in the ring (eq. (10)), his the cross-sectional height, and I is the cross-sectional moment of inertia.

(94) As shown in FIG. 14, we design a load cell similar to the straight-beam load cell shown in FIGS. 12a and 12b but now with curved beams instead of straight beams. The load cell is loaded by a tensile or compressive force, P. We derive the theory for the load cell in compression mode. The theory derived here could be straightforwardly altered for tension mode. The load cell consists of four symmetrical quadrants. In this derivation, we consider the bottom right quadrant, which extends from .sub.r=0 to .sub.r=/2. .sub.r is the angle along the undeflected curved beam with respect to the vertical.

(95) $\begin{array}{cc}l\left(\mathrm{Surf}\left({}_s\right)\right)=l\left(\mathrm{Ring}\left({}_R\right)\right).fwdarw.{}_R=\frac{l\left(\mathrm{Surf}\left({}_S\right)\right)}{R_0},& \left(20\right)\end{array}$

(96) where we assume that the ring arc length equals R.sub.0.sub.R. The flexible curved beams have mean radii R.sub.0. The curved beams have cross-sectional height h which may vary along the angle . We may choose to keep h constant along .sub.R or vary the height along .sub.R according to

(97) $\begin{array}{cc}h=h_0+\frac{h_f-h_0}{{\left(\frac{}{2}\right)}^q}{}_R^q,& \left(21\right)\end{array}$

(98) where q is an arbitrary power. Eq. (21) is valid from 0.sub.R/2 and then symmetrical in the other load cell quadrants. The outer rigid surface has shape S.sub.out(.sub.s) and the inner rigid surface has shape S.sub.In(.sub.s, In), defined in polar coordinates. The outer and inner surfaces have curvatures .sub.S and .sub.S,In, respectively. The rigid surfaces have monotonically increasing curvatures,

(99) $\frac{{}^{2}S}{{}_S^2}0.$

(100) The load cell may deflect up to a total distance .sub.max, after which overstops prevent further deflection from additional force. The load cell may be fabricated with gaps if the fabrication technique cannot allow the curved beam root to meet the rigid surface at a point.

(101) Below, we derive the theory for the bottom right quadrant of the circular load cell in compression, shown in FIG. 15. When P is applied to the entire load cell, P/2 is applied to each load cell quarter due to horizontal symmetry of the load cell quarters. When the quarter load cell deflects by /2, the entire load cell deflects by due to vertical symmetry of the load cell quarters.

(102) The equation for the internal moment in the z direction along the curved beam as a function of the angle with respect to the vertical, .sub.R, is

(103) $\begin{array}{cc}M=\frac{{\mathrm{PR}}_0}{2}\left(1-\sin {}_R-M_D\right),& \left(22\right)\end{array}$

(104) where M.sub.D is the moment in the z-direction acting at the top of the load cell.

(105) Next, we determine the relationship of the applied force P/2, moment M.sub.D, and contact point .sub.Rc by simultaneously solving two equations. First, the rotation of the ring at point D with respect to point B must be 0 due to symmetry. That is,

(106) 0$\begin{array}{cc}{}_D={}_c+{}_{{}_{\mathrm{Rc}}}^{\text{/}2}{R}_0{}_R={}_c+{}_{{}_{\mathrm{Rc}}}^{\text{/}2}\frac{M}{\mathrm{EI}}{R}_0{}_R=0.& \left(23\right)\end{array}$

(107) .sub.c is the change in angle of the ring at the contact point:

(108) $\begin{array}{cc}{}_c={}_{\mathrm{Rc}}-{\tan }^{-1}\left(\frac{y}{x}{.Math.}_{{}_{\mathrm{Sc}}}\right),& \left(24\right)\end{array}$

(109) where .sub.Rc is the angle of the contact point on the undeflected ring (before the force is applied) and

(110) ${\tan }^{-1}\left(\frac{y}{x}{.Math.}_{{}_{\mathrm{Sc}}}\right)$
is the angle of the surface at the contact point, .sub.Sc (one way to find the surface angle with respect to the horizontal,

(111) $\frac{y}{x}{.Math.}_{{}_{\mathrm{Sc}}},$
is to convert S(.sub.Sc) from polar to Cartesian coordinates).

(112) For the ring, the internal energy from .sub.R=0 to the [unknown] contact point .sub.R=.sub.Rc, depends on how much the ring curvature changes to match the surface curvature to which it is tangent. This internal energy due to bending is

(113) $\begin{array}{cc}{U}_1=\frac{E}{2}{}_0^{{}_{\mathrm{Rc}}}{I\left(\frac{1}{R_0}-{}_S\left({}_S\right)\right)}^2{R}_0{}_R,& \left(25\right)\end{array}$

(114) where E is the curved beam elastic modulus, and I is the cross section moment of inertia (which may be a variable function along .sub.Rc, i.e. if the cross-section height is defined by eq. (21)). The surface curvature .sub.S(.sub.S) can be converted to a function of .sub.R using eq. (20).

(115) From the [unknown] contact point .sub.R=.sub.Rc to .sub.R=/2, the internal energy depends on the internal moment in the ring that causes bending. This internal energy component is

(116) $\begin{array}{cc}{U}_2=\frac{1}{2E}{}_{{}_{\mathrm{Rc}}}^{\text{/}2}{\mathrm{IM}}^2{R}_0{}_R,& \left(26\right)\end{array}$

(117) where the internal moment M is defined in eq. (22). The total internal energy in the ring is
U=U.sub.1+U.sub.2.(27)

(118) Next, we minimize the internal energy U with respect to the contact point .sub.R. That is, we solve

(119) $\begin{array}{cc}\frac{U}{{}_R}=0.& \left(28\right)\end{array}$

(120) To find the relationship of the applied force P/2, moment M.sub.D, and contact point .sub.Rc, we may simultaneously solve eq.s (23) and (28) for a fixed force and geometric parameters.

(121) To find the deflection of the load cell, we rearrange Castiglaino's first theorem into

(122) $\begin{array}{cc}\frac{}{2}={}_0^{P\text{/}2}\frac{\frac{U\left(F\right)}{F}}{F}F,& \left(29\right)\end{array}$

(123) where the internal energy is a function of the dummy variable, the applied force, F.

(124) Again, the stiffness of the load cell is K=dP/d.

(125) Finally, the equations for stress in the load cell are similar to those of the vibrating spring and straight beam load cell. For the curved beam segment in contact with the rigid surface, the normal stress is

(126) $\begin{array}{cc}=\frac{\mathrm{Eh}}{2},& \left(30\right)\end{array}$

(127) where =R.sub.0.sub.S(.sub.S) is the required change in the beam curvature for it to be tangent to the surface. The normal stress in the free segment of the curved beam is

(128) $\begin{array}{cc}=\frac{\mathrm{Mh}}{2I},& \left(31\right)\end{array}$

(129) where M is a function of .sub.R defined in eq. (22) and h may be the function of .sub.R defined in eq. (21)

(130) A fuller mathematical analysis underpinning the present invention may be found in the provisional application referred to earlier and in J. M. Kluger et al, Robust Energy Harvesting from Walking Vibrations by Means of Nonlinear Cantilever Beams, Journal of Sound and Vibrations (2014). The contents of this reference is incorporated herein by reference in its entirety. The numbers in square brackets refer to the references listed herein.

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