Ring-based stiffening flexure applied as a loadcell with high resolution and large force range

Jocelyn M. KlugerGraduate Research Assistant

Department of Mechanical EngineeringMassachusetts Institute of Technology

Cambridge, MA 02139Email: jociek@mit.edu

Alexander H. Slocum ∗

ProfessorDepartment of Mechanical EngineeringMassachusetts Institute of Technology

Cambridge, MA 02139Email: slocum@mit.edu

Themistoklis P. SapsisAssociate Professor

Department of Mechanical EngineeringMassachusetts Institute of Technology

Cambridge, MA 02139Email: sapsis@mit.edu

This paper presents the theory of nonlinear (stiffening) flex-ures applied to a load cell to achieve both large force rangeand large resolution. Low stiffness at small forces causeshigh sensitivity while high stiffness at high forces preventsover-straining. With a standard 0.1 µm deflection sensor, thenonlinear load cell may detect 1% changes in force over 5 or-ders of force magnitude. In comparison, a traditional linearload cell functions over only three orders of magnitude. Wephysically implement the nonlinear flexure as a ring that in-creasingly contacts rigid surfaces with carefully chosen cur-vatures as more force is applied. We analytically describethe load cell performance as a function of its geometry. Wedescribe methods for manufacturing the flexure from a mono-lithic part or multiple parts. We experimentally verify thetheory for two load cells with different parameters.

1 IntroductionCompliant mechanisms, or flexures, are machine joints

that transfer motions and forces without friction, resultingin little wear or backlash and high precision. Flexure ap-plications include prosthetics (Jeanneau et al.) [1], nano-positioning for semiconductor fabrication (Teo et al.) [2], gy-roscope acceleration detection (Thomas) [3], and energy har-vesting devices (Kluger et al.) [4]. This paper focuses on theuse of flexures as load cells (Kluger et al.) [5], although theanalysis and fabrication methods may be relevant to manyother applications.

Load cells are often implemented as S-beams or disksthat deform under a force and cause a deflection or straintransducer to send a corresponding electric signal (Smith)

∗Address all correspondence to this author.

[6]. Load cells themselves have a large set of applicationsranging from material strength testing to prosthetic limbsensing (Sanders et al.) [7], monitoring infusion pumps de-livering drugs (Mokhbery) [8], agricultural product sorting(Change and Lin) [9], and human-robot collision force sens-ing (Cordero et al.) [10]. Traditional linear load cells can bedesigned for almost any force capacity from 1.0× 10−1 −2.5 × 106 N and withstand 50-500% overload capacity bythe use of overstops [6]. Because traditional load cells de-form linearly, they have constant force measurement resolu-tion over their entire force range.

There are several challenges to designing a load cell.The load cell should have minimal mass, volume, hystere-sis, and parasitic load sensitivity [6]. The most critical chal-lenge is the trade-off between force sensitivity and range: Itis desirable to maximize load cell strain or deflection in or-der to increase force measurement resolution by the strain ordeflection sensor, which typically resolves 14-bits between0 and its maximum rated measurement [11–13]. Simultane-ously, one wants to maximize the load cell’s functional forcerange and protect it from overloading, which requires limit-ing the strain.

A common approach for overcoming this design chal-lenge is using multiple linear springs with varied stiffnessesin series (Chang and Lin [9], Storace and Sette [14], andSuzuki et al. [15]). Overload stops prevent the weakersprings from deflecting too far, after which the stiffer springscontinue to deflect. Using this approach, [14] was able tomeasure weights over a range of 1 g to 30 Kg. A challengewith this approach is that multiple springs and transducersresult in a bulky and expensive device.

Our approach for designing a load cell with high force

resolution and capacity is to use a nonlinear stiffening mech-anism rather than multiple linear ones (Kluger et al.) [16]. Anonlinear load cell may have a low stiffness at low forces,and therefore its deflection and strain will be very sensitiveto the applied force. When used with a constant-resolutiondeflection sensor, this allows the load cell to have high forceresolution at small forces. High stiffness at large forces pro-tects the load cell from over-straining. The design may bevolume compact and inexpensive due to requiring only onenonlinear spring and sensor per device.

This paper describes a nonlinear stiffening load cellwhich uses curved beams that increasingly contact surfaceswith carefully chosen curvatures as more force is applied,as shown in Fig. 1. We derived the design starting from anonlinear cantilever-surface spring described by Timosheko[4, 17]. This load cell has high resolution (within 1% of theforce value) over a large range (5 orders of magnitude). Anabsolute linear encoder senses the nonlinear deflection. InSection 2, we develop the force-deflection theory. In Section3, we describe two methods for manufacturing the load cellmechanical components from one monolithic part or multi-ple parts. In Section 4, we describe the load cell performancesensitivity to geometric parameters. In Section 5, we exper-imentally verify the theory for a monolithic and multi-partload cell. We describe conclusions in Section 6.

2 Theoretical modelingWe consider the nonlinear ring flexure shown in Fig. 1.

The flexure deflects nonlinearly as increasing force causesan increasing length of the ring to wrap along the surfacestarting from θR = 0. As the free length of the ring shortens,the flexure stiffens.

We found that flexure stiffening rate may be favorablydecreased, as described in Section 4, when the ring thicknessis tapered; that is, the ring thickness symmetrically increasesin each quadrant from the root, θR = 0, to θR = π/2 accord-ing to

t(θR) = ti +(t f − ti)(

θR

π/2

)q

, (1)

where ti is the ring thickness in the flexure plane at θR = 0,t f is the thickness at θR = π/2, and q is a chosen power pa-rameter. The variable thickness causes variable ring rigidity,

EI(θR) = Eb(t (θR))

3

12. (2)

We determine the ring deflection as a function of force,P, in four main steps:

1. Express the internal loading along the ring as a functionof the applied force, P.

2. Determine the free segment’s boundary conditions foreach applied force, P. By ”boundary conditions,” werefer to the two unknown parameters that appear in the

(a)

(b)

Fig. 1: A nonlinear load cell flexure cut from a flat plate withan applied load P: (a) entire load cell with a linear encoderand (b) free body diagram of one symmetric flexure quadrantfor compression mode

strain energy expression: the ring-surface contact angle,θRC(P), and the reaction moment, MD(P), as labeled inFig. 1b. θRC is the coordinate along the ring where thedeformed ring stops conforming to the surface shape.

3. Express the strain energy in the ring as a function of theapplied force, P.

4. Use Castigliano’s first theorem to calculate the cumula-tive ring deflection for increasing force, P.

In our derivation, we consider a compressive loadingand use the sign convention shown in Fig. 1b. The ten-sile loading theory matches the compression theory when

the force P is negative-valued and the surface mean radiusis smaller than the ring mean radius, S < R.

We make several simplifying assumptions in our model.First, we assume that the flexure deflects only in-plane. Sec-ond, we assume that once a ring segment wraps around thesurface, it conforms to the surface shape and does not liftaway from the surface for larger forces. Third, in this paper,we consider only a surface with a constant radius, S. Fu-ture work may extend the theory for surface curvatures thatchange with θS.

2.1 Internal loading along the ringFirst, we describe internal bending moments, and forces

along the ring. For the ring segment in contact with the sur-face, θR < θRC, the ring curvature changes to match the meansurface curvature,

∆κ =1S− 1

R. (3)

The mean surface radius, S, is a theoretical design parameter.The physically fabricated outer surface radius So(θS) differsfrom S to account for the tapered ring thickness. So(θS) isS plus half the ring thickness, t(θR), at the ring location, θR,that contacts the surface at θS. We find the ring location, θR,that contacts a certain surface location, θS, by equating thecurves’ arc lengths,

RθR = SθS. (4)

The physically fabricated outer surface radius is

S(θS) = S+t( SθS

R )

2, (5)

where the ring thickness is defined in Eq. 1. Subtractionis used instead of addition for the inner surface radius. Theload cell nonlinear deflection is symmetric in compressionand tension modes if the inner and outer rigid surfaces causethe same magnitude ∆κ as the ring wraps around them.

Along the free ring segment, θRC < θR < π/2, the inter-nal bending moment is

M(θR) = MD − P2

R(1− sin(θR)) , (6)

where MD is a reaction moment at θR = π/2, as shown inFig. 1.

Along the entire ring, the force normal to the ring’scross-section, is

N(θR) =−P2

sin(θR). (7)

The shear force parallel to the cross-section is

V (θR) =−P2

cos(θR). (8)

2.2 Equations to determine the free ring segmentboundary conditions

Next, we determine the contact angle, θRC(P), and re-action moment, MD(P), for a given force, P. We use Euler-Bernoulli beam theory relating a thin ring’s internal momentto the change in curvature,

M = EI∆κ. (9)

First, we consider θRC(P). For the ring segment in con-tact with the surface, θR < θRC, we assume that the ring cur-vature changes to match the surface mean curvature. At thecontact angle, the ring change in curvature is continuous be-cause the surface does not impose an applied moment on thering. We apply Euler-Bernoulli beam theory, Eq. 9, at thecontact angle to equate the ring change in curvature requiredfor surface tangency and the change in curvature due to theinternal moment,

M(θRC) = EI(θRC)∆κ(θRC), (10)

where EI(θR) is the ring cross-section rigidity defined in Eq.2, ∆κ(θR) is the change in curvature defined in Eq. 3, andM(θR,MD,P) is the internal moment defined in Eq. 6.

Second, we consider MD(P). As shown in Fig. 1, thering cannot rotate about the z axis at point D due to symme-try. Counterclockwise rotation of the ring at point D due tointernal loading is

φD =∫

π/2

0∆κRdθR = 0. (11)

Substituting Eq. 3, 6, and 9 into Eq. 11, the rotation atD is

φD = ∆κS RθRC +∫

π/2

θRC

M(θR,MD,P)EI(θR)

RdθR = 0. (12)

We solve Eq.s 10 and 12 for θRC(P) and MD(P) usingthe numeric solver fsolve in Matlab for each force, P.

2.3 Complementary strain energy in the ringComplementary strain energy in the ring is the summa-

tion of internal bending, shear, and normal complementaryenergies [18],

U =UBend +UShear +UNormal. (13)

While bending dominates the load cell internal energy forsmall forces, shear and normal energy become significant asthe ring free segment shortens. The complementary bendingenergy along the entire thin ring is

UBend = 4(∫

θRC

0

EI(∆κ)2

2RdθR +

∫π/2

θRC

M2

2EIRdθR

). (14)

The factor of 4 accounts for the four ring quadrants. Com-plementary energy due to the internal shear force is

UShear =∫ 2π

0

6V 2

10GARdθR, (15)

where GA is the ring cross-section’s shear stiffness. Comple-mentary energy due to the normal force is

UNormal =∫ 2π

0

N2

2AERdθR. (16)

2.4 Castigliano’s first theorem to determine deflectionWe use Castigliano’s first theorem to calculate the cu-

mulative ring deflection as increasing force is applied [18],

δ =∫ P

0

∂U(F)∂FF

dF, (17)

where the complementary strain energy, U , in the ring isdefined in Eq. 13.

3 Fabrication MethodsWe consider two different load cell designs. We fab-

ricated both load cells using an Omax MicroMAX waterjetmachine with the parameters listed in Table 1 [19].

The first design is a multi-part load cell assembled fromtwo inner blocks, two outer blocks, and two flat spring steelbeams bent into rings, all bolted together, as shown in Fig.2a. The spring steel ring has a constant thickness. The sec-ond design is a monolithic load cell where the blocks andflexible ring are cut from a single sheet of 7071 aluminumand the ring has variable thickness along its length (0.5 mmat the flexure root and 10 mm at 90-degrees), as shown inFig. 2b. As shown in Fig. 2c, we cut gaps at the surfaceroots and inserted curved blocks to extend the surface curveto the root. This was required because the waterjet cannotcut a kerf smaller than 0.4 mm, and for the chosen parame-ters, the load cell performance is quite sensitive to a gap atthe root: with a gap (no block), the load cell deflects withtwo linear regions due to the ring pivoting about the surfacestart-point rather than wrapping along the surface starting atthe root. We experimentally shows this effect in [5].

The two designs have different advantages. The mainadvantage of the multi-part load cell is that if the flexure

breaks, then only the flexure needs to be replaced, withoutneeding to replace the blocks. If part of the monolithic loadcell breaks, then the entire device needs to be replaced. An-other advantage of the multi-part load cell is that it is lesscostly to waterjet: cutting out the ring width from a springsteel sheet requires less precision than cutting out the ringthickness because the load cell performance is much less sen-sitive to errors in ring width than errors in ring thickness. Athird advantage of the multi-part load cell is that the springsteel ring has a higher yield stress than an aluminum ring.Fourth, the minimum ring thickness and minimum differencebetween ring and surface radii is not limited by the water-jet’s kerf and accuracy. Our optimization procedures for theload cell showed that minimizing the difference in ring andsurface radii increases the load cell range because then thering undergoes less bending. Minimizing the ring thicknessallows the load cell to be sized smaller for a given initialstiffness.

The main advantage of the monolithic load cell is thatwaterjetting the ring thickness allows the ring to have a vari-able thickness along its length, which allows the ring stiff-ness to increase with less stress than the shortening mecha-nism alone. Furthermore, although the monolithic load cellrequires root blocks adhered to the surfaces for assembly, itdoes not require bolts, which risk becoming loose. Finally,the monolithic load cell ring does not have a pre-stress beforeany load is applied, as does the multi-part load cell fabricatedby bending a flat beam into a ring, which limits the maximumcompressive force that can be applied to the load cell beforeyield.

Other features that improve the load cell performanceare overstops to prevent overstraining and plates to preventout-of-plane over-straining. The load cell size can be scaled,as described in [5].

In terms of performance, both load cell types can be de-signed with comparable force sensitivity, range, and size, asdiscussed in Section 4.

4 Performance sensitivity to parametersLoad cell performance is a tradeoff between flexure

stiffness, which limits how finely a deflection sensor mayresolve changes in force, and flexure stress, which limits theforce capacity.

The load cell stiffness is

K =∂P∂δ

. (18)

If a linear encoder deflection sensor can resolve 0.1 µm de-flections and we would like to resolve a force to within 1% itsvalue, then we require the flexure stiffness to remain below

K ≤ 105P. (19)

Extensive details on the stiffness limitations for a certainforce resolution may be found in [5].

(a) (b) (c)

Fig. 2: Fabricated load cells: (a) multi-part assembled load cell experiment during maximum tension, (b) monolithic loadcell with tapered ring thickness and root inserts in compression, and (c) close-up view of root inserts

Table 1: Fabricated load cell parameters

Parameter Multi-part load cell Monolithic load cell

Ring mean radius, R (mm) 74.67 80.00

Inner surface mean radius, Si (mm) 70.00 77.00

Outer surface mean radius, So (mm) 80.00 83.24

Ring width, b (mm) 12.70 6.35

Ring thickness at root, ti (mm) 0.508 0.50

Ring thickness at θR = π/2, t f (mm) 0.508 10.00

Ring polynomial for thickness variation, q - 3

Ring material blue tempered steel 1095 aluminum 7075-T651

Ring elastic modulus, E (GPa) 205 72

Ring yield stress (MPa) 1800 420

The stress along the ring due to bending is [18]

σ =Et ∆κ

2. (20)

Eq. 20 neglects stress due to the shear and normal forcesbecause we found that these stresses contribute less than 2%to the ring’s von Mises equivalent stress.

When the ring is fabricated by bending a flat beam into

a circle, the pre-stress added to the load cell bending stress is

σPre =Et2R

. (21)

Both the multi-part and monolithic load cells can be de-signed with comparable force sensitivity, range, and size.

While we want the ring to be a weak spring for highsensitivity at small forces, we want it to be a stiff spring tolimit bending stress at large forces. As more force is appliedto the load cell, two mechanisms cause the ring stiffness to

0 1 2 3 4 5 60

200

400

600

800

1000

Deflection, δ (mm)

For

ce, P

(N

)

Multi−partexperimental

load cell, Si or S

o

Multi−partload cell:

1.01Si or 0.99S

o

Monolithicexperimental

load cell, Si or S

o

Monolithicload cell:

1.01Si or 0.99S

o

0 0.5 1 1.50

200

400

600

800

1000

contact angle, θRC

(rad)

For

ce, P

(N

)

Monolithic experimentalload cell, S

i or S

o

Monolithic load cell:1.01S

i or 0.99S

o

Multi−part experimentalload cell, S

i or S

o

Multi−part load cell:1.01S

i or 0.99S

o

(a) (b)

102

104

106

108

10−2

10−1

100

101

102

103

For

ce, P

(N

)

Stiffness, K (N/m)

Stiffne

ss lim

it for

1%

forc

e re

solut

ion

with 1

0−7 m

sens

or re

solut

ion

Multi−part experimentalload cell, S

i or S

o

Monolithic load cell: 1.01Si or 0.99S

o

Monolithic experimental load cellS

i or S

o

Multi−part load cell:1.01S

i or 0.99S

o

0 0.5 1 1.5−1500

−1000

−500

0

500

angle, θR (rad)

Str

ess

whe

n P

= 1

,000

N (

MP

a)

0.99

S o1.

01S i

Multi−part load cell compression mode

Multi−part load cell tension mode

Monolithic load celltension mode

Monolithic load cellcompression mode

So

Si S

i

So

0.99So1.01S

i

(c) (d)

Fig. 3: Load cell performance sensitivity to 1% increase in surface radius: (a) force versus deflection, (b) force versus contactangle, (c) force versus stiffness, and (d) stress along the ring inner radius when force P = 1,000 N. Monolithic load cell:black. Multi-part load cell: gray. Experimental parameters: solid line. Inner surface radius Si increased by 1%: dashed line.

increase: shortening of the free length, and increase in theaverage thickness of the free length (for the monolithic vari-able thickness ring only). For the two designs consideredin this paper, the variable thickness mechanism allows thering stiffness to increase with less stress than the shorteningmechanism alone, as shown in Fig.s 3c and d.

A main limitation for both load cell designs was theminimum allowable ring radius. The multi-part load cell ra-dius was limited in order to limit pre-stress when bendingthe spring steel beam into a circle. The multi-part load cellhas a -680 MPa pre-stress at all points due to bending theflat spring steel beam into a circle, which limits the allow-able ring force in compression before yield. On the otherhand, the pre-stress greatly increases the force range in ten-sion, as shown in Fig. 3d. The monolithic load cell radiuswas limited to decrease the initial stiffness. The load cell’s

initial stiffness can be decreased (i.e. force sensitivity canbe increased) by decreasing the ring thickness or its radius.The monolithic load cell ring thickness was limited by thewaterjet kerf and accuracy. cut by the waterjet was. For bothload cell designs, sensitivity at high forces is not a problem:if the load cell is sensitive at small forces, then its stiffnessincreases gradually enough so that a constant-resolution sen-sor can resolve larger forces to a higher relative resolutionthan smaller forces, as shown in Fig. 3c.

Fig. 3 illustrates the load cell sensitivity to fabricationerrors by plotting the effect of increasing the inner surfaceradius, Si, or decreasing the outer surface, So, by 1% for bothload cell designs. Fig. 3a shows that a 1% increase in surfaceradius decreases the deflection caused by a 1,000 N force by0.69 mm (20%) for the monolithic load cell and by 0.89 mm(15%) for the multi-part load cell. As shown in Fig. 3d, this

corresponds to a maximum stress decrease from 193 MPa to159 MPa (18%) for the monolithic load cell and from -1363MPa to -1296 MPa (5%) for the multi-part load cell.

These parameter sensitivity plots are caveats for loadcell design when using overstops or when designing for fa-tigue life: If the multi-part (monolithic) load cell is designedfor 1.00So and overstops for 1,000 N, but 0.99So is fabri-cated, then the overstops will not engage until 5400 N (1903N) is applied and the stress exceeds 1,525 MPa, a 13% in-crease (202 MPa, a 4.7% increase, not shown). If the mono-lithic load cell is designed for 1.01Si or 0.99So but 1.00Si or1.00So is fabricated, then overstops will stop the load cell de-flection at 9.65 N for the multi-part load cell and at 345 N forthe monolithic load cell rather than 1,000 N.

These different behaviors are due to how the multi-partload cell force-deflection curve is initially much flatter andthen approaches a steeper stiffness asymptote than the mono-lithic load cell with a tapered ring, which has a more gradualstiffness increase.

We note that while the 1% change in surface radius has asignificant effect on the load cell’s final deflection and stress,it has a small effect on the ring contact angle and stiffness ata given force, as shown in Fig.s 3b and c.

For designing a load cell with a range of 0.01-1,000 Nand 1% force sensitivity throughout that range, the mono-lithic tapered-ring load cell has the favorable property thatits stress remains low at large forces, but machine kerf lim-itations set a lower-bound to the load cell size for satisfyingthe initial stiffness. The multi-part load cell has the favor-able property that the ring thickness and difference in surfaceradii and ring radii may all be made much smaller than a ma-chine’s kerf, which reduces load cell stress at large forces.Despite this property, though, the multi-part load cell’s re-liance on the free length shortening mechanism alone insteadof also the free length thickening mechanism causes the loadcell stress to be very high at large forces, which limits theload cell range. Future work will include considering multi-part load cells that use surfaces with elliptical shapes ratherthan circular shapes, as this may reduce the multi-part loadcell’s stress at large forces.

5 Experimental verificationWe performed quasi-static force versus deflection tests

to verify the flexure theory described in Section 2 and showthe effectiveness of the two fabrication methods described inSection 3. Fig. 4 compares the force versus deflection resultsto the theory. The two slightly different experimental set-ups for the multi-part and monolithic load cells are shown inFig. 2. Both tests used an ADMET eXpert 5000 force testermachine with ±5× 10−5 mm resolution that compressed ortensioned the load cell at a rate of 0.05 mm/s, an InterfaceSM-25 Interface load cell with a 100 N capacity for smallforces, and an Interface SM-250 load cell with a 1,000 Ncapacity for large forces. The nonlinear load cell bottom wasbolted to the tabletop. The multi-part nonlinear load cell topwas bolted to the Interface load cell. A clevis pin attached tothe Interface load cell went through a hole near the top of the

monolithic nonlinear load cell.We used spacer blocks to determine a reference point

for the load cell zero deflection. For the multi-part load cell,the zero-deflection point was determined by resting a spacerblock on the bottom inner rigid block. The ADMET headtraveled downwards until the Interface load cell measured asharp force increase to 5 N, which signified that the spacebetween the top and bottom inner rigid blocks equaled thespacer block height of 17.3569 mm ± 0.0056 mm, basedon micrometer measurements. Data from these proceduresshowed that the ADMET head traveled 0.0259 ± 0.0139 mmafter the force started to increase.

For the monolithic load cell, the zero-deflection pointwas determined by resting two spacer blocks on the bottominner rigid block. Due to gravity, the top inner block restedon the spacer blocks when the clevis pin, loose in the holes,was not supporting it. The ADMET head traveled upwardsuntil the Interface load cell measured an increased force of0.1 N, signifying that the ADMET head was supporting thenonlinear load cell with a spacing between the inner andouter surfaces equal to the spacer block height. The spacerblocks had a height of 3.2310 ± 0.0036 mm, based on mi-crometer measurements.

During the tests, the ADMET head slipped at largeforces. To account for this, we conducted force-versus-deflection tests in both tension and compression mode withthe Interface load cell bolted directly to the bottom fixture todetermine ADMET slip during the multi-part load cell testand with the clevis pinned through a hole in a sufficientlyrigid block (δ(P = 1,000N)<< 0.001 mm) to determine theADMEt slip during the monolithic load cell test. The meanexperimental deflection from five trials was determined foreach test configuration and direction. This deflection as afunction of force was subtracted from the experimental non-linear load cell deflection.

ADMET backlash and load cell taper contributed to anadditional subtraction of 0.2095 ± 0.0890 mm from the de-flection for the multi-part load cell. ADMET backlash, clevispin slip, and load cell taper contributed to an additional sub-traction of 0.2193 ± 0.1745 mm from the deflection for themonolithic load cell.

The above sources of deflection uncertainty in the non-linear load cell force-versus-displacement data include thespacer block height, the ADMET displacement resolutionfrom both the nonlinear load cell test and the ADMET sliptests, and the offset due to initial ADMET backlash, load celltaper, and slip. These sources sum to a zero-deflection pointmeasurement uncertainty of ± 0.0948 mm for the multi-partload cell and ± 0.1782 mm for the monolithic load cell,which are indicated by horizontal bars in Fig. 4. Deflec-tion measurement relative to the zero-deflection point have± 0.0001 mm uncertainty.

We performed five trials for both load cells in both com-pression and tension modes, with a 100 N load cell and a1,000 N load cell, and determined the experimental forcemean and standard deviation as a function of load cell de-flection.

As shown in Fig. 4, the force-versus-deflection experi-

0 1 2 3 4 5 6

10−2

10−1

100

101

102

103

displacement (mm)

For

ce (

N)

TheoryCompression− 100 N load cellCompression− 1,000 N load cellTension− 100 N load cellTension− 1,000 N load cell

103

104

105

106

107

10−2

10−1

100

101

102

103

Stiffness (N/m)

For

ce (

N)

Stiffne

ss lim

it for

1% fo

rce re

solut

ion w

ith

10−7 m

sens

or re

solut

ion

TheoryCompression− 100 N load cellCompression− 1,000 N load cellTension− 100 N load cellTension− 1,000 N load cell

(a) (b)

0 1 2 3 4

10−2

10−1

100

101

102

103

displacement (mm)

For

ce (

N)

TheoryCompression− 100 N load cellCompression− 1,000 N load cellTension− 100 N load cellTension− 1,000 N load cell

103

104

105

106

10−2

10−1

100

101

102

103

Stiffness (N/m)

For

ce (

N)

Stiffness

limit f

or

1% force

reso

lution w

ith

10−7 m

senso

r reso

lution

TheoryCompression− 100 N load cellCompression− 1,000 N load cellTension− 100 N load cellTension− 1,000 N load cell

(c) (d)

Fig. 4: Experimental force, stiffness, and deflection behavior for load cells shown in Fig. 2: (a) and (b) multi-part load cell,(c) and (d) monolithic load cell. The horizontal and vertical bars indicate measurement uncertainty.

mental results agree well with the theory over the 0.01-1,000N range.

The multi-part load cell mean tests results had the fol-lowing maximum deviations from the theory: +28% at 18 Nfor the 100 N load cell compression test, +18% at 39 N forthe 1,000 N load cell compression test, +54% at 0.011 N forthe 100 N load cell tension test, and +43% at 428 N for the1,000 N load cell tension test. The multi-part load cell testhad the following maximum deviations due to random erroramong the five experimental trials: 30% at 0.011 N for the100 N load cell compression test, 3.6% at 15 N for the 1,000N load cell compression test, 150% at 0.011 N for the 100 Nload cell tension test, and 7.5% at 0.2 N for the 1,000 N loadcell tension test.

The monolithic load cell mean tests results had the fol-lowing maximum deviations from the theory: +91% at 0.38N for the 100 N load cell compression test, +46% at 18 N

for the 1,000 N load cell compression test, +45% at 0.01 Nfor the 100 N load cell tension test, and +18% at 7 N for the1,000 N load cell tension test. The monolithic load cell testhad the following maximum standard deviations among thefive experimental trials: 396% at 0.01 N for the 100 N loadcell compression test, 13.3% at 13.3 N for the 1,000 N loadcell compression test, 37% at 0.01 N for the 100 N load celltension test, and 14% at 0.085 N for the 1,000 N load celltension test.

The errors at small forces may reflect the 100 N Inter-face load cell resolution limitations and zero-deflection un-certainty. The errors at large forces may reflect how dis-placement uncertainties lead to large force errors when theforce-deflection gradient (stiffness) is large. Furthermore, asillustrated in Section 4, small fabrication errors in the sur-face shape may result in large force-versus-deflection errors.Fabricating the monolithic load cell root inserts too large

may have caused the load cell to be stiffer than predictedfor forces less than 13 N, which is when the ring contacts theroot inserts.

While we note the discrepancies above between the the-ory and experiment force versus deflection relationships, theexperimental stiffness versus force plots in Fig.s 4b and dverify that both load cells have nonlinear stiffness curve thatsatisfies the 1% force measurement resolution from 0.01-1,000 N. These experimental stiffness curves were deter-mined by taking the gradient of the data moving averageforce and displacement in 0.05 mm segments.

Finally, in [5], we show that load cells using the com-bined flexible-element and rigid surface have minimal hys-teresis.

6 ConclusionsWe showed how a load cell with increasing stiffness may

be designed with a larger force measurement resolution andforce range than a traditional linear load cell. We physicallyimplemented a stiffening load cell by designing flexible ringsthat increasingly contact rigid surfaces as additional force isapplied. As the ring contacts the rigid surfaces, two mecha-nisms siffen the load cell: a shortening of the free ring seg-ment’s length and an increase in the free ring segment’s av-erage thickness (for the ring with a tapered thickness).

We investigated parameters that allow the nonlinear loadcell to measure forces with resolutions of 1% of the appliedforce over a 5-orders-of-magnitude force range, 0.01-1,000N. High resolution was achieved by designing the nonlinearload cell’s stiffness to remain below values that allow a sen-sor with 0.1 µm resolution to detect changes larger than 1%the applied force.

We described methods for fabricating the load cell frommultiple parts or from a monolithic part. The advantage ofthe waterjetted monolithic load cell is that its variable thick-ness ring allows for highly variable stiffness without largering stresses. The advantage of the multi-part load cell isthat if the ring breaks, it may be replaced without replacingthe entire load cell.

We experimentally verified the nonlinear load cell the-ory and showed the effectiveness of the fabrication methodfor the two different load cell designs.

The nonlinear load cell currently has several disadvan-tages compared to traditional linear load cells. If the ringthickness is large, then a large ring radius is required so thatthe initial load cell stiffness is low enough for ±0.0001 Nforce resolution when 0.01 N is applied. The thin ring width,b, makes the load cell susceptible to out-of-plane parasiticmotions. We found that the monolithic load cell had verylittle resistance to out-of-plane bending while the multi-partload cell had more resistance due to the multi-part load cell’slarger ring width. The load cell requires a high-cost absolutelinear encoder (on the order of $700) rather than a standardstrain gauge (on the order of $30) because the absolute de-flection of the nonlinear load cell must be known in order toknow the applied force.

Future work on this project will address these disadvan-tages and make other improvements: making the load cellout of a thinner spring steel beam will allow a smaller loadcell. Using surfaces with curves of changing radii (such asellipses) could reduce stress in the ring. Furthermore, wewill consider using springs shaped with roll annealed steelto eliminate prestress in the spring steel ring. Manufacturingthe load cell from two or more well-spaced or perpendicu-lar planes rather than a single plane may reduce sensitivityto out-of-plane parasitic loads. We will add a linear encoderand test the functioning of the complete load cell. While thispaper focused on a 1% force resolution over 0.01-1,000 N,the load cell design constraints may be adjusted for differ-ent ranges, resolutions, and sizes. Finally, we note that thedesign concepts described in this paper may apply to manyother applications besides load cells.

AcknowledgementsWe are grateful to Peter Liu for his assistance fabricat-

ing the load cells and to Wesley Cox for his assistance fab-ricating the load cells and performing experiments. The Mi-croMAX, named as the 2016 R&D 100 Finalist, was devel-oped at OMAX under the suport of an NSF SBIT Phase IIgrant. We also gratefully acknowledge the National ScienceFoundation for support of JMK through the Graduate Re-search Fellowship Program under Grant No. 112237, and theMIT Energy Initiative through the project ’Efficient nonlin-ear energy harvesting from broad-band vibrational sourcesby mimicking turbulent energy transfer mechanisms’. Theresearch results in this manuscript are protected under patentUS 9382960.

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