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Theses and Dissertations

2004-11-19

Compliant Mechanisms to Perform Bearing and Spring Function Compliant Mechanisms to Perform Bearing and Spring Function

in High Precision Applications in High Precision Applications

Jesse R. Cannon Brigham Young University - Provo

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BYU ScholarsArchive Citation BYU ScholarsArchive Citation Cannon, Jesse R., "Compliant Mechanisms to Perform Bearing and Spring Function in High Precision Applications" (2004). Theses and Dissertations. 229. https://scholarsarchive.byu.edu/etd/229

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1CHAPTER 1 INTRODUCTION

1.1 BACKROUND

Bearings and springs are traditional mechanical elements commonly used in

machine design. Springs are used to provide energy storage or reaction to a force or

torque. Bearings use either sliding contact or rolling elements to constrain lateral

translation while allowing rotation between two rigid parts. Sliding contact bearings such

as bushings are composed of as few as two parts, making them simple to manufacture and

assemble. There can be a good deal of energy loss, however, due to friction between the

sliding surfaces. Rolling-element bearings have much lower friction, yet typically have

from 10 to 30 parts, increasing manufacturing costs and assembly time. Both types of

bearings usually need lubrication to reduce friction and remove heat. They also suffer

from backlash [1].

An increasing number of mechanical systems are being designed on the micro and

meso scales. Assembly and maintenance become increasingly difficult as the size of

mechanisms decrease, and the minimum size of bearings and springs is limited. The

backlash of bearings also limits their usefulness in applications where high precision and

1

repeatability are needed. At small scales and for high precision applications, alternative,

non-traditional elements are needed.

1.2 RESEARCH MOTIVATION

An example that demonstrates these needs is Sandia National Laboratory’s

Stronglink assembly, shown in Figure 1.1. The Stronglink is a high precision safety

and arming device used to provide mechanical isolation between electrical inputs

and a desired output, preventing unauthorized or unintentional detonation of a

weapon [2]. The Stronglink is a meso-scale device. The entire assembly is 1.186

inches in diameter, and its components are measured in fractions of an inch.

Future Stronglink designs will be done at even smaller scales.

Some of the assembly components include two solenoid actuated rotors (the drive

and gate rotor), and two pawls (the drive pawl and gate pawl). Both the rotors and the

Figure 1.1: Drive components in the Stronglink assembly.

2

pawls have helical extension springs to provide a restoring torque, and the rotors use

rolling-element bearings to constrain lateral translation. Functional specifications for the

rotors and pawls are given in Table 1.1.

Due to the nature of the Stronglink’s function as a safety and arming device,

every component in the assembly must have very high reliability. Because of the

assembly methods required to install them, the helical springs create a major reliability

concern. The springs have wire diameters ranging from 50 to 100 micrometers. The

helical coils are wound using semi-automated production equipment. The hooks at the

end of the springs, however, are formed by hand with tweezers and pliers, and the springs

are manually assembled into the Stronglink. These manual operations can nick or flatten

the wire, creating areas of high stress that may lead to failure [2].

The bearings used in the Stronglink are the smallest commercially available

miniature bearings with an outer diameter of 0.1 inches. As the size of the Stronglink

decreases in future designs, the size of the bearings are not able to decrease with it.

Table 1.1: Functional specifications for the (a) pawls and (b) rotors.

(a)

Rotation Min. Torque Max. Torque Drive Pawl 17 degrees 0.00026 in-lb 0.00052 in-lb Gate Pawl 9 degrees 0.0003 in-lb 0.0006 in-lb

(b)

Lateral Load Max. Lateral Displacement

Rotation Min. Torque Max. Torque

Drive Rotor 0.8 lb 0.001 in 8 degrees 0.0136 in-lb 0.0272 in-lb Gate Rotor 0.02 lb 0.001 in 15 degrees 0.00103 in-lb 0.00205 in-lb

3

Alternative solutions—components that keep the same basic functions of a bearing and a

spring, while avoiding the inherent disadvantages of each—are needed.

This research will use the Sandia Stronglink as an example and case study for the

methods and components developed in this thesis.

1.3 THESIS OBJECTIVES

The objective of this thesis is to develop reliable and scalable compliant

components to replace bearings and helical springs. Components replacing springs must

be able to produce specified torque/motion requirements. Components replacing bearings

must permit sufficient motion about the axis of rotation, bear specified loads in the lateral

directions, and fit within roughly the same design space as a bearing. Additionally, all

components will be designed to be manufactured using planar fabrication processes.

Practical application of the components will be demonstrated by their use in Sandia

National Laboratory’s Stronglink assembly.

Compliant mechanisms are good candidates for fulfilling these objectives.

Compliant mechanisms gain at least some of their mobility from the deflection of flexible

members [3] instead of from the relative motion between separate, rigid parts. They can

often be made in one piece, reducing the amount of assembly required, and they avoid the

friction, wear, and backlash associated with movable joints. These characteristics help to

increase the mechanism’s reliability, and make them easy to miniaturize.

4

1.4 DESIGN PROCESS

The first step in the design process is concept generation. From the list of

potential solutions, each was evaluated based on its ability to meet the functional

requirements of bearings and springs. Five concepts were chosen for further study. These

concepts fall into three categories: mechanisms that replace 1) the helical spring, 2) the

bearing, and 3) both the helical spring and the bearing.

The general method for evaluating each concept begins by creating an analytical

model to predict the movement, stresses, and resulting torque of the mechanism. These

models are used to determine whether the mechanism allows sufficient motion, meets the

specified torque/motion requirements, is able to bear the specified lateral loads, and fits

within the allotted design space. Components used in the Stronglink case study will

undergo only a few cycles in their life, and therefore none of the analysis includes

fatigue.

The analytical models are then validated by creating a CAD model, FEA model,

and a macro-scale prototype. The CAD model verifies the geometry of the design, the

FEA model verifies the stress and torque predictions, and testing of the macro-scale

prototype validates the torque and motion predictions.

Building and testing of meso-scale prototypes of selected concepts will be

conducted by Sandia National Laboratories.

The remainder of this chapter will review the manufacturing processes to be used

in fabricating the designs, and will then provide a review of the literature pertaining to

this research.

5

1.5 MANUFACTURING PROCESSES

Designs in this thesis have been developed on three different scales. Prototypes of

concepts are done on the macro scale, designs for Sandia’s Stronglink mechanism are

done on the meso scale, and application of concepts on the micro scale have also been

investigated. Table 1.2 lists the processes, materials, and design parameters associated

with each of these scales.

1.5.1 Computer Numerical Control Milling

Computer Numerical Control (CNC) milling was used to create macro-scale

prototypes. A two dimensional drawing is exported from a CAD model as a DXF file,

and imported into a Computer-Aided-Manufacturing (CAM) program. The CAM

program uses the drawing to create tool paths, and allows the user to specify

manufacturing parameters. This information is exported from the CAM program as an

NC file, which is imported into the CNC program. The CNC program controls the mill

and creates the part.

Table 1.2: Manufacturing processes and parameters.

Scale Macro Meso Micro Process CNC Milling Micro Wire

EDM SUMMiT V

Material Polypropylene Titanium 6Al 4V Polycrystalline Silicon

E [Mpsi (GPa)] 0.2 (1.4) 16.5 (114) 24.5 (169) Sy [kpsi (MPa)] 5 (34) 120 (825) 261 (1800) Min. Feature Thickness [in (mm)] - 0.002 (0.0508) 0.000039 (0.001) Kerf Width [in (mm)] 0.0938 (2.381) 0.003 (0.0762) 0.000039 (0.001)

6

1.5.2 Micro Wire EDM

Micro wire electrical discharge machining (EDM) will be used by Sandia

National Laboratories to create parts for the Stronglink mechanism. In the micro wire

EDM process, a voltage difference is created between the workpiece and a wire thread.

The wire is brought up to the workpiece and an arc develops. This arc removes material,

cutting a path as the wire moves through the material. The workpiece material must be

conductive; titanium was used for the designs for Sandia. Three-dimensional machining

is possible though more difficult, and planar geometries are preferred [2].

1.5.3 SUMMiT V Fabrication

Microelectromechanical systems (MEMS) integrate mechanical and electrical

components on the micro scale [3]. The Sandia Ultra-planar, Multi-level MEMS

Technology 5 (SUMMiT V) fabrication process is a surface micromachining process

developed by Sandia, and is used to manufacture MEMS devices. It uses processes and

technologies similar to those developed for integrated circuit fabrication. Multiple layers

are deposited and then patterned using planar lithography. Five planar layers of

polycrystalline silicon are available for design. The first layer is grounded to the

substrate. The remaining layers can contain independent planar mechanisms, or can be

connected to form more complex three-dimensional geometries. SUMMiT V fabrication

has the potential of obtaining high volume, low cost production [4].

7

1.5.4 Design Constraints

The minimum feature thickness and kerf width parameters constrain the geometry

of a part. The thickness of a flexure, for example, cannot be less than the minimum

feature thickness. Kerf width constrains the allowable distance between features of a part.

Spaces between features must be greater than or equal to the kerf width. For CNC

milling, the kerf width is equal to the diameter of the end mill. For micro wire EDM it is

equal to the diameter of the wire plus an offset from that diameter, and can vary from

0.0024 up to 0.0036 inches, depending on the thickness of the material being cut. For a

0.114 inch thick part (the thickness of the Stronglink’s rotors) the wire kerf is estimated

to be 0.003 inches. The kerf width for the SUMMiT V process is not associated with a

tool dimension, but is the minimum distance needed to ensure that separate features do

not fuse together.

The minimum inside radius of a part is also constrained by the kerf width, and

must be greater than or equal to one half the kerf width.

1.6 REVIEW OF LITERATURE

1.6.1 COMPLIANT REVOLUTE JOINTS

Compliant components that can replace bearings fall into the class of mechanisms

known as compliant revolute joints. Moon, Trease, and Kota [5] have suggested five

criteria be used in comparing different types of compliant revolute joints. They are (1)

range of motion, (2) amount of axis drift, (3) ratio of off-axis stiffness to stiffness about

8

the axis of rotation, (4) stress concentration effects, and (5) compactness of the joint.

They provide a qualitative comparison of some of the existing compliant revolute joints,

given in Table 1.3.

Table 1.3: Benchmarked flexible revolute joints. (Table taken from [5].)

9

Notch-type joints, as shown in Table 1.3(a), were first described by Paros and

Weisbord [6]. Table 1.3(b) [7], 1.3(c) [8], 1.3(d), 1.3(e) [9], 1.3(f) [10], and 1.3(g) [7] all

show methods that use leaf springs to create a revolute joint. Table 1.3(h) and 1.3(i) [9]

use circular leaf springs. The split-tube flexure in Table 1.3(j), introduced by Goldfarb

and Speich [11], and the cross-type joint in Table 1.3(k), introduced by Moon, Trease,

and Kota [5], gain their motion from deflection of elements in torsion.

To replace a bearing, a compliant component should permit a large range of

motion about the axis of rotation, have a high stiffness in the lateral directions, and be

compact enough to fit within the same design space as a bearing. These objectives

correspond to the criteria 1, 4, and 5 of Table 1.3. The only joints that are at least

‘normal’ or above on all three of these criteria are the two torsional joints, in Tables 1.3(j)

and 1.3(k), and none of them are ‘good’ in all three. One of the contributions of this

thesis will be to develop new types of compliant revolute joints that perform well in at

least these three, if not all five, criteria.

1.6.2 THE PSEUDO-RIGID-BODY MODEL

Howell [3] presents a method for analyzing systems that undergo large, nonlinear

deflections. The pseudo-rigid-body model (PRBM) treats compliant segments, such as

the beam in Figure 1.4(a), as if it were two rigid links pinned together, with a torsional

spring at the pinned joint, as shown in Figure 1.4(b). This approximation can give very

accurate results, predicting the position of the end of the beam with an error of less than

0.5%.

10

(a)

(b)

Figure 1.2: (a) Cantilevered segment with forces at the free end, and (b) its pseudo-rigid-body model equivalent.

The PRBM allows a compliant mechanism to be analyzed as if it was a rigid-link

mechanism. Traditional rigid-link mechanism theory can be used to determine the motion

and force-deflection relationship of a compliant mechanism. More details on the PRBM

can be found in [3].

11

12

2CHAPTER 2 THE SERPENTINE FLEXURE

The remainder of this thesis will discuss each of the five concepts that were

selected for further development. In this chapter a scalable, planar, and space efficient

spring concept is introduced. The serpentine flexure gets its name from its winding,

snake-like pattern [12], and is an alternative to traditional spring elements, such as helical

springs or straight flexures. The design, application, and analysis of the serpentine flexure

are investigated, and its performance in the Stronglink case study is discussed.

2.1 DESIGN CONCEPT

The serpentine flexure is a compliant member that can replace helical springs or

straight compliant flexures, and can be made at small scales using planar fabrication

techniques. The serpentine flexure uses the same basic concept as a cantilever beam

(Figure 2.1(b)), but utilizes the available design space more efficiently by having the

beam weave back and forth between the fixed and free ends, as shown in Figure 2.1(b).

This effectively increases the beam’s length, decreasing the stiffness and maximum

stress, and increasing the maximum possible deflection.

13

(a) (b)

Figure 2.1: (a) A cantilever beam, and (b) a serpentine flexure.

2.2 CASE STUDY

Currently, the Stronglink’s drive and gate pawls use helical springs to provide a

restoring torque. Preliminary concepts for replacing these springs used straight cantilever

beams attached to the pawls to provide the necessary torque. Due to space constraints

limiting the length and manufacturing constraints limiting the minimum thickness of the

beams, the designs were not flexible enough to meet the required torques and had safety

factors very close to 1. The serpentine flexure overcomes these problems. The serpentine

flexure uses the same design space as a cantilever beam, but its larger effective length

increases the flexibility and the safety factor.

The serpentine flexure can be integrated with other parts, greatly reducing

assembly requirements. In the Stronglink application, one end of the flexure is attached to

the pawl, and the flexure’s free end is constrained by contact with a pin. Manufacturing

processes and parameters for the macro-scale prototypes and for Sandia’s meso-scale

applications are given in Table 1.2. Functional specifications for the Stronglink pawls are

listed in Table 1.1(a). In addition to these requirements, the pawl/flexure part must be

balanced about its center of rotation, should not have self contact or contact with parts

14

other than the pin constraining its end, and the drive pawl must be able to be manually

disengaged.

2.3 ANALYSIS

2.3.1 Drive Pawl Application

To determine the minimum and maximum torque positions of the drive pawl

serpentine flexure, the angle between the flexure’s undeflected and deflected states was

measured at four key positions of the pawl, as shown in Figure 2.2. These measurements

Figure 2.2: Angular deflection at four key positions.

15

show that the minimum angular deflection (θ), and therefore the minimum torque, occur

at Position 2, and the maximum deflection and torque occur at Position 4.

A finite element analysis (FEA) model of the flexure was created and the

geometric parameters manipulated to achieve the desired torque characteristics. The FEA

mesh included 23,400 nodes, used the PLANE183 element, a higher order 2-D element,

and used a large displacement static analysis. The free end of the flexure was constrained

to allow translation only in the direction tangent to pin at the point of contact, and the

fixed end was constrained to allow only rotation about the center of rotation of the pawl.

A force was applied to create a torque, and the resulting deflection and stress results were

obtained. The deflection was analyzed at each of the key positions by

superimposing the FEA deflection results over the CAD model to ensure that the

flexure did not interfere with other parts. Next a scaled plastic prototype was constructed

and used to validate the FEA results. Figure 2.3 shows that the FEA and prototype

results are very similar, and that both have no interferences.

2.3.2 Gate Pawl Application

Whereas the drive rotor experiences both translation and rotation, the gate pawl

only rotates. This creates a simpler analysis, as only two positions need to be considered.

A FEA model was used to analyze the torque characteristics of the flexure, using the

same procedure used for the drive pawl FEA analysis. The FEA deflection results were

then superimposed over the CAD model and compared to the scaled prototype

deflections.

16

(a) (b)

Figure 2.3: Drive pawl (a) FEA results and (b) prototype results.

Figures 2.4 and 2.5 show that the FEA and prototype results are very similar, and

that there is no interference between parts.

17

(a) (b)

Figure 2.4: Gate pawl (a) FEA results and (b) prototype results for the minimum deflection positions.

(a) (b)

Figure 2.5: Gate pawl (a) FEA results and (b) prototype results for the maximum deflection positions.

2.4 RESULTS

The FEA results predict that the drive pawl serpentine flexure’s torque is 0.00026

in-lb at minimum deflection (14.5 degrees of rotation), and 0.00056 in-lb at maximum

deflection (32.5 degrees of rotation). The gate pawl serpentine flexure’s torque is

18

predicted to be 0.0003 in-lb at the minimum deflection (8 degrees of rotation), and

0.0006 in-lb at maximum deflection (17 degrees of rotation).

2.5 CONCLUSIONS

2.5.1 Drive Pawl Application

Figure 2.6 gives the final geometry of the drive pawl serpentine flexure. Since the

flexure does not go the full depth of the pawl, the pawl could be manually disengaged by

Figure 2.6: Drive pawl serpentine flexure.

19

inserting a pin underneath the flexure to push on the pawl. The design meets the torque

specifications, does not interfere with other parts during the range of its motion, is

balanced about its center of rotation, and has a safety factor against yielding of 3.0.

2.5.2 Gate Pawl Application

Figure 2.7 gives the final geometry of the gate pawl serpentine flexure. The

flexure has the same depth as the pawl, 0.035 inches. The design meets the torque

specifications, does not interfere with other parts during the range of its motion, is

balanced about its center of rotation, and has a safety factor against yielding of 4.6.

Figure 2.7: Serpentine flexure geometry

20

3CHAPTER 3 THE COMPLIANT ROLLING-

CONTACT ELEMENT

3.1 DESIGN CONCEPT

Mankame and Ananthasuresh [13] have shown that contact-aided compliant

mechanisms—compliant mechanisms that use contact interactions with external bodies or

with different parts of the mechanism itself—can have enhanced functionality when

compared with traditional compliant mechanisms. The flexible members used in

traditional compliant mechanisms are typically constrained by boundary conditions at

their ends. Examples include a cantilever beam with a force at the free end, a beam fixed

at one end and guided at the other, or a beam that is pinned at both ends [3]. Beams that

contact other elements along their length have the potential of creating favorable

behaviors that might not be achieved by beams constrained only at their ends.

Contact interactions can be used to improve the joint’s ability to bear lateral loads.

Guérinot [14] has demonstrated that the contact of rigid elements in a joint can divert a

compressive load away from buckling-prone flexible segments. Contact interactions can

also increase the maximum possible deflection of a compliant beam [15].

21

Figure 3.1: Rolling-contact joints used in a robot finger. (Figure taken from [16])

A potential application of these principles is the rolling-contact joint. Rolling-contact

joints consist of two rigid links that are constrained to roll without slipping over each

other. Figure 3.1 shows an example of rolling-contact joints used in a robot finger [16].

Designs such as these use gear teeth to impose a no-slip condition, and additional

linkages to hold the gear teeth in contact with each other. Chironis and Sclater [17] give

some examples of compliant rolling-contact joints, shown in Figure 3.2. These impose

no-slip contact by connecting the two rigid links using flexible strips. The flexures

contact the surface profiles of the link along their length, and are attached to the links at

the flexure ends. The multiple layers of the flexures alternate directions. The

configuration shown in Figure 3.2(a) is used to replace rack and pinion gears in hard

drives to obtain high precision movements. The configuration shown in Figure 3.2(b)

uses circular surface profiles for both links.

(a) (b)

Figure 3.2: Compliant rolling-contact joint with links having (a) a circular profile and a straight profile, and (b) two circular profiles. (Figures taken from [17])

22

Figure 3.3: The CORE.

In the configurations shown in Figure 3.2, the two links and the flexures are all

separate parts that must be assembled. The compliant rolling-contact element (CORE),

shown in Figure 3.3, is a variation on these designs that reduces the assembly

requirements by making each layer of the CORE monolithic, with the flexures and links

manufactured as one part. A similar monolithic design has been developed independently

by Jeanneau, Herder, Laliberte, and Gosselin [18].

3.2 MANUFACTURING AND ASSEMBLY

On the macro and meso scales, the components of the CORE are manufactured in

the configuration shown in Figure 3.4(a), using the manufacturing processes and

parameters listed in Table 1.2. Straight cuts adjacent to the ends of the flexure remove a

small section of the contact surface, but are needed in order to form the flexure. The

width of these cuts is slightly greater than the kerf width. Three of these components are

stacked together, as shown in Figure 3.4(b), and the lower surfaces are fixed together.

23

(a) (b)

Figure 3.4: (a) Manufactured configuration of a single CORE component, and (b) three components stacked together.

Deflection of the flexures onto the surfaces allows the upper surfaces to align and be

fixed together in the operational configuration shown in Figure 3.3.

On the micro scale, assembly becomes very difficult. Fortunately, the kerf width

for surface micromachining is very small. The CORE is able to be manufactured in the

operational configuration, leaving a 1 µm gap between the surfaces and the flexures, as

shown in Figure 3.5.

Figure 3.5: Micro-scale CORE design.

24

3.3 ANALYSIS

3.3.1 The Bernoulli-Euler Equation

Contact of the beams with the surfaces creates well defined moments in the beams

due to the relationship between the radius of curvature of a beam and the moment. This

relationship is derived from the Bernoulli-Euler equation, which states

dsdEIM θ

= (3.1)

where E is the modulus of elasticity, and I is the moment of inertia of the cross section.

dsdθ is the curvature, defined as the rate of change of the angle of the beam, θ , per unit of

arc length, s. Defined in this way, the Bernoulli-Euler equation is valid for large

deflections. Assumptions made when using equation 3.1 are that 1) the material is

linearly elastic, homogeneous, and isotropic, 2) the transverse-shear component of

deflection is small compared to that due to bending [1], and 3) the beams are thin, so that

the neutral axis can be assumed to be coincident with the centroidal axis even for initially

curved beams [3]. The effective radius of curvature of the beam is equal to

dsdR

/1'

θ= (3.2)

Substituting this into equation (3.1) gives

'REIM = (3.3)

25

(a) (b)

Figure 3.6: (a) Beam constrained by contact surface and (b) its equivalent beam with an applied moment. (Adapted from [3])

When a beam is in contact with a surface that has a constant radius of curvature,

the radius of curvature of the beam is also constrained to be constant, as in Figure 3.6(a).

Equation (3.3) predicts that this constant radius of curvature throughout the beam will

create a moment that is also constant along the beam length. This means the segment

has the same behavior as a beam with a moment applied to the free end, as shown in

Figure 3.6(b).

Some beams may have an initial curvature in their undeflected state. The effective

radius of curvature in equation (3.3) takes into account any initial curvature of the beam

using the equation

1

0

11'−

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

RRR

s

(3.4)

where Rs is the radius of curvature of the surface constraining the beam’s shape, R0 is the

beam’s initial radius of curvature, and the beam’s thickness is assumed to be small

compared to Rs.

26

Counter-Clockwise Path = Positive Radius of Curvature

Clockwise Path = Negative Radius of Curvature

Figure 3.7: Sign conventions for radii of curvature applied to CORE flexures.

3.3.2 Sign Convention

Equation (3.2) predicts that the radius of curvature is proportional to the rate of

change of the angle of the beam per unit of arc length. If the angle of the beam is

increasing (along the path of the beam starting at the grounded end) then the radius of

curvature will be positive. The path of the beam will be counter-clockwise. If the angle of

the beam is decreasing then the radius of curvature will be negative, and the path will be

clockwise, as shown in Figure 3.7.

3.3.3 Torque

Equation (3.3) can be used to find the torque acting on the CORE. If the lower

surface is grounded, then the torque on a single layer of the upper surface is equal to the

sum of the moments in the upper and the lower segments of the flexure, as shown in

Figure 3.8. This can be simplified to

27

(a) (b) (c)

Figure 3.8: (a) Moment in the upper segment of the flexure, (b) moment in the lower segment of the flexure, and (c) torque on the upper surface.

⎟⎟⎠

⎞⎜⎜⎝

⎛+= ''

11

LU RREIT (3.5)

where T is the torque, is the effective radius of curvature of the upper segment of the

flexure, and is the effective radius of curvature of the lower segment of the flexure.

The total torque is then found by summing the torques acting on each layer of the CORE.

'UR

'LR

The CORE flexures for the macro and meso scale designs are initially straight

( 0R = ∞) therefore, by equation (3.4), R` is equal to Rs. The signs are different for the

radii of the upper and the lower surfaces, so

''LU RR −= (3.6)

Substituting equation (3.6) into equation (3.5) and simplifying gives

0=T (3.7)

The micro scale configuration has an initial radius of curvature equal to .

Substituting this into equation (3.4) yields

sR−

2

' sRR = (3.8)

28

for a deflected section of the flexure. The deflected section can be on the upper or lower

segment, depending on the direction that the upper surface is displaced. The opposite

segment is undeflected ( 'R = ∞), and the inverse of the effective radius of curvature will

be 0. Substituting equation (3.8) into equation (3.5) gives

sREIT 2

= (3.9)

Equation (3.9) predicts a constant torque throughout the rotation of the CORE.

3.3.4 Stress

The maximum stress in a rectangular beam in bending is equal to

I

Mh2max =σ (3.10)

where h is the thickness of the beam. Because the moment is constant along the length of

a beam in contact with a constant radius surface, equation (3.10) predicts that the stress

will also be constant along its length. This constant stress is given by substituting

equation (3.3) into equation (3.10):

'2max REh

=σ (3.11)

Thus the maximum stress in the CORE depends only on the modulus of elasticity

of the chosen material, the thickness of the flexure, and the effective radius of curvature.

29

Figure 3.9: Geometric parameters defining a point P on the CORE.

3.3.5 Position Analysis

The motion of given point P on the upper surface of the CORE is defined by a

cycloid, similar to the path of a point on a wheel as it rolls without slipping along the

ground. The x and y coordinates of point P are given by the following equations

902 12 −= θθ (3.12)

221 sincoscos2 θθθ baRx sP ++= (3.13)

221 cossinsin2 θθθ baRy sP −+= (3.14)

where 2θ , 3θ , a, and b are defined in Figure 3.9.

3.4 RESULTS

Figure 3.10 shows the cycloidal path followed by an arbitrary point P on the

CORE. Though this path is not circular, it can be seen that, for a finite range of motion, a

30

0

1

2

3

4

-5 -4 -3 -2 -1 0 1 2 3

xy

Figure 3.10: Motion path of a point on the CORE.

section of the curve can closely approximate a circular path. The CORE could be used as

a revolute joint, but would experience some axis drift. The magnitude of this axis drift

would depend on the operational range of motion of the CORE.

A macro-scale prototype of the CORE was made, and performed as expected. A

micro-scale design has been submitted for fabrication using the SUMMiT V process.

3.5 CONCLUSIONS

The CORE bearing can be made using planar fabrication methods. It can bear

high loads with low friction, making it an ideal candidate for replacing a bearing. It does

experience axis drift, though for a limited range of motion this drift can be very small.

The macro and meso scale configurations analyzed in this chapter have no reaction

torque. The initial radius of curvature in the CORE flexures could be adjusted to create a

torque that is constant throughout the range of the CORE’s motion, as was done in the

micro scale configuration.

31

32

4CHAPTER 4 THE CORE BEARING

The previous chapter introduced the compliant rolling-contact element (CORE),

an element that allows large deflections and has cycloidal motion. Creatively combining

the CORE with other elements allows it to be used in mechanisms that obtain purely

rotational motion. This chapter introduces the CORE bearing as a potential replacement

for a traditional bearing.

4.1 DESIGN CONCEPT

Rigid roller elements can be combined with flexible beam elements to obtain pure

rotation. One example, shown in Figure 4.1, is a rotary bearing developed by Sandia

Figure 4.1: A contact aided rotary bearing. (Figure taken from [19])

33

National Laboratories [19]. Lateral loads are transmitted through the roller elements in

the bearing, while a single flexible band is used to hold the elements in place. The CORE

bearing design uses similar principles to obtain rotation. By using the CORE flexures as

its flexible beam elements, however, it is able to be manufactured as one single part.

The CORE bearing imitates a planetary gear system with three planets, a sun, and

a ring, as shown in Figure 4.2. CORE flexures replace the gear teeth, providing a no-slip

condition. These flexures also create a restoring torque. Flexure 1 refers to the flexures

connecting the sun and the planets, and Flexure 2 refers to the flexures connecting the

planets and the ring. The sun is able to rotate relative to the ring without any sliding

between surfaces, minimizing frictional energy losses. Lateral loads applied to the ring

are transmitted through the planets to the sun, constraining lateral motion.

Sun

Planet

Ring

Flexure 1

Flexure 2

PlanetPlanet

h1

Rp

Rs

Rp

RpRrh2

b = out-of-plane thickness

(a) (b)

Figure 4.2: (a) CORE design, and (b) design parameters.

34

4.2 MANUFACTURING AND ASSEMBLY

The CORE is manufactured in one piece. Figure 4.3 shows two possible

configurations. In the configuration shown in Figure 4.3 (a), Flexure 1 is initially straight,

and Flexure 2 has an initial curvature equal to that of the ring. The ring is sectioned into

three parts. This configuration requires that the flexures be wound onto the sun and

planets, and then that the ring section be fixed together. This could be done either by

placing the assembled CORE into its mating hole, by having a separate constraining ring

that fits around it, or by including latches on the ring sections that lock the sections

together.

Figure 4.3(b) shows a configuration that does not require any assembly. This is

configuration is more practical for meso or micro scale applications. In the manufactured

position, there are spaces between the flexures and each of the parts that they contact.

These spaces are at least as wide as the kerf width. The profiles of the sun and ring are

(a) (b)

Figure 4.3: (a) A manufactured position that requires assembly and (b) a manufactured position with no assembly required.

35

Figure 4.4: Drawing of micro-scale CORE bearing ( mOD µ1000= ).

designed so that once the bearing is rotated past the manufactured position the flexures

come into contact with the circular sections of the ring, planets, and sun, eliminating

spaces between the parts. Figure 4.4 shows the drawing of a micro-scale design using this

configuration.

4.3 ANALYSIS

4.3.1 Effective Radii of Curvature

The effective radii of Flexures 1 and 2 when in contact with the planet are found

using equation (3.4):

1

0

'1

1

11−

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

RRR

pp (4.1)

1

0

'2

2

11−

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

RRR

pp (4.2)

36

where and are the initial radii of Flexures 1 and 2, respectively. The effective

radius of Flexure 1 when in contact with the sun is

10R20R

1

0

'1

1

11−

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

RRR

ss (4.3)

and the effective radius of Flexure 2 when in contact with the ring is

1

0

'2

2

11−

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

RRR

rr (4.4)

For the configuration of Figure 4.2(a), equations (4.1), (4.2), (4.3), and (4.4) simplify to

pp RR ='1 (4.5)

1

'2

11−

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

rpp RR

R (4.6)

ss RR ='1 (4.7)

∞=rR '2 (4.8)

and for the configuration of Figure 4.2(b) these equations simplify to

∞=pR '1 (4.9)

∞=pR '2 (4.10)

1

'1

11−

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

pss RR

R (4.11)

1

'1

11−

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

rpr RR

R (4.12)

37

4.3.2 Torque on Planets

The resulting torque for the device is found by assuming that the ring is fixed and

solving for an output torque on the sun. First, the torque on the planet must be found. The

planet has two CORE joints; one connecting it to the ring, and one connecting it to the

sun. The torque on the planet is equal to the sum of the torque produced by each CORE

joint. These torques are found using equation (3.5), and their sum is equal to

⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟

⎟⎠

⎞⎜⎜⎝

⎛+= '

2'2

2'1

'1

11111

rpspp RR

EIRR

EIT (4.13)

where I1 and I2 are the moments of inertia of the flexures and are defined as

12

31

1bhI = (4.14)

12

32

2bhI = (4.15)

When determining the signs of the radii, the ends of the flexures attached to the

planet should be considered as the grounded ends.

4.3.3 Torque on Sun

Planetary gear theory provides the equation necessary to find the torque applied on the

sun by the planet when the ring is fixed:

)22()2(

psp

psspps RRR

RRRTT

+

+= (4.16)

where Tp is the torque on the planet, and Tps is the torque applied by the planet on the sun.

38

Finally, multiplying by np, the number of planets, gives the combined torque due

to all the planets. The total output torque, then is

pspout TnT = (4.17)

4.4 RESULTS

Macro-scale prototypes of both the assembly-required configuration of Figure

4.3(a) and the no-assembly configuration of Figure 4.3(b) were built to demonstrate the

CORE bearing concept, and performed as expected. Figure 4.5 shows the no-assembly

required prototype.

Both configurations were analyzed on the meso scale. For the given design space

and manufacturing constraints, the safety factor is less than one, and the CORE bearing

would fail. The safety factor can be raised by extending the design space and allowing a

larger outer diameter for the CORE bearing, or by using an alternative fabrication process

that allows a smaller flexure thickness than micro wire EDM.

Figure 4.5: Prototype of no-assembly CORE bearing.

39

LIGA (a German acronym for lithography, electroplating, and molding) is another

process available at Sandia National Laboratories. Analysis was done to determine if this

process would present a feasible alternative. The material used in LIGA is Nickel

manganese, with a Young’s modulus of 196 GPa and a yield strength of 800 MPa.

Design parameters and results for the designs are given in Tables 4.1 and 4.2,

respectively.

Reasonable safety factors are possible using LIGA. The torque is constant, and is

close to 0.0001 in-lb for both configurations. This is over two orders of magnitude below

the maximum required torque needed to replace the spring in the Stronglink components.

4.5 CONCLUSIONS

The CORE bearing minimizes frictional losses, allows up to 120 degrees of pure

rotation without any axis drift, has a high lateral stiffness, and meets all the requirements

for replacing a bearing. Designs for Sandia’s Stronglink cannot be made using micro wire

Table 4.1: Design parameters (in inches).

Design Configuration Rr Rp Rs h1 h2

Assembly Required 0.05 0.0167 0.0167 0.0001 0.00016 No Assembly Required 0.05 0.00167 0.0167 0.00005 0.00016

Table 4.2: Results

Design Configuration Torque (in-lb) Safety Factor

Assembly Required 0.00009 1.3 No Assembly Required 0.00011 1.3

40

EDM, but LIGA, a fabrication process available at Sandia, appears to be a feasible

alternative. The CORE bearings provide a constant torque. This torque is not high enough

to replace the springs in the Stronglink, but could be used as a spring in other

applications.

41

42

5CHAPTER 5 THE ELLIPTICAL CORE BEARING

5.1 DESIGN CONCEPT

The elliptical CORE bearing, shown in Figure 5.1, is a variation of the CORE

bearing. It is similar to a planetary gear and, like the CORE bearing, has CORE flexures

connect the sun to the planets, providing a no-slip condition between them. The

(a) (b)

Figure 5.1: (a) Elliptical CORE bearing and (b) the kinematic model of the bearing.

43

difference in the elliptical CORE bearing is that the sun and planet profiles are

elliptical instead of circular. This is done to reduce the stresses in the CORE flexures and

to create a more compact device that is better able to meet the given space constraints.

The elliptical CORE bearing uses the elliptical gearing concept. Two ellipses,

pinned at their foci as shown in Figure 5.2(a), are able to rotate a full 360 degrees while

maintaining a no-slip contact [17]. The distance between the two pinned foci remains

constant. If one of the ellipses is grounded, then as the non-grounded ellipse rolls on the

grounded ellipse, its foci will travel in a circular path, as shown in Figure 5.2(b).

The elliptical CORE bearing places one of the foci of the sun ellipse at the center of the

device (Point O in Figure 5.1(b)). One of the foci of the planet ellipse then revolves about

the device’s center. The kinematic model would include a link (the planet arm)

connecting the foci of the sun and planet ellipses, as shown in Figure 5.1(b). The contact

surface between the planet and ring is a circular profile. The center of the ring is at Point

O, and the center of the circular section of the planetary gear coincides with the focus of

the ellipse.

(a) (b)

Figure 5.2: (a) Elliptical gears pinned at foci and (b) with one of the gears grounded.

44

Figure 5.3: A passive joint between the planet and ring.

A no-slip condition can be obtained between the planets and ring by using gear

teeth, as shown in Figure 5.1(a), or they could be connected by CORE flexures. CORE

flexures at this interface could have high stresses due to the small radius of curvature of

the circular planet surface. Another variation of the design could use a passive joint, as

shown in Figure 5.3. With a passive joint, the condition between the planets and ring

would change from that of rolling without slipping to a pinned condition. This would

change the kinematics of the joint and introduce some friction, but would avoid high

stresses and provides a simpler design than the gear teeth configuration.

The CORE joints in the design can provide a torque when the device is rotated,

and the in-plane translation is constrained.

5.2 MANUFACTURING AND ASSEMBLY

To form the CORE joints, the elliptical CORE bearing is made in multiple layers.

Figure 5.4 shows one layer of the passive joint configuration in its manufactured position.

To assemble the bearing, the initially straight flexures are deflected onto the surfaces of

45

Figure 5.4: A single layer of the elliptical CORE bearing in its manufactured position.

the sun and planets, and the multiple layers are fixed together with the direction of the

flexures alternating between each layer. These are then placed inside of the ring.

5.3 ANALYSIS

5.3.1 Definition of an Ellipse

An ellipse is defined by the geometric parameters a and b, as shown in Figure

5.5(a). In the standard position, the major axis of an ellipse is along the x axis, and is

equal to 2a. The minor axis is along the y axis, and is equal to 2b. The foci lie on the

major axis of the ellipse, and are a distance a from the point (0, b). By definition, a point

lies on the ellipse if

aFPdPFd 2),(),( 21 =+ (5.1)

46

(a) (b)

Figure 5.5: (a) Defining the foci of an ellipse, and (b) defining a point on the ellipse. (Figures taken from [20])

where F1 and F2 represent the foci of the ellipse, as shown in Figure 5.5(b). From this

definition, the equation of an ellipse in standard position can be found [20] to be

21

2

2

1 ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

axby (5.2)

The parameter c is the distance from the origin to the foci, and is equal to

22 bac −= (5.3)

5.3.2 The Elliptical Gearing Concept

The elliptical gearing concept follows from the definition of an ellipse. Figure 5.6

shows two ellipses that contact each other at a point P, and are symmetric about an axis

passing through P and tangent to both ellipses. By symmetry

),(),( 11 BA FPdPFd = (5.4)

),(),( 22 BA FPdPFd = (5.5)

47

Figure 5.6: Elliptical gear concept.

According to equation (5.1), therefore

aFPdPFd BA 2),(),( 21 =+ (5.6)

aFPdPFd AB 2),(),( 21 =+ (5.7)

The distance between the opposite foci of two ellipses as they roll without slip, therefore,

is constant, and is equal to 2a. For the elliptical CORE bearing, therefore

aLp 2= (5.8)

where is the length of the planet arm in the kinematic model. pL

5.3.3 Position Analysis

The initial position of the ellipses is defined as the position when the point of

contact between the two ellipses is at (0, b). The angle of the planet arm, measured from

the horizontal, is

⎟⎠⎞

⎜⎝⎛= −

ab

p1sin

0θ (5.9)

48

For a point of contact with the coordinate (x, y), where x is given and y is defined by

equation (5.2), the angle of the planet arm from the horizontal is

⎟⎠⎞

⎜⎝⎛

−= −

xcy

p1tanθ (5.10)

for cx < . The angular displacement from the initial position, therefore, is given by

0ppp θθθ −=∆ (5.11)

If the elliptical CORE bearing uses a passive joint between the planets and the

ring, then the angular displacement of the planet arm is equal to the angular displacement

of the ring. Equation (5.11), therefore, defines the total rotation of the ring relative to the

sun.

5.3.4 Stress Analysis

The internal moment in the beam can be found using the Bernoulli-Euler

equation:

dsdEIM θ

= (5.12)

Substituting this into equation (3.10), the equation for the maximum stress in a

rectangular beam, results in

⎟⎠⎞

⎜⎝⎛=

dsdEh θσ

2 (5.13)

The curvature, dsdθ , for an ellipse is not constant. It can be found for a twice

differentiable curve, however, using the equation [20]

49

0

20

40

60

80

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

x

d υ/d

s

Figure 5.7: The curvature of an ellipse ( 03.0=a ) as a function of x.

[ ] 2/32)'(1

''

y

ydsd

+=

θ (5.14)

Substituting equation (5.2) into equation (5.14) and simplifying gives

2/32

2

22 11

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛−+

=

ax

aba

bdsdθ (5.15)

The curvature increases as x increases, as shown in Figure 5.7. The maximum stress in

the CORE flexures occurs when x is greatest. The maximum value of x occurs at xf,

the x coordinate where the flexures attach to the ellipse. Decreasing xf will decrease the

stress, but will also decrease the range of motion of the bearing.

5.3.5 Torque

In Chapter 3 it was shown that when the CORE flexures are manufactured

initially straight, the CORE joint doesn’t have a restoring torque. The initial curvature of

the flexures could be adjusted to create a restoring torque if a torque is desired.

50

5.3.6 Arc Length of an Ellipse

The length of the CORE flexures must be equal to the arc length of the elliptic

surface that they will contact. The general formula for the arc length of a differentiable

curve is [20]

∫ +=b

a

ys 2)'(1 (5.16)

The length of the elliptical CORE bearing flexures, therefore, is equal to

∫ +=fx

f yl0

2)'(12 (5.17)

Substituting equation (5.2) into equation (5.16) and simplifying gives

∫ −−−

=fx

f xaxabal

022

2222 )/1(2 (5.18)

This can be solved using Jacobi’s elliptic integral of the second kind [21]. Elliptic

integrals are a class of integrals used in solving many engineering problems, such as the

non-linear deflection of a cantilever beam [3]. These integrals were given their name

because they first arose while studying the problem of how to compute the arc length of

an ellipse [22]. For more details on elliptic integrals see [3, 21-23]. Equation (5.18) can

also be solved numerically.

5.4 STRONGLINK CASE STUDY

The parameters for the drive rotor design are listed in Table 5.1, and the results in

Table 5.2. t is the thickness of the CORE flexure, and is the safety factor. The gate fn

51

Table 5.1: Drive rotor elliptical CORE bearing design parameters.

a b fx t 0.03 0.004 0.014 0.002

Table 5.2: Drive rotor elliptical CORE bearing results.

fl fn θ∆ 0.028 1.14 8

rotor needs 15 degrees of rotation, almost twice as much as the drive rotor. Using the

micro wire EDM manufacturing constraints, the safety factor for this design would be

below 1. Using an alternative fabrication process such as LIGA would allow the CORE

flexure thicknesses to decrease, and greater rotation for the gate rotor design could be

obtained.

5.5 CONCLUSIONS

The elliptical CORE bearing’s design has been discussed, and a method of

analysis presented. The device is scalable, able to be made by planar fabrication

processes, can bear lateral loads, and has pure rotation without axis drift. It is a good

candidate for replacing traditional bearings.

The elliptical CORE bearing is much more compact than the CORE bearing. The

tradeoff is that it has less rotation. As the size of the elliptical CORE bearing decreases,

its maximum rotation decreases as well.

52

For the Stronglink application, the elliptical CORE bearing is able to obtain

sufficient rotation to replace the drive rotor bearing. It could also replace the gate rotor

bearing if it were made using an alternate fabrication process, such as LIGA.

53

54

6CHAPTER 6 THE COMPLIANT CONTACT-AIDED

REVOLUTE JOINT

The CORE design in Chapter 3 obtained increased deflection and lateral

stiffness by the use of contact interactions. In the CORE design, flexures

contacted a constant radius surface along their entire length. In this chapter a

design is introduced that uses contact interactions along only a partial section of

a cantilever beam. Figure 6.1 shows a concept presented by Chironis and Sclater

[17], where a cantilever beam is placed adjacent to a constant radius surface. As

the beam deflects, a portion of the beam comes into contact with the contact surface.

This type of contact interaction, as with that of the CORE design, can increase the

deflection and lateral stiffness of a joint. The application and analysis of this concept will

be investigated in this chapter.

Figure 6.1: A leaf spring contacting a curved surface. (Figure taken from [17])

55

6.1 DESIGN CONCEPT

The compliant contact-aided revolute (CCAR) joint, shown in Figure 6.2, consists of

multiple flexures. These flexures are fixed at one end (Point B), and free at the other

(Point A). The free ends of the flexures are constrained by the cam surface of the

gauge pin. The cam surface has a constant radius of curvature, Rc, with its center located

at the Point C. The contact surfaces also have a constant radius of curvature, Rs, and

interact with the flexures. Deflection of the flexures allows the CCAR joint to rotate, and

provides a restoring torque. As the flexures deflect they come into contact with the

contact surfaces. The contact surfaces constrain the flexures to have a constant radius of

curvature along the contacting segments of each flexure.

Contact Surfaces

Gauge Pin with Cam Surfaces

Flexures

b = depth into the page

(a) (b)

Figure 6.2: (a) CCAR joint design and (b) design parameters.

56

6.2 MANUFACTURING AND ASSEMBLY

For a beam that is either initially straight or has a constant initial curvature, and is

in contact with a surface of constant curvature, the following principle applies: if a

section of the contact surface is removed, the beam segment corresponding to that section

will retain the shape of the contact surface so long as a subsequent segment of the beam

is still in contact with the surface.

The proof of this principle is as follows: the radius of curvature in the contacting

segment is constrained to be constant, as shown in Figure 6.3(a). Equation (3.3) predicts

that this constant radius of curvature will create a moment that is also constant along the

segment length. For an initially straight beam ( = ∞), equation (3.4) predicts that R` is

equal to R

0R

s, therefore

sREIM = (6.1)

This moment is transferred to the end of the segment corresponding to the removed

contact surface, as shown in Figure 6.3(b). With an applied moment, the effective radius

(a) (b)

Figure 6.3: (a) Segment of beam in contact with surface, and (b) segment of beam corresponding to removed contact surface.

57

of curvature will be constant along the length of this segment of the beam. By rearranging

equation (3.3), the effective radius of curvature of the segment is found to be

MEIRb = (6.2)

Substituting equation (6.1) into equation (6.2) and simplifying gives

sb RR = (6.3)

The segment of the beam corresponding to the removed contact surface has the

same constant radius of curvature as the section that is in contact with the surface.

This principle is important because the kerf width for the micro wire EDM

process, as given in Table 1.2, makes the configuration shown in Figure 6.2 difficult to

manufacture. The spaces between the contact surfaces and the flexures are smaller than

the wire kerf.

Figure 6.4(a) shows and alternative configuration that meets all the manufacturing

constraints. All spaces between features are 0.0034 inches or larger, the smallest inside

(a) (b)

Figure 6.4: (a) Manufactured position and (b) assembled position of CCAR joint.

58

radius is 0.0017 inches, and the minimum thickness of any feature is 0.002 inches. Figure

6.4(b) shows the assembled position. The exterior flexures deflect, allowing the sections

to come together and be held in place by a latch.

Once assembled, the beams and contact surfaces in this configuration have the same

geometry as those of the configuration in Figure 6.2, except that sections of the contact

surfaces are removed, as shown in Figure 6.5. When the beam in Figure 6.5(a) is in

contact with the surface subsequent to the removed sections, it has the same behavior as

the beam shown in Figure 6.5(b), where the contact surface extends to the base of the

beam. The beams also have the same behavior at the beginning of rotation before either

beam makes initial contact with the surfaces. The only difference between the two

configurations is that the beam in Figure 6.5(b) will initiate contact with the surface

earlier in the rotation of the joint than the beam in Figure 6.5(a). The behavior will vary

slightly during the period.

The exterior beams connecting the sections of the CCAR joint are initially curved,

and then constrained to be straight after assembly. Equation (3.4) for these beams

simplifies to

extRRext 0

' = (6.4)

(a) (b)

Figure 6.5: (a) Section of joint with cuts, and (b) equivalent section without cuts.

59

where is the initial curvature of the exterior beams. Equation (3.11) becomes ext

R0

ext

ext REhext

0max 2

=σ (6.5)

where hext is the thickness of the exterior beam.

A latch holds the joint together once assembled. The beam on the latch deflects a

distance of yl during assembly. Using linear elastic beam equations, the stress in this

beam is found to be

2max 23

l

ll

LhEy

latch=σ (6.6)

where hl is the thickness of the latch beam, and Ll is its length. and , the safety

factors against yielding for the exterior beams and for the latch, respectively, can be close

to one because these beams only deflect during assembly, and then are fixed in place by

the mating hole in the rotor.

extfnlatchfn

The kerf width for the micro-scale design is small enough that the CCAR joint

can be manufactured in its operational configuration.

6.3 ANALYSIS

As the CCAR joint rotates, the flexures come into contact with the opposing

contact surfaces. Figure 6.6 represents a flexure in contact with a constant radius surface..

The length of beam in contact with the surface is measured by the variable sθ , which

increases with the rotation of the joint. This contact interaction will be developed by first

analyzing the contacting segment of the beam, and then analyzing the non-contacting

60

non-contacting segment

contacting segment

Figure 6.6: A beam in contact with a surface of constant radius.

6.3.1 Contacting Segment Analysis

The contacting segment’s radius of curvature is constrained by the contact

surface, and the moment in this segment of the beam is given by equation (6.1). Initially,

as the CCAR joint begins to rotate, the moment at the base of the beam will be less than

the moment in equation (6.1). In this condition, the beam acts as a traditional cantilever

beam, with sθ being equal to zero. Contact with the surface is initiated once the moment

at the base of the beam becomes equal to the moment in equation (6.1), and the length of

the contacting segment will increase with additional rotation after initial contact occurs.

This means the segment has the same behavior as a beam with a moment applied to the

free end, as shown in Figure 6.7.

The maximum stress for the beam is found in the contacting segment, and can be

found using equation (3.11).

61

Figure 6.7: Segment with curvature constrained by the contact surface (dashed lines) and an equivalent beam with an applied moment.

6.3.2 Non-Contacting Segment Analysis

The non-contacting segment can be treated as if it were rigidly attached at its base

to the contact surface, as shown in Figure 6.8, with the condition that the moment at the

base of the non-contacting segment equal the moment at the end of the contacting

segment, as defined by equation (6.1). The non-contacting segment can be modeled

Figure 6.8: Pseudo-rigid-body model of beam.

62

using the pseudo-rigid-body model (PRBM) [3]. The length of the non-contacting

segment of the beam is

Rll sfsf θθ −=)(' (6.7)

fl is the total length of the beam, including the contacting segment, and equals

2/hRRl ccrf −−+= δ (6.8)

where andrR cδ are geometric parameters defined in Figure 6.2(b).

The base of the non-contacting segment is at an angle of sθ from the vertical, and

the PRBM parameters φ and Θ are measured from this orientation. The position of the

base is given by

sb Rx θsin)(= (6.9)

)cos1)(( sb Ry θ−= (6.10)

where xb and yb are the x and y positions as measured from the base of the contacting

segment of the beam.

6.3.3 Kinematic Analysis

Using the pseudo-rigid-body model for the beam, the mechanism can be modeled

as a rigid-body four-bar mechanism, as shown in Figure 6.9. The cam surface of the

gauge pin can be modeled kinematically as if there were a link connecting Point A and

Point C. This is link 4. Link 3 is the PRBM link of the non-contacting segment of the

beam, and link 2 connects the PRBM link to the center of the mechanism. Both link 2 and

link 3 change lengths as sθ increases. Link 1 connects the center of the mechanism with

63

(a) (b)

Figure 6.9: (a) Non-contacting segment of beam and gauge pin, and (b) equivalent rigid-link four-bar mechanism.

the center of the cam surface arc. The following parameters are used in the derivation of

the link lengths and angles, as shown in Figure 6.10:

221 )( bbr yxR +−=δ (6.11)

Figure 6.10: Kinematic analysis parameters.

64

'2 )1( flγδ −= (6.12)

⎟⎟⎠

⎞⎜⎜⎝

⎛−

= −

br

b

xRy1tanα (6.13)

The link lengths are given by the following equations:

cr δ=1 (6.14)

)cos(2 2122

212 sr θαδδδδ +−+= (6.15)

)'(3 flr γ= (6.16)

2/4 hRr c −= (6.17)

The angle 2θ is related to the total rotation of the mechanism, tθ , by the equation

⎟⎟⎠

⎞⎜⎜⎝

⎛+−+= − )sin(sin

2

212 st r

θαδαθθ (6.18)

The angles 3θ and 4θ can be found using the closed-form equations for a four-bar

mechanism. The force angle and pseudo-rigid-body angle are calculated as

ts θθθφ −−= 4 (6.19)

1803 −−−=Θ ts θθθ (6.20)

and the torque experienced by the mechanism is

4sinθδ cb FnT = (6.21)

where nb is the number of beams in the mechanism, and F is the force acting on the beam

as predicted by the PRBM.

65

6.3.4 Sensitivity Analysis

The output torque of the CCAR joint is a function of the flexure thickness, h, and

the radius of the cam surface, Rc. The minimum tolerance of ± 0.0002 inches is assumed

to be a surface profile tolerance [24], so maximum change in the radius of the cam

surface, ∆Rc, is 0.0002 inches. The flexure thickness, however, involves two surfaces;

therefore the change in flexure thickness, ∆h, is 0.0004 inches.

The nominal condition is shown in Figure 6.11(a). The length of the flexure is

decreased by twice the tolerance value, or 0.0004 inches, creating a space between the

flexure tip and the cam surface. This space in the nominal condition permits assembly of

the CCAR joint in the upper bound condition (Figure 6.11(b)), the condition in which the

torque will be highest. In this condition the flexure is thicker, the radius of the cam

surface is smaller, and the space between the two parts is eliminated. The upper bound

for the torque is defined as

),( ccnomupper RRhhTTnom

∆−∆+= (6.22)

(a) (b) (c)

Figure 6.11: Flexure end and cam surface on pin in the (a) nominal condition, (b) upper bound condition, and (c) lower bound condition.

66

where hnom is the nominal thickness of the flexure, is the nominal radius of the cam

surface. In the lower bound condition, shown in Figure 6.11(c), the flexure thickness is

smaller, the radius of the cam surface is larger, and the space between the flexure tip and

the cam surface is equal to four times the tolerance value, or 0.0008 inches. The lower

bound for the torque is defined as

nomcR

),( ccnomlower RRhhTTnom

∆+∆−= (6.23)

The space between the flexure tip and the cam surface results in zero torque

during the first degrees of rotation. The flexure contacts the cam surface after the joint

rotates 0t

θ degrees, where 0t

θ is defined as

⎥⎦

⎤⎢⎣

⎡−

−−+= −

)(2)(cos

321

24

232

211

0 rrrrrrr

tθ (6.24)

r1, r2, r3, and r4 are parameters in the kinematic model of the joint, and are functions of h

and Rc.

6.3.5 Lateral Load Analysis

Both of the rotors will have lateral loads. The magnitudes of these loads are listed

in Table 1.1(b). Some of the beams are placed into compression, and must be able to

support the load without buckling.

At maximum rotation, the contacting segment of the beam extends to Point D on

the contact surface, as defined by the parameter 2sθ (see Figure 6.2(b)). This parameter

must be defined by the position of the flexure in the upper bound condition, since the

flexure contacts the highest percentage of the surface in this condition. The

67

profile of the contact surface after this point is no longer a constant radius, but is a curve

defined by the linearized equation for the deflection of a cantilever beam

)3(6

2

xLEIxF

y p −= (6.25)

where the origin is located at Point A, the x-axis is tangent to constant radius section of

the contact surface, and Fp is equal to

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛= '

1' f

p lREIF (6.26)

In the absence of lateral loads, the curvature of this profile matches the curvature

of the non-contacting segment of the beam without exerting any force on it. When lateral

loads are present this section of the contact surface prevents the beam from deflecting

past its maximum rotation position. The contact surface provides a stop for the beam,

limiting the lateral translation and helping to prevent buckling. Once a beam is in contact

with the surface its boundary condition becomes similar to that of a fixed-pinned beam.

This condition is over eight times more resistant to buckling than the fixed-free condition

[3]. A conservative approximation assumes that the load is borne by a single beam, in

which case the critical load is

2

22

fcr l

EIP π= (6.27)

Some lateral displacement can occur when the beam is not yet in contact with the

entire contact surface. The maximum lateral displacement condition, shown in Figure

6.12, occurs when the compressive forces cause the beam’s end to slide along the cam

surface to its initial position, and deflect the beam until it contacts the entire surface. The

angle of rotation when this condition is possible is given by

68

Secondary Stop

Beam contacting entire surface.

Beam end at initial position on cam surface.

Figure 6.12: Normal position (dashed lines) and maximum displacement position.

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−+

++=

−−

cc

fpsfbl Rh

EIlFly2/

)]3/(tansin[sin

2'12

'1

δθ

θ (6.28)

and the resulting lateral displacement is

}cos)2/()]3/(tancos[{ 2'12

'lccfpsfbrl RhEIlFlxR θδθδ −++++−= − (6.29)

Should the lateral displacement ever reach the maximum allowable value a

secondary stop, shown in Figure 6.12, is provided between the contact surface and the

outside pin surface. This would prevent additional displacement, though it would also

introduce friction between the two surfaces.

6.4 NUMBER OF FLEXURES

Determining the number of flexures for a CCAR joint depends primarily on the

rotation and the torque characteristics desired for the design. As the radius of the cam

surface, Rc, increases, the maximum possible rotation of the joint increases as well.

However, this also decreases the thickness of the material between the pin’s cam

surfaces, as shown in Figure 6.13. If the minimum thickness is too small for a given value

69

Figure 6.13: Effect of cam surface radius on pin.

of Rc, the number of flexures can be decreased, thereby increasing the minimum

thickness. Fewer flexures, however, will also decrease the torque. Conversely, adding

flexures will increase the torque, but may require that the value of Rc be decreased,

decreasing the rotation of the joint. The general guideline, therefore, is that more flexures

should be used for designs requiring high torque and small rotation, and fewer flexures

for designs requiring low torque and large rotation.

6.5 MODEL VALIDATION

Validation of the PRBM analysis was done using finite element analysis (FEA),

and by testing a macro-scale prototype. The torque-deflection curve for the macro-scale

design was found using the PRBM analysis, as discussed in Section 6.3. The torque-

deflection curve was also found using FEA, as shown in Figure 6.14. The FEA mesh

Figure 6.14: FEA model with contact elements.

70

included 2671 node order 2-D, 8-node

element. The PLANE 183 element has large deflection and large strain capabilities, is

well suited to modeling irregular meshes, and has two degrees of freedom at each node

(translation in the x and y directions) [25]. Two degrees of freedom were sufficient for the

CCAR joint analysis, and at least two dimensions were necessary for the creation of

contact elements. TARGE169 and CONTA172, 2-D contact elements, were used to

model the interaction between the flexure and the pin, and between the flexure and the

contact surface.

The flexure was constrained to have no translation at its fixed end, the contact

surface was constrained to have no translation along its bottom edge, and the section of

the pin was constrained to rotate about the pin’s center. A force was then applied to the

pin, creating a torque, and the resulting deflections and stresses were computed.

Finally, a physical prototype (Figure 6.15) was manufactured and tested. Figure

6.16 shows the results. The PRBM and FEA predictions are very close, and have a

maximum error of only 5.4%. More variation (up to 35.6% error) is seen in the measured

values. This variation is most likely due to the variation and nonlinearity of the material

s, and used the PLANE183 element, a higher

Figure 6.15: Prototype of the CCAR joint.

71

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 5 10 15 20 25 30Degrees of Rotation

Torq

ue (i

n-lb

)

PRBM PredictionFEA PredictionMeasured Values

Figure 6.16: Results of PRBM predictions, FEA predictions, and prototype testing.

properties of polypropylene, and to machining tolerances. These results give confidence

to the PRBM predictions.

6.6 RESUL

TS

Table 6.1 lists the dimensions used for the drive and gate rotor meso-scale

designs, and Figure 6.17 shows the scaled polypropylene prototypes of these designs. The

drive rotor CCAR joint has more flexures to increase its torque, and the gate rotor design

has less in order to increase its rotation.

Table 6.1: Design dimensions (all values are in inches and degrees).

Rr Rp Rc Rs c h b rQ n

δ Drive Rotor 9 325 522 0.01045 0321 033 0.05 0.01 0.00 0.2 0.0 0.1 0.0 6 Gate Rotor 0.05 0.019 0.0055 0.1915 0.01045 0.002 0.0521 0.0022 4

72

Figure 6.17: Prototypes of the (a) drive rotor CCAR joint, and (b) gate rotor CCAR joint.

Table 6.2 lists the results. The nominal torque curves closely match the torque

requirements. The nominal maximum torque is equal to the desired maximum torque

for both the designs. The nominal minimum torque for the drive rotor is 98% of the

desired value, and for the gate rotor it is 94% of the desired value. The upper and lower

bounds for the maximum torque range from 56% to 194% of the desired value for the

drive rotor, and 44% to 195% of the desired value for the gate rotor.

Table 6.2: (a) Drive rotor results, and (b) gate rotor results.

(a)

Min Torque (in-lb)

Max Torque (in-lb)

Min Rotation Position

(deg)

Max Rotation Position

(deg)

% of Surface

Contacted

lδ (in)

Safety Factor

Nominal 0.0133 0.0272 14.85 22.85 26.0 0.0010 1.15 Upper Bound 0.0230 0.0527 14.85 22.85 64.4 0.0006 1.02 Lower Bound 0.0061 0.0152 14.85 22.85 0 0.0014 1.35

(b)

Min Torque (in-lb)

Max Torque (in-lb)

Min Torque Position

(deg)

Max Torque Position

(deg)

% of Surface

Contacted

lδ (in)

Safety Factor

Nominal 0.00097 0.00205 32 47 0 0.0019 1.39 Upper Bound 0.00191 0.00400 32 47 18.4 0.0014 1.17 Lower Bound 0.00040 0.00090 32 47 0 0.0024 1.99

73

Figures 6.18 and 6.19 show the nominal torque-deflection curve as well as the

lower and upper bounds for the torque. The nominal values in these designs are more

linear than in previous designs. As the flexures contact the adjacent contact surfaces,

the torque-deflection relationship becomes very nonlinear. In these designs, however, the

radii of the contact surfaces were decreased, resulting in the flexures contacting a smaller

percentage of the surface at the maximum rotation of the joint (indicated by the “% of

Surface Contacted” values in Table 6.2). The torque curves, therefore, are more linear.

This is necessary because the flexures always contact a higher percentage of the surface

in the upper bound condition than in the nominal condition. If the flexures contact a high

percentage of the surface in the nominal condition, then the upper bound condition

flexures will contact even more of the surface and cause the torque curve to become very

nonlinear, resulting in very high torque.

0

0.01

0.02

0.03

0.04

0.05

0.06

0 5 10 15 20 25Degrees of Rotation

Torq

ue (i

n-lb

)

Nominal Values

Lower Bound

Upper Bound

Figure 6.18: Effect of tolerances on the drive rotor CCAR joint.

74

0

0.001

0.002

0.003

0.004

0.005

0 5 10 15 20 25 30 35 40 45 50

Degrees of Rotation

Torq

ue (i

n-lb

)

Nominal Values

Lower Bound

Upper Bound

Figure 6.19: Effect of tolerances on the gate rotor CCAR joint.

If the flexures in the upper bound condition contact the entire surface before

reaching the maximum rotation position, then the joint will be unable to complete its

specified rotation. This is illustrated in Figure 6.20. The design used in Figure 6.20 has a

large contact surface radius, and in the nominal condition the flexures contact 66% of the

surface at the maximum rotation of 22.85 degrees. In the upper bound condition,

0

0.02

0.04

0.06

0.08

0.1

0.12

0 5 10 15 20 25Degrees of Rotat ion

Torq

ue (i

n-lb

)

Nominal Values

Lower Bound

Upper Bound

Figure 6.20: Design with high percentage of flexure length in contact with surface.

75

however, the flexure contacts 100% of the surface at only 21.35 degrees of rotation. The

torque is very high, and the joint is unable to rotate the full 22.85 degrees.

By lowering the contact surface radius in the drive and gate rotor design this

problem is avoided. The cost of doing so is that the safety factors are lowered as well.

The lowest safety factor is in the upper bound condition of the drive rotor, which has a

safety factor against yielding that is just above 1. Exceeding the yield strength of the

material, however, would only result in some local deformation in the flexures, and not in

catastrophic failure. Flexure deformation would lower the output torque of the joint. In

the upper bound condition this would bring the output torque closer to the desired torque

values, and would not be detrimental to the design. The lower bound condition, where

decreased torque would be most problematic, also has the highest safety factor.

Table 6.2 lists the values for lδ , the maximum lateral displacement. These values

do not represent the actual displacement due to the lateral loads given in Table 1.1, but

represent the maximum possible displacement before the contact surfaces constrain

further deflection and lateral displacement. The actual displacement is most likely less

than this value. The drive rotor joint has a nominal maximum lateral displacement equal

to the maximum allowable displacement specified in Table 1.1. The nominal maximum

lateral displacement for the gate rotor joint is almost twice the maximum allowable value.

Should the actual lateral displacement reach the maximum allowable value, the

secondary stop between the outer surface of the pin and the inner surface of the joint

would prevent further displacement. The nominal distance between these two surfaces is

equal to the maximum allowable displacement, 0.001 inches. If the surfaces contact,

76

some of the load would be transferred from the flexures to the surface, constraining

further deflection but introducing some friction.

6.7 CONCLUSIONS

A unique joint capable of replacing both bearings and springs has been presented,

and a model for predicting its behavior developed and validated. The contact surfaces in

the CCAR joint allow for larger deflections than could be obtained without contact

interactions, and give it a high lateral stiffness. The nominal torque values closely match

the torque requirements, the nominal lateral deflection is within the allowable range, and

the available design space is not exceeded. Nominal safety factors against yielding are

1.15 for the drive rotor, and 1.39 for the gate rotor.

77

78

7CHAPTER 7 CONCLUSIONS

Four unique compliant components for replacing bearings and/or springs have

been presented. Table 7.1 rates each of the components for replacing bearings by the

Moon, Trease, and Kota criteria [5] discussed in Chapter 2. Contact interactions help all

of the components to have a large range of motion, low stress concentrations, and high

off-axis stiffness. All of the components have pure rotation without axis drift except the

CORE, which does experience some axis drift due to its cycloidal motion. All of

them are compact enough to fit within the tight design space available for Sandia’s

application except the CORE bearing. The CORE bearing can be made more compact,

however, by using a different manufacturing process such as LIGA.

Table 7.2 lists which components meet the requirements for the Stronglink

application. All of the components are scalable, they all fit within the design space

(assuming the CORE bearing uses the LIGA process), they are all able to be

manufactured by planar fabrication methods, and can all obtain the required rotation.

Though the serpentine flexure is unable to bear significant lateral loads, it can meet the

torque requirements, and is a good candidate for replacing springs. The CORE, CORE

bearing, and elliptical CORE bearing can all bear high lateral loads, and are good

79

Table 7.1: Flexible revolute joints.

(X: Poor, ∆: Normal, : Good)

Ran

ge o

f M

otio

n

Axi

s Drif

t

Stre

ss

Con

cent

ratio

n

Off

-Axi

s St

iffne

ss

Com

pact

ness

CORE

∆

CORE Bearing ∆

Elliptical CORE

Bearing

CCAR Joint

candidates for replacing bearings. Though they aren’t able to meet the torque

requirements in the Stronglink application, they can have a resulting torque and may be

able to replace springs in other applications. The CCAR joint meets all of the

requirements, and can replace both bearings and springs.

In summary, all of the components are good candidates for replacing bearings

and/or springs, and all of the components meet the objectives of this thesis.

80

Table 7.2: Functional requirements.

X: Doesn’t meet criterion. ∆: Meets criterion in some applications, but not in the Stronglink application. : Meets criterion in the Stronglink application.

Scal

able

Fits

in D

esig

n Sp

ace

Plan

ar

Fabr

icat

ion

Obt

ains

Req

uire

d R

otat

ion

Can

Rep

lace

B

earin

gs

Can

Rep

lace

Sp

rings

Serpentine Flexure X

CORE

∆

CORE Bearing

∆

Elliptical CORE

Bearing

∆

CCAR Joint

81

82

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85