[PPT] - Flexure Mechanisms: Why? Design Principles for Precision PowerPoint Presentation

SLIDE 1

1 Design Principles: Flexure Mechanisms

Design Principles for Precision Mechanisms

3. Flexure Mechanisms

2 Design Principles: Flexure Mechanisms

Flexure Mechanisms: Why?

Miniaturization
No friction
Controlling DOFs

3 Design Principles: Flexure Mechanisms

Flexure Mechanisms: Limitations

Limited stroke
Stress
Stiffness/Energy

4 Design Principles: Flexure Mechanisms

Deflection formula or Euler-Bernoulli equations

2

1 2 2 1 3 6 1 2 2 1 1 2 2 1 2 2

1 1 1 ) ( 1 C x C Mx x F Lx F EI f C Mx x F FLx EI dx df M x L F EI x M EI dx f d

F, f

M, ) ( ) (

2 1

C

x f C x dx df EI FL EI ML EI FL EI ML f 2 3 2

2 3 2

x

SLIDE 2

5 Design Principles: Flexure Mechanisms

f

P

x F A B l L l L C

Deflection of beam-end about instant centre of rotation P

.

l) 3(2L l 3L l = x x f EI l l L F EI Fl f EI l l L F EI Fl

)

( 2 3 2

2 3 2

1)

Force F in B; L =l : x = 2/3·l 2) When L (load is moment), then x l/2

6 Design Principles: Flexure Mechanisms

M

z

F

x

F P

2 l 2 l

F L

A single flexure may serve as a hinge when it is in line with the dominant load

7 Design Principles: Flexure Mechanisms

Loadability and stiffness

l EA cy

3

12 l EI cz

8 Design Principles: Flexure Mechanisms

Equal stiffness in every direction?

direction principal 2 direction principal sin cos sin ) ( sin sin sin cos ) ( cos cos cos

2 2 2 2 2 2 2 2 2 2 2 2 2 2 y x y x x y x x

c F f c F f c c F f f c F f c F f c F f c F

and

,

f

t independen is , then c, c c If

y x

c F f

SLIDE 3

9 Design Principles: Flexure Mechanisms

P

l

l a P

y z x

2

2 2 2 2 2 1 1 1 1

3 3 1 1 3 3 1 1 4 a a l K a a l K M k

z z

I

E K z

1

for beams with cross sections with approximately the same thickness as width. I E K z

)

1 (

2

for beams with cross sections with large width to thickness ratio.

Generic model of a cross flexure

1

l

2

l

10 Design Principles: Flexure Mechanisms

Symmetric cross flexure l EI M k l Et I t M L EI M

tot LS in LS in

2 2 ) ( 2

1 1 1

Symmetry

small relatively

l

t

11 Design Principles: Flexure Mechanisms

Asymmetric cross flexure

Cross spring hinge

centre line sheet flexure non symmetric embodiment symmetric embodiment

12 Design Principles: Flexure Mechanisms

Stiffness reduction due to eccentric tensile load

The displacement x of point A:
The compliance in A is then:

SLIDE 4

13 Design Principles: Flexure Mechanisms

Stiffness reduction due to eccentric tensile load

Stiffness ratio

14 Design Principles: Flexure Mechanisms

Stiffness reduction due to eccentric tensile load

Stiffness ratio

For a beam with a square cross-section and tensile force with an eccentricity a/h* = 0.5, the tensile stiffness drops by a factor of 4!

15 Design Principles: Flexure Mechanisms

The over-constraint of a cross flexure

16 Design Principles: Flexure Mechanisms

P

x y z

A B

Resolving the overconstraint of a cross flexure

SLIDE 5

17 Design Principles: Flexure Mechanisms

Examples of cross flexure

Philips Lighting version

Embodiment of cross flexure

18 Design Principles: Flexure Mechanisms

Adjustment of two angles (1/2) A B C Two cascaded cross flexure mechanisms based on wire flexures

19 Design Principles: Flexure Mechanisms

Adjustment two angles (2/2) easy to manufacture...

20 Design Principles: Flexure Mechanisms

http://www.flexpivots.com/ Riverhawk Company

Riverhawk Company Headquarters is located in New Hartford, New York, USA Mailing Address 215 Clinton Road New Hartford, New York, 13413

Commercial flexures

SLIDE 6

21 Design Principles: Flexure Mechanisms

Injection moulding of flexures in plastic (1/2) All hinges are “through-injected”

Injection molded plastic part with wire flexure functionality

22 Design Principles: Flexure Mechanisms

Injection moulding of a cross flexure in plastic (2/2)

A A

cross section A-A Symmetric injection molded cross flexure

23 Design Principles: Flexure Mechanisms

Injection moulding of flexures in plastic

Polypropylene linkage as a station scale indicator coagulation at the hinge! Division (pref. at the clamp section) enables “through-injection”of all hinges!

24 Design Principles: Flexure Mechanisms

L

x y

F

Flexure parallel guiding Shortening effect!

SLIDE 7

25 Design Principles: Flexure Mechanisms

x

F

M

x

l l l l I E M F

2 3

4 6 6 12

l I E , ,

3

12 l I E x F cx

General equation:

One sheet flexure under parallel guiding Stiffness

26 Design Principles: Flexure Mechanisms

One sheet flexure under parallel guiding The maximum bending stress (parallel guiding) occurs at the clamp:

2 max ,

3 l x h E

b

27

Design Principles: Flexure Mechanisms

2 F K 2 F K M M

1

M M

2

b a l L l K M

2

N

1

N

Force F evokes reactions N1 and N2

N

N N M M M K K K

2

1 2 1 2 1

2
a

F b N M

2 K M = EI M 2EI K =

2

b

a l F N

2

28 Design Principles: Flexure Mechanisms

Sheet flexure parallel guiding

3

12 2 l EI cz

L

P centre of compliance

SLIDE 8

29 Design Principles: Flexure Mechanisms

L

x y

F

cy

h

x

0.2

0.4 0.6 0.8 1 1.2 5 10 15 20 25 30

, y y

c c h x

Normal range

Supporting stiffness decreases with increasing deflection

Position dependent guiding stiffness

30 Design Principles: Flexure Mechanisms h x

,

y y

c c

x

d c

c

1000

x

d c

c 100

x

d c

c 10

x

d c

c

x

d c

c

100 10-1 10-2 10-3 10-4 5 10 15 20 25 30

Supporting stiffness also a function of drive stiffness

31 Design Principles: Flexure Mechanisms

Overconstraint in direction

One DOF () is constrained twice
One DOF remains: x

32 Design Principles: Flexure Mechanisms

Relief of overconstraint in direction by additional rotational freedom in flexure

SLIDE 9

33 Design Principles: Flexure Mechanisms

Principle of a wire flexure parallel guiding

34 Design Principles: Flexure Mechanisms

Second order shortening of sheet (wire) flexures l z y

2

5 .

l

z y

2

6 .

35

Design Principles: Flexure Mechanisms

x xH y

B

x xH y

B H

Compensating the “shortening effect”

Intermediate body

l

x y

2

6 .

36

Design Principles: Flexure Mechanisms

Sheet and wire flexures with reinforced mid-sections

Limited capacity in compression Buckling

Reinforced mid-section

SLIDE 10

37 Design Principles: Flexure Mechanisms

x

p p p l p l p l p l I E M F

3 2 3 2 3 2 3 3

1 4 1 1 6 1 1 6 1 1 12 1

Stiffness matrix of a reinforced flexure

x

F

M

I

E,

l p

I E,

l

38 Design Principles: Flexure Mechanisms

Sheet and wire flexures with reinforced mid-sections

2 2 3 3 2 max

4 3 l EI F l Ebt c l Etb c l Etz

k z y

2

2 3 3 2 max

36 2 . 1 3 3 l EI F l Ebt c l Etb c l Etz

k z y

39

Design Principles: Flexure Mechanisms

cy h

x

l

p l

0.2 0.4 0.6 0.8 1 1.2 5 10 15 20 25 30 , y y

c c h x

4

.

p

5 . 6 . 7 . 8 . 9 . 0.2 0.4 0.6 0.8 1 1.2 5 10 15 20 25 30

, y y

c c h x

4

.

p

5 . 6 . 7 . 8 . 9 .

Normal stiffness cy of a reinforced flexure

40 Design Principles: Flexure Mechanisms

Basic formula for stiffness calculations on flexures

SLIDE 11

41 Design Principles: Flexure Mechanisms

folding line

F F Wire flexure from a (single) piece of strip

42 Design Principles: Flexure Mechanisms

Wire flexure with reinforced mid-section

43 Design Principles: Flexure Mechanisms

Examples of wire flexures

Polypropylene injection moulded product Flattened steel wire

44 Design Principles: Flexure Mechanisms Pre-stressed rod Pre-stressed rod

Circumventing the wire clamping issue

Preload springs between pulleys

SLIDE 12

45 Design Principles: Flexure Mechanisms

constrains only ONE DOF!

Folded sheet flexure

46 Design Principles: Flexure Mechanisms

Folded sheet flexure

47 Design Principles: Flexure Mechanisms

Folded sheet flexure

48 Design Principles: Flexure Mechanisms

Folded sheet flexure

SLIDE 13

49 Design Principles: Flexure Mechanisms 50 Design Principles: Flexure Mechanisms h D

,

x

,

y

,

z

T

thickness t

D h h

D

Hole flexures for enabling small rotations (< 1)

51 Design Principles: Flexure Mechanisms

D T h

,

y

t m

,

x

,

z

used area Et cxx Et czz

2

12 Eth k E

T

t h 6

2

x

xx x

F ht

Dimensioning hole flexures

52 Design Principles: Flexure Mechanisms

F

thickness t

D p Example of dimensioning hole flexures

SLIDE 14

53 Design Principles: Flexure Mechanisms

Orientation of a hole flexure

y z

c c D h

:

high relatively

y z

c c D h

:

small relatively

x z

y z

c c D h

:

small relatively

54 Design Principles: Flexure Mechanisms

Cross pivot embodiment with hole flexures

55 Design Principles: Flexure Mechanisms

Flexure guiding for a relatively large stroke (1/2)

l Z y

H 2

6 .

2

1 Z Z : if ed accomplish be can

n

compensati shortening equal, are springs wire the

f

lengths When the

H

x xH y

B

ancillary body

H B H H

56 Design Principles: Flexure Mechanisms

Elastic guiding for a relatively large stroke (2/2)

stiffness difference due to weight load

H H T T B z l

SLIDE 15

57 Design Principles: Flexure Mechanisms

a y y z a l y l y z l y

H C H c

2 2

Linear guiding on basis of symmetry

“Aristoteles Calibrating Devices” (Arcade) rod

K

L

58 Design Principles: Flexure Mechanisms

Spatial embodiment linear guiding (Arcade)

Additional springs for constant elastic energy: fe = 0.3 Hz

L y

L

E H A1 A2 A3 D2 D1 D3 B1 C1 Bi Ci

59 Design Principles: Flexure Mechanisms

Arcade

60 Design Principles: Flexure Mechanisms

Hole flexure applications

SLIDE 16

61 Design Principles: Flexure Mechanisms

Convert this idea into a correct notch hinge mechanism. The right-hand wall is fixed. The blue part is adjustable by a screw (dark blue box) in the height direction. The blue part also has to sustain an external force F

F

A screw constraining only the longitudinal direction

62 Design Principles: Flexure Mechanisms

Adjusting mechanism with correct hinge orientation

63 Design Principles: Flexure Mechanisms

Adjustable slit

For an optical instrument Adjustable in width from 0 to 1 mm, symmetrical with its bisector Mechanism’s principle

64 Design Principles: Flexure Mechanisms

Adjustable slit

Mechanism’s principle Embodiment design

SLIDE 17

65 Design Principles: Flexure Mechanisms

Symmetric slit mechanism

Input: S Output: M D P D B 150 mm

66 Design Principles: Flexure Mechanisms

P

screw 1 screw 2

Mirror angle adjustment mechanism

Screw 2 for course adjustment Screw 1 for fine adjustment

67 Design Principles: Flexure Mechanisms

in

u

ut

u l x

in
ut

u l x u

3

Reduction of input motion by deflection ratio

Elastic Adjustment

68 Design Principles: Flexure Mechanisms

guiding roller guiding roller

Tilt adjustment of roller by elastic/plastic deformation

Elastic Adjustment

SLIDE 18

69 Design Principles: Flexure Mechanisms

2

l

a

Torsion flexure hinge from folded strip

6 l/a : preferebly

70

Design Principles: Flexure Mechanisms

Torsion flexure from strip without the “bending-over-disadvantage”

71 Design Principles: Flexure Mechanisms

Torsional stiffening due to “cross-section warping”

Stiffening factor

Qt 1 2 5

b l l b

72 Design Principles: Flexure Mechanisms

P A B

Universal joint function in flexures

SLIDE 19

73 Design Principles: Flexure Mechanisms

Elastic ball joint No common pole…..

74 Design Principles: Flexure Mechanisms

A B C

D

E B A C E D

Flexure based dish antenna adjustment in meridional () and azimuthal () directions

75 Design Principles: Flexure Mechanisms 76 Design Principles: Flexure Mechanisms

Clamping flexures with high tensile forces at varying angles

R Et E R t R t l l 2 2

2 1

R

t

SLIDE 20

77 Design Principles: Flexure Mechanisms

Flexures in tube walls: hollow tie rod Equivalent to double cardan coupling h

A high value for h gives a high y-stiffness

78 Design Principles: Flexure Mechanisms

stiffness low : , , stiffness high : , ,

y

z x

Concertina bush

h

Compliance in z, and through slots in tube wall

79 Design Principles: Flexure Mechanisms

Straight guiding flexure mechanism in tube wall

80 Design Principles: Flexure Mechanisms

Spring structures with negative spring stiffness (1/2) L L

F

x

3 max , 3

9 . 8 210 L L Eh l EI c

b

F
x
F

x

vmax

vmax l l l l v 3 4 1 1

max

SLIDE 21

81 Design Principles: Flexure Mechanisms

Spring structures with negative spring stiffness (2/2)

3 max , 3

7 . 12 576 L L Eh l EI c

b

82

Design Principles: Flexure Mechanisms

F g F Pendulum and inverse-pendulum

83 Design Principles: Flexure Mechanisms

Lateral stiffness preloaded tensile members

v

F : preload

l ?

x

c x d l x Fv d

v

F

v

F ~ l F x l x F c

v v x

d

d g

l g m c fe

x

2

1 2 1

Pendulum frequency

84 Design Principles: Flexure Mechanisms v

F

3

3 l EI cb

F ?

tr

c l

3

3 l EI l F c c c

v b tr

F

?

tr

c l

v

F

3

12 l EI l F c c c

v b tr

3

12 l EI cb

Lateral stiffness of a compressively loaded flexure

SLIDE 22

85 Design Principles: Flexure Mechanisms

l I E , ,

L c c

:

Preload : Stiffness

x

tot

c

L

Flexure parallel guidance with preload for negative stiffness in x-direction

86 Design Principles: Flexure Mechanisms

Flexure parallel guidance with preload for negative stiffness in x-direction and 1 DOF coupling to functional parallel guiding

87 Design Principles: Flexure Mechanisms