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

Flexure Mechanisms: Why? Design Principles for Precision • Miniaturization Mechanisms • No friction • Controlling DOFs 3. Flexure Mechanisms Design Principles: Flexure Mechanisms 1 Design Principles: Flexure Mechanisms 2 Deflection formula or Euler-Bernoulli equations Flexure Mechanisms: Limitations x M, � 2 1 1 d f � � � � • Limited stroke � � � � ( ) M x F L x M 2 dx EI EI F, f • Stress � � 1 df � � 2 � � � � 1 FLx F x Mx C 2 1 • Stiffness/Energy dx EI 2 3 ML FL � � � � 1 f � 2 � 3 � 2 � � 1 1 1 2 3 f F Lx F x Mx C x C EI EI 2 6 2 1 2 EI 2 ML FL � � � 2 EI EI df � � � � ( 0 ) 0 0 x C 1 dx � � � � ( 0 ) 0 0 f x C 2 Design Principles: Flexure Mechanisms 3 Design Principles: Flexure Mechanisms 4

Deflection of beam-end about instant centre of rotation P A single flexure may serve as a hinge when it is in line with the dominant load � � 2 � Fl F L l l � � � 2 EI EI � � � 3 2 F Fl F L l l l L F � � z L � f l x 3 2 P EI EI f A � � � � B f x M C � ( ) l 3L l � x = . � L F 3(2L l) P 1) Force F in B; L = l : x = 2/3· l F x When L � � (load is moment), then x � l /2 2) 2 l 2 l Design Principles: Flexure Mechanisms 5 Design Principles: Flexure Mechanisms 6 Loadability and stiffness Equal stiffness in every direction? 12 EI c z � 2 2 � cos ( ) � � F � � 3 � � � � � 2 2 � l cos cos cos F c f f 2 x c EA x c y � 2 2 � sin ( ) � � � � F l � � � � � 2 2 � sin sin sin F c f f y 2 c x � � � � � 2 � 2 � cos sin � � � � � f F � � � � 2 2 c c x y � � F � 0 principal direction f c x � � � F � � � principal direction f � 2 � c y F � � � � � � � If c c c, then , is independen t of , and f x y c Design Principles: Flexure Mechanisms 7 Design Principles: Flexure Mechanisms 8

Symmetric cross flexure Generic model of a cross flexure � � P P a � l l 2 l y l 1 z x EI � � M 1 in LS L �� � ( 2 1 ) � t l M t � � � � � � Et 1 1 M � � 1 � � in LS � � � � � � � 2 � � � � 2 4 � 1 3 3 1 3 3 � k a a a a � � � � � 1 1 � 2 2 relatively small � 2 � K l K l � I l 1 1 2 2 z z � 1 2 � Symmetry M EI K z � � � � for beams with cross sections with approximately the same thickness as width. tot k E I � � l � � 2 ( 1 ) � K z for beams with cross sections with large width to thickness ratio. � E I Design Principles: Flexure Mechanisms 9 Design Principles: Flexure Mechanisms 10 Asymmetric cross flexure Stiffness reduction due to eccentric tensile load Cross spring hinge • The displacement x of point A: non symmetric embodiment symmetric embodiment • The compliance in A is then: centre line sheet flexure Design Principles: Flexure Mechanisms 11 Design Principles: Flexure Mechanisms 12

Stiffness reduction due to eccentric tensile load Stiffness reduction due to eccentric tensile load • Stiffness ratio • 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! Design Principles: Flexure Mechanisms 13 Design Principles: Flexure Mechanisms 14 Resolving the overconstraint of a cross flexure The over-constraint of a cross flexure B y x z A P Design Principles: Flexure Mechanisms 15 Design Principles: Flexure Mechanisms 16

Examples of cross flexure Adjustment of two angles (1/2) C Embodiment of cross flexure Two cascaded cross flexure A mechanisms based on wire flexures B Philips Lighting version Design Principles: Flexure Mechanisms 17 Design Principles: Flexure Mechanisms 18 Adjustment two angles (2/2) Riverhawk Company Headquarters is easy to manufacture... located in New Hartford, New York, Commercial flexures USA Mailing Address 215 Clinton Road New Hartford, New York, 13413 http://www.flexpivots.com/ Riverhawk Company Design Principles: Flexure Mechanisms 19 Design Principles: Flexure Mechanisms 20

Injection moulding of flexures in plastic (1/2) Injection moulding of a cross flexure in plastic (2/2) Injection molded plastic Symmetric injection molded cross flexure part with wire flexure functionality A A cross section A-A All hinges are “through-injected” Design Principles: Flexure Mechanisms 21 Design Principles: Flexure Mechanisms 22 y Injection moulding of flexures in plastic Flexure parallel guiding x Polypropylene linkage as a station scale indicator L F coagulation at the hinge! Shortening effect! Division (pref. at the clamp section) enables “through-injection”of all hinges! Design Principles: Flexure Mechanisms 23 Design Principles: Flexure Mechanisms 24

One sheet flexure under parallel guiding One sheet flexure under parallel guiding M General equation: � � �� � � � � 12 6 � � The maximum bending stress F l x E I � � � � � � � � � � � (parallel guiding) occurs at the F 3 6 4 2 � � � � � � M l l l � clamp: x , , E I l � � � � 3 E h x � � , max 2 b l Stiffness � F E I � � � � � � 0 12 c x � 3 x l Design Principles: Flexure Mechanisms 25 Design Principles: Flexure Mechanisms 26 Force F evokes reactions N 1 and N 2 Sheet flexure parallel guiding b L L a � N N 1 2 M K � 2 K � 2 F F K M � M � M M P l l 1 2 � � � 12 K K K EI 2 � � 1 2 K � M K � � � � � � 2 = = 0 M c z � � M M M 3 2EI EI 2 l 1 2 � � � � b N N N � � � 2 1 2 N F l a centre of compliance � � � � � 2 0 � M N b F a Design Principles: Flexure Mechanisms 27 Design Principles: Flexure Mechanisms 28

Position dependent guiding stiffness Supporting stiffness also a function of drive stiffness Supporting stiffness decreases with increasing deflection 10 0 � � c d c x � 1000 c d c Normal range x � 100 c d c x y 10 -1 F c x y c y � 10 c c d c , 0 x y 1.2 � 0 c d c � x x 10 -2 L 1 h 0.8 � c y 0.6 10 -3 c , 0 0.4 y 0.2 10 -4 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 � x � x h h Design Principles: Flexure Mechanisms 29 Design Principles: Flexure Mechanisms 30 Overconstraint in � direction Relief of overconstraint in � direction by additional rotational freedom in flexure One DOF ( � ) is constrained twice - - One DOF remains: x Design Principles: Flexure Mechanisms 31 Design Principles: Flexure Mechanisms 32

Second order shortening of sheet (wire) flexures Principle of a wire flexure parallel guiding 2 z � � 0 . 5 y l 2 z � � 0 . 6 y l Design Principles: Flexure Mechanisms 33 Design Principles: Flexure Mechanisms 34 Sheet and wire flexures with reinforced mid-sections Compensating the “shortening effect” � Limited capacity in compression � Buckling � y � y � � l 2 � � � 0 . 6 y x x x x H x H B B H Intermediate body Reinforced mid-section Design Principles: Flexure Mechanisms 35 Design Principles: Flexure Mechanisms 36

Sheet and wire flexures with reinforced mid-sections 3 Etz � � max 2 Stiffness matrix of a reinforced flexure l 3 Etb � c y l 3 Ebt � � 1 . 2 c z 3 l M � 2 36 EI � F k 2 � l � F x � � 1 12 1 6 � E , l I � � � � � � �� � � 3 � 3 2 � 3 � 1 1 F x l p l p � � � � � � � � � � E I � � � � � 2 � p � 1 6 1 4 � � � p p � � � M l � � � � 3 � � Etz � � 2 1 3 1 3 � � � � l p l p max 2 l Etb � c y l E , I 3 Ebt � c z 3 l � 2 4 EI � F k 2 l Design Principles: Flexure Mechanisms 37 Design Principles: Flexure Mechanisms 38 Basic formula for stiffness calculations on flexures Normal stiffness c y of a reinforced flexure c y � 1.2 1.2 x 0 0 . . 9 9 1 1 0 0 . . 8 8 0 0 . . 7 7 0 0 . . 6 6 h 0 0 . . 5 5 0.8 0.8 � � 0 0 . . 4 4 p p c c 0.6 0.6 y y c c p � 0.4 0.4 , , 0 0 l y y l 0.2 0.2 0 0 0 0 5 5 10 10 15 15 20 20 25 25 30 30 � � x x h h Design Principles: Flexure Mechanisms 39 Design Principles: Flexure Mechanisms 40

F Wire flexure with reinforced mid-section folding line Wire flexure from a (single) piece of strip F Design Principles: Flexure Mechanisms 41 Design Principles: Flexure Mechanisms 42 Examples of wire flexures Circumventing the wire clamping issue Preload springs between pulleys Pre-stressed rod Pre-stressed rod Polypropylene injection moulded product Flattened steel wire Design Principles: Flexure Mechanisms 43 Design Principles: Flexure Mechanisms 44