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Design Innovation Paper

Gravity-Insensitive Flexure Pivot Oscillators

[+] Author and Article Information
M. H. Kahrobaiyan

Instant-Lab,
École Polytechnique Fédérale de Lausanne (EPFL),
Microcity, Rue le la Maladière 71b,
Neuchâtel CH-2000, Switzerland
e-mail: mohammad.kahrobaiyan@epfl.ch

E. Thalmann

Instant-Lab,
École Polytechnique Fédérale de Lausanne (EPFL),
Microcity, Rue le la Maladière 71b,
Neuchâtel CH-2000, Switzerland
e-mail: etienne.thalmann@epfl.ch

L. Rubbert

INSA de Strasbourg,
Université de Strasbourg,
24 Bld de la Victoire,
Strasbourg 67084, France
e-mail: lennart.rubbert@insa-strasbourg.fr

I. Vardi

Instant-Lab,
École Polytechnique Fédérale de Lausanne (EPFL),
Microcity, Rue le la Maladière 71b,
Neuchâtel CH-2000, Switzerland
e-mail: ilan.vardi@epfl.ch

S. Henein

Instant-Lab,
École Polytechnique Fédérale de Lausanne (EPFL),
Microcity, Rue le la Maladière 71b,
Neuchâtel CH-2000, Switzerland
e-mail: simon.henein@epfl.ch

1Corresponding author.

Contributed by the Design Innovation and Devices of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received June 9, 2017; final manuscript received March 28, 2018; published online May 11, 2018. Assoc. Editor: Oscar Altuzarra.

J. Mech. Des 140(7), 075002 (May 11, 2018) (9 pages) Paper No: MD-17-1398; doi: 10.1115/1.4039887 History: Received June 09, 2017; Revised March 28, 2018

Classical mechanical watch plain bearing pivots have frictional losses limiting the quality factor of the hairspring-balance wheel oscillator. Replacement by flexure pivots leads to a drastic reduction in friction and an order of magnitude increase in quality factor. However, flexure pivots have drawbacks including gravity sensitivity, nonlinearity, and limited stroke. This paper analyzes these issues in the case of the cross-spring flexure pivot (CSFP) and presents an improved version addressing them. We first show that the cross-spring pivot cannot be simultaneously linear, insensitive to gravity, and have a long stroke: the 10 ppm accuracy required for mechanical watches holds independently of orientation with respect to gravity only when the leaf springs cross at 12.7% of their length. But in this case, the pivot is nonlinear and the stroke is only 30% of the symmetrical (50% crossing) cross-spring pivot's stroke. The symmetrical pivot is also unsatisfactory as its gravity sensitivity is of order 104 ppm. This paper introduces the codifferential concept which we show is gravity-insensitive. It is used to construct a gravity-insensitive flexure pivot (GIFP) consisting of a main rigid body, two codifferentials, and a torsional beam. We show that this novel pivot achieves linearity or the maximum stroke of symmetrical pivots while retaining gravity insensitivity.

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References

Privat-Deschanel, P. , and Focillon, A. , 1877, Dictionnaire Général Des Sciences Techniques Et Appliquées, Garnier Frères, Paris, France.
Barrot, F. , Dubochet, O. , Henein, S. , Genequand, P. , Giriens, L. , Kjelberg, I. , Renevey, P. , Schwab, P. , Ganny, F. , and Hamaguchi, T. , 2014, “ Un Nouveau Régulateur Mécanique Pour une Réserve de Marche Exceptionnelle,” Actes de la Journée d'Etude de la Société Suisse de Chronométrie, pp. 43–48.
Bateman, D. A. , 1977–1978, “ Vibration Theory and Clocks,” Horological J., 120–121(Pt. 7).
Eastman, F. S. , 1935, Flexure Pivots to Replace Knife Edges and Ball Bearings, an Adaptation of Beam-Column Analysis (Experiment Station series), University of Washington, University of Washington, Seattle, WA.
Eastman, F. S. , 1937, “ The Design of Flexure Pivots,” J. Aeronaut. Sci., 5(1), pp. 16–21. [CrossRef]
Wittrick, W. H. , 1951, “ The Properties of Crossed Flexure Pivots, and the Influence of the Point at Which the Strips Cross,” Aeronaut. Q., 2(4), pp. 272–292. [CrossRef]
Henein, S. , and Kjelberg, I. , 2015, “ Timepiece Oscillator,” Swiss Center for Electronics and Microtechnology, Neuchâtel, Switzerland, U.S. Patent No. 9207641B2. https://patents.google.com/patent/US9207641
Kahrobaiyan, M. , Rubbert, L. , Vardi, I. , and Henein, S. , 2016, “ Gravity Insensitive Flexure Pivots for Watch Oscillators,” Actes du Congrès International de Chronométrie, Montreux, Switzerland, pp. 49–55.
Hongzhe, Z. , Dong, H. , and Shusheng, B. , 2017, “ Modeling and Analysis of a Precise Multibeam Flexural Pivot,” ASME J. Mech. Des., 139(8), p. 081402. [CrossRef]
Merriam, E. G. , and Howell, L. L. , 2016, “ Lattice Flexures: Geometries for Stiffness Reduction of Blade Flexures,” Precis. Eng., 45, pp. 160–167. [CrossRef]
Cosandier, F. , Henein, S. , Richard, M. , and Rubbert, L. , 2017, The Art of Flexure Mechanism Design, EPFL Press, Lausanne, Switzerland.
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Awtar, S. , Slocum, H. , and Sevincer, E. , 2007, “ Characteristics of Beam-Based Flexure Modules,” ASME J. Mech. Des., 129(6), pp. 625–639. [CrossRef]
Plainevaux, J. E. , 1956, “ Etude des Déformations d'une Lame de Suspension Élastique,” Nuovo Cimento, 4(4), pp. 922–928. [CrossRef]
Hongzhe, Z. , and Shusheng, B. , 2010, “ Stiffness and Stress Characteristics of the Generalized Cross-Spring Pivot,” Mech. Mach. Theory, 45(3), pp. 378–391. [CrossRef]
Hongzhe, Z. , and Shusheng, B. , 2010, “ Accuracy Characteristics of the Generalized Cross-Spring Pivot,” Mech. Mach. Theory, 45(10), pp. 1434–1448. [CrossRef]
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Figures

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Fig. 1

(a) Rigid pivot watch time base [1] and (b) flexure pivot watch time base [2]

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Fig. 2

CSFP: (a) three-dimensional view, (b) top view, and (c) demonstrator

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Fig. 3

New GIFP: (a) modelization, (b) demonstrator, and (c) transparent view

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Fig. 4

A CSFP subjected to a rotation of angle θ around axis x. (a) Deflections of the beams and (b) an exploded view to indicate reaction forces and moments of the beams.

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Fig. 5

A cantilever beam under axial load P, shear force F, and bending moment M with axial shortening ξ, lateral deflection f, and slope θ at its mobile extremity

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Fig. 6

Normalized nominal stiffness k¯c,0 of CSFP versus geometric parameter δc. Analytical and FEA results are shown.

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Fig. 7

The codifferential. (a) Three-dimensional view and (b) side view under rotation θ about the u-axis.

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Fig. 8

Normalized nominal stiffness k¯g,0 of GIFP versus geometric parameter δg. Analytical and FEA results are shown.

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Fig. 9

(a) CSFP gravity sensitivity εc versus δc = dc/Lc and (b) GIFP gravity sensitivity εg versus δg = dg/Lg for different normalized gravity loads at an angle φ = 45 deg and a rotation θ = 0.1 deg. Analytical and FEA results are shown.

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Fig. 10

Gravity sensitivity ε of CSFP and GIFP versus rotation angle θ for normalized gravity load N¯=0.2 and geometric parameter δg = δc = 0.127

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Fig. 11

Relative nonlinearity μ of CSFP and GIFP versus their geometric parameter δ. Analytical and FEA results are shown.

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Fig. 12

Geometry of a cantilever beam with rectangular cross section

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Fig. 13

Normalized bending strokes Sb* of CSFP and GIFP versus geometric parameter δ. Analytical and FEA results are shown. The maximum admissible stress of the beams in the FEA was chosen such that the angle of the pivot stays below 20 deg.

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