Research Papers: Design of Mechanisms and Robotic Systems

Eliminating Underconstraint in Double Parallelogram Flexure Mechanisms

[+] Author and Article Information
Robert M. Panas

Materials Engineering Division,
Lawrence Livermore National Laboratory,
7000 East Ave, L-229,
Livermore, CA 94551
e-mail: panas3@llnl.gov

Jonathan B. Hopkins

Mechanical and Aerospace Engineering,
University of California, Los Angeles,
Los Angeles, CA 90095
e-mail: hopkins@seas.ucla.edu

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received July 8, 2014; final manuscript received May 18, 2015; published online July 17, 2015. Assoc. Editor: Oscar Altuzarra.The United States Government retains, and by accepting the article for publication, the publisher acknowledges that the United States Government retains, a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for United States government purposes.

J. Mech. Des 137(9), 092301 (Jul 17, 2015) (9 pages) Paper No: MD-14-1409; doi: 10.1115/1.4030773 History: Received July 08, 2014

We present an improved flexure linkage design for removing underconstraint in a double parallelogram (DP) linear flexural mechanism. This new linkage alleviates many of the problems associated with current linkage design solutions such as static and dynamic performance losses and increased footprint. The improvements of the new linkage design will enable wider adoption of underconstraint eliminating (UE) linkages, especially in the design of linear flexural bearings. Comparisons are provided between the new linkage design and existing UE designs over a range of features including footprint, dynamics, and kinematics. A nested linkage design is shown through finite element analysis (FEA) and experimental measurement to work as predicted in selectively eliminating the underconstrained degrees-of-freedom (DOF) in DP linear flexure bearings. The improved bearing shows an 11 × gain in the resonance frequency and 134× gain in static stiffness of the underconstrained DOF, as designed. Analytical expressions are presented for designers to calculate the linear performance of the nested UE linkage (average error < 5%). The concept presented in this paper is extended to an analogous double-nested rotary flexure design.

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Grahic Jump Location
Fig. 1

Flexure DP with nested UE linkage. This linkage selectively removes the underconstraint inherent in the DP design by linking the motion of the intermediate and final stage.

Grahic Jump Location
Fig. 2

(a) DP linear bearing with components labeled. Possible motions for the structure are shown in an equivalent linkage model, drawn from pseudo-rigid-body models [1,12,14] in (b). The solid arrows show the nominal translational motion, with the kinematic errors canceled by geometry reversal. The dotted arrows show the y-axis DOF for the structure observed at large displacements, which is revealed as a drop in axial stiffness.

Grahic Jump Location
Fig. 3

(a) Double tilted-beam linear bearing shown on left with instant centers identified. The two possible motions for the structure are shown in equivalent linkage models in (b). The solid arrows show the nominal translational motion where the final stage does not rotate. Note that the kinematic errors are not canceled in this translation. The dotted arrows show the extra DOF of the structure where the final stage rotates without translating.

Grahic Jump Location
Fig. 4

(a) Exact constraint folded flexure utilizing an external linkage. The possible motion for the structure is shown in the equivalent linkage model in (b). The solid arrows show the nominal translational motion, which is equivalent to the DP flexure bearing alone.

Grahic Jump Location
Fig. 5

(a) Exact constraint folded flexure design with an improved, nested linkage for UE. The possible motion for the structure is shown in the equivalent linkage model in (b). The solid arrows show the nominal translational motion, which is equivalent to the DP flexure bearing alone.

Grahic Jump Location
Fig. 6

(a) Double-nested rotary flexure, (b) similar flexure but with a UE linkage inserted, and (c) the final stage is forced to rotate twice as many radians as the intermediate stage in the same direction

Grahic Jump Location
Fig. 7

FEA of the nested UE linkage design undergoing loading along the former DOF of the intermediate stage. A 1 N load is applied in the x-axis on the intermediate stage (red arrow), while the final stage and grounds are held in place. The UE linkage rigid body is warped by the loading, with the top horizontal beam in the linkage forming an “S” shape, and both of the side beams bowing in the same direction. The motion is amplified to clearly indicate the warping.

Grahic Jump Location
Fig. 8

Flexure DP with nested linkage, attached to optical table. This general setup was used to capture the static and dynamic properties of the main DOF as well as the underconstrained DOF.

Grahic Jump Location
Fig. 9

The Bode gain plot for the unaugmented DP flexure is compared to that of the DP flexure with the nested linkage. The intermediate stage resonance has been shifted from 60 Hz to 650 Hz, while the main stage resonance has been changed by only ≈5%.

Grahic Jump Location
Fig. 10

Logarithmic plot of axial stiffness versus nondimensional displacement, showing large displacement effects. Both simple DP and DP with UE stage displacements were simulated using large displacement FEA with displacement (hollow points) and force (solid points) control on the final stage. The DP results are compared to models presented in literature (LD DP Theory) [33].




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