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Research Papers: Design of Mechanisms and Robotic Systems

Synthesis of C0 Path-Generating Contact-Aided Compliant Mechanisms Using the Material Mask Overlay Method

[+] Author and Article Information
Prabhat Kumar

Mechanical Engineering,
Indian Institute of Technology Kanpur,
Kanpur 208016, India
e-mail: kprabhat@iitk.ac.in

Roger A. Sauer

Mechanical Engineering,
AICES, RWTH Aachen University,
Schinkelstrasse 2,
Aachen 52056, Germany
e-mail: sauer@aices.rwth-aachen.de

Anupam Saxena

Mechanical Engineering,
Indian Institute of Technology Kanpur,
Kanpur 208016, India
e-mail: anupams@iitk.ac.in

1Corresponding author.

2A small positive ϵ is chosen to circumvent singularity in the stiffness matrix during the finite-element analysis.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received July 11, 2015; final manuscript received April 8, 2016; published online April 27, 2016. Assoc. Editor: Charles Kim.

J. Mech. Des 138(6), 062301 (Apr 27, 2016) (9 pages) Paper No: MD-15-1484; doi: 10.1115/1.4033393 History: Received July 11, 2015; Revised April 08, 2016

Contact-aided compliant mechanisms (CCMs) are synthesized via the material mask overlay strategy (MMOS) to trace desired nonsmooth paths. MMOS employs hexagonal cells to discretize the design region and engages negative circular masks to designate material states. To synthesize CCMs, the modified MMOS presented herein involves systematic mutation of five mask parameters through a hill climber search to evolve not only the continuum topology but also to position the rigid, interacting surfaces within some masks. To facilitate analysis with contact, boundary smoothing is performed by shifting boundary nodes of the evolving continuum. Various geometric singularities are subdued via hexagonal cells, and the V-notches at the continuum boundaries are alleviated. Numerous hexagonal cells get morphed into concave subregions as a consequence. Large deformation finite-element formulation with mean-value coordinates based shape functions is used to cater to the generic hexagonal shapes. Contact analysis is accomplished via the Newton–Raphson (NR) iteration with load incrementing in conjunction with the augmented Lagrange multiplier method and active set constraints. An objective function based on Fourier shape descriptors (FSDs) is minimized subject to suitable design constraints. Two examples of path-generating CCMs are presented, their performance compared with a commercial software and fabricated to establish the efficacy of the proposed synthesis method.

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Figures

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Fig. 1

Hexagonal cells ΩH are used to discretize the design domain Ω shown with superposed negative circular masks ΩM. Masks remove the material beneath them and also generate rigid contact surfaces. They are characterized via five parameters (xm, ym, rm, sm, and fm). sm = 1 indicates that the mth mask suspends a contact surface (circular regions in gray) within it while with sm = 0 the material is removed. ρ(ΩH)=0 (cells whose centroids are inside any mask) implies that the hexagonal subregion has no material, while ρ(ΩH)=1 (cells whose centroids are outside any mask) suggests that the cell has the desired material. Boundary(ies) of the continuum generated via ρ(ΩH)=1 is (are) smoothened. Circular regions within masks with sm = 1 interact (shown with double head arrows) with the continuum to render the desired path. Fixed boundary(ies) (∂ΩF), input (∂ΩI), and output (∂Ωo) conditions are also depicted.

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Fig. 5

Example of single and multiple contact analysis with a cantilever beam. Depending on the nature of contact, back tracking (a) of the output port (the beam tip) or a kink (b) is observed. (a) Deformed configuration with a single contact surface and (b) deformed configuration with two contact surfaces.

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Fig. 2

Boundary smoothing to facilitate contact analysis. (a) Midpoints of boundary edges are joined with straight lines and boundary nodes shifted along their shortest perpendicular distances on these segments. (b) Boundary smoothing with β = 10 steps.

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Fig. 3

Each cell is divided into six subregions and integration over each subregion is performed to evaluate the element stiffness matrix k

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Fig. 4

Contact formulation of the two bodies Ωs and Ωcs in their current configuration. Normal triad on Ps are nis and that on Pm are nim, where i = 1, 2, and 3.

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Fig. 6

Evaluation of FSDs

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Fig. 10

Different deformed configurations are shown in (a)–(c). (d) Compares the desired path (black), the actual path (green/ grey solid), and that using abaqus (red/grey dotted) with neo-Hookean material and four-noded plain strain (CPE4I) elements.

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Fig. 11

Different deformed configurations of the fabricated prototype for Example 1 are shown in (a)–(c). (d) The traced path.

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Fig. 12

Undeformed (red/light grey), deformed (blue/dark grey), contact surfaces (black circles), the desired path (black curve) and the actual path (green/light grey curve) are shown. Active contact surfaces are enclosed within the dashed–dotted circle.

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Fig. 13

Different deformed configurations are shown in (a)–(c). (d) Comparison of the desired path (black), the actual path (green/grey solid), and that using abaqus (red/grey dotted) with neo-Hookean material and four-noded plane strain (CPE4I) elements.

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Fig. 14

Different deformed configurations of the fabricated prototype for Example 2 are shown in (a)–(c). (d) The traced path.

Grahic Jump Location
Fig. 7

Ω is discretized via hexagonal cells ΩH. Negative circular masks ΩM are employed to remove the material and to encompass contact surfaces within some. Contact analysis with minimization of FSDs based objective is performed to achieve the path close to that desired.

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Fig. 8

Design specifications for the synthesis of CCMs

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Fig. 9

Undeformed (red/light grey), deformed (blue/dark grey), contact surfaces (black circles), the desired path (black curve) and the actual path (green/grey curve) are shown. Active contact surfaces are enclosed within the dashed–dotted circle.

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