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Research Papers: Design of Mechanisms and Robotic Systems

Computational Synthesis of Large Deformation Compliant Mechanisms Undergoing Self and Mutual Contact

[+] Author and Article Information
Prabhat Kumar

Mechanical Engineering,
Indian Institute of Technology Kanpur,
Kanpur 208016, India
e-mail: prabhatkumar.rns@gmail.com

Anupam Saxena

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

Roger A. Sauer

Aachen Institute for Advanced Study in
Computational Engineering Science (AICES),
RWTH Aachen University,
Templergraben 55,
52056 Aachen, Germany

1Corresponding author.

2Present Address: Department of Precision and Microsystems Engineering, Faculty of 3mE, Delft University of Technology, Mekelweg 2, Delft 2628 CD, The Netherlands.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received May 28, 2018; final manuscript received July 29, 2018; published online October 10, 2018. Assoc. Editor: Nam H. Kim.

J. Mech. Des 141(1), 012302 (Oct 10, 2018) (13 pages) Paper No: MD-18-1399; doi: 10.1115/1.4041054 History: Received May 28, 2018; Revised July 29, 2018

Topologies of large deformation contact-aided compliant mechanisms (CCMs), with self and mutual contact, exemplified via path generation applications, are designed using the continuum synthesis approach. Design domain is parameterized using honeycomb tessellation. Assignment of material to each cell, and generation of rigid contact surfaces, are accomplished via suitably sizing and positioning negative circular masks using the stochastic hill-climber search. To facilitate contact analysis, boundary smoothing is implemented. Mean value coordinates are employed to compute shape functions, as many regular hexagonal cells get degenerated into irregular, concave polygons as a consequence of boundary smoothing. Both geometric and material nonlinearities are considered. The augmented Lagrange multiplier method with a formulated active set strategy is employed to incorporate both self and mutual contact. Synthesized contact-aided compliant continua trace paths with single, and importantly, multiple kinks and experience multiple contact interactions pertaining to both self and mutual contact modes.

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Figures

Grahic Jump Location
Fig. 1

A set of hexagonal cells ΩH discretize the design space Ω. Negative circular masks ΩM (circles) superposed on Ω help determine the material state of ΩH. Five variables (xp, yp, rp, sp, fp) define each mask. (xp, yp) and rp are center coordinates and the radius of the pth mask. sp = 1 (b) implies a rigid contact surface (dark, filled circular regions) of radius fprp is generated within the pth mask while with sp = 0 (a), no contact surface is generated. ρH) = 0 (ΩH⊂ any ΩM) implies a void material state while ρH) = 1 indicates full material state. Rigid contact surfaces can interact with the continuum. In addition, surfaces (e.g., Γs1 and Γs2) of the continuum may interact in self-contact mode.

Grahic Jump Location
Fig. 2

V-notches furnish jumps in boundary normals (Γcm) (a) which are subdued via boundary smoothing (c) to facilitate contact analysis. (b) depicts the way boundary smoothing is performed. (d) shows that for higher values of β, some elements may experience flipping. (a) Body Bm without boundary smoothing, (b) boundary smoothing scheme, (c) body Bs with boundary smoothing; considering β = 1, and (d) higher values of β can result in significant distortion of elements, and also, element flipping.

Grahic Jump Location
Fig. 3

A schematic diagram for the contact formulation: (a) deformed configurations, (b) contact description with tractions,(c) closest point evaluation, and (d) contact discretization, piecewise linear segments approximate interacting boundaries

Grahic Jump Location
Fig. 4

Schematic describing self-contact when two surface regions Γs1 and Γs2 of the same body penetrate each other. The hatched portion shows that the top part of the body has penetrated into the bottom part. Arrows are marked on the boundary to indicate orientation and to differentiate between two interacting surfaces. For point xs∈Γs1, point xp∈Γs2 is the nearest neighbor. xp1∈Γs1 or xp2∈Γs1 are also candidate nearest neighbors since normal gaps g1·np1 and g2·np2 are negative in both cases. Such candidate nearest neighbors are discarded noting that the dot products between normals, e.g., nps·np1 and nps·np2 are positive.

Grahic Jump Location
Fig. 5

A schematic flowchart for a single design iteration within the proposed optimization approach

Grahic Jump Location
Fig. 6

Design specification and specified paths for four different CCMs: (a) design specification for all examples with the initial guess for mask parameters, and (b) desired output paths for all four CCMs are depicted

Grahic Jump Location
Fig. 7

Final solutions for CCM I–IV: left column: final continua with positions of masks and mutual contact surfaces shown, middle and right columns: comparison of paths generated and final deformed configurations. Active contact regions are depicted within dashed circles.

Grahic Jump Location
Fig. 8

Deformed configurations of CCMs at stages A, B and C (Fig. 7): (a) CCM I, (b) CCM II, (c) CCM III, and (d) CCM IV

Grahic Jump Location
Fig. 9

(a) Prototypes of CCMs I–IV in their undeformed (left) and deformed (middle and right) configurations (b) Paths from the prototypes (PP) compared with those traced by the CCMs (AP) in simulation. Left-right: CCMs I–IV.

Grahic Jump Location
Fig. 10

Paths traced by CCMs I–IV meshed with different number of hexagonal cells (inset). (d) desired path is rotated for better comparison. SP represents the desired path. (a) CCM I, (b) CCM II, (c) CCM III, and (d) CCM IV.

Grahic Jump Location
Fig. 11

Paths and CPU time with various Gauss points, Key: GP–Gauss points, CT(s)–CPU time in seconds

Grahic Jump Location
Fig. 12

Computational cost and convergence history with INTEL CORE(TM) i5-6000 CPU at 2.70 GHz. Solution (a) is obtained after 6000 function evaluations. A feasible candidate continuum, that having the required input/output ports, some fixed boundaries, and that is well connected, is not available for the first 880 iterations. The overall synthesis takes about 12 h. (a) CCM after 6000 iterations and (b) convergence history.

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