0
Research Papers: Design of Mechanisms and Robotic Systems

Topology Optimization of Planar Gear-Linkage Mechanisms

[+] Author and Article Information
Neung Hwan Yim

WCU Multiscale Design Division,
School of Mechanical and
Aerospace Engineering,
Seoul National University,
Seoul 151-742, South Korea
e-mail: leem0925@snu.ac.kr

Seok Won Kang

WCU Multiscale Design Division,
School of Mechanical and
Aerospace Engineering,
Seoul National University,
Seoul 151-742, South Korea
e-mail: mugens92@snu.ac.kr

Yoon Young Kim

WCU Multiscale Design Division,
School of Mechanical and
Aerospace Engineering,
Institute of Advance Machines and Design,
Seoul National University,
1 Gwanak-ro, Gwanak-gu,
Seoul 151-742, South Korea
e-mail: yykim@snu.ac.kr

1Corresponding author.

Paper presented at ASME 2018 IDETC/CIE (Paper No: DETC2018-85298).Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received June 12, 2018; final manuscript received November 26, 2018; published online February 11, 2019. Assoc. Editor: Joo H. Kim.

J. Mech. Des 141(3), 032301 (Feb 11, 2019) (18 pages) Paper No: MD-18-1443; doi: 10.1115/1.4042212 History: Received June 12, 2018; Revised November 26, 2018

Topology optimization for mechanism synthesis has been developed for the simultaneous determination of the number and dimension of mechanisms. However, these methods can be used to synthesize linkage mechanisms that consist only of links and joints because other types of mechanical elements such as gears cannot be simultaneously synthesized. In this study, we aim to develop a gradient-based topology optimization method which can be used to synthesize mechanisms consisting of both linkages and gears. For the synthesis, we propose a new ground model defined by two superposed design spaces: the linkage and gear design spaces. The gear design space is discretized by newly proposed gear blocks, each of which is assumed to rotate as an output gear, while the linkage design space is discretized by zero-length-spring-connected rigid blocks. Another set of zero-length springs is introduced to connect gear blocks to rigid blocks, and their stiffness values are varied to determine the existence of gears when they are necessary to produce the desired path. After the proposed topology-optimization-based synthesis formulation and its numerical implementation are presented, its effectiveness and validity are checked with various synthesis examples involving gear-linkage and linkage-only mechanisms.

Copyright © 2019 by ASME
Your Session has timed out. Please sign back in to continue.

References

Zhang, C. , Norton, P. R. , and Hammonds, T. , 1984, “Optimization of Parameters for Specified Path Generation Using an Atlas of Coupler Curves of Geared Five-Bar Linkages,” Mech. Mach. Theory, 19(6), pp. 459–466. [CrossRef]
Dooner, D. , 1999, “A Geared 2-dof Mechanical Function Generator,” ASME J. Mech. Des., 121(1), pp. 65–70. [CrossRef]
Nokleby, S. B. , and Podhorodeski, R. P. , 2001, “Optimization-Based Synthesis of Grashof Geared Five-Bar Mechanisms,” ASME J. Mech. Des., 123(4), pp. 529–534. [CrossRef]
Beiner, L. , 2001, “A Three-Gear Linkage for Generating Dwell Motions,” ASME J. Mech. Des., 123(4), pp. 637–640. [CrossRef]
Pennock, G. R. , and Sankaranarayanan, H. , 2003, “Path Curvature of a Geared Seven-Bar Mechanism,” Mech. Mach. Theory, 38(12), pp. 1345–1361. [CrossRef]
Liu, J. , Chang, S. , and Mundo, D. , 2006, “Study on the Use of a Non-Circular Gear Train for the Generation of Figure-8 Patterns,” Proc. Inst. Mech. Eng., Part C, 220(8), pp. 1229–1236. [CrossRef]
Chen, D.-Z. , Shieh, W.-B. , and Yeh, Y.-C. , 2008, “Kinematic Characteristics and Classification of Geared Mechanisms Using the Concept of Kinematic Fractionation,” ASME J. Mech. Des., 130(8), p. 082602. [CrossRef]
Mundo, D. , Gatti, G. , and Dooner, D. , 2009, “Optimized Five-Bar Linkages With Non-Circular Gears for Exact Path Generation,” Mech. Mach. Theory, 44(4), pp. 751–760. [CrossRef]
Buśkiewicz, J. , 2010, “Use of Shape Invariants in Optimal Synthesis of Geared Five-Bar Linkage,” Mech. Mach. Theory, 45(2), pp. 273–290. [CrossRef]
Coros, S. , Thomaszewski, B. , Noris, G. , Sueda, S. , Forberg, M. , Sumner, R. W. , Matusik, W. , and Bickel, B. , 2013, “Computational Design of Mechanical Characters,” ACM Trans. Graphics, 32(4), p. 83. [CrossRef]
Sandhya, R. , Kadam, M. , Balamurugan, G. , and Rangavittal, H. , 2015, “Synthesis and Analysis of Geared Five Bar Mechanism for Ornithopter Applications,” Second International and 17th National Conference on Machines and Mechanisms, Kanpur, Indian, Dec. 16–19.
Kamesh, V. V. , Rao, K. M. , and Rao, A. B. S. , 2017, “An Innovative Approach to Detect Isomorphism in Planar and Geared Kinematic Chains Using Graph Theory,” ASME J. Mech. Des., 139(12), p. 122301. [CrossRef]
Kawamoto, A. , 2004, “Generation of Articulated Mechanisms by Optimization Techniques,” Ph.D. thesis, Technical University of Denmark, Lyngby, Denmark. http://orbit.dtu.dk/files/3025833/Generation%20of%20Articulated%20Mechanisms%20by%20Optimization%20Techniques.pdf
Kawamoto, A. , Bendsøe, M. P. , and Sigmund, O. , 2004, “Planar Articulated Mechanism Design by Graph Theoretical Enumeration,” Struct. Multidiscip. Optim., 27(4), pp. 295–299. [CrossRef]
Kawamoto, A. , Bendsøe, M. P. , and Sigmund, O. , 2004, “Articulated Mechanism Design With a Degree of Freedom Constraint,” Int. J. Numer. Methods Eng., 61(9), pp. 1520–1545. [CrossRef]
Sedlaczek, K. , and Eberhard, P. , 2009, “Topology Optimization of Large Motion Rigid Body Mechanisms With Nonlinear Kinematics,” ASME J. Comput. Nonlinear Dyn., 4(2), p. 021011. [CrossRef]
Ohsaki, M. , and Nishiwaki, S. , 2009, “Generation of Link Mechanism by Shape-Topology Optimization of Trusses Considering Geometrical Nonlinearity,” J. Comput. Sci. Technol., 3(1), pp. 46–53. [CrossRef]
Yoon, G. H. , and Heo, J. C. , 2012, “Constraint Force Design Method for Topology Optimization of Planar Rigid-Body Mechanisms,” Comput.-Aided Des., 44(12), pp. 1277–1296. [CrossRef]
Heo, J. C. , and Yoon, G. H. , 2013, “Size and Configuration Syntheses of Rigid-Link Mechanisms With Multiple Rotary Actuators Using the Constraint Force Design Method,” Mech. Mach. Theory, 64, pp. 18–38. [CrossRef]
Kim, S. I. , and Kim, Y. Y. , 2014, “Topology Optimization of Planar Linkage Mechanisms,” Int. J. Numer. Methods Eng., 98(4), pp. 265–286. [CrossRef]
Kim, S. I. , Kang, S. W. , Yi, Y. S. , Park, J. , and Kim, Y. Y. , 2017, “Topology Optimization of Vehicle Rear Suspension Mechanisms,” Int. J. Numer. Methods Eng., 113(8), pp. 1412–1433.
Kim, Y. Y. , Jang, G.-W. , Park, J. H. , Hyun, J. S. , and Nam, S. J. , 2007, “Automatic Synthesis of a Planar Linkage Mechanism With Revolute Joints by Using Spring-Connected Rigid Block Models,” ASME J. Mech. Des., 129(9), pp. 930–940. [CrossRef]
Kang, S. W. , Kim, S. I. , and Kim, Y. Y. , 2016, “Topology Optimization of Planar Linkage Systems Involving General Joint Types,” Mech. Mach. Theory, 104, pp. 130–160. [CrossRef]
Han, S. M. , Kim, S. I. , and Kim, Y. Y. , 2017, “Topology Optimization of Planar Linkage Mechanisms for Path Generation Without Prescribed Timing,” Struct. Multidiscip. Optim., 56(3), pp. 501–517. [CrossRef]
Kim, B. S. , and Yoo, H. H. , 2012, “Unified Synthesis of a Planar Four-Bar Mechanism for Function Generation Using a Spring-Connected Arbitrarily Sized Block Model,” Mech. Mach. Theory, 49, pp. 141–156. [CrossRef]
Kim, B. S. , and Yoo, H. H. , 2014, “Unified Mechanism Synthesis Method of a Planar Four-Bar Linkage for Path Generation Employing a Spring-Connected Arbitrarily Sized Rectangular Block Model,” Multibody Syst. Dyn., 31(3), pp. 241–256. [CrossRef]
Kim, B. S. , and Yoo, H. H. , 2015, “Body Guidance Syntheses of Four-Bar Linkage Systems Employing a Spring-Connected Block Model,” Mech. Mach. Theory, 85, pp. 147–160. [CrossRef]
Nam, S. J. , Jang, G.-W. , and Kim, Y. Y. , 2012, “The Spring-Connected Rigid Block Model Based Automatic Synthesis of Planar Linkage Mechanisms: Numerical Issues and Remedies,” ASME J. Mech. Des., 134(5), p. 051002. [CrossRef]
Kang, S. W. , and Kim, Y. Y. , 2018, “Unified Topology and Joint Types Optimization of General Planar Linkage Mechanisms,” Struct. Multidiscip. Optim., 57(5), pp. 1955–1983. [CrossRef]
Bendsøe, M. P. , and Sigmund, O. , 1999, “Material Interpolation Schemes in Topology Optimization,” Arch. Appl. Mech., 69(9–10), pp. 635–654.
Svanberg, K. , 1987, “The Method of Moving Asymptotes—A New Method for Structural Optimization,” Int. J. Numer. Methods Eng., 24(2), pp. 359–373. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

(a) A typical geared mechanism we aim to synthesize by the proposed topology optimization method and (b) the proposed block ground model consisting of two design spaces, the linkage design space SL and the gear design space <r>SG (the superscript r in <r>SG denotes the gear ratio)

Grahic Jump Location
Fig. 2

Representation of various joints in the SL-based SBM. The stiffness of the block-connecting and anchoring springs takes on the bound value (kmin and kmax) to simulate various joint states.

Grahic Jump Location
Fig. 3

Representation of gear and link components in several simple mechanisms using the new notations, <>B(l) and <r>B(l)

Grahic Jump Location
Fig. 4

(a) A geared five-bar mechanism (GFBM). (b) The representation of the GFBM by the developed modeling approach. (c) The representation of the GFBM in terms of M-blocks, a floating block, and a gear block.

Grahic Jump Location
Fig. 5

Various configurations inducing the deformation of springs. Each configuration consisting of (a) B(l), B(m) (an adjacent block of B(l)), and the block-connecting spring kCi; (b) B(l), the ground, and the anchoring spring kAj; and (c) B(l), <r>G(l), and a set of gearing springs denoted by <r>kGw.

Grahic Jump Location
Fig. 6

The synthesis of a geared five-bar mechanism having an output gear with a gear ratio of 1 by the proposed formulation: (a) the target mechanism, (b) its representation in terms of rigid and gear blocks, (c) the ground model consisting of SL⊕<1>SG⊕<0.5>SG employed for the synthesis, (d) the convergence history of η¯ and ψ¯max=maxt*∈{1,2,…,T}ψt*, and (e) the values of the design variables (ξi) at convergence

Grahic Jump Location
Fig. 7

The synthesis of a geared five-bar mechanism having an output gear with the gear ratio of 0.5: (a) the target mechanism, (b) its representation in terms of rigid and gear blocks, (c) the ground model consisting of SL⊕<1>SG⊕<0.5>SG employed for the synthesis, (d) the convergence history of η¯ and ψ¯max=maxt*∈{1,2,…,T}ψt*, and (e) the values of the design variables (ξi) at convergence

Grahic Jump Location
Fig. 8

The evolution history of the synthesized mechanism expressed by the proposed block ground model for the synthesis problem defined in Fig. 6. The numbers in the figure represent the block number.

Grahic Jump Location
Fig. 9

The evolution history of the synthesized mechanism expressed by the proposed block ground model for the synthesis problem defined in Fig. 7. The numbers in the figure represent the block number.

Grahic Jump Location
Fig. 10

The synthesis of a geared five-bar mechanism having an output gear with the gear ratio of 1 located at the block 2 location. (a) The target mechanism and (b) its representation in terms of rigid and gear blocks.

Grahic Jump Location
Fig. 11

The evolution history of the synthesized mechanism expressed by the proposed block ground model for the synthesis problem defined in Fig. 10

Grahic Jump Location
Fig. 12

The synthesis of (a) a linkage-only mechanism and (b) a geared mechanism with two gear trains (i.e., two output gears). The ground model consisting of SL⊕<1>SG⊕<0.5>SG is used in applying the proposed topology optimization based synthesis method.

Grahic Jump Location
Fig. 13

The convergence histories of η¯ and ψ¯max=maxt*∈{1,2,…,T}ψt* and the values of the design variables (ξi) at convergence for (a) the problem depicted in Fig. 12(a) and (b) the problem in Fig. 12(b). The design variables having intermediate values are associated with the anchoring springs connected to floating blocks. Therefore, they do not affect the synthesized mechanism configurations.

Grahic Jump Location
Fig. 14

The evolution history of the synthesized mechanism expressed by the proposed block ground model for the synthesis problem defined in Fig. 12(a)

Grahic Jump Location
Fig. 15

The evolution history of the synthesized mechanism expressed by the proposed block model for the synthesis problem defined in Fig. 12(b)

Tables

Errata

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In