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Research Papers: Mechanisms and Robotics

A Building Block Approach to the Conceptual Synthesis of Compliant Mechanisms Utilizing Compliance and Stiffness Ellipsoids

[+] Author and Article Information
Charles J. Kim

Bucknell Universitycharles.kim@bucknell.edu

Yong-Mo Moon

Worcester Polytechnic Institutemoon@wpi.edu

Sridhar Kota

University of Michigankota@umich.edu

The forces and displacements are “parallel” if they may be related as $ũ=λf̃$. In this case the scalar multiplier $λ$ is the corresponding eigenvalue.

$ϕc$ and $n3$ are not relevant since they are not initially specified; i.e., they can take on any value and satisfy the initial problem specifications. It is also assumed that the initial problem specifications are posed with attention given to the available compliance ellipsoid magnitudes in the building block library. The available magnitudes may be readily determined using parametrized design maps similar to those shown in the Appendix. As discussed in Sec. 23, modifying the in-plane thickness effectively scales the magnitude of a building block compliance ellipsoid while leaving the other ellipsoid characteristics largely unchanged. Thus, it is assumed that the building block magnitude may be tuned to match the specified compliance ellipsoid magnitude.

J. Mech. Des 130(2), 022308 (Jan 02, 2008) (11 pages) doi:10.1115/1.2821387 History: Received December 19, 2006; Revised March 30, 2007; Published January 02, 2008

Abstract

In this paper, we investigate a methodology for the conceptual synthesis of compliant mechanisms based on a building block approach. The building block approach is intuitive and provides key insight into how individual building blocks contribute to the overall function. We investigate the basic kinematic behavior of individual building blocks and relate this to the behavior of a design composed of building blocks. This serves to not only generate viable solutions but also to augment the understanding of the designer. Once a feasible concept is thus generated, known methods for size and geometry optimization may be employed to fine-tune performance. The key enabler of the building block synthesis is the method of capturing kinematic behavior using compliance ellipsoids. The mathematical model of the compliance ellipsoids facilitates the characterization of the building blocks, transformation of problem specifications, decomposition into subproblems, and the ability to search for alternate solutions. The compliance ellipsoids also give insight into how individual building blocks contribute to the overall kinematic function. The effectiveness and generality of the methodology are demonstrated through two synthesis examples. Using only a limited set of building blocks, the methodology is capable of addressing generic kinematic problem specifications for compliance at a single point and for a single-input, single-output compliant mechanism. A rapid prototype of the latter demonstrates the validity of the conceptual solution.

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Figures

Figure 2

CDB

Figure 3

CBB

Figure 4

Deformable body with fixed boundary condition along Ω and applied load at point P

Figure 5

C̃p maps the unit force sphere to the compliance ellipsoid. PCV, SCV, and TCV correspond to the semiaxes of the compliance ellipsoid.

Figure 6

(a) Compliance ellipsoid with PCV aligned with the ux-uθ plane makes an angle ψ with the uθ axis. The compliance ellipsoid is cut by a plane perpendicular to the PCV. Both the SCV and TSV lie in this plane. This plane is also the ux-uy plane rotated by ψ about the uy axis. (b) ϕ is measured from the ux′ axis to the SCV.

Figure 7

Flowchart of proposed synthesis methodology

Figure 8

(a) The problem specifications given as a translation and rotation and (b) transformed to the target compliance ellipsoid

Figure 9

Error between the target compliance ellipsoid and the ellipsoid properties of various geometries of the CDB (l1=60mm). Minimum error=0.064.

Figure 10

Two dimensional representation of decomposition

Figure 11

(a) Possible combinations of ψs and ϕs for the target stiffness ellipsoid; (b) possible combinations of n2s and n3s for the target stiffness ellipsoid

Figure 1

Construction of basic building blocks. The CDB and the CBB are higher order abstractions of a cantilevered beam.

Figure 12

(a) Conceptual design matching desired ellipsoid characteristics. (b) The design displaces along the desired direction under a unit load. (c) Alternate solutions may be obtained by varying free choices (ϕc and n3c).

Figure 13

The general compliant mechanism synthesis problem may be addressed by submechanisms synthesized using the methodology presented in Sec. 4.

Figure 14

The general planar compliant mechanism problem is decomposed into three single-point subproblems for the (i) input constraint, (ii) coupling, and (iii) output constraint. Solutions for the subproblems are obtained using the method presented for the single-point synthesis.

Figure 15

(a) Finite element analysis of composed mechanism. (b) Close-up view of output displacement shown. (c) Physical prototype of mechanism.

Figure 16

Maximum variation values from the mean of compliance ellipsoid parameters while varying h between 5mm and 15mm. Values are shown for α∊[0deg,170deg] (l1=60mm, l2norm=1, b=5mm, E=25GPa).

Figure 17

ψc for different length scales of the CDB (b=5mm, h=1mm, E=2.5GPa)

Figure 18

n2c for different length scales of the CDB (b=0.5mm, h=1mm, E=2.5GPa)

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