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RESEARCH PAPERS

Design of a Compliant Mechanism to Modify an Actuator Characteristic to Deliver a Constant Output Force

[+] Author and Article Information
C. B. Pedersen

Department of Engineering, University of Cambridge, Trumpington Street, Cambridge, CB2 1PZ, United Kingdomcbwp2@eng.cam.ac.uk

N. A. Fleck

Department of Engineering, University of Cambridge, Trumpington Street, Cambridge, CB2 1PZ, United Kingdomnafl@eng.cam.ac.uk

G. K. Ananthasuresh

Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560012, Indiasuresh@mecheng.iisc.ernet.in

J. Mech. Des 128(5), 1101-1112 (Dec 06, 2005) (12 pages) doi:10.1115/1.2218883 History: Received June 26, 2005; Revised December 06, 2005

Topology and size optimization methods are used to design compliant mechanisms that produce a constant output force for a given actuator characteristic of linearly decreasing force versus displacement. The design procedure consists of two stages: (i) topology optimization using two-dimensional (2-D) continuum parametrization, and (ii) size optimization of the beam-element abstraction derived from the continuum topology solution. The examples considered are based upon electrostatic microactuators used widely in microsystems. The procedure described here provides conceptual as well as practically usable designs for compliant transmission mechanisms with a constant output force characteristic. For any given topology design, the maximum achievable constant force over a given displacement range is determined. Ideal rigid-body and spring-equipped mechanisms are analyzed and their features are used to compare with the compliant solutions obtained.

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Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 1

(a) Examples of actuators and their characteristics. (b) The generation of a desired constant force characteristic by a compliant transmission mechanism.

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Figure 2

Spring mechanism with linear kinematics

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Figure 3

(a) The characteristic of the actuator of constant stiffness K for two different stall forces P1 and P2. (b) The actual transformed output forces of the compliant mechanism are Fj. In the optimization the errors ε between maximized constant forces Fj* and the actual output forces Fj are minimized in given collocation points of vi. These collocation points vi are indicated with numbered circles. The prescribed displacements vi of the collocation points lead through the equilibrium to the corresponding displacements of the actuator indicated by circles on the actuator response.

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Figure 4

The design domain for the compliant transmission mechanism. The mechanism is fully clamped on foundations of length s. (a) The actuator is modeled by the force P. At the output point the prescribed displacement v is applied in the x1-direction leading to the reaction force F. The mechanism is to be designed to make F constant for a prescribed maximum output displacement, vmax. (b) A small transverse force Q is applied as an independently loadcase in the x2-direction leading to the transverse stiffness k=Q∕w where w is the displacement. The transverse stiffness is an imposed constraint on the compliant mechanism.

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Figure 5

Flow diagrams for (a) the topology optimization procedure and (b) the size optimization procedure.

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Figure 6

Identification procedure for converting the components in the topology optimized structures into components consisting of beam elements. The beam model is used for further size optimization.

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Figure 7

(a) Mechanism A is designed to have three constant output forces using three stall forces (J=3). (b) Mechanism B is designed to have one constant force using one stall force (J=1). The circles on the output responses indicate the collocation points where the forces are measured. The circles on the input responses indicate the input response for the corresponding collocation point. (vmax=2.0μm, K=10N∕m).

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Figure 8

Effect of actuator stiffness K upon the optimized output force. The mechanisms are designed using different values of the actuator stiffness K. (J=1, P2=0.10mN, I=5, vmax=2.0μm).

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Figure 9

The effect of the output stroke and the number of collocation points upon the optimized design. Mechanisms C and D are designed for vmax=10μm. Mechanism C is designed using five collocation points (I=5) whereas mechanism D is designed using nine collocation points (I=9). The circles on the output responses indicate the collocation points where the forces are measured. The circles on the input responses indicate what the input response is for the corresponding collocation points. (J=1, P2=0.10mN, K=10N∕m, vmax=10.0μm).

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Figure 10

The input displacement u and the energy efficiency η=∫0vFdv′∕∫0uPdu′ as a function of output displacement v. The mechanisms B and C are compared with the rigid-body mechanism. Mechanism C is also compared to a size optimized beam model solution.

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Figure 11

Effect of maximum output displacement vmax and the required transverse stiffness k upon the size optimization for a mechanism with parent topology C. The black elements of the initial mechanism are interpreted to be most rigid units (hmin=4.0μm) and the gray elements are interpreted to be the most flexible parts (hmin=0.2μm). (J=1, P2=0.10mN, K=10N∕m, F2*=0.0080mN, I=5, hmin=0.2μm).

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Figure 12

Conversion of the parent topology optimized mechanism C in Fig. 9 into a beam model. Symmetry is applied. The black elements of the initial mechanism are interpreted to be the most rigid parts (hmin=4.0μm) and the gray elements are interpreted to be the most flexible parts (hmin=0.2μm). The mechanisms are shape optimized using the prescribed constant forces (a) F2*=0.0080mN, (b) F2*=0.0010mN, (c) F2*=0.0040mN, (d) F2*=0.0120mN, and (e) F2*=0.0160mN. (J=1, P2=0.10mN, K=10N∕m, I=5, vmax=10.0μm).

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Figure 13

Interpretation of parent topology optimized mechanism D in Fig. 9 into three alternative different beam models where symmetry is applied. The black elements of the initial mechanism are interpreted to be most rigid parts (hmin=4.0μm) and the gray elements are interpreted to be the most flexible parts (hmin=0.2μm). (J=1, P2=0.10mN, K=10N∕m, F2*=0.0080mN, I=5, vmax=10.0μm).

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Figure 14

(a) Continuum model of the mechanism in Fig. 1 (F2*=0.0120mN and vmax=10.0μm). (b) Continuum model of the mechanism in Fig. 1 (F2*=0.0080mN, vmax=2.0μm and k=20N∕m). (c) Continuum model of the mechanism in Fig. 1 (F2*=0.0080mN and vmax=10.0μm).

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