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Research Papers

# Curve Decomposition for Large Deflection Analysis of Fixed-Guided Beams With Application to Statically Balanced Compliant Mechanisms

[+] Author and Article Information
Charles Kim1

Associate Professor of Mechanical Engineering, Bucknell University, Lewisburg, PA 17837charles.kim@bucknell.edu

Donna Ebenstein

Assistant Professor of Biomedical Engineering, Bucknell University, Lewisburg, PA 17837donna.ebenstein@bucknell.edu

Some numerical solvers index the argument of the elliptical integral with the square of the argument (i.e., $sin2α2$) Caution must be exercised in determining the argument to these functions.

i.e., along a straight line with no change in orientation. These boundary conditions are distinct from Qiu et al.  [11] where rotation is permitted on the moving end of the beam.

Note that these deformation modes differ from those found in Qiu et al.  [11] due to the rectilinear boundary condition on the guided end.

Equation (13) is equivalent to Eqn. (21) when $φ*=π2$.

1

Corresponding author.

J. Mechanisms Robotics 4(4), 041009 (Sep 17, 2012) (9 pages) doi:10.1115/1.4007488 History: Received August 18, 2011; Revised August 03, 2012; Published September 17, 2012; Online September 17, 2012

## Abstract

Statically balanced compliant mechanisms require no holding force throughout their range of motion while maintaining the advantages of compliant mechanisms. In this paper, a postbuckled fixed-guided beam is proposed to provide the negative stiffness to balance the positive stiffness of a compliant mechanism. To that end, a curve decomposition modeling method is presented to simplify the large deflection analysis. The modeling method facilitates parametric design insight and elucidates key points on the force–deflection curve. Experimental results validate the analysis. Furthermore, static balancing with fixed-guided beams is demonstrated for a rectilinear proof-of-concept prototype.

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## Figures

Figure 1

Fixed-guided beam. The angle ζ represents the angle between the horizontal at the fixed end and the line segment between the fixed and guided ends.

Figure 4

Deformed curves consisting of two inflection points have the same curvature at the fixed and free ends. The curves can assume two configurations (a) and (b), but can both be rearranged as a curve consisting of four elastica curves.

Figure 5

Locus of end-displacements available with a single inflection point. Dots correspond to θ*→α.

Figure 7

Normalized force–deflection curves

Figure 8

Normalized moment versus ζ. The normalized moment is always greater at the branch transition than at the bistable configuration.

Figure 9

Test setup. (a) Top view. (b) Perspective view shows fixed-guided beams and parallelogram constraint more clearly. All structural parts are composed of aluminum. (c) Schematic of test set-up from side-view.

Figure 10

Comparison of uncorrected and pile-up corrected modulus values at each peak load. Mean values of uncorrected and corrected modulus for the pooled data (over all loads) are shown as dashed lines for comparison.

Figure 11

Comparison of force–displacement behavior from experiment and theory. The experiment and theory show close agreement. Note that the experimental curve was obtained by subtracting out the stiffness of the rectilinear constraint. Values for modulus of elasticity are theory (102 Gpa), theory high (105 GPa), and theory low (99 GPa)

Figure 12

Force–displacement for the statically balanced double-parallelogram mechanism. The overall stiffness and force is very close to zero throughout the range of motion.

Figure 2

Figure 3

Deformation with a single inflection point at C. The curvature and internal moment at A and E are equal and opposite. The curve can decomposed as two elastica curves between BC and DC and two partial elastica curves between AB and DE.

Figure 6

If snap-through occurs, four key points may be identified and related to the curve geometry.

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