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Research Papers: Design Automation

Design of Nonlinear Springs for Prescribed Load-Displacement Functions

[+] Author and Article Information
Christine Vehar Jutte

Department of Mechanical Engineering, University of Michigan, 2231 G.G. Brown Building, 2350 Hayward Street, Ann Arbor, MI 48109-2125cvehar@umich.edu

Sridhar Kota

Department of Mechanical Engineering, University of Michigan, 2231 G.G. Brown Building, 2350 Hayward Street, Ann Arbor, MI 48109-2125kota@umich.edu

J. Mech. Des 130(8), 081403 (Jul 16, 2008) (10 pages) doi:10.1115/1.2936928 History: Received September 25, 2007; Revised April 21, 2008; Published July 16, 2008

A nonlinear spring has a defined nonlinear load-displacement function, which is also equivalent to its strain energy absorption rate. Various applications benefit from nonlinear springs, including prosthetics and microelectromechanical system devices. Since each nonlinear spring application requires a unique load-displacement function, spring configurations must be custom designed, and no generalized design methodology exists. In this paper, we present a generalized nonlinear spring synthesis methodology that (i) synthesizes a spring for any prescribed nonlinear load-displacement function and (ii) generates designs having distributed compliance. We introduce a design parametrization that is conducive to geometric nonlinearities, enabling individual beam segments to vary their effective stiffness as the spring deforms. Key features of our method include (i) a branching network of compliant beams used for topology synthesis rather than a ground structure or a continuum model based design parametrization, (ii) curved beams without sudden changes in cross section, offering a more even stress distribution, and (iii) boundary conditions that impose both axial and bending loads on the compliant members and enable large rotations while minimizing bending stresses. To generate nonlinear spring designs, the design parametrization is implemented into a genetic algorithm, and the objective function evaluates spring designs based on the prescribed load-displacement function. The designs are analyzed using nonlinear finite element analysis. Three nonlinear spring examples are presented. Each has a unique prescribed load-displacement function, including a (i) “J-shaped,” (ii) “S-shaped,” and (iii) constant-force function. A fourth example reveals the methodology’s versatility by generating a large displacement linear spring. The results demonstrate the effectiveness of this generalized synthesis methodology for designing nonlinear springs for any given load-displacement function.

Copyright © 2008 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

The solid black curve defines a nonlinear spring’s load-displacement function and strain energy absorption rate

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Figure 2

Two 1D spring models (cantilever beams with transverse end loads) and their corresponding load-displacement functions. (a) has a free end, and (b) has a horizontally constrained end. The constraint on (b) induces greater nonlinearities in the response.

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Figure 3

A typical spring design generated from the design parametrization used for this nonlinear spring synthesis methodology

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Figure 4

Comparison of a nonlinear spring (coincident input/output point) to a nonlinear compliant mechanism (separate input and output points). Unlike a compliant mechanism, the spring has a specified strain energy absorption rate.

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Figure 5

A prescribed nonlinear load-displacement function with (a) the shape function, (b) the load-range, and (c) the displacement-range

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Figure 6

A prescribed load-displacement curve with target points (A–D). The double arrows are the errors that determine the evaluated design’s SFE.

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Figure 7

When a spring yields before reaching the applied load, the SFE is only evaluated over the spring’s elastic range

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Figure 8

A negative slope in the load-displacement function indicates snap-through buckling

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Figure 9

The topology is parametrized using a branching network of nine compliant beams (splines) that connect the input to various ground points

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Figure 10

Each spline is represented as a cubic B-spline whose shape is controlled with five control points

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Figure 11

(a) A looped B-spline and (b) a reordering of the control points to avoid loops

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Figure 12

Splines P1 and P1-S2

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Figure 13

Design space specifications for the example problems

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Figure 14

Prescribed J-curve with three target points (A, B, and C)

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Figure 15

Generated J-curve nonlinear spring

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Figure 16

Load-displacement function for the resulting J-curve nonlinear spring, SFE=0.88%

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Figure 17

FEA stress contours for J-curve

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Figure 18

Prescribed S-curve with three target points (A, B, and C)

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Figure 19

Generated S-curve nonlinear spring

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Figure 20

Load-displacement function for the resulting S-curve nonlinear spring, SFE=1.21%

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Figure 21

FEA stress contours for S-curve

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Figure 22

Prescribed constant-force curve with three target points (A, B, and C)

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Figure 23

Generated constant-force nonlinear spring

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Figure 24

Load-displacement function for the resulting constant-force nonlinear spring (solid line) and its scaled design (dashed line), SFE=5.34%

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Figure 25

FEA stress contours for constant-force spring

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Figure 26

Constant-force spring prototype; (a) undeformed and (b) deformed configurations

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Figure 27

Comparison of load-displacement plots between the constant-force prototype and FEA results

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Figure 28

Generated planar linear spring

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Figure 29

Load-displacement function for the resulting large displacement linear spring with nine target points

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Figure 30

FEA stress contours for large displacement linear spring

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