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Research Papers

Design of Single, Multiple, and Scaled Nonlinear Springs for Prescribed Nonlinear Responses

[+] Author and Article Information
Christine Vehar Jutte

Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109cvehar@umich.edu

Sridhar Kota

Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109kota@umich.edu

J. Mech. Des 132(1), 011003 (Dec 17, 2009) (10 pages) doi:10.1115/1.4000595 History: Received November 16, 2008; Revised September 21, 2009; Published December 17, 2009; Online December 17, 2009

Nonlinear springs enhance the performance of many applications including prosthetics, microelectromechanical systems devices, and vibration absorption systems. This paper describes a comprehensive approach to developing compliant elements of prescribed nonlinear stiffness. It presents a generalized methodology for designing a single planar nonlinear spring for a prescribed load-displacement function. The spring’s load-range, displacement-range, and nonlinear behavior are matched using this methodology, while also addressing stress, material, stability, and space constraints. Scaling guidelines are included within the optimization to relax the constraints on the solution space. Given the nonlinear nature of the spring designs, this paper further investigates their function in new configurations. Compliant structures with customized elastic properties are constructed by exploiting symmetry and by arranging nonlinear springs in series and/or in parallel. Scaling guidelines are used to meet new design specifications. The guidelines allow adjustment of load-range, displacement-range, material, and the overall footprint while preserving the spring’s nonlinear behavior without violating stress constraints. Various examples are provided throughout the paper to demonstrate the implementation and merit of these design approaches.

Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 4

The shape function error (SFE) is evaluated at target points (A–D) along the prescribed shape function. To meet the stress constraint, the generated spring design is only evaluated over its allowable stress range. Subsequent scaling procedures enable the generated design to meet the prescribed load-range and, in certain cases, the displacement-range.

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Figure 5

A plot illustrating how scaling (i) enables a smaller-scaled problem (dotted curve) to have a range of acceptable displacements and (ii) converts the smaller-scaled design into a final design (solid line) that meets the required load-range (Ftarget) and displacement-range (dtarget)

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Figure 6

A nonlinear spring (unit cell) arranged in a two-dimensional array

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Figure 7

Tiered arrangements of a J-curve spring

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Figure 8

Load-displacement responses for the J-curve unit cell and its tiered arrangements

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Figure 9

Design space specifications for the original and smaller-scaled problem

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Figure 10

Load-displacement functions for the generated J-curve design (small-scale) and its final scaled design. The final design meets the specifications for the shape function (J-curve), load-range (10 N), and displacement-range (20 mm).

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Figure 11

Final J-curve design in its undeformed and deformed configurations

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Figure 12

Load-displacement functions for the generated S-curve design (small-scale) and its final scaled design. The final design meets the specifications for the shape function (S-curve), load-range (75 N), and displacement-range (80 mm).

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Figure 13

Final S-curve design in its undeformed and deformed configurations

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Figure 14

Instron 8516 test data of the seat assembly with the foam (34)

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Figure 15

Final prototype without the foam seat cushion and covering (34)

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Figure 1

An illustration of the design problem, including design specifications and a potential solution

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Figure 2

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 3

Overview of generalized synthesis methodology using a constant-force spring example in Ref. 29

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