Technical Brief

Tailoring Strongly Nonlinear Negative Stiffness

[+] Author and Article Information
F. Liu

Department of Automotive Engineering,
Yancheng Institute of Technology,
Yancheng 224051, China
e-mail: fuhaoliu@mvrlab.com

S. Theodossiades

Department of Mechanical Engineering,
Loughborough University,
Loughborough LE11 3TU, UK
e-mail: S.Theodossiades@lboro.ac.uk

D. M. McFarland

Department of Aerospace Engineering,
University of Illinois at Urbana-Champaign,
Urbana, IL 61801
e-mail: dmmcf@illinois.edu

A. F. Vakakis

Department of Mechanical Science and Engineering,
University of Illinois at Urbana-Champaign,
Urbana, IL 61801
e-mail: avakakis@illinois.edu

L. A. Bergman

Department of Aerospace Engineering,
University of Illinois at Urbana-Champaign,
Urbana, IL 61801
e-mail: lbergman@illinois.edu

1Corresponding author.

Contributed by the Design Innovation and Devices of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received September 20, 2012; final manuscript received October 2, 2013; published online December 11, 2013. Assoc. Editor: Diann Brei.

J. Mech. Des 136(2), 024501 (Dec 11, 2013) (7 pages) Paper No: MD-12-1465; doi: 10.1115/1.4025794 History: Received September 20, 2012; Revised October 02, 2013

Negative, nonlinear stiffness elements have been recently designed as configurations of pairs or groups of linear springs. We propose a new design of such a system by using a single linear spring with its moving end rolling on a path described by an equation of varying complexity. We examine the effect that the selection of the path has on the size of the deflection regime where negative stiffness is achieved. The stability properties of the equilibrium positions of the system are also investigated, highlighting the influence that the complexity of the path equation brings. The latter naturally affects the characteristics of the forcing functions around these positions. It is demonstrated that the properties of the system can be tailored according to the nature of the equation used and we show how essentially nonlinear negative stiffness elements, (i.e., with no linear parts) can be designed. These results provide a useful standpoint for designers of such systems, who wish to achieve the desired properties in reduced space, which is a common requirement in modern applications.

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Fig. 1

Schematic representation of the proposed design exhibiting negative stiffness

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Fig. 2

Examined paths of motion for the free end of the linear spring: (a) y = c1x, (b) y = c2x2, and (c) y = c3x3

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Fig. 3

Force-deflection characteristics of the system for the examined paths of motion of the linear spring's free end: (a) y = c1x, (b) y = c2x2, and (c) y = c3x3

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Fig. 4

Nondimensional stiffness of the examined system, corresponding to the cases of Fig. 3: (a) y = c1x, (b) y = c2x2, and (c) y = c3x3

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Fig. 5

Comparison of the nondimensional stiffness characteristics of the examined system to those of two oblique springs [10] for (a) γ = 0.1 and (b) γ = 0.4

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Fig. 6

(a) Force and (b) stiffness characteristics of the system when y = c3x3 (c3 = 100, 300, and 500)

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Fig. 7

Space of the equilibrium positions y∧2 and y∧3, when y = c1x

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Fig. 8

∂2P/∂y∧2-y∧ plots of the system when y = c1x (c1 = 1) for γ = 0.4 and γ = 0.6

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Fig. 9

Stiffness variation around equilibrium position y∧=0 with respect to (a) γ and (b) c1

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Fig. 10

Stiffness variation around equilibrium position y∧ = -1-(γ+x∧)2 with respect to (a) γ and (b) c1



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