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Research Papers

# The Relation Between Helical Spring Compliances With Free and Fixed End Rotations

[+] Author and Article Information
S. J. Burns

Department of Mechanical Engineering, Materials Science Program, Hajim School of Engineering and Applied Sciences,  University of Rochester, Rochester, NY 14627-0133burns@me.rochester.edu

J. Appl. Mech 78(6), 061005 (Aug 24, 2011) (5 pages) doi:10.1115/1.4003739 History: Received June 24, 2010; Revised February 28, 2011; Published August 24, 2011

## Abstract

A helical spring that is constrained to no rotation has a compliance that is typically more than 95% of the compliance of springs constrained to free rotation when restricted to symmetric wires made from materials with Poisson’s ratio between 0 and 1/2. It is shown that the shape of the spring wire can be designed so the spring will not twist when it is extended nor extend when it is twisted. The constrained spring versus a freely rotating spring with the helix angle equal to $π/4$ has the largest reduction in compliance in the limits of beam theory. Spring compliances for torsion and extension with quite complex helical spring geometries are found to be related by a dimensionless ratio of compliances in a very simple equation that only depends on Poisson’s ratio and the helical, spring angle, $ψ$. Springs made from materials with negative Poisson’s ratio, however, can have a very substantial reduction in compliance; the no rotation compliance is zero when Poisson’s ratio is −1. There are large changes in spring compliances for springs with geometric coils that are elongated rectangles or flattened ellipses.

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Copyright © 2011 by American Society of Mechanical Engineers
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## Figures

Figure 4

(a) The term 1 − λ on a symmetric spring versus Poisson’s ratio for a 45 deg spring. (b) A plot of λ for a symmetric spring with ν = 0.5 versus the spring angle, ψ. (c) A plot of λ for an asymmetric spring with ν = 1/31 versus the spring’s shape factor b/h. λ equals 1 when b/a = 1+2ν. (d) λ versus Poisson’s ratio for selected values of b/h equal to 1.0, 0.9, 0.8, 0.7, and 0.6. It is seen that as b/h is decreased λ is further from 1.

Figure 3

A schematic of the load, P, and displacement δ of the load point with the torque, T, and the rotation θ of the applied torque is shown relative to a fixed point

Figure 2

(a) The components of the moment, PR, on a section of the cut spring; one component aligns with the axis of the wire while the second component is perpendicular to the wire axis. (b) The components of the torque T on a section of the cut spring; one component aligns with the axis of the wire while the second component is perpendicular to the wire axis. Note that the two bending components in (a) and (b) are in opposite directions.

Figure 1

(a) Schematic of a spring with a positive helix. (b) Free body diagram of (a) at the cut section there is a shear force equal to P, a moment in the rotational eθ direction of magnitude PR and a moment in the ez direction equal to −T.

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