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Research Articles

Effects of the Nonlinear Elastic Behavior of Bicycle Chain on Transmission Efficiency

[+] Author and Article Information
James B. Spicer

Department of Materials Science
and Engineering,
The Johns Hopkins University,
3400 North Charles Street,
Baltimore, MD 21218
e-mail: spicer@jhu.edu

Manuscript received November 30, 2011; final manuscript received August 3, 2012; accepted manuscript posted August 23, 2012; published online January 22, 2013. Assoc. Editor: Wei-Chau Xie.

J. Appl. Mech 80(2), 021005 (Jan 22, 2013) (8 pages) Paper No: JAM-11-1455; doi: 10.1115/1.4007431 History: Received November 30, 2011; Revised August 03, 2012; Accepted August 23, 2012

A model for nonfrictional power loss in derailleur-type, bicycle chain drives is developed to identify factors that influence transmission efficiency. Existing treatments of chain drive efficiency consider frictional losses but these do not explain the measured tension dependence of power losses and efficiencies for derailleur-type systems. Based on a nonlinear, spring-mass, mechanical transmission line, the model developed in this work shows that losses can be related to harmonic generation and dispersion in the chain. The nonlinear response leading to harmonic generation results from elastic contact at pin-bushing interfaces while dispersion is related to the periodic nature of the chain construction. Using this approach, the tension-dependence of power loss and efficiency are modeled and the influences of various chain-related characteristics on efficiency are assessed. If Hertzian contact descriptions are used, then the dependence of loss and efficiency on pin-bushing clearance, contact length and modulus can be estimated. Modeled results agree with experiment and show that power loss decreases with increasing chain tension and that efficiency varies nearly linearly with the reciprocal of the chain tension under operational conditions that are typical for bicycle chain drives. Significant increases to the power transmission efficiency of bicycle chain drives in derailleur-based systems could be achieved by altering the geometries and materials of current chain components.

FIGURES IN THIS ARTICLE
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Copyright © 2013 by ASME
Topics: Chain , Bicycles , Bushings , Tension
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References

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Figures

Grahic Jump Location
Fig. 1

Schematic illustration of the nonlinear, spring-mass system used to model the elastic response of bicycle chain. The masses correspond to chain half-links (assumed to have equal masses) and the springs correspond to contacts at pin-bushing interfaces.

Grahic Jump Location
Fig. 3

Measured chain extension as a function of applied load (points) for a commercially-available bicycle chain compared to the results of modeling chain stiffness using Hertzian contact mechanics (solid curve). Equation (14) was fit to the measured chain response by adjusting values for physical parameters.

Grahic Jump Location
Fig. 2

Schematic illustration of chain link components (some are omitted for clarity of illustration) and the corresponding geometrical quantities used to describe Hertzian contact at the pin-bushing interface

Grahic Jump Location
Fig. 4

Chain transmission efficiency as a function of the reciprocal of the chain tension derived from Eqs. (13) and (15). Efficiency is shown for various pin-bushing clearance parameters. Measurements of commercially-available chains indicate that pin-bushing clearances can be within the range shown, 0.00125–0.01 mm−1.

Grahic Jump Location
Fig. 5

Variation of chain transmission efficiency as a function of the Young's modulus of the pin and bushing—both are assumed to have the same stiffness. Efficiency is shown for three different chain tensions. For these results, the Poisson ratio has been fixed at 0.29 and the pin-bushing clearance parameter was 0.003 mm−1.

Grahic Jump Location
Fig. 6

Measured chain extension as a function of applied load (points) for a commercially-available bicycle chain compared to the results of modeling chain stiffness using Hertzian contact mechanics (solid curve) or using a hyperbolic fit (dashed curve). Either model can be used to represent the measured chain behavior but each yields distinct values for the chain stiffnesses.

Grahic Jump Location
Fig. 7

Chain transmission efficiency as a function of the reciprocal of the chain tension. Data points represent actual measurements performed on a derailleur-type chain drive operating in the offset condition for the sprocket combinations noted, 52-21 and 52-11 (number of teeth on the proximal-distal sprockets) [7]. Solid curves represent calculations of efficiency using Eq. (13) along with chain stiffnesses based on the hyperbolic chain extension model.

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