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A design sensitivity analysis of bicycle stability and experimental validation

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Abstract

In this study, the sensitivities of weave and capsize speeds with respect to design parameters were calculated on the basis of the linear forms of uncontrolled-bicycle dynamic equations. The significance of the design parameters and the way in which the stable speed range is changed were determined by analyzing the sensitivity curves. Among the seven important parameters (out of 25), head angle was found to be the most dominant, followed by the diameter of front wheel, mass, and moment of inertia, demonstrating the importance of front side design to bicycle stability. The procedure for predicting the stable speed range using sensitivity information was also investigated. When a single parameter was changed, the stable range was determined by that parameter, whereas when multiple parameters were changed, the stable range was determined by summing all the contributions from each parameter. The weave speeds in the nominal and changed configurations yielding the lowest values were measured using an experimental bicycle of variable configuration. A comparison of the measured and predicted values of weave speeds showed a good correlation, demonstrating the validity of the sensitivity-based stability analysis. This study used a novel procedure for predicting the stable speed range through the sensitivity analysis of design parameters related to bicycle stability and validation of the stable speed range with experiment.

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Abbreviations

ϕ :

Roll angle

δ :

Steel angle

q :

Generalized coordinate vector

v :

Forward velocity

R rw :

Radius of the rear wheel

θ :

Angular velocity

b :

Design parameter vector

M :

Symmetric mass matrix

v C 1 :

Damping matrix linear in forward speed v

gk 0 :

Stiffness matrix

v 2 K 2 :

Stiffness matrix quadratic in the forward speed

f :

Applied torque vector

Vw:

Weave speed

Vc :

Capsize speed

α:

Head angle

ε :

Caster offset

w:

Wheel base

m fw :

Mass of front wheel

m rw :

Mass of rear wheel

m ff :

Mass of front frame

m rf :

Mass of rear frame

d rf :

Distance from rear frame mass center to rear wheel

h rf :

Mass of rear frame

d ff :

Distance from front frame mass center to rear wheel

h ff :

Height of front frame mass center

Axx, Ayy, Azz :

Mass moments of inertia of rear wheel

Bxx, Byy, Bzz :

Mass moments of inertia of rear frame

Cxx, Cyy, Czz :

Mass moments of inertia of front frame

Dxx, Dyy, Dzz :

Mass moments of inertia of front wheel

D fw :

Diameter of front wheel

Dyy :

Front wheel moment of inertia with respect to hub

λ :

Eigenvalue

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Authors and Affiliations

  1. School of Mechanical and Biomedical Engineering, Kangwon National University, Chuncheon City, Kangwon-do, 24341, Korea

    Sheng-peng Zhang & Tae-oh Tak

Authors
  1. Sheng-peng Zhang
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  2. Tae-oh Tak
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Corresponding author

Correspondence to Tae-oh Tak.

Additional information

Sheng-peng Zhang is a Ph.D. student and received his M.S. degree from the Department of Mechanical and Biomedical Engineering at Kangwon University in 2015. His research interests include multi-body dynamics and vehicle dynamics and control.

Tae-oh Tak received his B.S. and M.S degrees in Mechanical Design and Production Engineering from the Seoul National University in Korea in 1982 and 1984, respectively. He obtained his Ph.D. degree from the University of Iowa, USA in 1990. He is currently a Professor at the Department of Mechanical and Biomedical Engineering at Kangwon National University in Chuncheon, Korea. His research interests are multibody dynamics and sensitivity analysis.

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Zhang, Sp., Tak, To. A design sensitivity analysis of bicycle stability and experimental validation. J Mech Sci Technol 34, 3517–3524 (2020). https://doi.org/10.1007/s12206-020-0803-2

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  • DOI: https://doi.org/10.1007/s12206-020-0803-2

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