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The Lagrangian Method for a Basic Bicycle

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DOI: 10.4236/jamp.2014.24007    6,406 Downloads   9,166 Views   Citations
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ABSTRACT

The ground plan in order to disentangle the hard problem of modelling the motion of a bicycle is to start from a very simple model and to outline the proper mathematical scheme: for this reason the first step we perform lies in a planar rigid body (simulating the bicylcle frame) pivoting on a horizontal segment whose extremities, subjected to nonslip conditions, oversimplify the wheels. Even in this former case, which is the topic of lots of papers in literature, we find it worthwhile to pay close attention to the formulation of the mathematical model and to focus on writing the proper equations of motion and on the possible existence of conserved quantities. In addition to the first case, being essentially an inverted pendulum on a skate, we discuss a second model, where rude handlebars are added and two rigid bodies are joined. The geometrical method of Appell is used to formulate the dynamics and to deal with the nonholonomic constraints in a correct way. At the same time the equations are explained in the context of the cardinal equations, whose use is habitual for this kind of problems. The paper aims to a threefold purpose: to formulate the mathematical scheme in the most suitable way (by means of the pseudovelocities), to achieve results about stability, to examine the legitimacy of certain assumptions and the compatibility of some conserved quantities claimed in part of the literature.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Talamucci, F. (2014) The Lagrangian Method for a Basic Bicycle. Journal of Applied Mathematics and Physics, 2, 46-60. doi: 10.4236/jamp.2014.24007.

References

[1] Gantmacher, F.R. (1975) Lectures in Analytical Mechanics, MIR.
[2] Astrom, K.J., Klein, R.E. and Lennartsson, A. (2005) Bicycle Dynamics and Control: Adapted Bicycles for Education and Research. Control Systems, IEEE, 25, 26-47. http://dx.doi.org/10.1109/MCS.2005.1499389
[3] Fajans, J. (2000) Steering in Bicycles and Motorcycles. American Journal of Physics, 7, 654-659.
http://dx.doi.org/10.1119/1.19504
[4] Schwab, A.L., Meijaard, J.P. and Kooijman, J.D.G. (2012) Lateral Dynamics of a Bicycle with a Passive Rider Model: Stability and Controllability. Vehicle System Dynamics, 50, 1209-1224.
http://dx.doi.org/10.1080/00423114.2011.610898
[5] Schwab, A.L., Meijaard, J.P. and Papadopoulos, J.M. (2005) Benchmark Results on the Linearized Equations of Motion of an Uncontrolled Bicycle. KSME International Journal of Mechanical Science and Technology, 19, 292-304.
http://dx.doi.org/10.1007/BF02916147
[6] Ambrosi, D. and Bacciotti, A. (2009) Stabilization of the Inverted Pendulum on a Skate. Differential Equations and Dynamical Systems, 17, 201-215. http://dx.doi.org/10.1007/s12591-009-0016-8
[7] Ambrosi, D. and Bacciotti, A. (2007) The Bicycle Stability as a Control Problem. Politecnico di Torino, Rapporto interno n. 33.
[8] Braun, M. (1978) Differential Equations and Their Applications. Springer-Verlag, New York.
[9] Ambrosi, D., Bacciotti, A. and Ropolo, G. (2008) La matematica della bicicletta. La Matematica nella Società e nella Cultura I, Serie I, 477-492.
[10] Doria, A., Formentini, M. and Tognazzo, M. (2012) Experimental Analysis of Rider Motion in Weave Conditions. Proceedings Experimental and Numerical Analysis of Rider Motion in Weave Conditions, Vehicle System Dynamics, 50, 1247-1260. http://dx.doi.org/10.1080/00423114.2011.621542
[11] Lowell, J. and McKell, H. D. (1982) The Stability of Bicycles. American Journal of Physics, 50, 1106-1112.
http://dx.doi.org/10.1119/1.12893

  
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