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|Title: ||Studies On The Dynamics And Stability Of Bicycles|
|Authors: ||Basu-Mandal, Pradipta|
|Advisors: ||Chatterjee, Anindya|
|Keywords: ||Bicycles - Dynamics|
Lagrange's Equations of Motion
Linearized Equations of Motion
Bicycles - Circular Motions
Newton-Euler Equations of Motion
|Submitted Date: ||Sep-2007|
|Series/Report no.: ||G22204|
|Abstract: ||This thesis studies the dynamics and stability of some bicycles. The dynamics of idealized bicycles is of interest due to complexities associated with the behaviour of this seemingly simple machine. It is also useful as it can be a starting point for analysis of more complicated systems, such as motorcycles with suspensions, frame ﬂexibility and thick tyres. Finally, accurate and reliable analyses of bicycles can provide benchmarks for checking the correctness of general multibody dynamics codes.
The ﬁrst part of the thesis deals with the derivation of fully nonlinear diﬀerential equations of motion for a bicycle. Lagrange’s equations are derived along with the constraint equations in an algorithmic way using computer algebra.Then equivalent equations are obtained numerically using a Newton-Euler formulation. The Newton-Euler formulation is less straightforward than the Lagrangian one and it requires the solution of a bigger system of linear equations in the unknowns. However, it is computationally faster because it has been implemented numerically, unlike Lagrange’s equations which involve long analytical expressions that need to be transferred to a numerical computing environment before being integrated. The two sets of equations are validated against each other using consistent initial conditions. The match obtained is, expectedly, very accurate.
The second part of the thesis discusses the linearization of the full nonlinear equations of motion. Lagrange’s equations have been used.The equations are linearized and the corresponding eigenvalue problem studied. The eigenvalues are plotted as functions of the forward speed ν of the bicycle. Several eigenmodes, like weave, capsize, and a stable mode called caster, have been identiﬁed along with the speed intervals where they are dominant. The results obtained, for certain parameter values, are in complete numerical agreement with those obtained by other independent researchers, and further validate the equations of motion. The bicycle with these parameters is called the benchmark bicycle.
The third part of the thesis makes a detailed and comprehensive study of hands-free circular motions of the benchmark bicycle. Various one-parameter families of circular motions have been identiﬁed. Three distinct families exist: (1)A handlebar-forward family, starting from capsize bifurcation oﬀ straight-line motion, and ending in an unstable static equilibrium with the frame perfectly upright, and the front wheel almost perpendicular. (2) A handlebar-reversed family, starting again from capsize bifurcation, but ending with the front wheel again steered straight, the bicycle spinning inﬁnitely fast in small circles while lying ﬂat in the ground plane. (3) Lastly, a family joining a similar ﬂat spinning motion (with handlebar forward), to a handlebar-reversed limit, circling in dynamic balance at inﬁnite speed, with the frame near upright and the front wheel almost perpendicular; the transition between handlebar forward and reversed is through moderate-speed circular pivoting with the rear wheel not rotating, and the bicycle virtually upright.
In the fourth part of this thesis, some of the parameters (both geometrical and inertial) for the benchmark bicycle have been changed and the resulting diﬀerent bicycles and their circular motions studied showing other families of circular motions.
Finally, some of the circular motions have been examined, numerically and analytically, for stability.|
|Appears in Collections:||Mechanical Engineering (mecheng)|
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