Hands-free circular motions of a benchmark bicycle

2007 ◽  
Vol 463 (2084) ◽  
pp. 1983-2003 ◽  
Author(s):  
Pradipta Basu-Mandal ◽  
Anindya Chatterjee ◽  
J.M Papadopoulos
Keyword(s):  
Nonlinear Equations ◽  
Dynamic Balance ◽  
Equations Of Motion ◽  
Ground Plane ◽  
Rear Wheel ◽  
Front Wheel ◽  
Static Equilibrium ◽  
Straight Line ◽  
Nonlinear Equations Of Motion

We write nonlinear equations of motion for an idealized benchmark bicycle in two different ways and verify their validity. We then present a complete description of hands-free circular motions. Three distinct families exist. (i) A handlebar-forward family, starting from capsize bifurcation off straight-line motion and ending in unstable static equilibrium, with the frame perfectly upright and the front wheel almost perpendicular. (ii) 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. (iii) 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. Small sections of two families are stable.

Direct Linearization of Continuous and Hybrid Dynamical Systems

2009 ◽  
Vol 4 (3) ◽  
Author(s):  
Julie J. Parish ◽  
John E. Hurtado ◽  
Andrew J. Sinclair
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Equations ◽  
Equilibrium Configuration ◽  
Equations Of Motion ◽  
Hybrid Dynamical Systems ◽  
Linearized Equations ◽  
Direct Linearization ◽  
Taylor Series Expansions ◽  
Nonlinear Equations Of Motion ◽  
And Control

Nonlinear equations of motion are often linearized, especially for stability analysis and control design applications. Traditionally, the full nonlinear equations are formed and then linearized about the desired equilibrium configuration using methods such as Taylor series expansions. However, it has been shown that the quadratic form of the Lagrangian function can be used to directly linearize the equations of motion for discrete dynamical systems. This procedure is extended to directly generate linearized equations of motion for both continuous and hybrid dynamical systems. The results presented require only velocity-level kinematics to form the Lagrangian and find equilibrium configuration(s) for the system. A set of selected partial derivatives of the Lagrangian are then computed and used to directly construct the linearized equations of motion about the equilibrium configuration of interest, without first generating the entire nonlinear equations of motion. Given an equilibrium configuration of interest, the directly constructed linearized equations of motion allow one to bypass first forming the full nonlinear governing equations for the system. Examples are presented to illustrate the method for both continuous and hybrid systems.

A New Displacement Form for the Nonlinear Equations of Motion of Shells of Revolution

1985 ◽  
Vol 52 (3) ◽  
pp. 507-509 ◽  
Author(s):  
J. G. Simmonds
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Equations ◽  
Equations Of Motion ◽  
Shells Of Revolution ◽  
Governing Equations ◽  
Hoop Strain ◽  
Axisymmetric Motion ◽  
Nonlinear Equations Of Motion ◽  
Surface Loads ◽  
Theory Of Shells

In the theory of shells of revolution undergoing torsionless, axisymmetric motion, an extensional and a bending hoop strain are introduced that are linear in the displacements, regardless of the magnitudes of the strains and the meridional rotation. The resulting equations of motion and boundary conditions are derived and some common conservative surface loads are listed along with their potentials. The governing equations appear to be the simplest possible in terms of displacements.

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