We present canonical linearized equations of motion for the Whipple bicycle model consisting of four rigid laterally symmetric ideally hinged parts: two wheels, a frame and a front assembly. The wheels are also axisymmetric and make ideal knife-edge rolling point contact with the ground level. The mass distribution and geometry are otherwise arbitrary. This conservative non-holonomic system has a seven-dimensional accessible configuration space and three velocity degrees of freedom parametrized by rates of frame lean, steer angle and rear wheel rotation. We construct the terms in the governing equations methodically for easy implementation. The equations are suitable for e.g. the study of bicycle self-stability. We derived these equations by hand in two ways and also checked them against two nonlinear dynamics simulations. In the century-old literature, several sets of equations fully agree with those here and several do not. Two benchmarks provide test cases for checking alternative formulations of the equations of motion or alternative numerical solutions. Further, the results here can also serve as a check for general purpose dynamic programs. For the benchmark bicycles, we accurately calculate the eigenvalues (the roots of the characteristic equation) and the speeds at which bicycle lean and steer are self-stable, confirming the century-old result that this conservative system can have asymptotic stability.

В работе представлены линеаризованные уравнения движения для модели велосипеда, впервые предложенной в работе [86]. Данная модель состоит из четырех продольно симметричных частей, соединенных между собой идеальными шарнирами: двух колес, рамы и переднего узла-руля и вилки. Колеса предполагаются осесимметричными бесконечно тонкими дисками, движущимися без проскальзывания по опорной поверхности. В остальном геометрия и распределение масс в модели предполагаются произвольными. Данная консервативная неголономная система имеет семь степеней свободы. В линеаризованных уравнениях движения из этих степеней свободы существенную роль играют только три:

A riderless bicycle can automatically steer itself so as to recover from falls. The common view is that this self-steering is caused by gyroscopic precession of the front wheel, or by the wheel contact trailing like a caster behind the steer axis. We show that neither effect is necessary for self-stability. Using linearized stability calculations as a guide, we built a bicycle with extra counter-rotating wheels (canceling the wheel spin angular momentum) and with its front-wheel ground-contact forward of the steer axis (making the trailing distance negative). When laterally disturbed from rolling straight, this bicycle automatically recovers to upright travel. Our results show that various design variables, like the front mass location and the steer axis tilt, contribute to stability in complex interacting ways.

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In this paper we present the linearized equations of motion for a bicycle as a benchmark. The results obtained by pencil-and-paper and two programs are compared. The bicycle model we consider here consists of four rigid bodies, viz. a rear frame, a front frame being the front fork and handlebar assembly, a rear wheel and a front wheel, which are connected by revolute joints. The contact between the knife-edge wheels and the flat level surface is modelled by holonomic constraints in the normal direction and by non-holonomic constraints in the longitudinal and lateral direction. The rider is rigidly attached to the rear frame with hands free from the handlebar. This system has three degrees of freedom, the roll, the steer, and the forward speed. For the benchmark we consider the linearized equations for small perturbations of the upright steady forward motion. The entries of the matrices of these equations form the basis for comparison. Three different kinds of methods to obtain the results are compared: pencil-and-paper, the numeric multibody dynamics program SPACAR, and the symbolic software system AutoSim. Because the results of the three methods are the same within the machine round-off error, we assume that the results are correct and can be used as a bicycle dynamics benchmark.