Applied Numerical Analysis. By C.F. Gerald. Pp. xii, 340. £3·60. 1970. (Addison-Wesley.)

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.

Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review

/pdf/linearized-dynamics-equations-for-the-balance-and-steer-of-a-1gm9tg8w8b.pdf

CFD analysis of journal bearing hydrodynamic lubrication by Bingham lubricant

The development of bicycles and motorcycles since the first patented running machine, or draisine, in 1817 is described. Bicycle modeling and control were also discussed. These models include: derivatives or simplifications of Whipple's bicycle dynamics model in which the lateral motion constraints at the road contact are nonholonomic, requiring special techniques to form correct equations of motion; and the Timoshenko-Young model in which the steer angle and speed completely determine the lateral motion of the base point of an inverted pendulum that represents the vehicle's roll dynamics

/pdf/bicycles-motorcycles-and-models-pu5ta7rwyx.pdf

Bicycles, motorcycles, and models

Analysis of finite magnetohydrodynamic journal bearings

Using linearized turbulent lubrication theory of Ng and Pan, the modified Reynolds equation has been solved by finite element method using Galerkin's technique. Static and dynamic performance characteristics of the noncircular (two-lobe) bearings have been studied, both for laminar and turbulent flow, in terms of load support, oil flow, fluid film stiffness coefficients, damping coefficients, and critical mass for various Reynolds numbers.

Performance Characteristics of Noncircular Bearings in Laminar and Turbulent Flow Regimes

The deformation of a bearing, which can be of the order of the fluid film thickness when the bearing operates at large eccentricities, may affect the performance appreciably, and all the more so if the viscosity of the lubricant changes with pressure. In this paper, matched solutions for pressure distribution and the bearing displacement field, taking into account the combined effects of bearing liner deformation as well as change in viscosity of the lubricant, are obtained by solving the respective governing equations using a suitable iteration scheme for eccentricity ratios up to unity. The performance characteristics of the journal bearing and their variations with bearing flexibility, are studied with reference to eccentricity ratio and a deformation coefficient which is a measure of the flexibility of bearing shell.

D.V. Singh

Papers

Analysis of finite magnetohydrodynamic journal bearings

Performance Characteristics of Noncircular Bearings in Laminar and Turbulent Flow Regimes

A Study of EHD Lubrication in a Journal Bearing with Piezoviscous Lubricants

Elastohydrodynamic effects in elliptical bearings

Finite element analysis of elliptical bearings lubricated by a non-newtonian fluid