On the nonsmooth dynamics of towed wheels
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RECENT ADVANCES IN NONLINEAR DYNAMICS AND VIBRATIONS
On the nonsmooth dynamics of towed wheels Mate Antali
. Gabor Stepan
Received: 4 November 2019 / Accepted: 20 August 2020 The Author(s) 2020
Abstract In this paper, a nonsmooth model of towed wheels is analysed; this mechanism can be a part of different kind of vehicles. We focus on the transitions between slipping and rolling in the presence of dry friction. The model leads to a three-dimensional dynamical system with a codimension-2 discontinuity. The systems can be analysed by means of the tools of extended Filippov systems. The essence of the calculation is to find the so-called limit directions, which show the possible directions of slipping-rolling transitions and their properties. By this method, four different scenarios are found. The results are compared to those from the creep models. Keywords Towed wheel Nonsmooth dynamics Dry friction Filippov systems
1 Introduction The towed wheels can be found in vehicles and machines, where the axle of the wheel can freely rotate with respect to the body of the vehicle. Simple everyday examples of towed wheels are the wheels of shopping carts and baby strollers, but the mechanism M. Antali (&) G. Stepan Department of Applied Mechanics, Budapest University of Technology and Economics, Budapest 1521, Hungary e-mail: [email protected]
of towed wheels cover the trailers [23], the aeroplane nose gears [22], and the front wheels of motorbikes [19]. The dynamical properties of these wheels are important from the point of view of comfort and safety. One of the fundamental phenomena of towed wheels is the self-excited oscillations called shimmy, which has been intensively studied in the literature (see [18, 21] and their references). There are many possibilities for modelling the tyre and the vehicle, and the resulting models often lead to complex mathematical models including, for example, time delay [3, 4]. In this paper, we try to focus on the simplest relevant models and try to understand the basic dynamical effects coming from the nonsmooth property of the friction forces between the wheel and the road. The resulting nonsmooth dynamical system is analysed by means of the tools developed by the authors [1] recently. The paper is organized as follows: In Sect. 2, the equations of motion of the system is derived for such coordinates that give the possibility to introduce several different contact models. In Sect. 3, we review and compare a few fundamental models of the tangential contact forces, including smooth and nonsmooth models. In Sect. 4, we analyse the towed wheel model with a linear creep force, which analysis serves as a reference when comparing it to the nonsmooth model. In Sect. 5, we present a throughout analysis of the wheel model with the spatial Coulomb friction, by
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Meccanica
using the recently developed calculation methods of this kind of nonsmooth system. Our purpose is to explore the possible scenarios when the system is switching betwe
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