rsta.royalsocietypublishing.org Research Cite this article: Takács D, Stépán G. 2013 Contact patch memory of tyres leading to lateral vibrations of four-wheeled vehicles. Phil Trans R Soc A 371: 20120427. http://dx.doi.org/10.1098/rsta.2012.0427 One contribution of 17 to a Theme Issue ‘A celebration of mechanics: from nano to macro’. Subject Areas: mechanical engineering, applied mathematics, mechanics Keywords: tyre, contact, time delay Author for correspondence: Dénes Takács e-mail: takacs@mm.bme.hu Electronic supplementary material is available at http://dx.doi.org/10.1098/rsta.2012.0427 or via http://rsta.royalsocietypublishing.org. Contact patch memory of tyres leading to lateral vibrations of four-wheeled vehicles Dénes Takács 1 and Gábor Stépán 2 1 Research Group on Dynamics of Machines and Vehicles, Hungarian Academy of Sciences, and 2 Department of Applied Mechanics, Budapest University of Technology and Economics, PO Box 91, Budapest 1521, Hungary It has been shown recently that the shimmy motion of towed wheels can be predicted in a wide range of parameters by means of the so-called memory effect of tyres. This delay effect is related to the existence of a travelling-wave-like motion of the tyre points in contact with the ground relative to the wheel. This study shows that the dynamics within the small-scale contact patch can have an essential effect on the global dynamics of a four-wheeled automobile on a large scale. The stability charts identify narrow parameter regions of increased fuel consumption and tyre noise with the help of the delay models that are effective tools in dynamical problems through multiple scales. 1. Introduction In engineering, the prediction of the dynamical behaviour of designed systems often requires the use of many degrees of freedom (d.f.) models leading to large systems of ordinary differential equations (ODEs) or partial differential equations (PDEs). The design of these systems needs approximate analytical models and solutions, too, in order to study the effects of certain parameters, to explore new physical phenomena of the systems, and also to check the correctness and precision of the corresponding numerical solutions. When analytical solutions to multiple scale problems are to be constructed, delay differential equations (DDEs) can be used efficiently. These DDEs can still capture the infinite dimensional structure of the corresponding phase spaces while using fewer 2013 The Author(s) Published by the Royal Society. All rights reserved.