Research Article Mathematical Formulation of Soft-Contact Problems for Various Rheological Models of Damper WiesBaw Grzesikiewicz 1 and Artur Zbiciak 2 1 Warsaw University of Technology, Faculty of Automotive and Construction Machinery Engineering, Institute of Vehicles, 84 Narbutta Str., 02-524 Warsaw, Poland 2 Warsaw University of Technology, Faculty of Civil Engineering, Institute of Roads and Bridges, 16 Armii Ludowej Ave., 00-637 Warsaw, Poland Correspondence should be addressed to Artur Zbiciak; a.zbiciak@il.pw.edu.pl Received 9 August 2017; Revised 30 December 2017; Accepted 31 January 2018; Published 27 February 2018 Academic Editor: Giorgio Dalpiaz Copyright © 2018 Wiesław Grzesikiewicz and Artur Zbiciak. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te paper deals with analysis of selected sof-contact problems in discrete mechanical systems. Elastic-dissipative rheological schemes representing dampers as well as the notion of unilateral constraints were used in order to model interaction between colliding bodies. Te mathematical descriptions of sof-contact problems involving variational inequalities are presented. Te main fnding of the paper is a method of description of sof-contact phenomenon between rigid object and deformable rheological structure by the system of explicit nonlinear diferential-algebraic equations easy for numerical implementation. Te results of simulations, that is, time histories of displacements and contact forces as well as hysteretic loops, are presented. 1. Introduction Impact is a short-lived phenomenon of energy exchange between colliding bodies. Te velocities of colliding bodies change rapidly and the reactions are impulsive in nature which means that the interactions are short-lived and reach large values. Classical theory of impact mechanics applied for rigid bodies assumes that the collision phenomenon results with discontinuous change of velocities and the reaction are modeled as impulses. However, such idealization may not be valid in many mechanical problems especially when we need to evaluate a history of reactions within a short time of collision. In the model of impact of two elastic spheres presented by Panovko and Stronge [1, 2] the local deformability of these spheres is assumed with use of nonlinear spring possessing the following property () fl  3/2 , where  denotes a parameter depending on the spheres’ radii and material. Te Hertz model of impact can be used for modelling of elastic collisions. Te analysis of impact and contact problems in discrete mechanical systems has received a great deal of attention in the literature [3–5]. Te mathematical description of such problems involves the notion of nonsmooth mechanics and needs a special numerical treatment [6–8]. In this paper we will apply rheological schemes represent- ing deformable dampers and unilateral constraints for model- ing of interaction between colliding bodies. Te phenomenon of contact between rigid objects and deformable rheological structures is sometimes called “sof contact” [9]. Te models we analyze in this paper allow evaluation of reactions during the time of collision. Te sof-contact models analyzed in our paper involve two cases of interaction between rigid bodies. Te frst case concerns the impact of two rigid bodies through a deformable built-in damper. In order to simplify mathemati- cal description we assumed that one of these bodies possesses an infnite mass. Te second case of interaction applying sof-contact models describes a rigid body interaction with discrete modelling of compliance for the contact region. In such models the compliance of the contact region is modelled with use of rheological schemes (see [2]). We will analyze both linear viscoelastic schemes and non- linear elastoplastic and viscoelastoplastic models of dampers. Hindawi Shock and Vibration Volume 2018, Article ID 8675016, 9 pages https://doi.org/10.1155/2018/8675016