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![Dependencies of linear and angular velocities on time in deceleration process for rigid (black line) and elastic (colored lines) materials. Parameters α=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = 1$$\end{document}, μ=0.2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu = 0.2$$\end{document}](publication/344157153/figure/fig3/AS:960321321713690@1605969853277/Dependencies-of-linear-and-angular-velocities-on-time-in-deceleration-process-for-rigid.png)
Dependencies of linear and angular velocities on time in deceleration process for rigid (black line) and elastic (colored lines) materials. Parameters α=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = 1$$\end{document}, μ=0.2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu = 0.2$$\end{document}
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This paper investigates the influence of the material properties on the deceleration dynamics of a deformable cylinder rolling with slipping on a half-space of the same material. The interaction of the cylinder and the half-space is described by the 2D quasistatic contact problem of viscoelasticity (Goryacheva: J Appl Math Mech 37(5):877–885, 1973;...
Citations
... The applications of the quasistatic solutions of the corresponding contact problems for elastic [47] and viscoelastic [2] contacting bodies to the dynamics of an infinite cylinder (with horizontal axis) that rolls or slides along a half-space which border is horizontal or inclined are investigated in [48][49][50][51][52][53]. The external field is gravity, the acceleration in z-direction is neglected based on investigations [54], so it is assumed that z ≡ 0 in Eq. (7). ...
... Dynamics of the elastic or viscoelastic cylinder of radius R rolling over the half-plane of the same material is studied in [49][50][51][52][53]. According to the quasistatic solutions [2,47], the contact region is divided in two subregions: stick and slip zones, which width and position depend on the velocity of the cylinder center ẋ and creep ratio δ = ωR−ẋ x . ...
... The case of a viscoelastic materials of the cylinder and foundation is considered in [52,53]. It is shown that deceleration consists of two phases: on the first stage the relative sliding velocity ωR −ẋ decreases almost linearly in time and this stage is governed primarily by friction coefficient. ...
This review presents a collection of the solved dynamic problems taking into account the normal and shear stress distributions in the contact region due to the deformation of contacting bodies. The considered dynamic problems differ in the number of degrees of freedom and the type of relative displacements of contacting bodies (2-D models with rolling and sliding, 3-D models with sliding and spinning, 3-D models with sliding, rolling and spinning). The contact mechanics solutions are used in formulation of the dynamic problems, which are studied based on the analytical or semi-analytical approach. The effects of the mechanical properties of the contacting bodies and the contact conditions on the dynamics are analyzed and discussed.
Introduction . Designing motion control systems for mobile robots requires the construction of mathematical models. Researchers have repeatedly addressed this topic. In particular, works have been published on the calculations of multiphysical processes, modeling the movement of various types of wheels under certain conditions. In addition, the dynamics of deformable contacting bodies during sliding, rolling and rotation, issues of autonomy and controllability of mobile robots were considered. Note, however, that the dynamics and positioning accuracy of wheeled robots is largely determined by friction. The literature does not present studies on the dynamics of a robot with a differential drive taking into account the interrelationships of sliding, spinning and rolling friction effects based on the theory of multicomponent friction. Research in this area can reveal new dynamic effects. Based on the data obtained in this way, it is possible to improve the accuracy of positioning in building mathematical control models. The presented work aims at investigating the movement of an automatic device with a differential drive taking into account three contact models: nonholonomic, Coulomb friction, and multicomponent models.
Materials and Methods . The scheme of a two-wheeled robot with differential drive and continuous movement on the support surface was adopted as the basic one. The movement of the device was provided through software control. The dynamics was described in the form of Appel equations. Mathematical models were used for calculations, taking into account friction in different ways. Coordination of the actions of the mechanism was formed at a dynamic level. The control actions were the moments of the wheel motors. When visualizing the models under study, the built-in numerical methods of the Wolfram Mathematica system were used with a minimum accuracy of 10 ⁻⁶ .
Results . When building a mathematical model, the equations for the angular velocities of the wheels were determined. The authors took into account the presence of a contact site and derived the equations of dynamics of a differential drive robot. The elements of the system were force and moment projections, indicators of platform spin, masses, angular accelerations, and inertia of the wheels. It was shown how control actions were formed within the framework of nonholonomic mechanics. The model of engines that created a moment of control on the driving wheels was described. The solution was derived as the relationship between the inductance of the conductors of electric motors and the operation of the power supply. Three models describing the dynamics of a differential drive robot were examined in detail. The first model was nonholonomic. The second and third included a system of equations for the dynamics of a differential drive robot for a general case with a contact platform. At the same time, in the second model, the switching time in the engine was ignored and the Coulomb friction was involved. In the third model, a parameter to determine the speed of transients in the engine was introduced, and Pade decomposition was involved. This was a model with multicomponent friction. The calculation results were shown in the form of graphs. On them, the studied models were visualized in the form of curves of different colors. Comparison of the graphs showed in which cases, after the completion of transients, the control provided the required accuracy. These were models 1 and 2. In model 3, the software control generated an error in the angular velocity of rotation of the platform. This error could not be predicted within the framework of the 1st and 2nd models. In all the systems considered, the sliding speed of the wheels in the transverse direction dropped to zero. The condition of continuous motion of the support wheel was obtained and validated.
Discussion and Conclusion . Software control is acceptable in models that do not take into account wheel friction during simultaneous sliding, spinning and rolling (general case of spatial motion). However, it is important to consider the relationship between these processes and multicomponent friction. This is required for the robot to perform program movements more accurately. It was established that software control in a model that takes into account the friction of spinning and rolling caused deviations from the program values of the angular velocity of the platform. The results obtained can be used in the building of a control system with predictive models.
