Submit Manuscript | http://medcraveonline.com Nomenclature m – Mass of a disc g – Gravity acceleration i - Index for axis ox or oy J - Mass moment of inertia of a disc J i - Mass moment of inertia of a disc around axis i l –Radius of the disc rolling along the curvilinear path R - Radius of a disc T - Load torque T am.i, T cti, T cr.i, T in.i - Torque generated by the change in the angular momentum, centrifugal, Coriolis and common inertial forces respectively, and acting around axis i T r.i, T pi – Resistance and precession torque respectively acting around axis i t -Time γ - Angle of inclination of a disc η – Coeffcient of correction ω - Angular velocity of a disc ω i - Angular velocity of precession around axis i Introduction Most of the textbooks of machine dynamics and publications that dedicated to gyroscope theory content the typical examples with solving of gyroscopic effects by defned analytical approaches. 1−3 However, the practice demonstrates that the known mathematical models for acting forces on the gyroscopic devices do not match their actual forces and motions. 4,5 Recent investigations in the area of the physical principles of gyroscopic effects have presented the new mathematical models of forces acting on a gyroscope. 6−8 It is defned that the action of the external load on a gyroscope generates several inertial resistance and precession torques based on the action of the rotating mass elements of the spinning rotor. The resistance torque is generated by the action of the centrifugal and Coriolis forces of the gyroscope’s mass elements. The precession torque is generated by the action of the common inertial forces of the gyroscope’s mass elements and by the change in the angular momentum of the spinning rotor. These torqueses are acting simultaneously, interdependently and strictly perpendicular to each other around their axes. Equations of inertial torques are represented in Table 1. 6 Table 1 contains the following symbols and components of the equations that marked by subscript signs indicating the axis of action: 2 2 J mR = is the rotor’s mass moment of inertia around the spinning axle; m is the mass of the rotor; R is the external radius of the rotor; i ω is the angular velocity of a rotor around axis i and ω is the angular velocity of a spinning rotor; T r.x is the resistance torque acting around axis ox, T py is the precession torque acting around axis oy;T ct.i , T in.i, T cr.i and T am.i is the torque acting around axis i that generated by the centrifugal, common inertial forces, Coriolisforces and the angular momentum respectively . This work presents the mathematical model for the free motion of the tilted rolling disc on the fat surface under the action of the eight inertial torques around two axes. Practically, the design of the disc can be the wheels, rims, hoops, discs, and similar designs that possess gyroscopic properties. Methods The simple design of the wheel or thin disc is unstable on the vertical plane, but a rolling motion demonstrates its stability and steering itself in case of the disc tilts. This tilts leads to the turn of the rolling disc to the direction of its fall. This motion of the inclined thin Int Rob Auto J. 2018;4(5):329‒332. 329 © 2018 Usubamatov. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and build upon your work non-commercially. Analysis of motion for a rolling disc on the fat surface Volume 4 Issue 5 - 2018 Ryspek Usubamatov Department of Automation& Robotics, Kyrgyz State Technical University, Kyrgyzstan Correspondence: Ryspek Usubamatov, Department of Automation& Robotics, Kyrgyz State Technical University, , 66 Aitmatov Avenue, KSTU, 720044 Bishkek, Kyrgyzstan, Tel +996 312 545125, Email rispek0701@yahoo.com Received: May 16, 2018 | Published: October 15, 2018 Abstract Background: Recent investigations in gyroscopic effects have demonstrated that their origin has more complex nature that represented in known publications. Actually, on a gyroscope are acting simultaneously and interdependently eight inertial torques around two axes. This torques is generated by the centrifugal, common inertial and Coriolis forces as well as the change in the angular momentum of the masses of the spinning rotor. The action of these forces manifests in the form of the inertial resistance and precession torques of gyroscopic devices. New mathematical models for the inertial torques acting on the spinning rotor demonstrate fundamentally different approaches and results of solving the problems of gyroscopic devices.The tendency in contemporary engineering is expressed by the increasing of a velocity of rotating objectslike turbines, rotors, discs and othersthat lead to the proportional increase of the magnitudes of inertial forces forming their motions. This work considers a typical example for computing of the inertial torques acting on the free rolling disc, which can be a bicycle wheel, rims, hoops, discs, and similar designs that express the gyroscopiceffects. Keywords: gyroscope theory, torques, motions, forces International Robotics & Automation Journal Research Article Open Access