Conservation laws for a spherical top on a plane with friction
International Journal of Non-Linear Mechanics, 2021, vol. 129, 103666, 5 pp.
Abstract
pdf (353.6 Kb)
This paper is concerned with the analysis of the influence of the friction model on the existence of additional
integrals of motion in a system describing the sliding of a spherical top on a plane. We consider a model in
which the friction is described not only by the force applied at the point of contact, but also by an additional
friction torque. It is shown that, depending on the chosen friction model, the system admits various first
integrals. In particular, we give examples of friction models in which either the Jellett integral or the Lagrange
integral or the area integral is preserved.
Keywords:
spherical top, friction model, friction, integrals of motion
Citation:
Kilin A. A., Pivovarova E. N., Conservation laws for a spherical top on a plane with friction, International Journal of Non-Linear Mechanics, 2021, vol. 129, 103666, 5 pp.
The influence of the first integrals and the rolling resistance model on tippe top inversion
Nonlinear Dynamics, 2021, vol. 103, no. 1, pp. 419-428
Abstract
pdf (494.9 Kb)
In this paper, we analyze the effect which
the choice of a frictionmodel has on tippe top inversion
in the case where the resulting action of all dissipative
forces is described not only by the force applied at the
contact point, but also by the additional rolling resistance
torque. We show that the possibility or impossibility
of tippe top inversion depends on the existence
of specific integrals of the motion of the system. In this
paper, we consider an example of the law of rolling
resistance by which the area integral is preserved in the
system.We examine in detail the case where the action
of all dissipative forces reduces to the horizontal rolling
resistance torque. This model describes fast rotations
of the top between two horizontal smooth planes. For
this case, we find permanent rotations of the system
and analyze their linear stability. The stability analysis
suggests that no tippe top inversion is possible under
fast rotations between two planes.
Keywords:
spherical tippe top, rolling resistance torque, integrals of motion, tippe top inversion, reduction, stability analysis, partial solutions, bifurcations
Citation:
Kilin A. A., Pivovarova E. N., The influence of the first integrals and the rolling resistance model on tippe top inversion, Nonlinear Dynamics, 2021, vol. 103, no. 1, pp. 419-428
Nonintegrability of the problem of a spherical top rolling on a vibrating plane
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2020, vol. 30, no. 4, pp. 628-644
Abstract
pdf (299.53 Kb)
This paper investigates the rolling motion of a spherical top with an axisymmetric mass distribution on a smooth horizontal plane performing periodic vertical oscillations. For the system under consideration, equations of motion and conservation laws are obtained. It is shown that the system admits two equilibrium points corresponding to uniform rotations of the top about the vertical symmetry axis. The equilibrium point is stable when the center of mass is located below the geometric center, and is unstable when the center of mass is located above it. The equations of motion are reduced to a system with one and a half degrees of freedom. The reduced system is represented as a small perturbation of the problem of the Lagrange top motion. Using Melnikov’s method, it is shown that the stable and unstable branches of the separatrix intersect transversally with each other. This suggests that the problem is nonintegrable. Results of computer simulation of the top dynamics near the unstable equilibrium point are presented.
Kilin A. A., Pivovarova E. N., Nonintegrability of the problem of a spherical top rolling on a vibrating plane, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2020, vol. 30, no. 4, pp. 628-644
Spherical rolling robots: Different designs and control algorithms
Robots in Human Life: Proceedings of the 23rd International Conference on Climbing and Walking Robots and the Support Technologies for Mobile Machines, CLAWAR 2020, 2020, pp. 195-202
Abstract
pdf (415.75 Kb)
This paper presents a review of papers devoted to the creation and investigation of spherical robots. Owing to their structural features, namely, geometric symmetry and resistance of actuating mechanisms and elements of the control system against aggressive environmental conditions, robotic systems of this type have a high potential of being used in problems of monitoring, reconnaissance, and transportation on Earth and other planets. A detailed description is given of the structures of prototypes of spherical robots which use different actuation principles: a spherical robot with an internal pendulum mechanism, a spherical robot with an internal omniwheel platform, a spherical robot with internal rotors, and a spherical robot of combined type. Experimental results are presented to give an estimate of the possibility and efficiency of controlled motion. The applied use of spherical robots depending on the type of the actuating mechanism is discussed.
Citation:
Karavaev Y. L., Mamaev I. S., Kilin A. A., Pivovarova E. N., Spherical rolling robots: Different designs and control algorithms, Robots in Human Life: Proceedings of the 23rd International Conference on Climbing and Walking Robots and the Support Technologies for Mobile Machines, CLAWAR 2020, 2020, pp. 195-202
Stabilization of a spherical robot rolling on an oscillating underlying surface
2020 International Conference Nonlinearity, Information and Robotics – IEEE, 2020, pp. 1-2
Abstract
pdf (1.09 Mb)
This paper is devoted to investigations of a spherical robot rolling on an oscillating underlying surface. The model of spherical robot of combined type is considered. Based on analysis of equations of motion taking into account the oscillation of the underlying surface, a control algorithm for stabilization of the spherical robot is proposed. The influences of oscillations in horizontal and vertical directions are evaluated. The design of the spherical robot and its control system for fabrication of a prototype are described.
Keywords:
spherical robot, nonholonomic model, oscillations of the underlying surface, feedback
Citation:
Karavaev Y. L., Kilin A. A., Klekovkin A. V., Pivovarova E. N., Stabilization of a spherical robot rolling on an oscillating underlying surface, 2020 International Conference Nonlinearity, Information and Robotics – IEEE, 2020, pp. 1-2
Speedup of the Chaplygin Top by Means of Rotors
Doklady Physics, 2019, vol. 64, no. 3, pp. 120-124
Abstract
pdf (486.65 Kb)
In this paper we consider the control of the motion of a dynamically asymmetric unbalanced ball
(Chaplygin top) by means of two perpendicular rotors. We propose a mechanism for control by periodically
changing the gyrostatic momentum of the system, which leads to an unbounded speedup. We then formulate
a general hypothesis of the mechanism for speeding up spherical bodies on a plane by periodically changing
the system parameters.
Qualitative Analysis of the Nonholonomic Rolling of a Rubber Wheel with Sharp Edges
Regular and Chaotic Dynamics, 2019, vol. 24, no. 2, pp. 212-233
Abstract
pdf (883.49 Kb)
This paper presents a qualitative analysis of the dynamics in a fixed reference frame of a wheel with sharp edges that rolls on a horizontal plane without slipping at the point of contact and without spinning relative to the vertical. The wheel is a ball that is symmetrically truncated on both sides and has a displaced center of mass. The dynamics of such a system is described by the model of the ball’s motion where the wheel rolls with its spherical part in contact with the supporting plane and the model of the disk’s motion where the contact point lies on the sharp edge of the wheel. A classification is given of possible motions of the wheel depending on whether there are transitions from its spherical part to sharp edges. An analysis is made of the behavior of the point of contact of the wheel with the plane for different values of the system parameters, first integrals and initial conditions. Conditions for boundedness and unboundedness of the wheel’s motion are obtained. Conditions for the fall of the wheel on the
plane of sections are presented.
Keywords:
integrable system, system with discontinuity, nonholonomic constraint, bifurcation diagram, body of revolution, sharp edge, wheel, rubber body model, permanent rotations, dynamics in a fixed reference frame, resonance, quadrature, unbounded motion
Citation:
Kilin A. A., Pivovarova E. N., Qualitative Analysis of the Nonholonomic Rolling of a Rubber Wheel with Sharp Edges, Regular and Chaotic Dynamics, 2019, vol. 24, no. 2, pp. 212-233
Controlled Motion of a Spherical Robot with Feedback. II
Journal of Dynamical and Control Systems, 2019, vol. 25, no. 1, pp. 1-16
Abstract
pdf (803.07 Kb)
In this paper, we develop a model of a controlled spherical robot of combined type moving by displacing the center of mass and by changing the internal gyrostatic momentum, with a feedback that stabilizes given partial solutions for a free system at the final stage of motion. According to the proposed approach, feedback depends on phase variables (current position, velocities) and does not depend on the specific type of trajectory. We present integrals of motion and partial solutions, analyze their stability, and give examples of computer simulations of motion with feedback that demonstrate the efficiency of the proposed model.
Ivanova T. B., Kilin A. A., Pivovarova E. N., Controlled Motion of a Spherical Robot with Feedback. II, Journal of Dynamical and Control Systems, 2019, vol. 25, no. 1, pp. 1-16
Control of the Rolling Motion of a Spherical Robot on an Inclined Plane
Doklady Physics, 2018, vol. 63, no. 10, pp. 435-440
Abstract
pdf (509.36 Kb)
In this work we consider the controlled motion of a pendulum spherical robot on an inclined plane. The algorithm for determining the control actions for the motion along an arbitrary trajectory and examples of numerical simulation of the controlled motion are given.
Citation:
Ivanova T. B., Kilin A. A., Pivovarova E. N., Control of the Rolling Motion of a Spherical Robot on an Inclined Plane, Doklady Physics, 2018, vol. 63, no. 10, pp. 435-440
Controlled Motion of a Spherical Robot of Pendulum Type on an Inclined Plane
Doklady Physics, 2018, vol. 63, no. 7, pp. 302-306
Abstract
pdf (568.83 Kb)
This paper is concerned with a model of the controlled motion of a spherical robot with an axisymmetric
pendulum actuator on an inclined plane. First integrals of motion and partial solutions are presented
and their stability is analyzed. It is shown that the steady solutions exist only at an inclination angle less than
some critical value and only for constant control action.
Citation:
Ivanova T. B., Kilin A. A., Pivovarova E. N., Controlled Motion of a Spherical Robot of Pendulum Type on an Inclined Plane, Doklady Physics, 2018, vol. 63, no. 7, pp. 302-306
Integrable Nonsmooth Nonholonomic Dynamics of a Rubber Wheel with Sharp Edges
Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 887-907
Abstract
pdf (2.02 Mb)
This paper is concerned with the dynamics of a wheel with sharp edges moving on a horizontal plane without slipping and rotation about the vertical (nonholonomic rubber model). The wheel is a body of revolution and has the form of a ball symmetrically truncated on both sides. This problem is described by a system of differential equations with a discontinuous
right-hand side. It is shown that this system is integrable and reduces to quadratures. Partial solutions are found which correspond to fixed points of the reduced system. A bifurcation analysis and a classification of possible types of the wheel’s motion depending on the system parameters are presented.
Keywords:
integrable system, system with a discontinuous right-hand side, nonholonomic constraint, bifurcation diagram, body of revolution, sharp edge, wheel, rubber model
Citation:
Kilin A. A., Pivovarova E. N., Integrable Nonsmooth Nonholonomic Dynamics of a Rubber Wheel with Sharp Edges, Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 887-907
Controlled Motion of a Spherical Robot with Feedback. I
Journal of Dynamical and Control Systems, 2018, vol. 24, no. 3, pp. 497-510
Abstract
pdf (606.17 Kb)
In this paper, we develop a model of a controlled spherical robot with an axisymmetric pendulum-type actuator with a feedback system suppressing the pendulum’s oscillations at the final stage of motion. According to the proposed approach, the feedback depends on phase variables (the current position and velocities) and does not depend on the type of trajectory. We present integrals of motion and partial solutions, analyze their stability, and give examples of computer simulation of motion using feedback to illustrate compensation of the pendulum’s oscillations.
Ivanova T. B., Kilin A. A., Pivovarova E. N., Controlled Motion of a Spherical Robot with Feedback. I, Journal of Dynamical and Control Systems, 2018, vol. 24, no. 3, pp. 497-510
Chaplygin Top with a Periodic Gyrostatic Moment
Russian Journal of Mathematical Physics, 2018, vol. 25, no. 4, pp. 509-524
Abstract
pdf (1.14 Mb)
In the paper, a study of rolling of a dynamically asymmetrical unbalanced ball (Chaplygin top) on a horizontal plane under the action of periodic gyrostatic moment is carried out. The problem is considered in the framework of the model of a rubber body, i.e., under the assumption that there is no slipping and spinning at the point of contact. It is shown that, for certain values of the parameters of the system and certain dependence of the gyrostatic moment on time, an acceleration of the system, i.e., an unbounded growth of the energy of the system, is observed. Investigations of the dependence of the presence of acceleration on the parameters of the system and on the initial conditions are carried out. On the basis of the investigations of the dynamics of the frozen system, a conjecture concerning the general mechanism of acceleration at the expense to periodic impacts in
nonholonomic systems is expressed.
Citation:
Kilin A. A., Pivovarova E. N., Chaplygin Top with a Periodic Gyrostatic Moment, Russian Journal of Mathematical Physics, 2018, vol. 25, no. 4, pp. 509-524
The Rolling Motion of a Truncated Ball Without Slipping and Spinning on a Plane
Regular and Chaotic Dynamics, 2017, vol. 22, no. 3, pp. 298-317
Abstract
pdf (399.6 Kb)
This paper is concerned with the dynamics of a top in the form of a truncated ball as it moves without slipping and spinning on a horizontal plane about a vertical. Such a system is described by differential equations with a discontinuous right-hand side. Equations describing
the system dynamics are obtained and a reduction to quadratures is performed. A bifurcation analysis of the system is made and all possible types of the top’s motion depending on the system parameters and initial conditions are defined. The system dynamics in absolute space is examined. It is shown that, except for some special cases, the trajectories of motion are bounded.
Keywords:
integrable system, system with discontinuity, nonholonomic constraint, bifurcation diagram, absolute dynamics
Citation:
Kilin A. A., Pivovarova E. N., The Rolling Motion of a Truncated Ball Without Slipping and Spinning on a Plane, Regular and Chaotic Dynamics, 2017, vol. 22, no. 3, pp. 298-317
Stability analysis of steady motions of a spherical robot of combined type
Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 4, pp. 611–623
Abstract
pdf (1.75 Mb)
The dynamics of a spherical robot of combined type consisting of a spherical shell and a pendulum attached at the center of the shell is considered. At the end of the pendulum a rotor is installed. For this system we carry out a stability analysis for a partial solution which in absolute space corresponds to motion along a circle with constant velocity. Regions of stability of a partial solution are found depending on the orientation of the spherical robot during the motion, its velocity and the radius of the circle traced out by the point of contact.
Pivovarova E. N., Stability analysis of steady motions of a spherical robot of combined type, Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 4, pp. 611–623
Stability analysis of steady motions of a spherical robot of combined type
Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 4, pp. 611–623
Abstract
pdf (1.76 Mb)
The dynamics of a spherical robot of combined type consisting of a spherical shell and a pendulum attached at the center of the shell is considered. At the end of the pendulum a rotor is installed. For this system we carry out a stability analysis for a partial solution which in absolute space corresponds to motion along a circle with constant velocity. Regions of stability of a partial solution are found depending on the orientation of the spherical robot during the motion, its velocity and the radius of the circle traced out by the point of contact.
Pivovarova E. N., Stability analysis of steady motions of a spherical robot of combined type, Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 4, pp. 611–623
Regular and chaotic dynamics in the rubber model of a Chaplygin top
Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 2, pp. 277-297
Abstract
pdf (2.3 Mb)
This paper is concerned with the rolling motion of a dynamically asymmetric unbalanced ball (Chaplygin top) in a gravitational field on a plane under the assumption that there is no slipping and spinning at the point of contact. We give a description of strange attractors existing in the system and discuss in detail the scenario of how one of them arises via a sequence of perioddoubling bifurcations. In addition, we analyze the dynamics of the system in absolute space and show that in the presence of strange attractors in the system the behavior of the point of contact considerably depends on the characteristics of the attractor and can be both chaotic and nearly quasi-periodic.
Keywords:
Chaplygin top, nonholonomic constraint, rubber model, strange attractor, bifurcation, trajectory of the point of contact
Citation:
Borisov A. V., Kazakov A. O., Pivovarova E. N., Regular and chaotic dynamics in the rubber model of a Chaplygin top, Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 2, pp. 277-297
Regular and Chaotic Dynamics in the Rubber Model of a Chaplygin Top
Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 885-901
Abstract
pdf (2.21 Mb)
This paper is concerned with the rolling motion of a dynamically asymmetric unbalanced ball (Chaplygin top) in a gravitational field on a plane under the assumption that there is no slipping and spinning at the point of contact. We give a description of strange attractors existing in the system and discuss in detail the scenario of how one of them arises via a sequence of period-doubling bifurcations. In addition, we analyze the dynamics of the system in absolute space and show that in the presence of strange attractors in the system the behavior of the point of contact considerably depends on the characteristics of the attractor and can be both chaotic and nearly quasi-periodic.
Keywords:
Chaplygin top, nonholonomic constraint, rubber model, strange attractor, bifurcation, trajectory of the point of contact
Citation:
Borisov A. V., Kazakov A. O., Pivovarova E. N., Regular and Chaotic Dynamics in the Rubber Model of a Chaplygin Top, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 885-901
Influence of rolling friction on the controlled motion of a robot wheel
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, pp. 583-592
Abstract
pdf (395.52 Kb)
This paper presents an experimental investigation of the influence of rolling friction on the dynamics of a robot wheel. The robot is set in motion by changing the proper gyrostatic momentum using the controlled rotation of a rotor installed in the robot. The problem is considered under the assumption that the center of mass of the system does not coincide with its geometric center. In this paper we derive equations describing the dynamics of the system and give an example of the controlled motion of a wheel by specifying a constant angular acceleration of the rotor. A description of the design of the robot wheel is given and a method for experimentally determining the rolling friction coefficient is proposed. For the verification of the proposed mathematical model, experimental studies of the controlled motion of the robot wheel are carried out. We show that the theoretical results qualitatively agree with the experimental ones, but are quantitatively different.
Keywords:
robot-wheel, rolling friction, displacement of the center of mass
Citation:
Pivovarova E. N., Klekovkin A. V., Influence of rolling friction on the controlled motion of a robot wheel, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, pp. 583-592
A model of a screwless underwater robot
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, pp. 544-553
Abstract
pdf (308.24 Kb)
The paper is devoted to the development of a model of an underwater robot actuated by inner rotors. This design has no moving elements interacting with an environment, which minimizes a negative impact on it, and increases noiselessness of the robot motion in a liquid. Despite numerous discussions on the possibility and efficiency of motion by means of internal masses' movement, a large number of works published in recent years confirms a relevance of the research. The paper presents an overview of works aimed at studying the motion by moving internal masses. A design of a screwless underwater robot that moves by the rotation of inner rotors to conduct theoretical and experimental investigations is proposed. In the context of theoretical research a robot model is considered as a hollow ellipsoid with three rotors located inside so that the axes of their rotation are mutually orthogonal. For the proposed model of a screwless underwater robot equations of motion in the form of classical Kirchhoff equations are obtained.
Keywords:
mobile robot, screwless underwater robot, movement in ideal fluid
Citation:
Vetchanin E. V., Karavaev Y. L., Kalinkin A. A., Klekovkin A. V., Pivovarova E. N., A model of a screwless underwater robot, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, pp. 544-553
Spherical Robot of Combined Type: Dynamics and Control
Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 716-728
Abstract
pdf (306.92 Kb)
This paper is concerned with free and controlled motions of a spherical robot of combined type moving by displacing the center of mass and by changing the internal gyrostatic momentum. Equations of motion for the nonholonomic model are obtained and their first integrals are found. Fixed points of the reduced system are found in the absence of control actions. It is shown that they correspond to the motion of the spherical robot in a straight line and in a circle. A control algorithm for the motion of the spherical robot along an arbitrary trajectory is presented. A set of elementary maneuvers (gaits) is obtained which allow one to transfer the spherical robot from any initial point to any end point.
Kilin A. A., Pivovarova E. N., Ivanova T. B., Spherical Robot of Combined Type: Dynamics and Control, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 716-728
Comments on the Paper by A.V. Borisov, A.A. Kilin, I.S. Mamaev "How to Control the Chaplygin Ball Using Rotors. II"
Regular and Chaotic Dynamics, 2014, vol. 19, no. 1, pp. 140-143
Abstract
pdf (170.77 Kb)
In this paper we consider the control of a dynamically asymmetric balanced ball on a plane in the case of slipping at the contact point. Necessary conditions under which a control is possible are obtained. Specific algorithms of control along a given trajectory are constructed.
Ivanova T. B., Pivovarova E. N., Comments on the Paper by A.V. Borisov, A.A. Kilin, I.S. Mamaev "How to Control the Chaplygin Ball Using Rotors. II", Regular and Chaotic Dynamics, 2014, vol. 19, no. 1, pp. 140-143
Comment on the paper by A.V. Borisov, A.A. Kilin, I.S. Mamaev “How to control the Chaplygin ball using rotors. II”
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 1, pp. 127-131
Abstract
pdf (234.11 Kb)
In this paper we consider the control of a dynamically asymmetric balanced ball on a plane in the case of slipping at the contact point. Necessary conditions under which a control is possible are obtained. Specific algorithms of control along a given trajectory are constructed.
Ivanova T. B., Pivovarova E. N., Comment on the paper by A.V. Borisov, A.A. Kilin, I.S. Mamaev “How to control the Chaplygin ball using rotors. II”, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 1, pp. 127-131
Determination of moments of inertia and the position of the center of mass of robotic devices
Bulletin of Udmurt University. Physics and Chemistry, 2014, no. 4, pp. 79-86
Abstract
pdf (311.48 Kb)
In this paper we describe an inertiameter, which is an experimental facility for determining the inertia tensor components and the position of the center of mass of compound bodies. An algorithm for determining these dynamical properties is presented. Using the algorithm obtained, the displacement of the center of mass and the tensor of inertia are determined experimentally for a spherical robot of combined type.
Keywords:
inertiameter, spherical robot, moment of inertia, center of mass
Citation:
Alalykin S. S., Bogatyrev A. V., Ivanova T. B., Pivovarova E. N., Determination of moments of inertia and the position of the center of mass of robotic devices, Bulletin of Udmurt University. Physics and Chemistry, 2014, no. 4, pp. 79-86
Dynamics and Control of a Spherical Robot with an Axisymmetric Pendulum Actuator
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, pp. 507-520
Abstract
pdf (582.48 Kb)
This paper investigates the possibility of the motion control of a ball with a pendulum mechanism with non-holonomic constraints using gaits — the simplest motions such as acceleration and deceleration during the motion in a straight line, rotation through a given angle and their combination. Also, the controlled motion of the system along a straight line with a constant acceleration is considered. For this problem the algorithm for calculating the control torques is given and it is shown that the resulting reduced system has the first integral of motion.
Keywords:
non-holonomic constraint, control, spherical shell, integral of motion
Citation:
Ivanova T. B., Pivovarova E. N., Dynamics and Control of a Spherical Robot with an Axisymmetric Pendulum Actuator, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, pp. 507-520
Stability analysis of periodic solutions in the problem of the rolling of a ball with a pendulum
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2012, no. 4, pp. 146-155
Abstract
pdf (688.03 Kb)
In the paper we study the stability of a spherical shell rolling on a horizontal plane with Lagrange’s gyroscope inside. A linear stability analysis is made for the upper and lower position of a top. A bifurcation diagram of the system is constructed. The trajectories of the contact point for diﬀerent values of the integrals of motion are constructed and analyzed.
Keywords:
rolling motion, stability, Lagrange’s gyroscope, bifurcational diagram
Citation:
Pivovarova E. N., Ivanova T. B., Stability analysis of periodic solutions in the problem of the rolling of a ball with a pendulum, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2012, no. 4, pp. 146-155