The Rolling Motion of a Rigid Body on a Plane and a Sphere. Hierarchy of Dynamics
نویسنده
چکیده
In this paper we consider cases of existence of invariant measure, additional first integrals, and Poisson structure in a problem of rigid body’s rolling without sliding on plane and sphere. The problem of rigid body’s motion on plane was studied by S.A.Chaplygin, P.Appel, D.Korteweg. They showed that the equations of motion are reduced to a secondorder linear differential equation in the case when the surface of dynamically symmetric body is a surface of revolution. These results were partially generalized by P.Woronetz, who studied the motion of body of revolution and the motion of round disk with sharp edge on the surface of sphere. In both cases the systems are Euler – Jacobi integrable and have additional integrals and invariant measure. It turns out that after some change of time defined by reducing multiplier, the reduced system is a Hamiltonian system. Here we consider different cases when the integrals and invariant measure can be presented as finite algebraic expressions. We also consider the generalized problem of rolling of dynamically nonsymmetric Chaplygin ball. The results of studies are presented as tables that describe the hierarchy of existence of various tensor invariants: invariant measure, integrals, and Poisson structure in the considered problems.
منابع مشابه
The Transient Dynamics of a Beam Mounted on Spring Supports and Equipped with the Nonlinear Energy Sink
The transient dynamics of a beam mounted on springer-damper support and equipped with a nonlinear energy sink (NES) is investigated under the effects of shock loads. The equations of motion are derived using the Hamilton’s principle leading to four hybrid ordinary and partial differential equations and descritized using the Galerkin method. An adaptive Newmark method is employed for accurate an...
متن کاملModeling and Simulation Research on Dynamics of Missiles with Deflectable Nose Control
The dynamic characteristics of deflectable nose missiles with rotary single-channel control were investigated in this paper. Firstly, the effective angle of attack, effective angle of sideslip and quasi-body coordinates were introduced based on the spin characteristics of the missile’s body. Secondly, considering the interaction between the missile’s nose and body, the rigid kinetic model of mi...
متن کاملDynamical Behavior of a Rigid Body with One Fixed Point (Gyroscope). Basic Concepts and Results. Open Problems: a Review
The study of the dynamic behavior of a rigid body with one fixed point (gyroscope) has a long history. A number of famous mathematicians and mechanical engineers have devoted enormous time and effort to clarify the role of dynamic effects on its movement (behavior) – stable, periodic, quasi-periodic or chaotic. The main objectives of this review are: 1) to outline the characteristic features of...
متن کاملManipulation Control of a Flexible Space Free Flying Robot Using Fuzzy Tuning Approach
Cooperative object manipulation control of rigid-flexible multi-body systems in space is studied in this paper. During such tasks, flexible members like solar panels may get vibrated that in turn may lead to some oscillatory disturbing forces on other subsystems, and consequently produces error in the motion of the end-effectors of the cooperative manipulating arms. Therefore, to design and dev...
متن کاملEffects of Flight Dynamics on Performance of One Axis Gimbal System, Considering Disturbance Torques
The gimbal stabilization mechanism system is used to provide the stability to an object mounted on the gimbal by isolating it from the base angular motion and vibration. In this paper the model of one axis gimbal system with dynamics flying object is introduced. The gimbal torque relationships are obtained using Newton’s second law equation on the assumption that gimbal is rigid body. The syste...
متن کامل