The dynamics of a rigid body colliding with a rigid surface

Basic investigation techniques, algorithms, and results are presented for nonlinear oscillations and stability of steady rotations and periodic motions of a rigid body, colliding with a rigid surface, in a uniform gravity field.      

Invariant tracking for planar rigid body dynamics

A quasi‐static state feedback is proposed that achieves stable tracking for planar motions of rigid bodies and is invariant with respect to the choice of the inertial frame.     

Absolute and relative choreographies in rigid body dynamics

For the classical problem of motion of a rigid body about a fixed point with zero area integral, we present a family of solutions that are periodic in the absolute space. Such solutions are known as choreographies. The family includes the well-known Delone solutions (for the Kovalevskaya case), some particular solutions for the Goryachev-Chaplygin case, and the Steklov solution. The “genealogy” of solutions of the family naturally appearing from the energy continuation and their connection with the Staude rotations are considered. It is shown that if the integral of areas is zero, the solutions are periodic with respect to a coordinate frame that rotates uniformly about the vertical (relative choreographies).      

Numerical integration of rigid body dynamics in terms of quaternions

Unit–quaternions (or Euler parameter) are known to be well–suited for the singularity–free parametrization of finite rotations. Despite of this advantage, unit quaternions were rarely used to formulate the equations of motion (exceptions are the works by Nikravesh [1] and Haug [2]). This might be related to the fact, that the unit–quaternions are redundant, which requires the use of algebraic constraints in the equations of motion. Nowadays robust energy consistent integrators are available for the numerical solution of these differential–algebraic equations (DAEs). In the present work a mechanical integrator for the quaternions will be derived. This will be done by a size–reduction from the director formulation of the equations of motion, which also has the form of DAEs. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Regularization of 2D frictional contacts for rigid body dynamics

Classic rigid body mechanics does not provide frictional forces acting in a 2D contact interface between two bodies during sticking. This is due to the statical undeterminacy related with this problem. Many technical systems, e.g. disk clutches, have such surface-to-surface contacts and it is sometimes desirable to treat them as rigid body systems despite the 2D contact. Alternatively it is possible to model the systems using elastic instead of rigid bodies, but this might lead to certain drawbacks. Here a new regularization model of such 2D contacts between rigid bodies is proposed. It is derived from a material model for elasto-plasticity in continuum mechanics. Only dry friction is taken into account. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Integrable billiards model important integrable cases of rigid body dynamics

Abstract—A generalized billiard is considered, in which a point moves on a locally flat surface obtained by isometrically gluing together several plane domains along boundaries being arcs of confocal quadrics. Under this motion, a point moves from one domain to another, passing through the glued boundaries. Many integrable cases of rigid body dynamics with appropriate parameter values at certain levels of integrals are modeled by classical or generalized billiards; in the paper, Liouville equivalence is proved by comparing Fomenko–Zieschang invariants.     

Adiabatic invariants,diffusion and acceleration in rigid body dynamics

The onset of adiabatic chaos in rigid body dynamics is considered. A comparison of the analytically calculated diffusion coefficient describing probabilistic effects in the zone of chaos with a numerical experiment is made. An analysis of the splitting of asymptotic surfaces is performed and uncertainty curves are constructed in the Poincaré–Zhukovsky problem. The application of Hamiltonian methods to nonholonomic systems is discussed. New problem statements are given which are related to the destruction of an adiabatic invariant and to the acceleration of the system (Fermi’s acceleration).     

Asymptotic stability and associated problems of dynamics of falling rigid body

We consider two problems from the rigid body dynamics and use new methods of stability and asymptotic behavior analysis for their solution. The first problem deals with motion of a rigid body in an unbounded volume of ideal fluid with zero vorticity. The second problem, having similar asymptotic behavior, is concerned with motion of a sleigh on an inclined plane. The equations of motion for the second problem are non-holonomic and exhibit some new features not typical for Hamiltonian systems. A comprehensive survey of references is given and new problems connected with falling motion of heavy bodies in fluid are proposed.      

The equations of motion of a rigid body without parametrization of rotations

New dynamic equations are proposed for a rigid body, without using local parametrization of the rotation group to describe the rotational part of the motion. A simple system of differential-algebraic equations, well suited for constructing the equations of motion of articulated bodies, is obtained.     

On the dynamics of a rigid body carrying a material point

In this paper we consider a system consisting of an outer rigid body (a shell) and an inner body (a material point) which moves according to a given law along a curve rigidly attached to the body. The motion occurs in a uniform field of gravity over a fixed absolutely smooth horizontal plane. During motion the shell may collide with the plane. The coefficient of restitution for an impact is supposed to be arbitrary. We present a derivation of equations describing both the free motion of the system over the plane and the instances where collisions with the plane occur. Several special solutions to the equations of motion are found, and their stability is investigated in some cases. In the case of a dynamically symmetric body and a point moving along the symmetry axis according to an arbitrary law, a general solution to the equations of free motion of the body is found by quadratures. It generalizes the solution corresponding to the classical regular precession in Euler??s case. It is shown that the translational motion of the shell in the free flight regime exists in a general case if the material point moves relative to the body according to the law of areas.     

Stochastic Volterra equations driven by fractional Brownian motion

This paper is devoted to study a class of stochastic Volterra equations driven by fractional Brownian motion. We first prove the Driver type integration by parts formula and the shift Harnack type inequalities. As a direct application, we provide an alternative method to describe the regularities of the law of the solution. Secondly, by using the Malliavin calculus, the Bismut type derivative formula is established, which is then applied to the study of the gradient estimate and the strong Feller property. Finally, we establish the Talagrand type transportation cost inequalities for the law of the solution on the path space with respect to both the uniform metric and the L₂-metric.     

Evolution equations driven by a fractional Brownian motion

In this paper we study nonlinear stochastic evolution equations in a Hilbert space driven by a cylindrical fractional Brownian motion with Hurst parameter and nuclear covariance operator. We establish the existence and uniqueness of a mild solution under some regularity and boundedness conditions on the coefficients and for some values of the parameter H. This result is applied to stochastic parabolic equation perturbed by a fractional white noise. In this case, if the coefficients are Lipschitz continuous and bounded the existence and uniqueness of a solution holds if . The proofs of our results combine techniques of fractional calculus with semigroup estimates.     

分数Brown 运动驱动的非 Lipschitz随机微分方程

讨论了一类带分数Brown 运动的非Lipschitz 增长的随机微分方程适应解的存在唯一性。关于分数 Brown 运动的随机积分有多种定义,本文使用一种广义 Stieltjes积分定义方法,利用这种积分的性质,建立了一类由标准 Brown 运动和一个 Hurst 指数H ∈(1/2,1)的分数Brown 运动共同驱动的、系数为非Lipschitz 增长的随机微分方程适应解的存在唯一性定理。     

A new mathematical construction of the general nonlinear ODEs of motion in rigid body dynamics (Euler’s equations)

It is shown that the three nonlinear dynamic Euler ordinary differential equations (ODEs), concerning the motion of a rigid body free to rotate about a fixed point, are reduced, by means of a subsidiary function which is to be determined, to three Abel equations of the second kind of the normal form. Based on a recently developed mathematical construction concerning exact analytic solutions of the Abel nonlinear ODEs of the second kind, we perform a new mathematical solution for the classical dynamic Euler nonlinear ODEs.     

Clebsch transformation of hydrodynamic equations

A geometrïc meaning of the Clebsch transformation of the hydrodynamic equations for an ideal fluid is established, as well as its relation to the reduction of differential 1-forms to the Darboux normal form. The corresponding kinematic conservation laws are described. The well-known conservation law for the number of quasiparticles is considered as a particular example.     

Stochastic evolution equations driven by Liouville fractional Brownian motion

Let H be a Hilbert space and E a Banach space. We set up a theory of stochastic integration of ℒ(H,E)-valued functions with respect to H-cylindrical Liouville fractional Brownian motion with arbitrary Hurst parameter 0 < β < 1. For 0 < β < ? we show that a function Φ: (0, T) → ℒ(H,E) is stochastically integrable with respect to an H-cylindrical Liouville fractional Brownian motion if and only if it is stochastically integrable with respect to an H-cylindrical fractional Brownian motion.     

Viability for differential equations driven by fractional Brownian motion

In this paper we prove a viability result for multidimensional, time dependent, stochastic differential equations driven by fractional Brownian motion with Hurst parameter , using pathwise approach. The sufficient condition is also an alternative global existence result for the fractional differential equations with restrictions on the state.     

Convergence of delay differential equations driven by fractional Brownian motion

In this note, we prove an existence and uniqueness result of solution for stochastic differential delay equations with hereditary drift driven by a fractional Brownian motion with Hurst parameter H > 1/2. Then, we show that, when the delay goes to zero, the solutions to these equations converge, almost surely and in L ^p , to the solution for the equation without delay. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann–Stieltjes integral.