On discontinuous motions in systems with unilateral constraints

Mechanical systems with a finite number of degrees of freedom, subject to one or more unilateral geometric constraints, are considered. Apart from the main forms of motion—flying, supported motion and non-degenerate collisions—such systems also show more complex, discontinuous motions, including infinitely many impacts in any neighbourhood of the starting time. These motions are possible not only when no continuous motions exist, but also together with continuous motions [1]. It is proved that, in the case of ideal constraints, if the reactions of the constraints at the starting time are non-zero, there cannot be any discontinuous motion. In systems with dry friction there is yet another type of discontinuity, namely, tangential impact at zero approach velocity. Sufficient conditions for continuity of the motion are derived for this case also. The conditions are verified with examples that use the usual models of impact.     

Mathematical Models for the Triaxial Attitude Control Testbed

The Triaxial Attitude Control Testbed has been developed as part of a research program at the University of Michigan on multibody rotational dynamics and control. In this paper, equations of motion are derived and presented in various forms. Actuation mechanisms are incorporated into the models; these include fan actuators, reaction wheel actuators and proof mass actuators that are fixed to the triaxial base body. The models also allow incorporation of unactuated auxiliary bodies that are constrained to move relative to the triaxial base body. The models expose the dynamic coupling between the rotational motion of the triaxial base body, the relative or shape motion of the unactuated auxiliary degrees of freedom, and dynamics associated with actuation mechanisms. Many different model simplifications and approximations are developed. Control models for the triaxial attitude control testbed are formulated that reflect specific assumptions.     

A charged composite particle in a constant electric field

We consider the motion of a composite charged particle in a constant electric field. Using the billiard formalism, we establish exact laws of motion for such a particle with a small number of internal degrees of freedom and propose using a generalized Schwarz principle to straighten trajectories in the field presence. Within the billiard formalism, we obtain regimes of motion of a composite particle with two internal degrees of freedom in a constant field.     

Resolución numérica de las ecuaciones del movimiento de edificios de varias plantas con no linealidades severas

This paper presents a new algorithm for solving the equations of motion of multi-storey buildings that incorporate frictional energy dissipators as seismic protection. The behavior of the dissipators is represented by Coulomb dry friction models; they introduce severe nonlinearities in the dynamic behavior of the structure every time that the contact conditions (stick or slip) change in the dissipators. These nonlinearities complicate the resolution of the equations of motion as it usually is described by lumped masses models whose degrees of freedom are the displacements of the floors and, as the stick or slip conditions change, the degrees of freedom must be modified: for blocking conditions they are only the displacements of the storeys while under sliding conditions the displacements of the dissipators have to be also considered. In previous articles the accuracy of the proposed algorithm has been verified by comparison with experimental results; as well, the computational efficiency of the algorithm has been confirmed by comparing the required resources (in terms of computation time and of memory allocation) with those of other algorithms. The objectives of this paper are to describe in detail the numerical solution of the equations of motion and present representative examples confirming the ability of the algorithm to reproduce the dynamic behavior of buildings with friction dissipators and reporting preliminarily about the usefulness of such devices to reduce the oscillations of the structure to be protected.     

Integrated motion measurement of rigid multibody systems

During the last years, integrated navigation systems based on gyros, accelerometers, and GPS receivers became powerful devices for the guidance of aircraft and ships. Comparable equipment using especially wheel sensors exists for cars. The kernel of such systems is a Kalman filter estimating the relevant vehicle motion. The filter design in turn requires a kinematical model to settle on the motion components considered and to describe the mechanical meaning of the measurements employed. Until now, usual models consider only one to six degrees of freedom of a single rigid body. The assumption of a solitary rigid body is not a consequence of the basic concept of integrated navigation; it reflects merely classical navigation requirements. In principle, determining the motion of multibody systems, representing certain vehicle types of varying shape, is possible if appropriate kinematical models and sensor arrangements are available. Based on the theory of integrated navigation systems, the paper outlines the fundamentals of designing integrated motion measurement systems for rigid multibody structures. The example of a double pendulum with a movable inertial support and with equipment of microelectromechanical inertial sensors and of small radar units illustrates this approach. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Gravity in the stabilized brane world model in the five-dimensional Brans-Dicke theory

We obtain and solve the linearized equations of motion for gravity and scalar fields in the world stabilized brane model in the five-dimensional Brans-Dicke theory. We segregate the physical degrees of freedom and find the mass spectrum of the Kaluza-Klein excitations and the coupling constants of Kaluza-Klein modes to matter on the brane with negative tension.     

Reduction Methods for FE Analysis in Nonlinear Structural Dynamics

The computation of the nonlinear motion of large structures with implicit time integration schemes is costly. In each time step a large system of linear equations needs to be solved several times. In finite element models often a fine discretization is necessary to represent the geometry and to yield accurate results for the stress field. But from experience it is known that only a small number of degrees of freedom is sufficient to account for the dominant parts of a dynamic motion. Similar to modal decomposition, methods were developed to reduce the number of degrees of freedoms in nonlinear problems. Even though it is not possible to decompose the motion into decoupled modes, a reduction of the number of degrees of freedom yields less computational effort in many cases. The choice of appropriate basis vectors is important. Often used are load-dependent ‘Ritz’ vectors, which should be updated during the computation to yield sufficient accuracy. Dominant modes, computed by a proper orthogonal decomposition of a previous calculation can be used for repeated analyses of the same system with different loads. Significant time savings can be achieved with reduction methods. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling of the robotic Powerball®: a nonholonomic,underactuated and variable structure-type system

The Powerball® is the commercial name for a gyroscopic device that is marketed as a wrist exerciser. The device has a rotor with two underactuated degrees of freedom, which can be actuated by the appropriate motion of human or robot wrist axes. After the initial spin, applying the appropriate motion and torques to the housing leads to a spin-up of the rotor. Finding these torques intuitively is an easy task for human operators, but a complex task for a technical consideration, for example, in robotics.

This article's main contribution is a novel dynamic model that considers friction effects. The presented model includes all three working principles of the device: free rotor mode and both modes of rotor rolling in the housing. The work introduces models with one and two degrees of freedom actuation, both of which are suitable for laboratory control experiments. An estimation of the friction is discussed, and both the simulation and the experimental results are presented to evaluate the models.     

The analysis of the degenerate case of oscillations of a mechanical system

In this paper, the nonlinear model of the mechanism with two degrees of freedom will be studied. An approximate analytical solution of the differential equation of motion in the series showed the presence of features in the aspiration of the mass of one of the bodies to zero. It also gives an algorithm for finding the points of degeneracy of communication between small perturbations of the function of the problem and the derivatives of these functions at a time. Copyright © 2016 John Wiley & Sons, Ltd.     

On the existence of time-optimal control of mechanical manipulators

This paper is concerned with the problem of the existence and structure of time-optimal control for models derived from Lagrange equations of motion of mechanical systems involving links. The condition which ensures the existence of time-optimal control is demonstrated. The study conducted in this paper involves a highly nonlinear mathematical model of a two-degree-of-freedom mechanical system. However, the procedure and the results presented in this paper can be extended to mechanical systems with any finite number of degrees of freedom.The authors wish to thank Professor D. G. Hull and the reviewers for their most valuable comments and suggestions.     

Baryon string model. II. Special solutions of classical three-string equations of motion

Conclusions We have considered the simplest solutions of the three-string equations of motion; these are solutions with a finite number of excited degrees of freedom. It is of interest to construct the quantum theory of such motions of the relativistic three-string. Quantum theory of a meson string with finite number of degrees of freedom was constructed in [4]. Quantization of finite-mode solutions of the baryon string model will be considered in the third paper of the present work.Institute of High Energy Physics, Serpukhov. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 64, No. 2, pp. 245–258, August, 1985.     

The use of splitting of the equations of motion in the numerical solution of problems of the dynamics of elastic systems

Linear elastic systems with a finite number of degrees of freedom, the initial equations of motion of which are constructed using the finite element method or other discretization methods, are considered. Since, in applied dynamics problems, the motions are usually investigated in a frequency range with an upper bound, the degrees of freedom of the initial system of equations are split into dynamic and quasi-dynamic degrees. Finally, the initial system of equations is split into a small number of differential equations for the dynamic degrees of freedom and into a system of algebraic equations for determining the quasi-static displacements, represented in the form of a matrix series. The number of terms of the series taken into account depends on the accuracy required.     

Nonlinear Dynamics of Marine Systems in Random Waves

The dynamics of ships or offshore structures is influenced by several different effects, some of which have a distinctly nonlinear characteristic. Even though in many situations the motion can sufficiently be described by linear models, nonlinear phenomena play a crucial role in the investigation of some more critical operating conditions: Large amplitude motions, sudden jumps in the dynamical behavior and sensitivity to the initial conditions are likely to occur under some circumstances. The response of floating systems such as moored buoys and barges in regular waves can be approximated by analytical or numerical techniques. These analyses reveal the characteristics of different periodic motions. In order to determine how these responses change under a more general forcing, the motion of floating structures under the influence of random disturbances is described by probability distributions. Different mathematical tools can efficiently be applied to models with few degrees of freedom. The localized statistical linearization used here is also promising for larger systems. Modelling aspects of offshore structures and random waves are discussed as well as the determination of probability distributions. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of the periodic motions of the vibro-impact systems

In many other papers a new direct method is developed for the study of vibro-impact system stability, a case in which the stability conditions are directly and rapidly obtained. This new method is stated and applied for systems with one or several degrees of freedom whose motion is analysed by Lagrange's equations. Thus, the equations in variations determined from the differential equations are considered, which by applying the boundary conditions lead by integration to the linearized equation in perturbations necessary to write the stability conditions. As example, is analysed in detail a particular case with one degree of freedom realised by the rod-crank mechanism. Also, a general system with two degrees of freedom is completely studied.     

Mathematical modeling of complex mechanical systems

Mathematical modeling of mechanical systems based on multibody system models is a well tested approach. Generating the equations of motion for complex multibody systems with a large number of degrees of freedom is difficult with paper and pencil. For this reason methods for automatic equation generation have been developed. Most methods result in numerical equations of motion without explicit information about the parameters. In this paper a method is described resulting in symbolic equations of motion. The method allows also the determination of the constraint forces which are important for design purposes. The inverse problem of dynamics is also easily solved.     

A Study of the Motions of an Autonomous Hamiltonian System at a 1:1 Resonance

We examine the motions of an autonomous Hamiltonian system with two degrees of freedom in a neighborhood of an equilibrium point at a 1:1 resonance. It is assumed that the matrix of linearized equations of perturbed motion is reduced to diagonal form and the equilibrium is linearly stable. As an illustration, we consider the problem of the motion of a dynamically symmetric rigid body (satellite) relative to its center of mass in a central Newtonian gravitational field on a circular orbit in a neighborhood of cylindrical precession. The abovementioned resonance case takes place for parameter values corresponding to the spherical symmetry of the body, for which the angular velocity of proper rotation has the same value and direction as the angular velocity of orbital motion of the radius vector of the center of mass. For parameter values close to the resonance point, the problem of the existence, bifurcations and orbital stability of periodic rigid body motions arising from a corresponding relative equilibrium of the reduced system is solved and issues concerning the existence of conditionally periodic motions are discussed.     

A Prediction Model for State Observation and Model Predictive Control of Biped Robots

Biped walking robots present a class of mechanical systems with many different challenges such as nonlinear multi-body dynamics, a large number of degrees of freedom and unilateral contacts. The latter impose constraints for physically feasible motions and in stabilization methods as the robot can only interact due to pressure forces with the environment. This limitation can cause the system to fall under unknown disturbances such as pushing or uneven terrain. In order to face such problems, an accurate and fast model of the robot to observe the current state and predict the state evolution into the future has to be used. This work presents a nonlinear prediction model with two passive degrees of freedom (dof), point masses and compliant unilateral contacts. We show that the model is applicable for real-time model predictive optimization of the robot's motion. Experiments on the biped robot LOLA [1] underline the effectiveness of the proposed model to increase the system's long term stability under large unknown disturbances. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An algorithm for normalizing Hamiltonian systems in the problem of the orbital stability of periodic motions

A constructive procedure is proposed for constructing equations of perturbed motion convenient for investigating the orbital stability of periodic motion in an autonomous Hamiltonian system with two degrees of freedom. An algorithm for normalizing these equations is described, and formulae for evaluating the coefficients of the normal form are presented. The results are used to investigate the stability of motion in certain special cases of the regular Grioli precession of a heavy rigid body with one fixed point.     

Efficient decoupling technique applied to the numerical time integration of advanced interaction models for railway dynamics

Railway interaction is characterised by the coupling between the train and the track introduced through the wheel/rail contact. The introduction of the flexibility in the wheelset and the track through the finite element (FE) method in the last four decades has permitted to study high-frequency phenomena such as rolling noise and squeal, whose origin lies in the strongly non-steady state and non-linear behaviour of the contact forces that arise from the small contact area. In order to address models with a large number of degrees of freedom, innovative Eulerian-modal models for wheelsets with rotation and cyclic tracks have been developed in recent years. The aim of this paper is to extend the resulting formulation to an uncoupled linear matrix equation of motion that allows solving each equation independently for each time step, considerably reducing the associated computational cost. The decoupling integration method proposed is compared in terms of computational performance with Newmark and Runge-Kutta schemes, commonly used in vehicle dynamics, for simulations with the leading wheelset negotiating a tangent track and accounting the rail roughness.     

Method for solving the Boltzmann kinetic equation for polyatomic gases

A method is proposed for computing the collision operator of a generalized Boltzmann kinetic equation with allowance for energy transfer from translational to vibrational or rotational degrees of freedom. The collision operator is computed using a projection method on a uniform velocity grid. The operator satisfies the mass, momentum, and energy conservation laws and vanishes for an equilibrium velocity distribution function. Approximate models are suggested that provide savings on the computation of rotational-translational relaxation. Numerical examples are presented.