Stability analysis of steady rotations of a rigid body bearing two-degree-of-freedom control moment gyros with dissipation in gimbal suspension axes

To study the stability of steady rotations of a control moment gyro system with internal dissipation, we use the Barbashin-Krasovskii theorem and the relation, established in [1], between the Lyapunov function and steady motions. Taking into account the special properties of the original problem, we reduce it to a lower-dimensional problem.We give a detailed presentation of an algorithm for analyzing the stability of steady motions of a gyrostat and use this algorithm to perform a complete study for two systems consisting, respectively, of one and two gyros whose gimbal axes are parallel to the principal axis of inertia of the system. Each steady motion is identified as either asymptotically stable or unstable. We find periodic motions that exist only in the presence of dynamic symmetry and which are regular precessions. For the system with two gyros, we prove the asymptotic stability of quiescent states and prove that in the angular momentum range where these states are defined the system does not have any other stable motions.     

On steady rotations of a rigid body bearing a single-axis powered gyroscope whose precession axis is parallel to a principal plane of inertia

The set of steady motions of the system named in the title is represented parametrically via the gyro gimbal rotation angle for an arbitrary position of the gimbal axis.We study the set of steady motions for a system in which the gyro gimbal axis is parallel to a principal plane of inertia as well as for a system with a dynamic symmetry. We determine all motions satisfying sufficient stability conditions. In the presence of dissipation in the gimbal axis, we use the Barbashin-Krasovskii theorem to identify each steady motion as either conditionally asymptotically stable or unstable.     

Amplitude modulated dynamics of a resonantly excited autoparametric two degree-of-freedom system

Forced, weakly nonlinear oscillations of a two degree-of-freedom autoparametric vibration absorber system are studied for resonant excitations. The method of averaging is used to obtain first-order approximations to the response of the system. A complete bifurcation analysis of the averaged equations is undertaken in the subharmonic case of internal and external resonance. The locked pendulum mode of response is found to bifurcate to coupled-mode motion for some excitation frequencies and forcing amplitudes. The coupled-mode response can undergo Hopf bifurcation to limit cycle motions, when the two linear modes are mistuned away from the exact internal resonance condition. The software packages AUTO and KAOS are used and a numerically assisted study of the Hopf bifurcation sets, and dynamic steady solutions of the amplitude or averaged equations is presented. It is shown that both super-and sub-critical Hopf bifurcations arise and the limit cycles quickly undergo period-doubling bifurcations to chaos. These imply chaotic amplitude modulated motions for the system.     

Lagrangian block diagonalization

Relative equilibria, i.e., steady motions associated to specified group motions, are an important class of steady motions of Hamiltonian and Lagrangian systems with symmetry. Relative equilibria can be identified by means of a variational principle on the tangent space of the configuration manifold. We show that relative equilibria can also be found by means of a variational principle on the configuration manifold itself. Formal stability of a relative equilibrium corresponds to definiteness of the second variation of the energymomentum functional, which is a specified combination of the total energy and the group momentum, on an appropriate subspace. We decompose this subspace into three subspaces by means of the Legendre transformation and the group action and show that the second variation block diagonalizes with respect to these subspaces. The techniques employed here are a generalization of the reduced energy-momentum method of Simoet al. (1991), which applies only to simple mechanical systems, to a more general class of conservative systems, including systems on which the symmetry group does not act freely. We briefly discuss a generalization of a result due to Patrick (1990) that provides conditions under which formal stability implies nonlinear orbital stability. Several simple examples, including natural mechanical systems, are used to illustrate the block diagonalization procedure.     

On the Self-Propelled Motion of a Rigid Body in a Viscous Liquid and on the Attainability of Steady Symmetric Self-Propelled Motions

In this paper we study the motion of a self-propelled rigid body through a Navier-Stokes fluid that fills all the three-dimensional space exterior to it. We formulate the problem and prove the existence of a weak solution that is defined globally in time, provided that the net flux across the boundary, of the prescribed boundary values for the velocity, is zero. It is these prescribed boundary values that propel the body, and the body is free to rotate during its motion. In the special case of a body which is symmetric about an axis, and propelled by symmetric boundary values, we obtain strong solutions representing translational motions in the direction of the axis. Further, we prove that for small Reynolds numbers every steady solution with such axial symmetry is attainable as the limit, as time tends to infinity, of a strong nonsteady solution which starts from rest.     

Attraction property of a gimbal gyro with an electric motor

We consider an unbalanced gimbal gyro with vertical outer suspension axis mounted on an immovable base in the gravity field and supplied with an electric motor. The equations of motion of such a system admit a family of solutions describing its steady-state motions (regular precessions and rotor uniform rotations). We show that, in the case of an isolated minimum of the reduced potential energy, the perturbed motions tend in the course of time to steady-state motions in the same family (even in the critical cases of stability).     

Stability of finite-amplitude and overstable convection of a conducting fluid through fixed porous bed

The stability of infinitestimal steady and oscillatory motions and finite amplitude steady motions of a conducting fluid through porous media with free boundaries which is heated from below and cooled from above is investigated in the presence of a uniform magnetic field. Infinitesimal steady motions are investigated using Liapunov method and its is shown that the principle of exchange of stability is valid only when P_m/P_r≤1 with a restricted value of the Hartmann number. It is shown that overstable motions are due to the zonal current induced by the magnetic field. Finite amplitude steady motions are investigated using Veronis [1] analysis and it is shown that for a restricted range of Hartmann numbers and porous parameter P_l, steady finite-amplitude motions can exist for values of the Rayleigh number smaller than that value corresponding to oscillatory motions. Since the Busse number is greater than the wave number the horizontal scale of the steady finite-amplitude motions is larger than that of the overstable motions.     

On steady motions of a rigid body bearing a two-degree-of-freedom control momentum gyroscope with precession axis arbitrarily oriented in the carrying body

Steady motions of a rigid body with a control momentum gyroscope are studied versus the gimbal axis direction relative to the body and the magnitude of the system angular momentum. The study is based on a formula that gives a parametric representation of the set of the system steady motions in terms of the rotation angle of the gimbal. It is shown that, depending on the values of the parameters, the system has 8, 12, or 16 steady motions and the number of stable motions is 2 or 4.     

Static and dynamic behaviors of belt-drive dynamic systems with a one-way clutch

This paper focuses on the nontrivial equilibrium and the steady-state periodic response of belt-drive system with a one-way clutch and belt flexural rigidity. A nonlinear piecewise discrete–continuous dynamic model is established by modeling the motions of the translating belt spans as transverse vibrations of axially moving viscoelastic beams. The rotations of the pulleys and the accessory are also considered. Furthermore, the transverse vibrations and the rotation motions are coupled by nonlinear dynamic tension. The nontrivial equilibriums of the belt-drive system are obtained by an iterative scheme via the differential and integral quadrature methods (DQM and IQM). Moreover, the periodic fluctuation of the driving pulley is modeled as the excitation of the belt-drive system. The steady-state periodic responses of the dynamic system are, respectively, studied via the high-order Galerkin truncation as well as the DQM and IQM. The time histories of the system are numerically calculated based on the 4th Runge–Kutta time discretization method. Furthermore, the frequency–response curves are presented from the numerical solutions. Based on the steady-state periodic response, the resonance areas of the dynamic system are obtained by using the frequency sweep. Moreover, the influences of the truncation terms of the Galerkin method, such as 6-term, 8-term, 10-term, 12-term, and 16-term, are investigated by comparing with the DQM and IQM. Numerical results demonstrate that the one-way clutch reduces the resonance responses of the belt-drive system via the torque-transmitting directional function. Furthermore, the comparisons in numerical examples show that the investigation on steady-state responses of the belt-drive system with a one-way clutch and belt flexural rigidity needs 16-term truncation     

Slip motion and stability of a single degree of freedom elastic system with rate and state dependent friction

We consider quasistatic motion and stability of a single degree of freedom elastic system undergoing frictional slip. The system is represented by a block (slider) slipping at speed V and connected by a spring of stiffness k to a point at which motion is enforced at speed V₀ We adopt rate and state dependent frictional constitutive relations for the slider which describe approximately experimental results of Dieterich and Ruina over a range of slip speeds V. In the simplest relation the friction stress depends additively on a term A In V and a state variable θ; the state variable θ evolves, with a characteristic slip distance, to the value ? B In V, where the constants A, B are assumed to satisfy B > A > 0. Limited results are presented based on a similar friction law using two state variables.Linearized stability analysis predicts constant slip rate motion at V₀ to change from stable to unstable with a decrease in the spring stiffness k below a critical value k_cr. At neutral stability oscillations in slip rate are predicted. A nonlinear analysis of slip motions given here uses the Hopf bifurcation technique, direct determination of phase plane trajectories, Liapunov methods and numerical integration of the equations of motion. Small but finite amplitude limit cycles exist for one value of k, if one state variable is used. With two state variables oscillations exist for a small range of k which undergo period doubling and then lead to apparently chaotic motions as k is decreased.Perturbations from steady sliding are imposed by step changes in the imposed load point motion. Three cases are considered: (1) the load point speed V₀ is suddenly increased; (2) the load point is stopped for some time and then moved again at a constant rate; and (3) the load point displacement suddenly jumps and then stops. In all cases, for all values of k:, sufficiently large perturbations lead to instability. Primary conclusions are: (1) ‘stick-slip’ instability is possible in systems for which steady sliding is stable, and (2) physical manifestation of quasistatic oscillations is sensitive to material properties, stiffness, and the nature and magnitude of load perturbations.     

Critical states of thin-walled hyperbolic gyros elastically connected to a rotating platform

We analyze effects related to the origination of critical states of thin-walled rigid and elastic axisymmetric hyperbolic gyros (shells) elastically connected by the smaller base to a rotating carrier body. We consider the cases of simple and compound rotations of the body and show that, in the case of simple rotation, the specific features of origination of critical states of solid and elastic shells are determined by the ratio of their axial moments of inertia. In the case of compound rotation, the first frequency of precession resonance vibrations of elastic shells generated by the gyroscopic forces of inertia is independent of the elastic compliance of the body-shell joint.     

Composition of nonlinear oscillations with evolution near the manifold of steady motions of reversible and dissipative systems

With some complications of proof, the normalization near a singular point (the zero of a vector field) can be used in the situation when singular points are not isolated. Moreover, there appears the possibility of partial normalization, namely, the possibility of correction of those variables that describe displacements along a manifold of equilibria. It is also possible to prove the existence of analytic integrals. Near the manifold of equilibria of motions for nonholonomic and, in general, reversible systems, the evolution along the manifold is found at higher approximations; however, this evolution is stopped when dissipation is added. The theory of steady and, in particular, stationary motions is well developed (uniform rotations or regular precessions of rigid bodies can be considered as examples of such motions). In addition to this theory, here the theory of normal forms is used for the first time to reduce the study of a neighborhood of an invariant manifold with periodic solutions to the problem of equilibria in the reduced system with a new independent angle-type variable.     

Three-dimensional numerical simulation of the wake flow of an afterbody at subsonic speeds

We numerically investigate the wake flow of an afterbody at low Reynolds number in the incompressible and compressible regimes. We found that, with increasing Reynolds number, the initially stable and axisymmetric base flow undergoes a first stationary bifurcation which breaks the axisymmetry and develops two parallel steady counter-rotating vortices. The critical Reynolds number (Re _cs) for the loss of the flow axisymmetry reported here is in excellent agreement with previous axisymmetric BiGlobal linear stability (BiGLS) results. As the Reynolds number increases above a second threshold, Re _co, we report a second instability defined as a three-dimensional peristaltic oscillation which modulates the vortices, similar to the sphere wake, sharing many points in common with long-wavelength symmetric Crow instability. Both the critical Reynolds number for the onset of oscillation, Re _co, and the Strouhal number of the time-periodic limit cycle, St_sat, are substantially shifted with respect to previous axisymmetric BiGLS predictions neglecting the first bifurcation. For slightly larger Reynolds numbers, the wake oscillations are stronger and vortices are shed close to the afterbody base. In the compressible regime, no fundamental changes are observed in the bifurcation process. It is shown that the steady state planar-symmetric solution is almost equal to the incompressible case and that the break of planar symmetry in the vortex shedding regime is retarded due to compressibility effects. Finally, we report the developments of a low frequency which depends on the afterbody aspect ratio, as well as on the Reynolds and on the Mach number, prior to the loss of the planar symmetry of the wake.     

Irreducible tensors and their applications in problems of dynamics of solids

One difficulty encountered in solving mechanical problems with complicated interaction is to express either the moments of forces or the force function via the phase variables of the problem. Here various transformations of coordinate systems are used, because interactions are determined by a relation between tensor variables one of which refers to the body and the other refers to the field. In this connection, the usual definition of a tensor in Cartesian coordinates is inconvenient because of the fact that the components of a tensor of rank l ≥ 2 can be arranged as several linear combinations that behave differently under rotations of the coordinate system. Naturally, one needs to define tensors in such a way that their components and linear combinations of these be transformed in a unified manner under rotations of the coordinate system. This requirement is satisfied by irreducible tensors. The mathematical apparatus of irreducible tensors was created to satisfy the requirements of quantum mechanics and turned out to be rather universal. As far as the author knows, this apparatus was first used in mechanics by G. G. Denisov and the author of the present paper [1]. Using this apparatus, one can see the clear physical meaning of complicated interactions, express these interactions in invariant form, easily perform transformations from one coordinate system to another coordinate system turned relative to the first, consider rather complicated types of interactions writing them in compact form explicitly depending on the phase variables of the problem, easily use the symmetry of both the rigid body and the force field structure, and perform the averaging procedure for the entire object rather than componentwise. The present paper further develops the paper [1]. We present a brief introduction to the theory of irreducible tensors. We show that the force function of various interactions between a rigid body and a force field can be represented as the scalar product of irreducible tensors. We study general properties of evolution motions of a rigid body in axisymmetric and nonsymmetric force fields under the action of moments caused by various harmonics of the force function.     

Dynamic analysis of planar rigid-body mechanical systems with two-clearance revolute joints

In this paper, the behavior of planar rigid-body mechanical systems due to the dynamic interaction of multiple revolute clearance joints is numerically studied. One revolute clearance joint in a multibody mechanical system is characterized by three motions which are: the continuous contact, the free-flight, and the impact motion modes. Therefore, a mechanical system with n-number of revolute clearance joints will be characterized by 3 ⁿ motions. A slider-crank mechanism is used as a demonstrative example to study the nine simultaneous motion modes at two revolute clearance joints together with their effects on the dynamic performance of the system. The normal and the frictional forces in the revolute clearance joints are respectively modeled using the Lankarani–Nikravesh contact-force and LuGre friction models. The developed computational algorithm is implemented as a MATLAB code and is found to capture the dynamic behavior of the mechanism due to the motions in the revolute clearance joints. This study has shown that clearance joints in a multibody mechanical system have a strong dynamic interaction. The motion mode in one revolute clearance joint will determine the motion mode in the other clearance joints, and this will consequently affect the dynamic behavior of the system. Therefore, in order to capture accurately the dynamic behavior of a multi-body system, all the joints in it should be modeled as clearance joints.     

Transition to escape in a system of coupled oscillators

We study a Hamiltonian system of coupled oscillators derived from two forced pendulums, connected with a torsional spring. The uncoupled limit is described by two identical oscillators, each possessing a homoclinic orbit separating bounded from unbounded motion. We focus on intermediate energy levels which lead to detained motions, defined as trajectories that, though unbounded as t → ∞, oscillate within the region defined by the homoclinic orbit of the unperturbed system for a long but finite time. We analyze the existence and behavior of these motions in terms of equipotential surfaces. These curves provide bounds on the motion of the system and are shown to be closed for low energies. However, above some critical energy level the equipotential curves become open. The detained trajectories are shown to arise from the region of phase space that was, for appropriate energies, stochastic. These motions remain within this region for long times before finally “leaking out” of the opening in the equipotential curves and proceeding to infinity.     

Nonlinear nonplanar dynamics of a parametrically excited inextensional elastic beam

The nonlinear dynamics of a clamped-clamped/sliding inextensional elastic beam subject to a harmonic axial load is investigated. The Galerkin method is used on the coupled bending-bending-torsional nonlinear equations with inertial and geometric nonlinearities and the resulting two second order ordinary differential equations are studied by the method of multiple time seales and by direct numerical integration. The amplitude equations are analyzed for steady and Hopf bifurcations. Depending on the amplitude of excitation, the damping and the ratio of principal flexural rigidities, various qualitatively distinct frequency response diagrams are uncovered and limit cycles and chaotic motions are found. In the truncated two-degree-of-freedom system the transition from periodic to chaotic amplitude-modulated motions is via the process of torus doubling and subsequent destruction of the torus.     

Thermocapillary convection in a three-layer system with the temperature gradient directed along the interfaces

The paper presents numerical simulations of the Marangoni–Bénard convection in a real symmetric three-layer system. The temperature gradient is directed along the interfaces. Nonlinear regimes of steady and oscillatory convective flows are investigated by means of the finite-difference method. Transitions between the motions with different spatial structures are studied. To cite this article: V. Shevtsova et al., C. R. Mecanique 333 (2005).     

Symmetry limit theory for cantilever beam-columns subjected to cyclic reversed bending

The behavior of a linear strain-hardening cantilever beam-column subjected to completely reversed plastic bending of a new idealized program under constant axial compression consists of three stages: a sequence of symmetric steady states, a subsequent sequence of asymmetric steady states and a divergent behavior involving unbounded growth of an anti-symmetric deflection mode. A new concept “symmetry limit” is introduced here as the smallest critical value of the tip-deflection amplitude at which transition from a symmetric steady state to an asymmetric steady state can occur in the response of a beam-column. A new theory is presented for predicting the symmetry limits. Although this transition phenomenon is phenomenologically and conceptually different from the branching phenomenon on an equilibrium path, it is shown that a symmetry limit may theoretically be regarded as a branching point on a “steady-state path” defined anew. The symmetry limit theory and the fundamental hypotheses are verified through numerical analysis of hysteretic responses of discretized beam-column models.     

Creeping flow around particles in a Bingham fluid

We revisit the variational setting of the creeping flow of Bingham fluid about a particle. We investigate two problems: the resistance and the mobility problem, which arise in the context of sedimentation. We present results and the general framework for uniqueness, symmetry and reversibility properties of solutions, and clarify the relation between resistance and mobility problems. We also consider general properties of the static stability limit (or load limit). In the second part of the paper we apply insights gained from the first part to the computation of 2D exterior flow around an infinite cylinder with elliptical cross-section. We study the static stability limit and find that the limiting flow solution approaches the perfectly plastic solid solution.