Stability of a periodic motion of a rod suspended by an ideal thread

We consider the plane motion of a rod suspended by an ideal thread in a homogeneous field of gravity. We study the nonlinear orbital stability problem for the translational periodic motion of the rod along the vertical. Depending on two dimensionless parameters of the problem, we make conclusions on orbital instability, stability for a majority of initial conditions, or formal stability.     

Cylindrical equilibrium surfaces of a capillary liquid and their stability

We consider the plane problem of the equilibrium of a capillary surface. We study the stability of a two-dimensional surface with respect to plane and spatial disturbances. We give data which can be used for deciding the question of the stability of any symmetric equilibrium surface in a field of gravitational forces and in conditions of weightlessness. We solve the problems of the stability of a liquid in a rectangular and a sectorial channel and also the problem of the separation of a plane drop from a horizontal wall.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 57–68, January–February, 1976.The authors are grateful to A. D. Myshkis and A. D. Tyuptsov for their evaluation and their useful comments.     

On the instability of a car in the vertical plane under rectilinear motion with friction forces taken into account

We consider the problem on the stability and instability of the equilibrium in the vertical plane for a wheeled vehicle performing a uniform rectilinear motion in the presence of rolling friction forces. We assume that the dependence of the rolling friction coefficients on the motion velocity is known and derive necessary and sufficient conditions on the system parameters under which such equilibria are stable.     

Nonlinear stability of flows of Jeffreys fluids at low Weissenberg numbers

We consider the stability of steady flows of viscoelastic fluids of Jeffreys type. For sufficiently small Weissenberg numbers, but arbitrary Reynolds numbers, it is proved that the flow is stable to small disturbances if the spectrum of the linearized operator is in the left half plane.     

Stability and stabilization of motion of a uniaxial wheel transport platform

We consider a uniaxial wheel transport platform with a single-degree-of-freedom gyroscope moving without slipping either on a plane nonrotating horizontal surface or on the spherical rotating Earth surface. We obtain a general mathematical model which, in a special case, coincides with the model in the form of Chaplygin equations, which permits obtaining a physical interpretation of the Chaplygin equations. In the case of stationary motion where only the balance weight is controlled, we find the minimum value of the gyro angular momentum that ensures the system stability. An example with parameters of the breadboard model is used to consider the problem of the stationary motion stability and stabilization without gyro; the control matrix minimizing the quadratic performance functional is obtained. The characteristic curves of the transient process in the system are given.     

Parametric Stability in a Sitnikov-Like Restricted P-Body Problem

We consider the dynamics of an infinitesimal particle under the gravitational action of P primaries of equal masses. These move in an elliptic homographic solution of the P-body problem and the infinitesimal particle moves along the straight line perpendicular to their plane of motion and passing through the common focus of the ellipses. In this work we consider the parametric stability of the infinitesimal mass located at the focus of the ellipses. We construct the boundary curves of the stability/instability regions in the plane of the parameters \(\mu \) and \(\epsilon \), which are the mass of each primary and the eccentricity of the elliptic orbit, respectively.     

The Analysis of Some Characteristic Equations Arising in Population and Epidemic Models

Epidemic models with a general infective period distribution are formulated as functional differential equations, as are population models with a general life span distribution. The analysis of the local stability properties of equilibria of such models leads to a characteristic equation involving the Laplace transform of the infective period (or life span) distribution. We obtain conditions under which all roots of the characteristic equation are in the left half plane, implying asymptotic stability of equilibrium, for every infective period distribution. We also consider the converse problem of describing when instability can occur for specific infective period distributions.     

Investigation of Forced Oscillations in Circular Cylindrical Vessels Separated by Diametrical Barriers

We consider nonlinear oscillations of an ideal incompressible liquid in a partially filled vertical semicircular cylindrical tank. We construct approximate periodic solutions for a four-mode system that describes nonlinear oscillations in a semicircular cylindrical tank under the action of a perturbation force in the plane of the barrier. We construct and investigate the domains of stability and instability for the physical processes considered. We perform a numerical realization of the method and analyze the hydrodynamic interaction of the liquid with the tank. The problem considered is of interest for the investigation of nonlinear processes in a liquid in the case of tanks with diametrical barriers.     

Parametric Stability in a P+2-Body Problem

We consider a planar restricted \(P+2\)-body problem where P bodies of equal masses located at the vertices of a regular polygon move in an homographic elliptic orbit and an additional mass is fixed at the center of the polygon. We study the equilibrium of the infinitesimal mass that lies on the half-line from the center of the polygon through the midpoint of its side, outside the unit circle. We study the parametric stability of this equilibrium constructing the boundary curves of the stability/instability regions in the plane of the parameters.     

Instability of an Unbounded Elastic Plate in a Gas Flow

The stability of an unbounded plane elastic plate in gas moving on one side of the plate and at rest on the other is analyzed. The gases are inviscid and in general different. The plate is under tension and has flexural stiffness. It is shown that the system is always unstable to plane sinusoidal perturbations with wave vector parallel to the velocity. As limiting cases, a tangential discontinuity between the two gases and unilateral flow past a plate with constant pressure on the opposite side are considered. In these cases, the conditions of stability to plane perturbations are non-trivial and are investigated below.     

Existence of Quasipattern Solutions of the Swift–Hohenberg Equation

We consider the steady Swift–Hohenberg partial differential equation, a one-parameter family of PDEs on the plane that models, for example, Rayleigh–Bénard convection. For values of the parameter near its critical value, we look for small solutions, quasiperiodic in all directions of the plane, and which are invariant under rotations of angle ${\pi/q, q \geqq 4}$ . We solve an unusual small divisor problem and prove the existence of solutions for small parameter values, then address their stability with respect to quasi-periodic perturbations.     

Effect of a permeable partition on convective instability of a horizontal liquid layer

We consider the effect of a thin permeable partition on the static stability of a horizontal liquid layer heated from underneath. The permeable partition is assumed to be plane and situated parallel to the boundary planes in the center of the layer. The resistance of the partition to the flow of liquid from one part of the layer to another leads to an increase in the static stability. We investigate the dependence of the minimum critical Rayleigh number-on the resistance of the partition and the form of the critical motions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 157–159, January–February, 1977.     

Stability of plane oscillations of a satellite in a circular orbit

We study motions of a rigid body (a satellite) about the center of mass in a central Newtonian gravitational field in a circular orbit. There is a known particular motion of the satellite in which one of its principal central axes of inertia is perpendicular to the orbit plane and the satellite itself exhibits plane pendulum-like oscillations about this axis. Under the assumption that the satellite principal central moments of inertia A, B, and C satisfy the relation B = A + C corresponding to the case of a thin plate, we perform rigorous nonlinear analysis of the orbital stability of this motion.In the plane of the problem parameters, namely, the oscillation amplitude ε and the inertial parameter, there exist countably many domains of orbital stability of the satellite oscillations in the linear approximation. Nonlinear orbital stability analysis was carried out in thirteen of these domains. Isoenergetic reduction of the system of equations of the perturbed motion is performed at the energy level corresponding to the unperturbed periodic motion. Further, using the algorithm developed in [1], we construct the symplectic mapping generated by the equations of the reduced system, normalize it, and analyze the stability. We consider resonance and nonresonance cases. For small values of the oscillation amplitude, we perform analytic investigations; for arbitrary values of ε, numerical analysis is used.Earlier, numerical analysis of stability of plane pendulum-like motions of a satellite in a circular orbit was performed in several special cases in [1–4].     

Satellite rotation stability at a Mercurian type resonance

We consider the satellite plane motion about the center of mass in a central Newtonian gravitational field in an elliptic orbit. This motion is described by a second-order differential equation known as the Beletskii equation. In the framework of the plane problem (under the assumption that the body vibrates in the unperturbed orbit plane), there exists a family of periodic solutions of the Beletskii equation near the 3: 2 resonance between the orbital revolution and axial rotation periods. A nonlinear stability analysis of these periodic solutions is carried out both in the presence of third- and fourth-order resonances and in their absence as well as on the boundaries of the stability regions in the first approximation. The problem is solved numerically. For fixed parameter values (the eccentricity of the center-of-mass orbit and the inertial parameter), the construction of a symplectic mapping of the equilibrium into itself is used to calculate the coefficients of the mapping generating function, which are further used to conclude whether the equilibrium is stable or not.     

On the stability of plane parallel viscoelastic shear flows in the limit of infinite Weissenberg and Reynolds numbers

Elastic effects on the hydrodynamic instability of inviscid parallel shear flows are investigated through a linear stability analysis. We focus on the upper convected Maxwell model in the limit of infinite Weissenberg and Reynolds numbers. We study the effects of elasticity on the instability of a few classes of simple parallel flows, specifically plane Poiseuille and Couette flows, the hyperbolic-tangent shear layer and the Bickley jet.The equation for stability is derived and solved numerically using the spectral Chebyshev collocation method. This algorithm is computationally efficient and accurate in reproducing the eigenvalues. We consider flows bounded by walls as well as flows bounded by free surfaces. In the inviscid, nonelastic case all the flows we study are unstable for free surfaces. In the case of wall bounded flow, there are instabilities in the shear layer and Bickley jet flows. In all cases, the effect of elasticity is to reduce and ultimately suppress the inviscid instability.     

Stability and Synchronization of a Ring of Identical Cells with Delayed Coupling

We consider a ring of identical elements with time delayed, nearest neighbour coupling. The individual elements are modelled by a scalar delay differential equation which includes linear decay and nonlinear delayed feedback. The linear stability of the trivial solution is completely analyzed and illustrated in the parameter space of the coupling strength and the coupling delay. Conditions for global stability of the trivial solution are also given. The bifurcation and stability of nontrivial synchronous solutions from the trivial solution is analyzed using a centre manifold construction.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday.     

Stability in a case of motion of a paraboloid over a plane

We solve a nonlinear orbital stability problem for a periodic motion of a homogeneous paraboloid of revolution over an immovable horizontal plane in a homogeneous gravity field. The plane is assumed to be absolutely smooth, and the body–plane collisions are assumed to be absolutely elastic. In the unperturbed motion, the symmetry axis of the body is vertical, and the body itself is in translational motion with periodic collisions with the plane.The Poincare´ section surfacemethod is used to reduce the problemto studying the stability of a fixed point of an area-preserving mapping of the plane into itself. The stability and instability conditions are obtained for all admissible values of the problem parameters.     

Relative displacements of the two sides of a plane crack

We consider a two dimensional deformation of an infinite anisotropic elastic material which contains a plane crack. When tractions are specified over the crack faces, we give a formula for the corresponding relative displacements of the two sides of the crack. We also consider the inverse problem of determining the elastic constants by measuring the tractions and the corresponding relative displacements on the crack.     

Quasiperiodic Collision Solutions in the Spatial Isosceles Three-Body Problem with Rotating Axis of Symmetry

We consider the spatial isosceles Newtonian three-body problem, with one particle on a fixed plane, and the other two particles (with equal masses) located symmetrically with respect to this plane. Using variational methods, we find a one-parameter family of collision solutions for this system. All these solutions are periodic in a rotating frame.     

Super-stable parallel flows of multiple visco-plastic fluids

We consider the stability of a multi-layer plane Poiseuille flow of two Bingham fluids. It is shown that this two-fluid flow is frequently more stable than the equivalent flow of either fluid alone. This phenomenon of super-stability results only when the yield stress of the fluid next to the channel wall is larger than that of the fluid in the centre of the channel, which need not have a yield stress. Our result is in direct contrast to the stability of analogous flows of purely viscous generalised Newtonian fluids, for which short wavelength interfacial instabilities can be found at relatively low Reynolds numbers. The results imply the existence of parameter regimes where visco-plastic lubrication is possible, permitting transport of an inelastic generalised Newtonian fluid in the centre of a channel, lubricated at the walls by a visco-plastic fluid, travelling in a stable laminar flow at higher flow rates than would be possible for the single fluid alone.