Stability of hinged viscoelastic rods under dynamic loading

Equations of Mathieu type describing the transverse vibrations of a straight viscoelastic rod compressed by a periodic longitudinal force are examined. The resonance regime is investigated by means of the method of averaging [2–4] in the case of a hinged viscoelastic rod acted upon by a periodic force.Institute of Cybernetics and Computer Center, Academy of Sciences of the Uzbek SSR, Tashkent. Translated from Mekhanika Polimerov, No. 6, pp. 1126–1129, November–December, 1973.     

A model of a reinforced tyre with a rod protector

A tyre design consisting of a steel-cord-reinforced rigid bond with sides connected to the wheel disc and a protector(tread) in contact with the road is examined. The tread is in the form of a set of rods connected by one end to the band, with the other end either free or in contact with the road. The rod end in contact with the road is acted upon by a force applied from the road, represented by a force normal to the road plane and a shear force due to dry friction. If the modulus of the shear force does not exceed the magnitude of the normal force multiplied by the dry friction coefficient, there is no slip at the contact point. In the opposite case, the rod end will be displaced along the road by an amount sufficient to distribute the normal and shear forces. The dynamics of longitudinal and transverse strains of the rods in contact with the road is analysed using the motion separation method in the quasi-static approximation. The behaviour of the tread rods as a function of the vertical displacement of the wheel centre is investigated, the contact area is found and the conditions are determined under which the contact area is divided into parts in which either slip of the rod ends occurs or does not occur, depending on the magnitude of the longitudinal displacement of the wheel centre or its turning relative to the horizontal axis. An analogue of a continuous model of a rod-like tread is considered, and the magnitudes of the forces and moments are found as a function of the wheel disc displacements. The equations of wheel rolling are obtained, and the conditions under which steady motions exist are found.     

The asymptotic stability of the equilibrium of non-autonomous systems

A method of investigating the stability of non-linear systems acted upon by unsteady perturbations is proposed, based on the use of Lyapunov's second method. The sufficient conditions for asymptotic stability of the solutions of non-autonomous systems in critical cases are obtained.     

The effective mass of a point mass moving along a string on a Winkler foundation

A simple in form and physically clear asymptotic solution of the problem of the motion (without friction) of a point mass acted upon by a specified external force on a string on a Winkler foundation is obtained, taking into account the wave drag on the motion. It is shown that the point mass moves along the string in the same way as a point with a variable velocity-dependent mass would move when acted upon solely by an external force (ignoring drag).     

The stability of a non-potentially loaded discrete-continuous gyroscopic system with internal and external friction

The problem of the stability of the axial rotations of a symmetric rotor by means of an elastic shaft in the form of a thin straight rod, when there is a follower force and linear dissipation, is considered. Linearized equations for the displacement of the rotation of the rotor and the transverse plane oscillations of the rod are presented in a complex form as well as the boundary conditions. Using the standard procedure of a Laplace transformation with respect to time, relations for the images are obtained which are then solved by means of quasipolynomials of the complex transform parameter, and an asymptotic analysis of them is carried out. The effect of a follower force is investigated and its critical values are determined. The results of mathematical modelling for specific numerical values of the parameters are presented.     

An analytical formula for an elastic thin rod shape under composite loading

The Euler problem on the stability of a thin elastic rod under compressing forces is widely known. In a number of papers this problem is generalized to finding the shape of such a rod under the simultaneous action of a compressing force and a torsional moment. The shape is determined by solving a complex nonlinear boundary value problem by numerical methods. In this paper, an approach providing a full analytical solution of the problem for a broad range of external conditions is discussed. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 7, pp. 35–42, 2005.     

Static and dynamic beam forms of the loss of stability of a long orthotropic cylindrical shell under external pressure

A cylindrical shell with end sections which are closed and supported by hinges, in accordance with the concepts of the rod theory, is considered to be under the action of an omnidirectional external pressure which remains normal to the lateral surface during the deformation process. It is shown that, for such shells, the previously constructed consistent equations of the momentless theory, reduced using the Timoshenko shear model to the one-dimensional equations of the rod theory, describe three forms of loss of stability: (1) static loss of stability, which occures through a bending mode from the action of the total end axial compression force since, under the clamping conditions considered, its non-conservative part cannot perform work on deflections of the axial line; (2) also a static loss of stability but one which occurs through a purely shear mode with the conversion of a cylinder with normal sections into a cylinder with parallel sloping sections and a corresponding critical load which is independent of the length of the shell; (3) dynamic loss of stability which occurs through a bending-shear form and can only be revealed by a dynamic method using an improved shear model.     

Time-invariant uniqueness for reacting viscous fluids near chemical equilibrium

A system of equations is investigated which models a reacting mixture of viscous fluids acted upon by an external body force. Through logarithmic convexity arguments the solution is shown to be unique when either the initial or the final data are given.     

The properties of solutions of a class of isoperimetric problems of stability optimization

A class of isoperimetric problems of stability optimization is considered. These arise, for example, when maximizing the Euler force in the destabilization of a column (rod) of varying cross-section and given volume (Lagrange's problem). It is well known that an extremum which depends on the form of the boundary conditions can be achieved for both simple and double eigenvalues. A class of problems is identified for which a global maximum is found at a simple eigenvalue. The possibility of achieving a local extremum for the first (simple) eigenvalue at stationary points is analysed qualitatively in terms of the parameter values and the form of the boundary conditions.     

Stability and control of transversal oscillations of a tethered satellite system

The tethered satellite system is characterized by weak nonlinearities but it practically works in conditions of internal resonance which produces unstable oscillations. The effect of a longitudinal control force is investigated. Since the displacement component in the orbit plane is always present in the motion due to the nonlinear coupling, the control force is assumed depending only on this component and also when a prevailing out-of-plane oscillation is considered. The harmonic balance method and numerical solutions of amplitude modulated equations are used to obtain stationary and nonstationary oscillations, respectively; the Floquet theory is followed in the stability analysis. The assumed control force is shown to be effective in reducing the primary and secondary instability regions of oscillations perturbed by internally resonant disturbance components.     

Buckling of vertical rod loaded by dead weight and tensile force subjected to its top end (applied to marine vertical oil pipelines—Risers)

The problem of buckling of a vertical rod loaded by dead weight and tensile force to its top end was examined. Four variants for attachment of the top end of the rod upon clamping the bottom end were analyzed. The problem was solved by individual consideration of the stretched and compressed zones of the rod and their sewing together. Critical values of the axial force were obtained, which permit prevention of buckling of a long rod by the action of dead weight. Analysis permits selection of the required tension of vertical pipelines for deep-water oil drilling and demonstration of the advantages of using composites (especially, carbon-fiber-reinforced plastics) as materials for manufacture of the elements of such structures.Institute of Polymer Mechanics, Latvian Academy of Sciences, LV-1006, Riga, Latvia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 4, pp. 527–539, July–August, 1997.     

Stability of a growing viscoelastic rod subjected to ageing

The stability of a compressed growing rod of viscoelastic material that possesses the property of ageing /1/ is investigated. In conformity with the Chatayev definition of the stability of dynamic systems and the Lyapunov method described in /2/, stability conditions are obtained for a rod growing during a finite time interval, and in finite and semi-infinite time intervals. Some results of a numerical analysis of the behaviour of such a rod are presented in /3/.     

Dynamics of a rod undergoing a longitudinal impact by a body

A longitudinal elastic impact caused by a body on a thin rod is considered. The results of theoretical, finite element, and experimental approaches to solving the problem are compared. The theoretical approach takes into account both the propagation of longitudinal waves in the rod and the local deformations described in the Hertz model. This approach leads to a differential equation with a delayed argument. The three-dimensional dynamic problem is considered in terms of the finite element approach in which the wave propagation and local deformation are automatically taken into account. A benchmark test of these two approaches showed a complete qualitative and satisfactory quantitative agreement of the results concerning the contact force and the impact time. In the experiments, only the impact time was determined. The comparison of the measured impact time with the theoretical and finite element method’s results was satisfactory. Owing to the fact that the tested rod was relatively short, the approximate model with two degrees of freedom was also developed to calculate the force and the impact time. The problem of excitation of transverse oscillation after the rebound of the impactor off the rod is solved. For the parametric resonance, the motion has a character of beats at which the energy of longitudinal oscillation is transferred into the energy of transverse oscillation and vice versa. The estimate for the maximum possible amplitude of transverse oscillation is obtained.     

Evolution of the precessional motion of unbalanced gyrostats of variable structure

The precessional motion of an unbalanced gyrostat of variable structure when acted upon by dissipative and accelerating external and internal moments, which depend on the angular velocities of the bodies (the carrier and the rotor) is considered. A qualitative method of analysing the phase space of non-autonomous dynamical systems is developed, based on the determination of the curvature of the phase trajectory. The motion is analysed and the conditions for obtaining the required modes of nutational-precessional motion of unbalanced gyrostats of variable structure are synthesized using this method. A number of cases of the motion of a gyrostat of variable structure, including free motion, motion when there are constant internal and reactive moments and, also, under the action of the moments of resistance forces, proportional to the angular velocities, is investigated. The possible evolutions in the above-mentioned cases of motion and the causes of these evolutions are determined. The conditions for evolution with a decreasing amplitude of the nutational oscillations are obtained.     

Modeling and analysis of a contact problem for a viscoelastic rod

We consider a nonlinear viscoelastic rod which is in contact with a foundation along its length and is in contact with an obstacle at its end. The rod is acted up by body forces and, as a result, its mechanical state evolves. Our aim in this paper is twofold. The first one is to construct an appropriate mathematical model which describes the evolution of the rod. The second one is to prove the weak solvability of the problem. To this end, we use arguments on second-order inclusions with multivalued pseudomonotone operators.     

The stability and stabilization of non-linear,non-stationary mechanical systems

Mechanical systems acted upon by extremely non-linear positional forces are considered. The decomposition method is used to determine the sufficient conditions for asymptotic stability of an equilibrium. Problems of stabilizing the equilibrium of non-linear, non-stationary systems with specified potential forces by adding forces of different structure are studied. For systems with a non-stationary, homogeneous, positive-definite potential, the possibility of stabilization by linear dissipative forces, uncharacteristic of linear systems, is established. For systems with an even number of coordinates n ≥ 4, in the presence of dissipative forces with complete dissipation, the possibility of vibrational stabilization by adding circular and gyroscopic forces with coefficients fluctuating about zero is demonstrated.     

The loss of stability of a two-step rod when it hits a rigid barrier

The critical precollision velocity of a two-step rod of finite length when it collides with a rigid obstacle, leading to a loss of its stability, is calculated by an analytical solution of the wave equation using d’Alembert's method. The critical force and velocity are calculated using Euler's formula for a static load.     

Neutral stability of compression solitons in the bending of a non-linear elastic rod

The spectral stability of compression solitons in non-linear elastic rods with respect to perturbations of the flexural mode of the oscillations of the rod is investigated. The system of equations of the isotropic theory of elasticity, taking account of the non-linear corrections corresponding to the interaction being studied, is used to describe the interaction of longitudinal and flexural waves in the rod. This system of equations describes long longitudinal-flexural waves of small but finite amplitude. It is shown that trapped flexural modes exist, which propagate together with a compression soliton. It is established that these modes, which are the least stable, do not increase with time.     

Control synthesis for a system with non-linear resistance

A dynamic controlled system with one degree of freedom is considered. The system is acted upon by a non-linear resistance, a controlling force and a disturbance. The control and the disturbance are of bounded magnitude. A feedback control is designed which takes the system to a prescribed state. The proposed control law has certain advantages compared with the alternative approach which totally ignores the disturbances. The control is synthesized for an arbitrary non-linear resistance. It is time-suboptimal and robust to variations of the parameters and disturbances.     

Investigation of the behaviour of a column beyond the elastic limit by methods of the technical theory of stability

The behaviour of a model column on a deformable base acted upon by compressive and lateral loads beyond the elastic limit is investigated using the methods of the technical theory of stability. The loading trajectory which leads to the greatest deflection is constructed for specified constraints on the absolute magnitude of the lateral load and a monotonic increase in the compressive load, and the possibility of the occurrence of a limiting state is investigated.