Vol 53, No 3 (2018)

Rolling without sliding of the pneumowheel, towed by means of a rigid device of length L, is considered. The angles of rotation of this device during vibrations are studied. Cases L ≈ L_* and L > L_* are distinguished, where L_* is the length at which the system is located at the boundary of stability. To describe the interaction of the running surface of the tire with the supporting plane, three well-known linear rolling models are used — the nonholonomic one, as well as the string and slip models. The case L > L_* is studied with the help of reduced mathematical models — modeling oscillators. The main results of the study: all three models indicate the same length L_* ≈ D/2 (D is the outer diameter of the free tire), in the parameter range studied the first two models actually coincide, with an imbalance of the pneumatic wheel and L ≈ L_*–which is often found in practice–parametric resonances occur if the Clarke–Dodge–Naibekken numbers DΩ/V_A (V_A is the speed of the hinge-hitch, Ω is the natural frequency of oscillations when the pneumatic wheel does not roll) belong to the natural series. In addition, the effect of the speed V_A on the natural frequency of oscillation is quantitatively described, and comments are made on the verification of shimming models and the relationship of mechanical permanent tires.

In the article, the Routh–Lyapunov method distinguishes stationary motions of a rigid body in the case of the existence of a partial Hess integral. In addition to already known, new families of stationary movements have been obtained.

An approach to constructing mathematical models of thermomechanical processes in a deformable body is considered by the rational thermodynamic relations of irreversible processes for a continuous medium with intrinsicstate parameters as well as the Eringen model for nonlocal theory of elasticity. The models take into account the effects of temporal and spatial nonlocality of a continuous medium. The temperature and stresses for the problem of pulsed heating in one-dimensional case are calculated.

The problem of stability of a rectangular plate, freely supported on two opposite sides, the third side of which is free, and the fourth side is either clamped or freely supported, is considered. The plate is compressed by the load applied on the free edge. For both cases of fixing the fourth side of the plate, conditions for the appearance of an instability localized in the vicinity of the free edge were obtained, both for the case of a conservative load and for the case of a tracking load.

The general properties of the relaxation curves with the initial stage of deformation and their dependence on the duration of the initial stage of deformation and the properties of the relaxation function are analytically studied. These curves are induced by the constitutive equation of viscoelasticity with an arbitrary relaxation. New accurate two-sided estimates for the relaxation curves and their absolute and relative deviations from the relaxation curves for instantaneous deformation are derived. The uniform convergence of the set of relaxation curves when the duration of the initial stage tends to zero is proved.

Effective universal two-sided estimates for the relaxation function (at any time instant) are obtained via the relaxation curves of the material during ramp-deformation. On their basis, simple and effective formulas for determining the relaxation function from the relaxation curves for ramp deformation obtained in the material tests have been proposed. The error estimations of these approximations are given. The uniform convergence of the set of approximations to the relaxation function when the duration of the initial stage tends to zero is proved. The higher accuracy of the estimates found and the proposed approximation in comparison with the known related approaches to the determination of the RF (relaxation function) is established.

The results of the analysis are useful for clarifying the many of the possibilities of the linear theory, the domain and indicators of its (non)applicability (and also of a number of its generalizing nonlinear constitutive equations of viscoelasticity, for example, proposed by Rabotnov,Ilyushin, Pobedrya, etc.) to improve the methods of selection, identification and adjustment of linear models. In particular, they are useful for obtaining reliable estimates of the lower bound for the observation window of the relaxation function from the experimental relaxation curve of the material in terms of the initial stage duration to clarify the “ten-times rule” and expand the observation window to the region of small time values.

In the problem of seismic vibrations of a segmental pipeline with damping joints, deformed by the law of linear viscoelasticity, an original analogy with a linear chain of concentrated masses was put forward. The constructed discrete system generalizes the monatomic lattice model in the sense that the viscoelastic interaction between the masses of the chain is considered and, moreover, the forced (and not own) oscillations of such a system are investigated. By the transition from a discrete system to a continuous one, the integro-differential oscillation equation of a segmented pipeline with viscoelastic joints in an elastically resisting medium is obtained. This equation is a generalization of the well-known Klein–Gordon differential equation describing the “constrained” vibrations of an elastic rod or string in a medium with elastic resistance. In addition, the equation gives the problem of seismic vibrations of a continuous pipeline from a polymer (viscoelastic) material.

Joint stationary seismic vibrations of a viscoelastic pipeline and soil were studied in an exact formulation and maximum stresses in the pipeline were found by solving the resulting integro-differential equation. The same stresses were found using the “hard pinch” engineering approach, according to which displacements and deformations in the seismic wave and pipeline are the same. By analyzing the stresses found under the viscoelasticity law in the form of the Kelvin–Voigt model relationship, it is established that the generally accepted position that the engineering approach gives an upper estimate for the stresses in the pipeline is valid only in the subsonic case (when the seismic wave velocity is lower than the wave velocity in the pipeline) and is not valid in the supersonic mode, when the exact theory can lead to stresses exceeding those calculated on the basis of the engineering approach.

The physical principles of the polarization-optical method for studying stresses that is commonly called the photoelasticity method are considered. Using the model of a linear oscillator it is established that the birefringence effect observed in some materials during their deformation can be explained by a shift in the eigenfrequencies of the oscillators, namely charged particles inside the deformed body.

This approach has been used in determining the parameters of the pair interaction potential, in particular, the Mie potential using the experimentally determined birefringence value. The expression for the shape of the strain-optical coefficient, that is, a quantity that serves to relate the difference in the principal deformations and the relative path difference is obtained.