The classical solution of one problem of an absolutely inelastic impact on a long elastic semi-infinite bar

In this article, we study the classical solution of the mixed problem in a quarter of a plane for a one-dimensional wave equation. On the bottom boundary, the Cauchy conditions are specified, meanwhile, the second of them has a discontinuity of the first kind at one point. The smooth boundary condition, which has the first and the second order derivatives, is set at the side boundary. The solution is built using the method of characteristics in an explicit analytical form. The uniqueness is proved and the conditions are established under which a piecewise-smooth solution exists. The problem with matcing conditions is considered.

Optimised design of systems and processes using algebraic inequalities

A method for optimising the design of systems and processes has been introduced that consists of interpreting the left- and the right-hand side of a correct algebraic inequality as the outputs of two alternative design configurations delivering the same required function. In this way, on the basis of an algebraic inequality, the superiority of one of the configurations is established. The proposed method opens wide opportunities for enhancing the performance of systems and processes and is very useful for design in general. The method has been demonstrated on systems and processes from diverse application domains. The meaningful interpretation of an algebraic inequality based on a single-variable sub-additive function led to developing a light-weight design for a supporting structure based on cantilever beams. The interpretation of a new algebraic inequality based on a multivariable sub-additive function led to a method for increasing the kinetic energy absorbing capacity during inelastic impact. The interpretation of a new inequality has been used for maximising the mass of deposited substance during electrolysis.

The effect of collimation on the dynamics of two meteoroid fragments of the same size

The article analyzes the problem of the supersonic flight of two meteoroid fragments of the same size in the framework of two-dimensional plane formulation using the multigrid method for modeling the dynamics of a system of bodies. Initially, the bodies followed each other with a slight displacement of the body located behind, perpendicular to the direction of motion. The variable parameter was the density of the body located behind. Collisions between bodies were calculated according to the formulas of perfectly inelastic impact without adherence of the bodies. It is shown that there are three different modes of system dynamics: spreading with forcing the leading body in the transverse direction, the oscillations of the lagging body in the trace of the leading body, and the gradual lagging the body located behind from the leading body. Depending on its density oscillations of the lagging body are either diverging in nature with its ejection on the head shock wave from the leading body, or of a damped nature with the ejection into the region of the far trace of the leading body. The configuration of the joint flight of bodies directly one after another is not realized.

Simulation of collisions between two identical meteoroid fragments arranged one behind the other

To calculate the dynamics of a system of meteoroid fragments, a simulation method has been developed with an algorithm for calculating collisions between individual bodies. The algorithm for calculating collisions allows one to simulate absolutely elastic, inelastic and absolutely inelastic impacts between individual bodies with the help of a given coefficient of the impact recovery. The impact recovery coefficient can be set separately for each collision based on the known characteristics of the colliding bodies. We carried out a numerical study of the problem of collisions between identical meteoroid fragments initially located one behind the other along the direction of motion. The study shows that bodies will periodically collide in the case of absolutely elastic impact; there is an equilibrium maximum distance between the bodies to which the system will evolve. In the case of an inelastic impact, the distance between the bodies decreases over time; the configuration evolves to the joint flight of the bodies located one right after another. The problem of an absolutely inelastic collision between the identical bodies located within a small initial distance and with a small deviation in position of the backward body shows that the location of the bodies directly behind each other is unstable to small oscillations and is not implemented numerically at large times.

Stochastic Responses of Lightly Nonlinear Vibroimpact System with Inelastic Impact Subjected to External Poisson White Noise Excitation

A procedure for analyzing stationary responses of lightly nonlinear vibroimpact system with inelastic impact subjected to external Poisson white noise excitation is proposed. First, the original vibroimpact system is transformed to a new system without velocity jump in terms of the Zhuravlev nonsmooth coordinate transformation and the Dirac delta function. Second, the averaged generalized Fokker-Planck-Kolmogorov (FPK) equation for transformed system under parametric excitation of Poisson white noise is derived by stochastic averaging method. Third, the averaged generalized FPK equation is solved by using the perturbation technique and inverse transformation of the Zhuravlev nonsmooth coordinate transformation to obtain the approximately stationary solutions for response probability density functions of original vibroimpact system. Last, analytical and numerical results for two typical lightly nonlinear vibroimpact systems are presented to assess the effectiveness of the proposed method. It is found that they are in good agreement and the proposed method is quite effective.