Problems with unilateral constraints are not uncommon in the practice of calculating  building construction and structures. Certain difficulties in solving them arise during contact friction, as well as the dynamic action of the load. It is known that such problems from a mathematical point of view s are not correct enough, their solution becomes more complicated and depends on the history of loading and deformation of the structure. At the same time, the ability to take into account the complex working conditions of the structure makes its calculation more complete and accurate. The paper considers the solution of  a dynamic contact problem on the basis of the finite element method and the step-by-step loading method. Unilateral constraints with Coulomb friction are modeled using contact finite elements of a frame-rod type. The method of compensating loads is applied in order to comply with the limitations under ultimate friction-sliding conditions. Based on the considered discrete contact model and the step-by-step analysis method, a numerical algorithm has been developed, which allows in one step-by-step process to integrate simultaneously the equations of motion and implement contact conditions with Coulomb friction. With the help of the proposed approach, numerical solutions of the problem pertaining to a structure contact with the base have been obtained and analyzed at various parameters of dynamic load. Comparison of the results with the solution obtained by the well-known iteration method on the ultimate friction forces allows to conclude about the efficiency and reliability of the developed algorithm under complex contact conditions and dynamic loading.

Commercial activity has always been influenced by the competitive environment and its spread to the online space is the next stage of development and a defining trend for the nearest time horizon. The changes in the business landscape influenced by COVID19 pose new challenges for marketers and entrepreneurs. It is necessary to use the forced sharp increase in online interaction with consumers. The course towards the digital economy determines the use of scientific, mathematical methods to optimize the target indicators of economic activity. These global shifts in business interactions are generating innovative tools for measuring business results and transforming old practices to meet new market realities. This is the basic condition for the sustainability of doing business in any industry. This study is devoted to the development of a theoretical description of the process of multi-stage interaction with a consumer pool. To solve this problem, a mathematical model has been developed, the basis of which is digital information interaction, starting from the stage of determining the target audience and ending with the complete completion of a commercial transaction. This article presents the results of modeling sales funnel, as the basis for the software of a modern market analyst, using a cross-system approach. In contrast to the classical sales funnel, the presented algorithms allow using the multidimensional conversion funnel not only for assessing business results for the reporting period. Thanks to the flow of model arguments in real time, it becomes possible to optimize the business process by moving to the concept of leading economic indicators.In practice, this means the ability to implement effective business planning on digital platforms. The arguments of the mathematical model are Internet statistics, the dynamics of consumer preferences, the history of the business process accumulated in the big data system. At the same time, the means of queuing theory, differential calculus, economic and mathematical modeling are involved, based on indicators such as KPI (Key Performance Indicators), CTR (click-through rate), CR (Conversion rate). This made it possible to formulate the concept of a digital twin of a commercial process and its transformation, convenient for practical applications, into a conversion funnel for embedding into algorithms implemented on a computer.

Актуальность поднимаемых в статье вопросов обусловлена широким применением программных комплексов для численного эксперимента в области механики и необходимостью верификации полученных результатов. В представленной работе отражены возможности программного пакета виртуального моделирования ADAMS для численного анализа динамических задач с учетом трения. В качестве объекта исследования рассмотрено устройство кривошипно-шатунного механизма, имеющего в своём составе фрикционную муфту. Для демонстрации возможностей моделирования задач с учетом трения в ПК ADAMS приведены различные примеры, верифицирующие аналитические решения с численными. Для исследования работы муфты была подтверждена зависимость момента сил трения от относительной скорости и смоделирована работа кривошипно-шатунного механизма. Отдельно была рассмотрена задача о получении коэффициентов трения качения, которые в результате моделирования в ПК ADAMS, оказываются значительно меньше своих предельных коэффициентов трения качения. The relevance of the issues raised in the article is due to the widespread use of software systems for numerical experiments in the field of mechanics and the need to verify the results obtained. Possibilities of the virtual modeling software package ADAMS for the numerical analysis of dynamic tasks taking into account friction are reflected in work. As an object of a research the device of the crank-conrod mechanism in corporating the frictional coupling is considered. For demonstration of tasks taking into account friction modeling opportunities in an ADAMS package various examples verifying analytical decisions with numerical are given. For a research of the coupling the dependence of the friction forces moment operation on relative speed was confirmed and operation of the crank-conrod mechanism is simulated. The task about receiving coefficients of swing friction which as a result of modeling in an ADAMS package, appear much less than the limit coefficients of swing friction was separately considered.