Simulation of oscillatory processes in the MVSTUDIUM package

The mathematical model and algorithms of oscillatory movements are considered. Various factors affecting the oscillatory process are considered. Oscillatory movements are constructed in the MVSTUDIUM modeling environment. The schemes of three computer models demonstrating oscillatory processes are determined: a model of a pendulum with a non-movable suspension point, a model of a pushing pendulum with friction force and a model of a breaking pendulum. Classes are being built to execute models with embedded properties, as well as with the ability to export the created classes to other models, and embed classes created by the program developer into the model. Creation of 2D and 3D models of oscillatory processes, an experiment behavior map and a virtual stand.

Analysis of the umbrella-type reflector opening process on a stand with an active gravity compensation system

During the verification of the functioning of the transformed structures in ground conditions, it is necessary to minimize the effect of gravity in order to exclude the occurrence of additional loads on the hinge assemblies and opening mechanisms. To perform this task, when testing a transformable umbrella-type reflector, stands with an active gravity compensation system are used, in which the gravity compensation force is applied to each spoke of the reflector. However, when compensating for the gravity spokes of the reflector, the fixing point of the suspension cable does not coincide with the center of mass of the spoke, which leads to the appearance of additional moments of forces acting on the suspended structure. Therefore, as an object of research, a part of the reflector was considered, consisting of a spoke, with cords of a formforming structure attached to it and a mesh. A 3D model has been developed, using which the positions of the center of mass of the structure under consideration were determined in the key phases of the reflector opening. A computational analysis of the driving forces and moments acting on the structure in the process of disclosure is carried out. The degree of influence of the suspension point position on the inaccuracy of gravity compensation has been established. The results of the analysis presented in the article can be used as initial data for the development of an algorithm for the operation of an active gravity compensation system, which will be able to take into account the position of the suspension point and the center of mass of the structure relative to the axis of rotation of the spoke during the opening of the reflector, by changing the gravity compensation force.

First integrals of the system “platform with pendulum” and controlling of it

The problem of controlling the movement of a movable platform with a hinged fixed pendulum is considered. This pair is interpreted as a system of stationary connected points one of which moves freely in a horizontal plane. In the first problem, a control is synthesized that ensures the movement of the platform in a limited area under any initial conditions. In the second problem, a control is synthesized that stabilizes the oscillations of the pendulum in a given fixed vertical plane relative to the suspension point. For the problem of control synthesis, the first integrals of a free system are found and used in the article.

The role of noise in the early universe

In this paper, we consider a quantum mechanical system to model the effect of quantum fields on the evolution of the early universe. The system consists of an inverted oscillator bilinearly coupled to a set of harmonic oscillators. We point out that the role of noise may be crucial in the dynamics of the oscillator, which is analyzed using the theory of harmonic oscillators with random frequency. Using this analogy, we argue that due to the fluctuations around its mean value, a positive vacuum energy density would not produce an exponentially expanding but an oscillating universe, in the same fashion that an inverted pendulum is stabilized by random oscillations of the suspension point (stochastic Kapitza pendulum). The results emphasize the relevance of noise in the evolution of the scale factor.

Physics of nonlinear oscillations with nonlocal variables

In cases where physical processes cannot be described by linear equations, and nonlinear equations are difficult to solve mathematically, we have to use approximate solutions to such problems. One such example is the description of the Kapitsa pendulum, which is a pendulum with a vibrating suspension point. In contrast to the previously known methods of describing such a problem, in this paper we propose to use additional variables in the form of higher derivatives, which allows us to obtain corrections that give a more detailed contribution to the description of this problem.

Experimental studies of load vibrations when moving the DEC-251 loading crane

Measurement of the parameters of vibrations of the load moved by a self-propelled crane with a flexible rope suspension when the crane moves along an unprepared construction site with irregularities is an urgent task, since it will allow using the obtained numerical values of the vibration parameters to improve the accuracy of the crane’s operation in terms of moving loads. Based on the solution of this problem, it is possible to create systems for automatic damping of cargo vibrations. This will reduce the time spent on performing a work step when moving a load. This also solves the problem of reducing the dynamic loads on the elements of the crane. The article discusses one of the methods for determining the angles of deviations of the point of the load and the point of suspension of the load on the boom when moving the DEK-251 mobile crane along the unevenness of the construction site using the projection-polynomial mathematical model of the optoelectronic system. As an example, the article presents a number of graphs of time dependences of changes in the values of the angles of deviations of the load and the point of suspension of the load when moving over the unevenness of the site of a crane with a boom length of 22 meters and an angle of inclination of the boom of 48 degrees. The cargo was at a height of 4.8 meters, the weight of the cargo was 200 kilograms. The graphical time dependences of the load fluctuations and the load suspension point in the longitudinal plane are given in the form of angles of deviations from the lens center, taking into account the microrelief. The data allows you to calculate the linear coordinates of objects in space. Moreover, the results were obtained taking into account the camera errors.

Asymptotic Stability of Solutions to a Class of Second-Order Delay Differential Equations

We consider a class of second-order nonlinear delay differential equations with periodic coefficients in linear terms. We obtain conditions under which the zero solution is asymptotically stable. Estimates for attraction sets and decay rates of solutions at infinity are established. This class of equations includes the equation of vibrations of the inverted pendulum, the suspension point of which performs arbitrary periodic oscillations along the vertical line.

MATHEMATICAL PENDULUM MODEL WITH MOBILE SUSPENSION POINT

Background. The new dynamic problem, which is posed and solved in this article, is a theoretical generalization of the well-known classical problem of free oscillations of a mathematical pendulum. In the proposed setting, it is new and relevant, and can be successfully used in such fields of technology as vibration protection, vibration isolation and seismic protection of high-rise flexible structures, long power lines, long-span bridges and other large-sized supporting objects. Objective. The aim of the work is to derive the equations of own oscillations of a new mathematical pendulum-absorber, to find a formula for determining the frequency of its small own oscillations and to establish those control parameters that allow you to tune the single-mass pendulum absorber to the frequency of the fundamental tone of the carrier object. Methods. To achieve this goal, we used the methods of analytical mechanics, namely, the Appel’s formalism, as well as the linearization of the obtained differential equations. Results. A mathematical model is constructed in the work that describes the own oscillations of a new-design mathematical pendulum with a movable (spring-loaded) suspension point with length L. The model is a system of differential equations obtained using the Appel’s formalism. Based on them, after linearization of nonlinear equations, a formula is established for the frequency of small own oscillations of a pendulum with a mobile suspension point. Conclusions. It is shown that the frequency of own oscillations of the new mathematical pendulum coincides with the frequency of own oscillations of the classical mathematical pendulum with an equivalent suspension length, which is equal to . In the case where the suspension point is fixed (k ® ¥), the frequency formula turns into a well-known formula for the frequency of small own oscillations of a classical mathematical pendulum . If the stiffness coefficient of elastic elements tends to zero (k ® 0), then the frequency w of the damper also tends to zero. An important structural feature of the proposed pendulum is noted, consisting in the fact that due to the appropriate choice of the three control parameters of the pendulum (k, L, m), its frequency in magnitude can be made any in the range from zero to .