The dynamics of a belt drive is considered in the framework of a nonlinear model of an extensible elastic string. Under the assumption of contour motion, in which the trajectories of particles of the string are pre-determined, we transform the general equations of string dynamics to the spatial description. Further simplification regarding the absence of slip of the belt on the surface of the pulleys leads to a new model with a discontinuous velocity field and concentrated contact forces. It is shown how the computed parameters of steady operation of the drive are influenced by the chosen strain measure for the string, and differences in the behavior of synchronous and friction belt drives are pointed out. For a sample set-up, we study the dependence of the maximal transmitted moment on the angular velocities of the pulleys and on the pre-tension of the belt.
The setting of a looped drive belt on two equal pulleys is considered. The belt is modelled as a Cosserat rod, and a geometrically nonlinear model with account for tension and transverse shear is applied. The pulleys are considered as rigid bodies, and the belt-pulley contact is assumed to be frictionless. The problem has two axes of symmetry; therefore, the boundary value problem for the system of ordinary differential equations is formulated and solved for a quarter of the belt. The considered part consists of two segments, which are the free segment without the loading and the contact segment with the full frictionless contact. The introduction of a dimensionless material coordinate at both segments leads to a ninth-order system of ordinary differential equations. The boundary value problem for this system is solved numerically by the shooting method and finite difference method. As a result, the belt shape including the rotation angle, forces, moments, and the contact pressure are determined. The contact pressure increases near the end point of the contact area; however, no concentrated contact force occurs.
Mathematics Subject Classification
Dynamics of a belt drive is analyzed using a nonlinear model of an extensible string at contour motion, in which the trajectories of particles of the belt are predetermined. The equations of string dynamics at the tight and slack spans are considered in a fixed domain by transforming into a spatial frame. Assuming the absence of slip of the belt on the surface of the pulleys, we arrive at a new model with a discontinuous velocity field and concentrated contact forces. Finite difference discretization allows numerical analysis of the resulting system of partial differential equations with delays. Example solution for the acceleration of a belt drive and an investigation of its frequency response depending on the velocity are presented and discussed.
The drive belt set on two pulleys is considered as a nonlinear elastic rod deforming in plane. The modern equations of the nonlinear theory of rods are used. The static frictionless contact problem for the rod is derived. The nonlinear boundary value problems for the ordinary differential equations are solved by the finite differences method and by the shooting method by means of computer mathematics. The belt shape and the stresses are determined in the nonlinear formulation which delivers the contact reaction and the contact area. The developed method allows performing calculations for any set of geometrical and stiffness parameters.
Для диагностики проводов линий электропередач (ЛЭП) создаются специальные машины-ав-томаты (ДМА). Их конструкция должна позволять двигаться с уклоном и преодолевать пре-пятствия. Но в работе ДМА возможны отказы из-за колебаний проводов. Даже при медленном движении машины могут возникать интенсивные колебания и связанные с ними значительные инерционные нагрузки. В статье представлена оригинальная ДМА и рассмотрены вопросы математического моделирования ее движения по проводам.МАШИНА-АВТОМАТ; ДИАГНОСТИКА ЛЭП; КОЛЕБАНИЯ СТРУНЫ; УРАВНЕНИЯ ЛАГРАНЖА; УПРУГАЯ НИТЬ; КОМПЬЮТЕРНАЯ МАТЕМАТИКА.In this article a novel inspection machine for high-voltage electrical line has been proposed and mathematical modeling issues of its movement along the conductors have been considered. In order to improve the mechanical mechanism and achieve dynamical stability to navigate through overhead electrical transmission lines, variation methods for problems of string have been considered. However, DMAs operation might be subject to failures because of the conductor vibration. Even in the case of slow movement of the machine, intensive vibrations and stipulated by them signifi cant inertial loading might occur. The behavior of the inspection machine while it is traveling through high-voltage electrical transmission lines has been studied.
Проведено комплексное численное моделирование внутрикамерных процессов, протекающих при выходе на расчётный режим работы ракетного двигателя твёрдого топлива. Рассматривается сопряжённая задача, включающая в себя: -нестационарное срабатывание воспламенительного устройства; -прогрев, воспламенение и последующее нестационарное горение заряда твёрдого топлива; -нестационарное трёхфазное гомогенно-гетерогенное течение продуктов сгорания в камере сгорания, сопле и за сопловым блоком ракетного двигателя; -разгерметизацию двигателя и вылет заглушки соплового блока. Приводятся результаты расчётов. Complex numerical modeling of the intrachamber processes, proceeding at the output settlement mode of operation of a solid propellant rocket engine, is carried out. We consider the coupled problem, which involves nonstationary operation of an ignition device; warming up, ignition and subsequent non-stationary burning of a propellant grain; non-stationary three-phase homogeneous-heterogeneous flow of the products of combustion in the combustion chamber, in the nozzle and behind the nozzle of the rocket engine; depressurization and moving of the nozzle block membrane. The results of modeling are given.
The drive belt set on two pulleys is considered as a plane elastic rod. The nonlinear theory of rods with tension is used. The static frictionless contact problem for the rod is formulated. The derived boundary value problem for the nonlinear ordinary differential equations is solved by the finite difference method and by the shooting method by means of computer mathematics. The belt shape and the stresses are determined. The contact reaction and the contact area are obtained in the solution. A benchmark study of extensible and inextensible models is performed.
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