Determining Energy-Optimal Upsetting Program of Compound Cylinders

Abstract

The upsetting of a compound cylinder consisting of a continuous inner cylinder and a hollow cylinder bordering it, with the heights of both cylinders coinciding, is considered. The steady creep of both cylinders is described by power relations with the same values of the exponent and various coefficients. It is assumed that both cylinders are deformed without barrel formation, the friction of the ends of the cylinders with the planes of the press is described by Coulomb’s law. The main interest is the value of the work of the external compressive force on the movements of the ends of the cylinder. The upsetting of a composite cylinder by the same value for the same duration by means of various loading programs is considered. The kinematic program (with a constant approach speed of the ends of the cylinder) and the power program(with a constant external compressive force) are considered. It is shown that the kinematic upsetting program is more efficient for energy costs than the power program. With the help of the variational approach, an optimal upsetting program is determined, at which the work of the external force is minimal. The calculations showed that the corresponding work of the external force is different from the work spent under the kinematic program, by the hundredth and thousandths of a percent. Therefore, in the technological processes of upsetting the cylinders, it is expedient to use the kinematic loading program.