We derive an equation for perturbations of differential systems preserving the shift operator along solutions of these systems on a symmetric time interval [−ω, ω]. In particular, such perturbations preserve mappings for the period [−ω, ω] of a periodic differential system. This simplifies the qualitative analysis of solution sets of differential systems.Along with the original differential systemwe consider the set of perturbed systemswhere α(t) is a continuous scalar odd function and ∆(t, x) is an arbitrary continuously differentiable vector function. Let us study the equivalence of the differential systems (1) and (2) in the sense of the coincidence of reflection functions [1, p. 11]. If the reflection functions of two systems coincide, then their shift operators [2, pp. 11-12] also coincide on a symmetric interval of the form [−τ, τ ] [1, p. 12]; and, therefore, the mappings for the period [−ω, ω] coincide for periodic systems. By [1, p. 11], a reflection function of system (1) satisfies the relation ∂F ∂t + ∂F ∂x X(t, x) + X(−t, F ) ≡ 0.
The possibility of forming parts by different methods is analyzed. The formation of defects is modeled and ways are proposed for eliminating them. A conclusion is reached as to the most efficient method of making the part that is discussed. Keywords: pneumo-thermal forming of sheetmetal parts, drop-hammer drawing, elastoforming, drawing in rigid dies.One of the most important problems in aeronautical and mechanical engineering is finding an efficient method for forming parts from flat-rolled products. The forming method is chosen on the basis of the experience and knowledge of the engineer, i.e., the effectiveness of the chosen solution depends on the human factor. Such dependence is unacceptable in modern industry. In this article, we will conduct a virtual search for an efficient manufacturing variant to make a part with the use of CAE systems.We chose a part ( Fig. 1) with a complex three-dimensional shape to analyze different forming methods. The double-curvature part contains two channels and two sides oriented in the same direction; the material of the part is alloy AMg6. The part's thickness is 0.8 mm.An efficient forming method will be sought in the PAM-STAMP 2G system developed by the French Company ESI Group to perform engineering analyses for sheet stamping.We will examine seven variants for forming the part. 1. Drawing out on a stamping hammer -the process most commonly used to make sheetmetal parts with a complex surface. The following assumptions were made during the modeling operation: a) the effect of deformation rate on stress is ignored; b) the end liners are modeled as having the properties of a material which can be described by the Mooney-Rivlin model; c) the end of impact is determined by the limit on the travel of the punch; d) the plastic part of the flow curve is expressed through the function in the Krupkowsky law (σ = K(ε 0 + ε p ) n , where σ is the running level of stress; K is the strength coefficient; ε 0 and ε p are the initial and running levels of plastic strain; and n is the strain-hardening coefficient); e) Hill model 48 is chosen to describe the yield surface; and f) the initial thickness of the semifinished product is 1 mm.
Results are presented from two experiments involving the pneumothermal forming of a cellular product of alloy OT4-1 in the superplastic regime. It was determined that the average velocity of the free part of the semifinished product must be taken into account when using a model in which the stresses depend directly on the deformation rate. Keywords: pneumothermal forming of disk-shaped parts, superplastic effect, temperature-speed conditions for deformation.The process of pneumothermal forming in the superplastic regime cannot be carried out without strictly adhering to the parameters specified for the forming operation. Those parameters are determined by modeling the process and constructing a graph which describes the change in pressure over time with allowance for the constancy of the deformation rate. The graph is usually calculated and plotted by using a model of the material in which the stresses depend directly on deformation rate [1]:where K is a proportionality factor; e is the deformation rate; m is the strain-hardening modulus; and σ is the stress. The behavior of the material during pneumothermal forming in the superplastic regime is described fairly simply by this model, but many factors are not accounted for. We chose a cellular 1-mm-thick part of alloy OT4-1 (Fig. 1) to study the effect of one of those factors -the average velocity of the free part of the semifinished product.The first stage of the investigation is the creation of a finite-element model of the semifinished product and the fixture. The model was constructed and the geometry of the elements in the forming operation (Fig. 2) was determined by using the CAD system Siemens NX.The software package PAM-STAMP 2G, created by the French company ESI Group, was used as the CAE system for modeling pneumothermal forming in the superplastic regime. In accordance with the aforementioned model, material OT4-1 was assigned the following parameters to perform the modeling operation in this software:• Young's modulus -112 GPa;
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