Model for Minimizing the Volume of Two-Bearing Shafts
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l for Minimizing the Volume of Two-Bearing Shafts B. M. Abdeeva, G. A. Gur’yanova, and E. A. Klimenkoa, * a
Serikbayev East Kazakhstan State Technical University, Ust-Kamenogorsk, Kazakhstan *e-mail: [email protected] Received July 16, 2018; revised December 27, 2018; accepted January 17, 2019
Abstract—A universal approach is proposed for optimizing the geometric parameters of two-bearing stepped shafts, so as to minimize the volume of a shaft consisting of three steps, with two adapters. The solution is appropriate for three design variants. A computational algorithm based on the model is derived for the optimization process. A numerical example is presented. Keywords: optimization, strength, geometric parameters, minimum volume, extremum, target function, stepped shafts DOI: 10.3103/S1068798X20080031
2 (1) 2 = zi (1 − yi + xi yi ) + kl . i =1 Here хi, х3, yi, and zi are variables (i = 1; 2) [5, 6] 2 0.25πLx3
xi = di d3−1, x3 = d3, yi = bi li−1, zi = li L−1.
(2)
They vary within the following ranges
C
d2(d02)
V = V (xi ,x3, yi ,zi )
d3(d03)
their steady values are х0i, х03, y0i, z0i. In addition, the design constant kl is such that 0 < kl < 1, while b1, b2, l1, l2, l3 = klL are linear dimensions of sections of the beam with diameters d1, d2, d3 = const (Figs. 1 and 2). We consider three versions of two-bearing steel beams (Fig. 2). The analysis is based on the following general assumptions [4] (previously employed in [5, 6]): (1) The material in the beams is uniform, continuous, isotropic, and linearly elastic (conforms to Hooke’s law). (2) The weight of the shaft–adapter system, the internal transverse forces, and the channels decreasing the stress concentration in cross sections С and D (Fig. 1) may be ignored. (3) The adapters A1 and A2 on the shafts may be assembled from spur gears and pulleys (Fig. 2). In the mathematical description of the three design variants (1–3 in Fig. 2), the following design parameters and forces are regarded as specified: d1(d01)
Today, mechanisms and machines must satisfy numerous requirements. On the one hand, they must be reliable; on the other, they must ensure peak performance and minimal resource consumption [1]. That explains the development of methods for optimizing systems so as to meet the strength, rigidity, and stability requirements [1–3]. This field lies at the intersection of the mechanics of deformable solids and optimal control theory [2–4]. Hundreds of studies have been published to satisfy the keen interest of engineers and designers in this topic. In fields such as manufacturing, aviation, shipbuilding, and instrument design, linear steel beams (solids of revolution) are widely used to transmit torques (torsional shafts and springs) and also radial transverse loads [4]. Smooth shafts are often replaced by stepped shafts so as to improve their performance and economic viability. In the present study (a continuation of [5]), we generalize the classical optimization problem [2] so as to permit the determination of the relative extremum Vmin = V
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