MODELING AND ANALYSIS OF PARAMETRIC OPTIMIZATION OF THE MASS OF A CLOSED PLANETARY MECHANISM FORMED BY TWO SIMPLE PLANETARY MECHANISMS OF JAMES

The mathematical model of estimation of a design mass of the closed planetary mechanism formed from two simple planetary mechanisms of James (mechanism of type ), taking into account their structural diagrams and design constraints, determined by the conditions of contact and bending strengths of external gearing of sun gears and satellites, is offered. A model is a dimensionless function (analogue of mass) of two variables – transmission relations of simple planetary mechanisms, and set of numerical parameters. As parameters of analogue of mass coefficients are chosen, characterizing the models of mass of gear wheels and carriers, structural and strength limitations of the external gearing of simple planetary mechanisms of the type , and also structure of these mechanisms. In the program Mathcad differential properties of the offered model and influence on position of minimum of analogue of mass are investigational depending on the numerical values of his parameters. Documents of the Mathcad program are presented that implement computer modeling of algorithms for parametric optimization of mass closed planetary mechanism, where the function of the analogue of the mass of the given mechanism is used as the objective function. A comparative analysis of minimizing the design mass of two kinematic schemes of planetary mechanisms is considered – closed planetary mechanism and in-line planetary of the type . Keywords: simple planetary mechanism of James, simple planetary mechanism type ; closed planetary mechanism; in-line planetary mechanism; mass of closed planetary mechanism; contact strength of gearing; bending strength of gearing; parametric optimization; parametric optimization of mass of planetary mechanism

ПІДБІР КІЛЬКОСТІ ЗУБЦІВ ПЛАНЕТАРНИХ МЕХАНІЗМІВ СХЕМ AA, II, AI, IA, І ЗА ДОПОМОГОЮ ПРОГРАМНОГО ЗАБЕЗПЕЧЕННЯ

Most of the methods for selecting the numbers of teeth of planetary mechanisms are based on general equations that take into account three main conditions for their existence: 1) ensuring a given gear ratio, 2) the condition of alignment, 3) the condition of assembly. The rest of the conditions are not included in the system of equations to be solved and are checked additionally. However, this approach has a number of disadvantages associated with the discreteness of the general equations depending on several parameters. Therefore, a method is proposed for selecting the number of teeth using equations obtained only when taking into account two mandatory conditions for the existence of planetary mechanisms: 1) ensuring a given gear ratio, 2) the condition of alignment. Equations for fitting by this method are presented. The remaining conditions for the existence of planetary gears are checked separately and as a result, unsuitable solutions are excluded. This approach ensures the selection of all existing solutions in a given range of teeth numbers with a relatively small number of iterations, but it is rational only when used on a computer. In this regard, the corresponding software has been written - the Planmex.exe program. A description of this program is given, which allows you to select the number of teeth of the wheels of the proposed schemes of planetary gears with a whole and fractional gear ratio, different modules of wheels on a double satellite, a different number of satellites in a given range of numbers of teeth, depending on the selected driving link. Three variants of optimization of the obtained results of selection of the number of teeth are proposed: 1) minimum mass, 2) maximum speed, 3) maximum efficiency. Coefficients are introduced to compare the results within each optimization option and equations are proposed to determine them. An example of the selection of the number of teeth for different modules of gears on a double satellite of the planetary mechanism of the AI scheme by the proposed method using the Planmex.exe program is given. The optimization of the obtained results is made according to three criteria, taking into account the region of existence of the given scheme.

Force analysis of the two-satellite planetary mechanism with elliptical gears

Non-circular gears can be used in modern machines and mechanisms for the implementation of various types of motion and have high strength and compactness compared to linkage mechanisms. This article presents the force analysis of non-circular gear on the example of the planetary mechanism with elliptical gears, providing the rotationally reciprocating motion of the impeller of the stirred tank. Based on the calculation schemes of the links, kinetostatic balance equations for each link of the mechanism are compiled and solved. Reaction forces in kinematic pairs and balancing moment on the input shaft of the mechanism are found. The results can be used in the synthesis and analysis of various machines with the proposed kinematic scheme of the mechanism.

KINETOSTATIC ANALYSIS OF DOUBLE-SATELLITE PLANETARY MECHANISM WITH ELLIPTICAL GEARWHEELS

Non-circular gears can be used in modern machines and mechanisms for the implementation of various types of output link movement and have high strength and compactness compared to linkage mechanisms. The article considers the problem of kinetostatic analysis of the planetary mechanism, which provides the rotationally reciprocating motion of the stirred tank impeller. Proposed mechanism is a two-satellite single-row planetary gear with two external gears, in which one pair of gears is elliptical gearwheels. There are constructed calculation schemes, kinetostatic balance equations are compiled and solved for each link of the mechanism. There are found reactions in kinematic pairs and a balancing moment on the input shaft of the mechanism, which are presented as functions of forces on the angle of rotation of the input link. The results can be used in the synthesis, analysis and design of various machines and mechanisms with the proposed kinematic scheme of planetary gear.