АНАЛИТИЧЕСКИЙ ПОДХОД ОПРЕДЕЛЕНИЯ УГЛОВЫХ КООРДИНАТ ЗВЕНЬЕВ ПЛАНЕТАРНЫХ МЕХАНИЗМОВ

The article is devoted to the development of a mathematical model of dynamic characteristics of a drive with a planetary gear mechanism. The subject of research is a drive with a planetary gear mechanism. The following issues were considered in the article: the synthesis of a planetary gear mechanism; the development of a mathematical model that describes the dynamic characteristics of the system; the determination of the dynamic characteristics of a drive with a planetary gear mechanism. Research is based on the method of determining the number of gear teeth; the method of determining the kinetic energy of the James gearbox; the method of determining the dynamic characteristics of an electric motor. The possible number of satellites is given in the article; the pitch radii of the wheels for a given modulus are defined; the moment of inertia of the mechanism reduced to the movable central wheel is determined; a mathematical model of the motion of a drive with a planetary gear mechanism is developed. Equations of motion of a drive with a planetary gear mechanism were obtained. Assuming, in a particular case, all the links of the drive with the planetary gear mechanism as rigid links, a mathematical model was developed for this system, considering the dynamic characteristics of an electric motor. A mathematical model was developed that describes the dynamic characteristics of the system. Analytical solutions for the developed mathematical model are given.

This paper considers methods for calculating gear transmissions of aircraft gearboxes, which can increase their load capacity and reduce mass and dimensions. In power gearboxes of aircraft engines, gears are used, obtained by a cutting tool with a profile angle a = 25° and 28°, or with a complex initial generating contour with a protrusion. Another way to increase the bearing capacity of gears is to use wheels cut with an offset (corrected wheels). Such gears have increased contact strength and bending strength. For a rational choice of displacement coefficients, it is necessary to use locking contours that are built for a pair of wheels being designed. The use of gears whose wheels are cut with an offset in planetary mechanisms requires a special approach to the synthesis (design) of such mechanisms. Optimal design of planetary mechanisms involves three main stages. The first stage is the selection of the optimal kinematic scheme of the gearbox that meets the required technical conditions. Analysis of existing kinematic schemes of aircraft gearboxes of domestic and foreign engines makes it possible to choose the optimal kinematic scheme. Such gearboxes include planetary mechanisms of simple and closed circuits – these are schemes АІ, та ( A – external, I – internal gearing). The next stage of design is the synthesis of the planetary mechanism itself, i.e. rational selection of the combined number of teeth that meet the kinematic and design requirements. To select the number of teeth that are cut with an offset, you can use "general" equations in which the condition of co-axiality is written with an additional parameter indicating the absence of co-axiality, with the subsequent solution of this dependence. The third stage of the synthesis of the planetary mechanism involves determining the optimal geometry of the gear meshing by selecting rational offset coefficients using blocking contours. Selection of offset coefficients for the scheme АІ and has its own characteristics and is considered separately for each of them. For the AI scheme, first the displacement coefficients are selected for the pair z1-z2 of the external gear, and then for the pair of wheels of the internal gear z3-z4 in such a way as to ensure the condition of co-axiality. The peculiarity of the scheme is that the satellite z2 is simultaneously in external gearing with the gear z1 and internal gearing with the wheel z3. To select the displacement coefficients for this scheme of the planetary mechanism, special (spatial) blocking circuits are constructed for all three wheels z1-z2-z3, which connect the three displacement coefficients x1, x2, x3 into a single system.

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

The article presents a method for comparative analysis of the bearing capacity of planetary mechanisms of various kinematic schemes using several CAD/CAE software environments. To implement the method of comparative analysis, the KOMPAS-3D and SolidWorks software environments were used. In the first environment, calculations were performed according to the method of the GOST 21354-87 standard. The initial data and results of calculations performed by the KOMPAS-GEARS software package are stored in the appropriate files. In the second environment, a parametric 3D geometric solid model of the kinematic scheme of the planetary mechanism was created. According to this model, such characteristics of gears as mass and axial moment of inertia were determined. These characteristics are used to perform optimization calculations of the mass and speed of the designed planetary mechanism. The study of the speed of the planetary mechanism can be carried out in SolidWorks Motion. The overall dimensions of the model allow the designer to assess the acceptability of the design constraints for given dimensions to the specified requirements. According to the method, a study was made of the suitability of gear designs that were designed for a closed planetary mechanism as gears of an open-loop sequential planetary mechanism. The closed planetary gear is formed by two simple James planetary gears. The circuities planetary gear is formed by the series connection of the same simple planetary gears. Two cases of loading gears for a given kinematic scheme of the planetary mechanism were studied. Keywords: simple James planetary mechanism (mechanisms type ); closed planetary mechanism; contourless planetary gear; moment of inertia of gear-wheel; gear mass; planetary gear mass; contact strength of the gearing; bending strength of the gearing; toothing resource; 3D geometric solid model of the kinematic scheme of the planetary mechanism

With the popularization of computer, the design of planetary gear trains is headed in the automatic direction. Based on the theoretical research of closed differential planetary gear transmission mechanism, a kind of software was developed to automatically calculate the tooth number with computer technology applied to the design of gear trains. Some research and analysis were done to the transmission mode of closed differential planetary gear transmission mechanism with the help of universal design method on gear trains. Five constraint conditions were summarized, which are gear ratio condition, adjacency condition, assembly condition, concentricity condition, and other conditions, and a optimum condition. With windows 7 as the platform, MyEclipse 8.5 as a tool, translating these conditional formulae into Java language, the software of the selection of tooth number for closed differential planetary gear transmission mechanism was developed. Multi-group datum of tooth number and also a set of optimal data can be obtained after running the software, which reduces the workload of manual calculation.

The article provides a kinematic diagram of a stationary planetary mechanism for driving a mixer working organ using epi and hypocyclic gears. The central gear wheel is made with double-sided gearing and is installed in a stationary position, while the pairwise satellites are in parallel gearing. When the carrier rotates, the satellites perform opposite types of trajectories - epicyclic and hypocyclic. In the course of geometric synthesis, a constructive condition for the alignment and proximity of the placement of the satellites in the planetary mechanism was ensured. As a synthesis method, a mathematical model was chosen in the form of a system of equations describing the relationship between the condition of alignment and the choice of the number of gear teeth for two types of trajectory of motion - epicyclic and hypocyclic. Based on the results of the obtained equations, graphs of the regularity of the change in the number of teeth of the central wheel with internal and external gearing for various modules were constructed, depending on the choice of the number of teeth of both satellites. Possible constructive designs of the synthesized mechanisms are shown and examples of the use of a mixer working organ in the mixer drive for high-quality mixing of a mixture of protein masses are given.

Processing of metals by pressure suggests using of various measuring devices. We developed and manufactured a device for measuring angular deformations upon elastic and plastic torsion of circular specimens. The design is based on the operation principle of the planetary mechanism without a movable solar central wheel. Design of the device is based on the Boyarshinov concept added with a round protractor with a graduated scale from 0 to 360° and rotating needle placed on the axis to indicate the angular displacement. The use of the planetary gear significantly improves the accuracy of measuring the angles of rotation of the cross sections of the sample relative to each other at a distance of the calculated length. To low the weight of the device, the main parts were made of aluminum alloy D16Т. Rated diameters of the central wheel and satellite are D1 = 220 mm, D3 = 20 mm, respectively; the linkage module m = 1 mm; the number of teeth on the wheel and satellite is 220 and 20, respectively. Laboratory tests of the device were carried out on a KM-50 torsional machine using a cylindrical sample with a diameter of 15 mm and a working length of 120 mm made of steel 40Kh. The results provided determination of the elastic shear modulus with a deviation of ~2.5% of the reference value. Experimental data were used to plot the torsion diagram and then to get shear diagram τ = τ(γ) according to P. Ludwik’s formula. This diagram was transformed into the hardening curve σ = σ(e) using von Mises theory of the plasticity. The obtained mechanical characteristics allowed us to draw a conclusion on the essential accuracy of measuring angular displacements using the developed device. Thus, the developed and easy to use device can be recommended as testing equipment for determination of the mechanical characteristics of materials under conditions of shear deformations.

Teeth matching conditions for planetary mechanism of single pin, dual pin and multiple pin planetary mechanisms are studied in details in this thesis. Single planetary mechanism teeth matching conditions include: concentric condition, homogeneity distribution condition, neighbor condition and gear ratio condition. With the increase of meshed gear pairs between sun gear and gear ring, the concentric condition will become wider and wider, while neighbor condition become stricter and stricter. Gear ratio condition is applied to check the deviation of actual and theoretical gear ratio of planetary mechanism under the teeth matching condition. If the deviation is large, adjustment could be made by revising structural parameters and number of planetary gear pairs participating in the meshing. Homogeneity distribution condition is most easily ignored, thus enough attention should be paid to it. In this thesis, the angle expression of planetary mechanism when each component is not interfered and homogeneously distributed is inputted its rotation speed equation, and after deduction, the homogeneity distribution condition is gained for various single planetary mechanisms, which will provide reference for engineers.

<p>Inspection of large space structures is imperative for long term mission success. One solution is to utilize a second free flying spacecraft capable of performing inspection in orbit. An Extended Kalman Filter (EKF) is used to perform estimations on the relative position, velocity, angular velocity, and attitude through the use of navigation markers. Simulation takes place using MATLAB to compare the true values with the estimated values using the EKF. The initial covariance (P), process noise covariance (Q), and measurement noise covariance (R) matrices were tuned for a space structure with three navigation points. The largest recorded errors over 100 iterations occurred during the initial estimation yielding 26.78 centimeters in relative position, 1.80 centimeters per second in relative velocity, 0.0444 radians per second in relative angular velocity, and a difference of 0.0172 in the unit vector of relative attitude. After allowing 20 seconds of settling time the maximum errors were reduced to 5.0 centimeters in relative position, 0.40 centimeters per second in relative velocity, 0.0046 radians per second in relative angular velocity, and a difference of 0.0020 in the unit vector of relative attitude. The paper also discusses the application of training algorithms to tune the EKF parameters for future consideration.</p>

<p>Inspection of large space structures is imperative for long term mission success. One solution is to utilize a second free flying spacecraft capable of performing inspection in orbit. An Extended Kalman Filter (EKF) is used to perform estimations on the relative position, velocity, angular velocity, and attitude through the use of navigation markers. Simulation takes place using MATLAB to compare the true values with the estimated values using the EKF. The initial covariance (P), process noise covariance (Q), and measurement noise covariance (R) matrices were tuned for a space structure with three navigation points. The largest recorded errors over 100 iterations occurred during the initial estimation yielding 26.78 centimeters in relative position, 1.80 centimeters per second in relative velocity, 0.0444 radians per second in relative angular velocity, and a difference of 0.0172 in the unit vector of relative attitude. After allowing 20 seconds of settling time the maximum errors were reduced to 5.0 centimeters in relative position, 0.40 centimeters per second in relative velocity, 0.0046 radians per second in relative angular velocity, and a difference of 0.0020 in the unit vector of relative attitude. The paper also discusses the application of training algorithms to tune the EKF parameters for future consideration.</p>

The task of parametric optimization on the criterion of total volume is considered for the two-stage planetary mechanism of type AI–II. The objective function of optimization is built as a sum of two dimensionless parametric functions of volume (analogs of volume) of the separate stages of planetary mechanism of type AI–II. Construction of analogs of volume of simple planetary mechanisms of type AI and II, corresponding the first and second to the stages of planetary mechanism of type AI–II, it is based on their presentation as conditional disks. Thus the volume of the conditional disk is equivalent to the volume of a simple planetary mechanism. The mathematical model of analogs of the volume is built taking into account the terms of the durability of the toothed hooking of the sun gear and planetary gears of simple planetary mechanism, and also taking into account an area possible his transmission relation. Considering terms of contact and bending durability of the toothed hooking, get the analogs of volume at a calculation on contact and bending to durability. Depending on the condition of durability (contact or bending) and type of simple planetary mechanism the analog of his volume appears as a parametric function of corresponding parameters of his kinematics diagram. As parameters transmission relation attitude of simple planetary mechanism, parameters of bringing a mechanism over, toward a conditional disk, and also the relation of reference diameters of planetary gears of the mechanism is chosen. An analysis over of influence of each is brought of parameters of the function of analog of volume on the pattern of behavior to it. On the basis of this analysis of function of analogs of volume of the stages of planetary mechanism of type AI–II appear as parametric functions of the guided parameters. For the simple planetary mechanism of type AI one guided parameter is chosen, namely transmission relation of mechanism. For the guided parameters of function of analog of volume for the simple planetary mechanism of type II a transmission relation and relation of reference diameters of planetary gears of mechanism are chosen, i.e. two guided parameters. Offered approach for the decision of optimization task, based on research of differential properties of objective functions of volume (analogs of volume) of the separate stages of planetary mechanism of type AI–II. It is given the example of the optimal designing of construction of minimum volume of planetary mechanism of type AI–II.

The article proposes a method for geometric calculation of gear links of planetary mechanisms in which the central wheels of external and internal gearing have the same number of teeth. The key to solving the problem is the existing possibility of a large positive displacement of the tool when machining the crown with internal involute teeth. The calculation used the formulas GOST 16532–70 and GOST 19274–73, arranged in a certain sequence. The planetary mechanisms under consideration underlie the new volumetric rotary hydraulic machines.

The subject matter of this article is the processes of synthesis of planetary gear mechanisms of AI type with uncoupled and coupled wheels to solve the problems of gear wheel design. The goal is to develop the synthesis of planetary mechanisms of AI type, considering the possible values of engagement angles and with predetermined quality indicators of gear pairs at the stage of design. The tasks are to develop a methodology of planetary mechanism synthesis of type AI, to find general equations, considering the dependence of the number of teeth on the engagement angles for mechanisms of this type with uncoupled and coupled wheels with additional synthesis parameters, which would give the possibility of performing the synthesis of the planetary mechanism to guarantee the necessary engagement angles of the cogwheels. The methods used are ones for solving the system of linear parametric equations, which is obtained from the general equations of the synthesis of planetary mechanisms. The following results were obtained: the general equations for the mechanism of type AI, which consider the parameters of the geometrical calculation of the gear pairs that make up the mechanism have been obtained. The research on received general equations has been carried out and the area of existence of planetary mechanisms of AI type for different engagement angles of pairs of coupled and uncoupled wheels has been constructed. From the results of synthesis, it can be seen that the range of existence of planetary mechanisms is significantly expanded, which makes it possible to realize a wider range of transmission ratio for each number of satellites and get additional numbers of teeth of the planetary mechanism, which cannot be found using the classical methods. Conclusions. The scientific novelty of the results obtained in the study consists of the following: the considered approach of synthesis with an account of engagement angles makes it possible not only to obtain additional combinations of numbers of teeth of planetary gear but also in further calculations to jointly use the existing locking contours for each pair of wheels, as well as to assess in advance the possibility of cutting the gears whose engagement angles will satisfy the selected parameter at the stage of synthesis of the planetary mechanism.

The article discusses the improvement of the methods of geometric design of positive-displacement hydraulic machines with floating satellites. At the first stage, based on standard methods, an “average” round-link planetary mechanism, having the number of teeth as in the designed hydraulic machine, is calculated. At the second stage, the same-type cyclic functions characterizing the paths of the center of the satellite in the coordinates associated with each of the central gearwheels are specified. At the third stage, the coordinate arrays of the satellite in a variety of its positions are calculated. The angular position of the satellite relative to one of the profiled gearwheels (for example, the sun gearwheel) provides equidistance of the centroid of this gearwheel to the path of the center of the satellite. The angular position of the satellite relative to the other profiled wheel is determined by its running around the first wheel. At the final stage of the design, the satellite is constructed in a variety of positions, and the profile of the corresponding non-circular gear rim is obtained as an envelope of the family of satellite curves-profiles.

Internal gears with small difference between teeth number are most commonly encountered in planetary and differential mechanisms (eg. on automatic transmissions for vehicles). One of the advantages of this gear is the high transmission ratio. Their parallel axes make this gear perfect for the cases when a small deviation of the distance between axes is required. Automatic transmission allows internal gear function to high speed. In this, we present a short overview of internal gear with small difference between teeth number.