Development of a mathematical model of dynamic characteristics of a drive with a planetary mechanism

. 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.


Introduction
Drives with planetary gear mechanisms are widely used in various branches of technology and industry. Simulation of dynamic characteristics and determination of the actual laws of motion of the working bodies play an important role in the design of such drives.
Numerous publications [1][2][3][4][5][6][7][8] are devoted to the study and analysis of various mechanisms of various functions. Analyzing numerous studies aimed at improving the operation of the mechanisms and their wide application, the theoretical foundations and designs of planetary and biplanetary drives of the working bodies of doughing machines were developed in [2]. Mathematical models for controlling the parameters and constraints of the friction mechanism were developed in [3]. The theory and designs of friction mechanisms with controlled friction were developed in [4,5]. In [6], the analytical implementation of the mathematical model of the controlled motion of the positioning mechanism was developed. A new coaxial eccentric indexing cam mechanism for highspeed automatic units was proposed [7]. A review of the development and state of the kinematics and dynamics of the cam mechanism was conducted [8].
Below, the equations of motion of the grinder drive with a planetary gear mechanism are proposed. Considering, in a particular case, that all the links of the drive with the planetary gear mechanism are rigid, a mathematical model was developed for this system, taking into account the dynamic characteristics of an electric motor. Analytical solutions for the developed mathematical model are given. The obtained research results were used in the design of a universal planetary mill.

Methods
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.

Materials
Determination of the actual laws of motion of the working body of a grinder is a necessary factor in its design. The working body of the grinder receives motion from an asynchronous electric motor through a planetary gearbox. To reduce the number of revolutions of the working disc of the grinder, it was proposed to install a James planetary gearbox. The kinematic diagram of the drive is shown in Fig. 1, where, 1 is a fixed central wheel, 2 is a satellite, 3 is a movable central wheel, 4 is a working body, H is a carrier.
Consider the methods for determining the number of teeth of a given gearbox [1]. Suppose that it is necessary to design a planetary gear that reproduces the gear ratio a u H =  (2) Considering this, we obtain ) 1 ( The number of teeth 3 z should be chosen so that there is no undercut or interference between the teeth. As is known, for normal gear wheels with an engagement angle  20 = α and a head height coefficient equal to one for external engagement, it is  We define the kinetic energy of the James gearbox as:  Let us determine the moment of inertia of the mechanism rated to wheel 3. In this case, the working disk is connected to the carrier, and wheel 3 is the driving link, then If we assume that all grinder links are rigid, then the dynamic model of the system is reduced to a single-mass model. The mathematical model for this system, considering the dynamic characteristics of the electric motor, has a well-known form [11]:  − ω H is the motor rated angular speed.
Taking into account the above, the equation of motion of the grinder drive can be written in the following form: Taking into account expression (9), equation (8) can be written in the following form: To solve equation (6), we use the Fourier series [2,3]. As is well known, if the force ( ) t Q is a periodic function of the time period ω π = τ / 2 , then under general assumptions, always satisfied in real dynamical systems, it can be expanded in a Fourier series: The individual terms of the series (11) are called harmonics. In linear systems, the principle of superposition is valid, according to which forced oscillations from each harmonic can be determined separately, and the results can be summed. Based on this, for the case of a harmonic exciting force, the expression for forced vibrations is written in the following form The dynamic factor g K and phase shift can be determined using the following formulas: ( ) In the grinder of mineral raw materials, considered in the article, an asynchronous electric motor is used with a power of N=18,5kW; rated speed of nH = 3000rpm. Let us determine the angular velocity of the electric motor