Mathematical model of a drive mechanism with a crank device of a crank press

. The most important link in the forging equipment is a crank mechanism. Their significant drawback is the unbalanced inertia forces of the moving masses of the crank mechanism, which cause vibration. The analysis of the phenomena occurring in the mechanism and the assessment of the technological process are based on the theory of chains, which allows analytically analyzing the dynamic characteristics of systems with a large number of degrees of freedom, based on the analysis of one structural element. The study of the process of force interaction inevitably comes down to the construction of a mathematical model of mechanisms, the formative movement of which leads to its formation. One of the partial systems makes an irregular programmed motion, meaning the crank drive mechanism. In addition, unwanted vibrations caused by kinematic excitation are superimposed on this drive. According to numerous papers on this topic, significant dynamic errors arise due to vibration accelerations. One of the main tasks in reducing the vibration activity and, accordingly, the level of acoustic emission of the process under study is to ensure the required law of motion of the instrument. On this basis, the study of the stability of formative movements is of particular importance. This question is complicated by the fact that in the processing, there is a change in the process parameters and, consequently, in the characteristics of the friction coupling. The latter circumstance presupposes the evolution of the system under study, and therefore the need for process control.


Introduction
The most important link in the forging equipment is a crank mechanism. Their significant drawback is the unbalanced inertia forces of the moving masses of the crank mechanism, which cause vibration. The analysis of the phenomena occurring in the mechanism and the assessment of the technological process are based on the theory of chains, which allows analytical-ly analyzing the dynamic characteristics of systems with a large number of degrees of freedom, based on the analysis of one structural element.
The study of the process of force interaction inevitably comes down to the construction of a mathematical model of mechanisms, the formative movement of which leads to its formation. One of the partial systems makes an irregular programmed motion, meaning the crank drive mechanism. In addition, unwanted vibrations caused by kinematic excitation Since the links of this mechanism perform complex flat movement in a vertical plane, it is necessary to take into account the work of gravity forces. In addition, the mechanism has a link that performs a plane-parallel motion. For such links, their inertial characteristics can be reduced to links that perform rotational and translational motion, that is, to a crank and frame. Using the mass substitution method, we get: (1) The electromechanical system cannot be described by an adequate model without taking into account the inertia of the processes occurring in the engine. The equations describing the motion of an asynchronous electric motor have the form:   Thus, the electromechanical system under investigation includes an electric motor described by a system of nonlinear differential equations of the first order, as well as a system of equations describing the mechanical part of the drive. For such problems, a special form of the Lagrange type II equations with "superfluous" coordinates is usually used (they are also called Ferrers equations). For the case in question, we have: We specify the absolute coordinate corresponding to the displacement at the beginning of the kinematic chain, and then enter the coordinates, moving along the kinematic chain. We write the basic kinematic relations: The angular coordinates of the corresponding sections in absolute motion, -the absolute coordinate of the mass. The last remark means that, with the exception of the generalized coordinates, relative coordinates are taken which are responsible for the deformation of the elastic elements. The equation of connection is written based on the condition that is a function of the position.
We differentiate this expression by time: As the "extra" coordinates, we take Due to the fact that the frame with moves in the vertical plane of Fig.1.2, it is convenient to present its movement in the following form: The common leg of the OAC and ABC rectangles is determined by the equality of the form: The transfer function of the drive mechanism can be represented as a function of the angle of rotation of the crank: Making obvious transformations, we have: For the second derivative of the transfer function, respectively, we obtain: To compile a mathematical model of the mechanical part of the drive, we express the kinetic and potential energy in terms of the accepted generalized coordinates, including the "redundant" ones: System dissipative function (Rayleigh function): The expression for virtual work can be written in the form: We write the expression of kinetic energy, taking into account the kinematic relations:   (8) It is necessary to consider the counting of potential energy from the position of stable equilibrium. In addition, we believe that all communications are stationary, the process is not considered as a component of the mechanism, but as an external influence. If we assume that the gear ratio does not depend on the speed of the links, the stiffness matrix of the approximating system of differential equations of the object under study can be taken as constant.

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Based on the matrix representation of the potential energy of the system and the fact that the stiffnesses of the bonds are constant, the coefficients of the stiffness matrix are obtained by equating the corresponding coefficients of the quadratic form representing the potential energy of the system, in the form: The coordinate entered into the expression of potential energy, but although it is rigidly connected with the coordinate by a nonlinear function, the change in the potential energy of the system under study is determined by the movement of inertial masses in the vertical plane.
Imagine the damping function in the matrix form: Based on the representation of the damping matrix in the form, we obtain the matrix in the form: The moments of friction forces in the supports neglected. Further actions are reduced to writing a system of differential equations of motion and the elimination of Lagrange multipliers. The total number of equations is. In the first part, in addition to the generalized forces, there is a summand. Let us write the equations that establish the connection between "extra" and independent coordinates.
Determine the coefficients of the equations of additional relationships: 8 For further discussion, it is convenient to present this equality in the matrix form: From the last equality follows: The definition of generalized forces is carried out, making the expression of the amount of work on virtual displacements: Finally, the system of equations of the forced motion of the mechanism can be written in a matrix form, highlighting the fifth equation, which is intended to determine the Lagrange multiplier: The last equation of the reduced system is used to determine the Lagrange multiplier: Finally, the system of equations of the power-saw bench drive is obtained in the form of matrix equality: Write the expression for the given value of the Lagrange multiplier: