DETERMINING THE STATE VARIABLES OF A DISCRETE PNEUMATIC DRIVE BY USING THE SECANT METHOD WHEN LINEARIZING A MATHEMATICAL MODEL

The pneumatic drive as a thermodynamic system is described based on the principles of "thermodynamics of a body of variable mass". In describing thermodynamic and gas-dynamic processes in pneumatic drives, two scientific schools have developed: researchers of the first school represent processes in the cavities of the drive as polytropic processes with a variable polytropic index, and the second consider processes based on the equation of energy (heat) balance of the gas-pneumatic drive in an open space. The authors investigate a nonlinear mathematical model obtained on the concepts of the second school. In the cycles of reducing the number of independent parameters that determine the nature of the transient process in the drive, a transition to a dimensionless form of equations is carried out based on the principle of minimizing dimensionless complexes (dynamic similarity criteria) that determine the nature of the transient process. The task of linearization of the nonlinear model was set with the purpose of obtaining analytical expressions for all state variables based on the linear model, which will allow avoiding the numerical step methods of integration of the original nonlinear model in calculations. It is shown that the replacement of nonlinear dependencies by the first terms of their expansion in a Taylor series (the tangent method), which is practiced for servo hydraulic pneumatic drives, leads to large errors in relation to discrete drives. The proposed linearization by the secant method with the choice of the optimal form of the sowing allowed to significantly increase the calculated accuracy of the linear mathematical model. and obtain the state variables of the pneumatic drive in analytical form. The calculations performed using the second-order linear model convincingly indicate that it is quite adequate to the calculated accuracy of the nonlinear mathematical model. In the entire range of the most probable parameters, the calculated accuracy of the second-order mathematical model is quite sufficient for practical use. Thus, the need to use step-by-step numerical methods for integrating the equations of a nonlinear model and organizing the computational process on a computer is eliminated.