MATHEMATICAL MODELING OF THE PRESSURE MEASUREMENT SYSTEM OF GAS-LIQUID MEDIA

n this paper, mathematical modelling of a mechanical system designed to measure the working medium in aircraft engines and consisting of a pipeline and a pressure sensor is carried out. On one side the pipeline is connected to the engine combustion chamber in which the pressure is measured, on the other side the pipeline is connected to a sensor that measures this pressure. The pipework is necessary to move the sensor some distance away from the engine in order to mitigate the effects of high temperatures and vibration accelerations on the sensor's sensitive element, which is an elastic plate. To describe the vibrations of the sensing element, a nonlinear model of the mechanics of a solid deformable body is used, which is a nonlinear system of partial differential equations to determine both transverse and longitudinal deformations. The mathematical model of the mechanical system takes into account the aerohydrodynamic and thermal effects on the plate. The aerohydrodynamic equations in the incompressible medium model are used to describe the motion of the working medium in the pipeline. Based on the small parameter method, asymptotic equations describing the joint motion of the working medium and the sensing element are obtained. Unsteady heat conduction equations are used to determine the temperature of the plate and the working medium in the pipeline. As a result, a mathematical model is constructed, which is a coupled nonlinear boundary value problem for the system of partial differential equations to determine five unknown functions – longitudinal and transverse components of the plate deformation, velocity potential, temperature of the working medium in the pipeline, temperature of the plate material. The Fourier method is used to solve the thermal problem. The Fourier and Bubnov–Galerkin methods are used to solve the differential equations describing the joint dynamics of the elastic plate and the working medium in the pipeline. As a result, the solution of the problem is reduced to the study of a system of ordinary differential equations relating the magnitude of pressure in the motor to the magnitude of deformation of the sensing element. With the help of the developed programme in Mathematica 12.0, a numerical experiment for specific parameters of the mechanical system is performed and graphs of the transverse and longitudinal components of the plate deformation at the midpoint of the plate are plotted. The results of the numerical experiment are used to analyse the frequency and amplitude of oscillations of the elastic element of the pressure sensor.