Mathematical modeling of the dynamics of the aeroelastic "pipeline - pressure sensor" system

This paper proposes mathematical models of mechanical systems "pipeline - pressure sensor" designed to control the pressure of the working medium in the combustion chambers of engines. In such systems, to mitigate the impact of vibration accelerations and high temperatures, the sensor is connected to the engine using a pipeline and is located at some distance from it. The movement of the working medium is described by linear models of fluid and gas mechanics. To describe the dynamics of an elastic sensitive element, linear models of the mechanics of a solid deformable body are used. Based on linear differential equations with partial derivatives, mathematical formulations of problems are proposed that correspond to three-dimensional models of pressure measurement systems in gas-liquid media for some pipeline cross-sectional shapes, namely, for a pipeline with a rectangular cross-section, with a section in the form of a sector and in the form of a ring. By introducing integral characteristics, the solution of problems is reduced to studying one-dimensional models. Equations have been obtained that make it possible to determine the pressure of the working medium in the combustion chamber at each moment of time by the value of deformation of the sensitive element of the sensor. Analytical and numerical-analytical methods for solving the corresponding initial-boundary value problems for systems of differential equations are proposed. In the analytical approach, the solution of the problem is reduced to solving a differential equation with a deviating argument. The numerical-analytical study of the problem is based on the application of the Galerkin method. Also, a numerical experiment was carried out and examples of calculating the deformation of the sensitive element of the sensor in the case of rigid fastening when specifying specific values of the mechanical parameters of the system are presented, also during setting the law of change in the excess pressure of the working medium in the engine.