Studying the nonlinear mathematical model of the mechanical system “pipeline – pressure sensor”

Abstract

Background. The primary element of the instrumentation for measuring the pressure of a gas-liquid medium is a sensor that supplies data on the pressure of the working medium, which determines the proper functioning of machines, mechanisms, and systems. Increasing the service life, reducing development time, and reducing the cost of sensors are one of the primary objectives. In this regard, mathematical modeling of the functioning of such systems plays an important role at the design stage of pressure measurement systems. To measure and control the pressure of the working gas-liquid medium in the combustion chambers of engines, a mechanical system “pipeline - pressure sensor” is used, in which, to reduce the effects of vibration accelerations and high temperatures, the sensor is connected to the engine via a pipeline and is located at some distance from it. The purpose of the work is to create a mathematical model of the “pipeline - pressure sensor” system and study it for the possibility of establishing a correspondence between the law of pressure change in the combustion chamber and the law of oscillation of the sensitive element of the pressure sensor. Materials and methods. To describe the movement of the working medium (in the ideal gas model), a nonlinear model of fluid and gas mechanics is used, under the assumption that the working medium is compressible. To describe the dynamics of the sensitive element of the sensor, a model is used, the basis of which is an ordinary differential equation that describes the oscillatory process of a single-mass system. Under these assumptions, a mathematical model of the mechanical system “pipeline – pressure sensor” was constructed. To solve the corresponding problem, the formulation of which contains a nonlinear partial differential equation, numerical and analytical solution methods based on the Galerkin method are proposed. Results. A nonlinear mathematical model of a system for measuring pressure in gas-liquid media has been developed. For the corresponding initial-boundary value problem, based on the Galerkin method, a method is proposed that makes it possible to reduce its study to solving the Cauchy problem for a system of ordinary differential equations. A numerical experiment is carried out and examples of calculating the dynamics of the sensor's sensitive element are presented. Conclusions. The proposed mathematical model makes it possible to determine the law of change in the deviation of the sensor's sensitive element depending on the law of change in pressure in the combustion chamber. The research results are intended for use at the design stage of pressure measurement systems.

Keywords

• pressure sensor
• pipeline
• sensitive element
• dynamics
• differential equations
• galerkin method