Nonlinear mathematical model of pressure measurement systems in gas-liquid media

Abstract

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. It determines the proper functioning of machines, mechanisms, and systems. Increasing the service life, reducing development time, and reducing the cost of sensors is one of the important tasks. Mathematical modeling of pressure measurement systems’ functioning plays an important role at the design stage of such systems. This article examines a nonlinear one-dimensional model of a mechanical system “pipeline – pressure sensor” designed to measure and control the pressure of the working gas-liquid medium in the combustion chambers of engines. In such a system, the sensor is connected to the engine via a pipeline and is located at some distance from it to reduce the impact of vibration accelerations and high temperatures. The purpose of the work is to study the dynamics and stability of joint oscillations of the elastic sensitive element in the pressure sensor and of the working medium in the pipeline for a given law of pressure change in the combustion chamber. The study is provided under the assumption that the working medium is ideal and compressible. To describe the movement of the working medium (gas or liquid), a nonlinear model of fluid and gas mechanics is used. Mathematical description of the process of interest includes an initial boundary value problem, whose formulation contains a nonlinear partial differential equation. To solve it, numerical-analytical method of solution based on the Galerkin method is proposed, which makes it possible to reduce the study of the problem to solving 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. 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.