Mathematical modelling of elastic vibrations arising from four-body impact interaction

Abstract

This paper presents a mathematical model of elastic vibrations arising from shock interaction in a four-body system. The resulting model has two applications: the study of linear elastic interaction and the study of nonlinear elastic interaction. The linear elastic body model is presented in analytical form. In the transition to nonlinear interaction, it is difficult to represent the model in analytical approximation. The application of numerical algorithms has made it possible to obtain solution of a nonlinear problem with a specified accuracy sufficient for the simulated process. The model is intended for the description of the behavior of the device for measuring the modulus of elasticity of soil by the stamp method. An algorithm is developed to obtain the displacement and acceleration of the falling load and the loading die. Under laboratory conditions, an experiment with a test rig capable of measuring the acceleration of a loading die during shock-elastic interaction with the falling load is conducted. The measuring part is a three-axis accelerometer sensor connected to a four-channel analog-digital converter, which transmits the data to a mobile device with the developed specialized software. The sampling frequency of the measuring system is 12 kHz for each of the four channels. Polyurethane was used as a nonlinear element of the system, the approximate stiffness of which was determined experimentally by measuring deformations under a given load. When comparing the experimental results with the simulation results, the numerical model parameters were optimized by an algorithm based on gradient descent, which was based on maximizing the correlation coefficient between the theoretical and experimental data.