Inverse problem of determining the time- dependent beam stiffness coefficient in the beam vibration equation using the finite difference method

Abstract

In this study, we consider the inverse problem of determining the time-dependent beam stiffness coefficient q(t) and transverse bending vibrations u(x,t) of a homogeneous beam using the finite difference method. The problem is governed by a fourth-order partial differential equation, which is discretized in space and time using numerical techniques to obtain an accurate and stable solution. An additional condition in the form of an integral is also included to determine q(t). The results obtained for different discretization grid points are compared, and a simple test example is presented to demonstrate the accuracy and agreement of the numerical solutions with analytical solutions. The paper provides a comprehensive overview of the methodology used to solve the inverse problem, and presents a detailed analysis of the results obtained.