Precision Comparison of Analytical and Semi-numerical Analytical Methods for Highly Non-linear Coupled Mechanical Problems

This paper investigates the application and suitability of the differential transform method (DTM), a powerful analytical solution method, and its enhanced version, the multi-step differential transform method (MsDTM) for highly nonlinear mechanical systems. DTM is a method based on truncated version of the Taylor series expansion. This truncated version causes the functions obtained with this method to deviate considerably from the actual values after a very short time interval for time-dependent systems. Therefore, the MsDTM, which is recommended to overcome this deviation problem, divides the examined time interval of this system by a sufficiently small-time step and applies DTM at each divided time interval. In this way, the time interval over which the results obtained with DTM converge to the correct value is increased. The important issue at this stage is how short the time step in MsDTM should be chosen. This is because although analytical methods have the advantage of obtaining the solution of the system as a function of time, analyzing the system in small time steps, i.e., using MsDTM, leads to errors due to discretization. However, through the ability to calculate derivatives and integrals of time-dependent functions representing each time interval, a more detailed analysis can be obtained compared to numerical methods. This paper’s main objective is to suggest a researcher or engineer to obtain a solution with high precision and accuracy when dealing with mechanical systems with linear and oscillatory motion with high non-linearity.