<![CDATA[This article deals with analytical and numerical methods, particularly focusing on their application in solving displacement problem using mathematical models. This is a study on the evaluation of material damping ratios obtained from experimental data and laboratory tests, an essential criterion to reproduce reality in virtual simulations done for our computational models. The methodology involves iteratively seeking optimal solutions through multiple simulations, providing a way to estimate dynamic responses accurately. The article concludes by emphasizing the importance of numerical-stepping methods in tackling challenging problems in dynamic response analysis. It highlights the significance of numerical-stepping methods for handling non-linear systems and time-varying excitations in dynamic response analysis. This research addresses fundamental issues in the context of applied mechanics. In particular, the article is intended to provide a few computational algorithms where respective graphs are generated, depicting the relationship between time and displacement. Some highly efficient numerical procedures can be developed for linear systems by interpolating the excitation over each time interval and developing the exact solution using a variety of five methods. The numerical results obtained are compared with the theoretical solution. Our research manages to evaluate the depreciation of the material through the models reviewed in the article and gives the results according to the applied models by comparing the results and calculating the absolute error.]]>
