An analytical approach to the non-oscillatory nonlinear mechanical systems having integral multiple roots and strong non-linearity

The existence of over-damped nonlinear differential equations results from a variety of engineering conundrums and physical natural occurrences. Non-oscillatory dynamics with forced over-damping are used in the simulation of nonlinear differential systems. For non-oscillatory nonlinear differential systems, it is possible to derive approximations of solutions using a variety of analytical methods, both with and without external forcing. This paper introduces a novel method for estimating solutions for highly nonlinear damped vibration systems subject to parameterized external forcing. The extended Krylov-Bogoliubov-Mitropolsky (KBM) technique and harmonic equilibrium (HM), which have both been previously developed in the literature, are the foundation of the suggested method. This method was initially created by Krylov-Bogoliubov to discover periodic details in second-order nonlinear differential equations. Several examples are provided to show how the suggested technique is applied. The process is fairly simple and straightforward, and using this formula, the result can be found with very marginal errors from the previous citations. The primary significance of this approach is in its ability to provide approximate analytical solutions of the first order that closely align with the findings obtained by numerical methods. These solutions are applicable to a variety of beginning scenarios and are distinct from those presented in earlier literature. Also, we illustrated the two-dimensional graph of all the solutions that we got in this article by using the data from the mentioned table. The results that we obtained from this method are effective and reliable for better measurements of strong nonlinearities.