Applying a New Trigonometric Radial Basis Function Approximation in Solving Nonlinear Vibration Problems

This study introduces a semi-analytical method called the New Trigonometric Radial Basis Function (NTRBF) approach to address the challenges of solving highly nonlinear differential equations in vibration problems. The method uses a particular trigonometric function to deal with differential equations in an extraordinary and original approach. This study conducted a comparative analysis of the proposed method against four different approaches, including the Global Residue Harmonic Balance Method for addressing circular sector oscillator problems, the Continuous Piecewise Linearization Method for solving highly nonlinear differential equation of a tapered beam, the Differential Transform Method for solving centrifugal rotating frame motion, and Akbari–Ganji’s Method to address Duffing-type nonlinear oscillator. These problems were solved under different conditions, and the resulting plots and tables represent both cumulative and maximum errors between the NTRBF and other methods. The numerical 4th-order Runge–Kutta method was used as a benchmark for accuracy in this comparative analysis. The outcomes prove the high accuracy and efficiency of the innovative technique and its unique capability in solving various nonlinear vibration problems.