The primary resonance of a Duffing oscillator with a restoring force of fractional-order derivatives by the extended Galerkin method

The nonlinear vibrations have wide appearances in many scientific and engineering problems to be solved by various techniques to satisfy requirements for solutions. With many different equations of nonlinear features, approximate methods have been suggested and tested to enable simple, accurate, and efficient solution procedures in dealing with increasingly complex problems. The introduction of fractional-order derivative as a restoring force in nonlinear vibration equations brought in advantages in dealing with some dissipation and excitation factors, but the complexity has also increased significantly, and the solution techniques are in strained testing for applicability and efficiency. To meet the challenges for newer solution techniques with a simpler procedure, the extended Galerkin method for nonlinear vibration analysis is applied to such problems with fractional-order derivatives through the expansion of integration interval to large number of periods, and approximate solutions of amplitudes are obtained and validated. This study demonstrates that the extended Galerkin method is an effective procedure for the nonlinear vibration equations with fractional-order derivative terms, expanding the application of the newer method and providing solutions with a simpler procedure.