Exact closed-form solution for the completely integrable helmholtz-duffing oscillator with applications to real-world problems

The Helmholtz-Duffing oscillator is an important vibration model that can be used to study many engineering systems and physical phenomena. In this article, we derived new exact closed-form solutions for the completely integrable Helmholtz Duffing oscillator subject to a constant force. The exact solution was derived naturally from the first integral of the governing differential equation. This approach is completely different from the ansatz method, used in previously published studies, where the exact solution was forced to take the form an initially assumed solution. Through various algebraic transformations and with the aid of standard elliptic integral tables, the present exact period was derived in terms of the complete elliptic integral of the first kind while the exact displacement was derived in terms of the Jacobi cosine function. Unlike the exact ansatz solutions, which have limited range of validity, the present exact solutions are applicable to all bounded periodic states of the Helmholtz-Duffing oscillator with constant force. The validity of the present exact solutions was verified using numerical solutions and published exact solutions in the form of the Weierstrass elliptic function. It was found that the numerical differentiation method produced significant errors for some system inputs and cannot be relied on as a benchmark solution. However, the present exact solutions are benchmark solutions that could be used to check the accuracy of new and existing approximate solutions. Finally, the application of the exact solutions to analyze some real-world problems was demonstrated.