Green’s function for uniform Euler–Bernoulli beams at resonant condition: Introduction of Fredholm Alternative Theorem

This paper deals with the dynamic analysis of Euler–Bernoulli beams at the resonant condition. The governing partial differential equation of the problem is converted into an ordinary differential equation by applying the well-known Fourier transform. The solution develops a Green’s function method which involves establishing the Green’s function of the problem, applying the pertinent boundary conditions of the beam. Due to the special conditions of the resonant situation, a significant obstacle arises during the derivation of the Green’s function. In order to overcome this hurdle, however, the Fredholm Alternative Theorem is employed; and it is shown that the modified Green’s function of the beam may still be achievable. Furthermore, the necessary requirement so that the resonant response will be found is introduced. A special case which refers to a case in the absence of resonance is also included, for some verification purposes.