The Fractional Derivative Expansion Method in Nonlinear Dynamics of Structures: A Memorial Essay

Analysis of Nonlinear Forced Vibrations of Fractionally Damped Suspension Bridges Subjected to the One-to-One Internal Resonance

“…: has been investigated in [4,6,24]. Since for finding the solution of equations (2) we will use the method of multiple time scales, where the functions i t e Z r are utilized as the main harmonic functions, then in order to carry out the calculations the following formulas will be utilized [27] 0 0 sin = ,…”

“…Let us substitute relations (16) A are 1 T -independent. Substituting then (16) and (24) in equations (15) and utilizing the standard procedure for eliminating secular terms, we obtain…”

“…The nonlinear suspension bridge model put forward allows one to obtain the damping coefficient dependent on the natural frequency of vibrations. The overview of the existing research of the internal resonance in suspension bridges could be found in [23,24]. In the present paper, the model proposed in [22] for the analysis of free damped vibrations is generalized to the case of nonlinear forced vibrations of suspension bridges, when the frequency of the external force is close to one of the natural frequencies of the vertical vibrations of the suspension combined system, which is subjected to the condition of the one-to-one internal resonance.…”