Application of Fractional Calculus for Analysis of Nonlinear Damped Vibrations of Suspension Bridges

Free damped vibrations of a suspension bridge with a bisymmetric stiffening girder are considered under the conditions of the internal resonance one-to-one, i.e., when natural frequencies of two dominating modes—a certain mode of vertical vibrations and a certain mode of torsional vibrations—are approximately equal to each other. Damping features of the system are defined by a fractional derivative with a fractional parameter (the order of the fractional derivative) changing from zero to one. It is assumed that the amplitudes of vibrations are small but finite values, and the method of multiple scales is used as a method of solution. It is shown that in this case the amplitudes of vertical and torsional vibrations attenuate by an exponential law with the common damping ratio, which is an exponential function of the natural frequency. Analytical solitonlike solutions have been found. A numerical comparison between the theoretical results obtained and the experimental data is presented. It is shown that the theoretical and experimental investigation agree well with each other at the appropriate choice of the parameters of the exponential function determining the damping coefficient.