APPLICATION OF FRACTIONAL CALCULUS FOR ANALYSIS OF NONLINEAR DAMPED VIBRATIONS OF SUSPENSION BRIDGES

Free damped vibrations of a suspension bridge with a bisymmetric stiff ening girder are considered under the conditions of the internal reson ance one-to-one, i.e., when natural frequencies of two dominating mode s-a certain mode of vertical vibrations and a certain mode of torsiona l vibrations-are approximately equal to each other. Damping features o f 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 o f solution. It is shown that in this case the amplitudes of vertical a nd torsional vibrations attenuate by an exponential law with the commo n damping ratio, which is an exponential function of the natural frequ ency. Analytical solitonlike solutions have been found. A numerical co mparison 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 coe fficient.