Oscillations of Suspension Bridges

Abstract

Abstract The author has developed a theory of oscillations of stiffened cable systems which is applicable to the design of suspension bridges. Dealing with free oscillations in the first part of the paper the author established a system of linear differential equations with variable coefficients taking into account the horizontal displacements and inertia forces, the change of thrust along the cable, the slope of the cable, and the torsional stiffness of the girder. An approximate integration by the Rayleigh energy method is shown and is numerically exemplified for the lowest antisymmetric mode. The results show that, in general, it is allowed to neglect horizontal displacements, change of thrust, and cable sag for the dynamic problem as well as for the static problem and they bring out the fact that for all practical proportions of relative cable sag and girder stiffness the frequencies and modes are essentially independent of the relative cable sag and dead load and only dependent upon the absolute cable sag h so that, for instance, for the lowest antisymmetric mode the equivalent pendulum length is 2h/π2. The second part of the paper treating self-excited oscillations has been prepared under the influence of observations, experiments, and conclusions reached in connection with an (official) report on the Tacoma bridge failure. This report demonstrated that the aerodynamic and structural damping—the first acting through the width of the roadway, the second by the internal friction—plays an important part in the safety of a suspension bridge. It showed further that the dangerous oscillations are of somewhat higher frequency than the free oscillations with logarithmic decrement a linear function of the reduced velocity so that not elastic instability under wind forces nor resonance from wind gusts but the bridge as a self-exciting mechanism absorbing energy out of a steady wind flow had to be assumed as the cause of the failure. This experimental linear decrement law is utilized by the author to derive in conjunction with the dynamic equations formulas intended to serve as the basis for the dynamic requirements for suspension bridges.