Nonlinear suspension bridge vibrations

SUMMARY Nonlinear coupled vertical and torsional vibrations of suspension bridges are investigated. The nonlinear equations of motion are derived using Hamilton 's principle and are nondimensionalized. Method of Multiple Scales, a perturbation technique, is applied to the equations to find approximate analytical solutions. The equations are not discritized as usually done, rather the perturbation method is applied directly to the partial differential equations. At the first order of approximation, which constitutes the linear part of the problem, it is observed that the transcendental equations yielding natural frequencies are in different forms for symmetric and asymmetric modes. Solutions are obtained for both cases and the natural frequencies are calculated. At the second order, the solutions of only asymmetric modes are found since the solution functions for the symmetric modes are extremely complicated. In the asymmetric modes the solution functions are derived seperately for the side and center spans. At the third order of approximation, the solvability conditions and the equations describing phase and amplitude modulations are derived. After that, free vibrations, and forced and damped vibrations are investigated seperately. Three different cases are considered in free vibrations: (1) the case in which the vertical vibration is dominating, (2) the case in which the torsional vibration is dominating, (3) the case in which the natural frequency of one of the vertical vibration modes is close to the natural frequency of one of the torsional vibration modes. For the first two cases, vertical displacement and torsional angle functions are derived, the dependence of nonlinear frequency on amplitude is described, and the criteria for the hardening and softening model behavior for vertical and torsional vibrations are established. In the third case, the change of amplitudes as a function of time is shown and steady state solutions are analyzed. In the forced-damped vibrations, no internal resonance and one-to-one internal resonance cases are considered. For the case of no internal resonance, steady state solution is analyzed, frequency-response equation is derived and the jump phenomenon in the frequency-response curves resulting from nonlinearity is explained. For the case of internal resonance, frequency-response equation is derived. Results of the analyses are applied to two suspension bridges. The natural frequencies of these bridges are calculated. The frequency-amplitude spectra for vertical-dominant and torsional-dominant vibrations are obtained. Coupled vibrational motions of closely spaced vertical and torsional modes are investigated. In free vibrations the change of amplitudes and phase as a function of time is shown graphically for various initial amplitude and phase values. In forced and damped vibrations for the case of no internal resonance, numerical integration of frequency-response equation is performed for various modes, damping coefficients, and amplitudes of excitation; related frequency-response curves are drawn. For the case of internal resonance, graphs of amplitude and phase modulations are drawn for various initial amplitude and phase values, amplitudes of excitation, and damping coeefficients. The results obtained from the graphs are discussed. Finally, the use of Artificial Neural Networks (ANN) as an alternative method to compute the natural frequencies of suspension bridges quickly and with small errors is studied. A general presentation of ANN is given and the characteristics of the network used in this study are described. The preparation of the input and output sets used, the training of the algorithm, and the comparison of results are explained. By the method developed, given the physical parameters, the natural frequencies of suspension bridges having various properties can be computed easily. XIV