Attenuation due to hysteretic damage in the free vibration of a beam

We present an asymptotic analysis of nonlinear free vibration of a beam with a damage plane represented by nonlinear hysteretic bending and shear springs. The perturbation parameter is the product of the ratio of the nonlinear to linear parts of the stiffness times the amplitude of the free vibration. The loss of energy and ensuing attenuation due to hysteresis is accounted for by reducing the amplitude of vibration after each cycle by an amount such that the loss in total system energy equals the work done to traverse the hysteresis loop. A new Fourier representation for each cycle of the hysteresis and the deflection solution is used for this purpose and leads to higher harmonics, an evolving complex stiffness and corrected natural frequency that are linked to the attenuation. The frequency increases to its linear value from an initially reduced value. The damage parameter, frequency shift and fundamental amplitudes are presented as functions of the initial damage parameter and time (cycles of vibration). The amplitudes of several of the higher harmonics are also presented as functions of time. Many of the results exhibit sufficient sensitivity with respect to the damage parameter that they should be able to be used to characterize the damage