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The energy dissipated in hysteretic damping is independent of cyclic frequency, which agrees with the experimental results of energy dispersion of many materials under cyclic loading. Despite these desirable properties, hysteretic damping suffers from a notable drawback; the direct integration solution of the equation of motion is divergent. In this paper, a virtual initial condition is proposed to address the instability of the solution. The new method develops virtual initial conditions associated to the real initial conditions, which make the direct integration solution eliminate the divergence term of the complementary solution and converge to the exact solution. The stability and accuracy of the proposed direct integration method are demonstrated to be effective in solving the divergence problem by the comparison of the numerical and theoretical solutions on the free and forced vibration of a system with hysteretic damping.

The Direct Integration Method with Virtual Initial Conditions on the Free and Forced Vibration of a System with Hysteretic Damping