The Time-Domain Solution of a Hysteretic Damping System in the Phase Plane
摘要
A single-degree-of-freedom(SDOF) system with hysteretic damping can be solved very easily in the frequency domain. However, the equation of motion in time-domain is nonlinear and has not been solved analytically so far. In this paper, the time-domain free-vibration of hysteretic damping SDOF system is analytically solved in the phase plane. The equation of motion is shown to be linear of only mass-spring system in each quadrant of the phase plane and has closed-form solution in a form of quarter-ellipse when the loss factor is smaller than one. The complete solution shows a trajectory of quadrant-decaying ellipse in the phase plane. The explicit formula for the decrement ratio and the damped natural frequency which depend on the loss factor are also obtained. As loss factor changes to value larger than one, the trajectory shows a hyperbolic curve in which the stable focus becomes the stable saddle point in the 2nd and 4th quadrants. In the case when the loss factor is equal to one, the system will have permanent deformation and the trajectories reduce to horizontal lines in the 2nd and 4th quadrants, respectively. The present paper extends the applicability of the hysteretic damping model from the frequency domain to the time domain. Thus, the general loadings, harmonic or inharmonic, can be treated in the time-doamain using the technique of convolution integral.