Hysteretic Loops in Correlation with the Maximum Dissipated Energy, for Linear Dynamic Systems

Abstract

This paper presents the outcomes of the theoretical and experimental research carried out on a real model at natural scale using Voigt–Kelvin linear viscoelastic type m, c, and k models excited by a harmonic force F(t) = F₀ sinωt, where F₀ is the amplitude of the harmonic force and ω is the excitation angular frequency. The linear viscous-elastic rheological system (m, c, k) is characterized by the fact that the c linear viscous damping—and, consequently, the fraction of the critical damping ζ—may be changed so that the dissipated energy can reach maximum $W_d^{\max }$ values. The optimization condition between the $W_d^{\max }$ maximum dissipated energy and the amortization ${\zeta }^0=\pm \left(1-{\Omega }^2\right)/2\Omega$ modifies the structure of the relation F = F(x), which describes the elliptical hysteresis loop F–x in the sense that it has its large axis making an angle less than 90° with respect to the x-axis in $\Omega <1$ ante-resonance, and an angle greater than 90° in post-resonance for $\Omega >1$ . The elliptical Q–x hysteretic loops are tilted with their large axis only at angles below 90°. It can be noticed that the equality between the arias of the hysteretic loop, in the two representations systems Q–x and F–x, is verified, both being equal with the maximum dissipated energy $W_d^{\max }$ .