Abstract Summary
In structural dynamics, it is well known that the motion of a single degree of freedom (SDOF) system with hysteresis subjected earthquake ground motions can be described by the equation m·a(t) + c·v(t) + H(x(t)) = -m·g(t) (plus the initial conditions v(0) and x(0)), where a(t), v(t), x(t), are the relative (to the ground) acceleration, velocity, and displacement, respectively; H(x(t)) is the hysteretic restoring force; g(t) is ground acceleration; and m and c are the mass and damping coefficient of the system, respectively. Because the external forcing term, -m·g(t), explicitly depends on the time (t), the dimension of the phase space of such a second order differential equation is not two, but three, meaning that it is a two-dimensional non-autonomous dynamical system. Consequently, any bi-dimensional (2D) plot of the variables involved in the description of this type of system is self-intersecting, such as the classical hysteresis curve [x(t), H(x(t))] or the phase portrait [x(t), v(t)]. The self-intersecting (or auto-intersecting) nature of these plots represents a barrier for a proper visualization of the dynamics of the system (including its hysteresis) and render them unsatisfactory, from a graphical perspective, for conducting qualitative evaluations. To cope with such a disadvantage, the previously mentioned equation of motion can be viewed as a three-dimensional (3D) autonomous system, which means that any plot including three variables is non-self-intersecting. Making use of such a property, this paper presents a series of novel 3D non-self-intersecting plots for describing the dynamics of SDOF systems encountered in structural engineering, such as, for instance, non-structural components anchored to reinforced concrete (RC) structures or elevated tanks supported by a steel structure, when subjected to ground motions (or any other external excitation explicitly dependent on the time). The 3D curves include plots incorporating the time as an additional variable, such as [x(t), H(x(t)), t] and [x(t), v(t), t], both of them proposed in a prior contribution by the author and collaborator published in 2021, as well as others of the type [v(t), H(x(t)), x(t)] and [a(t), v(t), x(t)], which depend implicitly on t, and resemble the 3D graph depicting the so-called ‘butterfly effect’ encountered by Edward Lorenz in meteorology. In addition, using data of experiments available in the literature, the paper uses the 3D curve [x(t), H(x(t)), t] for plotting the force-displacement hysteresis loops of structural members/assemblies subjected to quasi-static loading. Lastly, using animation videos, the evolution in time of the 3D curves, alongside its classical 2D projections in the three orthogonal planes, are planned to be shown during the presentation.
