MS15.9 - Nonlinear Dynamics and Dynamic Stability

In this paper, transverse and longitudinal vibrations and resonances in an elevator cable system induced by boundary excitations are studied. The dynamics can be described by an initial-boundary value problem for a (coupled system of) nonlinear wave-equation(s) on a relatively slowly time-varying spatial domain. It will be shown how boundary excitations and nonlinear terms influence transverse and longitudinal oscillations in the system. Firstly, due to the relatively slow variation of the cable length, a singular perturbation problem arises. By using an interior layer analysis many resonance manifolds are detected. Secondly, it will be shown that resonances in the system are caused not only by boundary disturbances but also by nonlinear interactions. Based on these observations, a multiple time-scales perturbation method is used to approximate the solution of the initial-boundary value problem analytically. It turns out that for special frequencies in the boundary excitations and for certain parameter values for the longitudinal stiffness and the conveyance mass, many oscillation modes jump up from small to large amplitudes in the transverse and longitudinal directions. Numerical simulations are presented to verify the obtained analytical results.

The non-linear dynamics of layered structures are highly dependent on the material and bonding between each layer; a lack of proper alignment, or delamination, could lead to deleterious outcomes including structural failure. Experiments on different types of thin-walled cylindrical shells have recently been carried out, showing that delamination can be detected by changes in the transmitted wave field [1]. The long longitudinal strain waves in a two-layered waveguide can be modelled by coupled Boussinesq equation [2]. In this talk, we will focus on the reduced case, when the lower layer of the waveguide to be significantly denser than the upper layer, leading to a system of Boussinesq-Klein-Gordon (BKG) equations [3]. We consider a two layered waveguide with a de-lamination in the centre and soft (imperfect) bonding either side of the centre. Direct numerical modelling is difficult and so I will use a semi-analytical approach using asymptotic methods, which leads to Ostrovsky equations in soft bonded regions and Korteweg-de Vries equations in the delaminated region. The semi-analytical approach and direct numerical simulations are in good agreement. We will also discuss how the dispersion relation is used to determine the wave speed and hence classify the length of the delamination, in addition to changes in the amplitude of the wave packet. These results can provide a tool to control the integrity of layered structures. We will also discuss recent results for the case when the materials of the upper and lower layers are similar, and thus the structure is modelled by coupled Boussinesq equations. Using a similar approach, the delamination length can be inferred from changes in the transmitted wave field. References [1] G.V. Dreiden, A.M. Samsonov, I.V. Semenova, A.G. Shvartz, Strain solitary waves in a thinwalled waveguide. Appl. Phys. Lett. 105, 211906 (2014). [2] K.R. Khusnutdinova, A.M. Samsonov, A.S. Zakharov, Nonlinear layered lattice model and generalized solitary waves in imperfectly bonded structures. Phys. Rev. E 79, 056606 (2009). [3] J.S. Tamber and M.R. Tranter, Scattering of an Ostrovsky wave packet in a delaminated waveguide. Wave Motion, 114, 103023 (2022).

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.

It is remarkable that trees with high slenderness ratios are able to survive regular strong wind events. This phenomenon is motivating the exploration of the inherent vibration mitigation mechanisms of trees. This study examines the role of trees’ hierarchical branching architecture on their modal properties. Its particular focus is on the modal frequencies and mode shapes of sympodial trees. This study idealizes trees as fractal structures with sympodial branching architecture and proposes a new group tree modeling approach to analyze their modal properties. Analytical closed-form solutions are derived to estimate the modal properties of trees. The analysis shows that sympodial trees localize vibrations on higher-order branches. Furthermore, the modal properties of trees with a specific fractal level could be self-similar, and repetitive and form recursive relations with that of the previous level. Overall, the results offer a possible explanation of how fractal branching architecture prevents trees from excessive vibration.