MS15.7 - Nonlinear Dynamics and Dynamic Stability

In the area of passive vibration mitigation, nonlinear energy sinks (NESs) have been extensively studied. They are an alternative to traditional tuned mass-dampers as they can mitigate vibrations over larger frequency ranges while being efficient in both transient and stationary regimes presenting period and/or non period responses. NESs are characterized by nonlinear stiffness properties (e.g., cubic stiffness), leading, in some cases, to irreversible transfers of energy between the main system and a NES. The literature on nonlinear absorbers contains several variations on the type of nonlinearity used. However, in all cases, nonlinear properties are mainly constant over time. In this work, a nonlinear absorber with a time-dependent nonlinear stiffness is considered. The nonlinearity, which can be adjusted in an acoustical application, follows a specific functional form, which is enforced through a control system. The objective of this work is to optimize the characteristics of the nonlinear absorber, including the time varying nonlinear stiffness. However, the optimal design of nonlinear absorbers is known to present several challenges. Specifically, their behavior can be acutely sensitive to uncertainties. In fact, it is well known that a NES exhibits a discontinuity in its response due to the presence of an activation threshold. This discontinuity is typically associated with significantly different dynamic behaviors that can be reached for small perturbations of the loading conditions or the design. This work presents a dedicated stochastic design optimization algorithm for the proposed absorber with time-varying nonlinear stiffness. It is tailored to identify and tackle the discontinuous behavior during the optimization process. The approach is based on support vector machine (SVM), clustering, Gaussian processes, and adaptive sampling. One of the key elements of the approach is the approximation of the absorber activation threshold through an SVM classifier. The stochastic design optimization enables the computation of probabilistic constraints, expected value and variance of the response. Both uncertainties in ``design” variables and loading conditions are included in the optimization. In addition, revealing fast and slow system dynamics which leads to detection of Slow Invar-iant Manifolds (SIMs), equilibrium and singular points are analytically/numerically derived through the Manevitch complexification approach. SIMs and characteristic points, corresponding to various designs, are used to physically interpret the optimal solutions and their robustness to perturbations in loading conditions

On the 17th of October 2021 a grandstand of the Goffert stadium partially collapsed while a crowd was jumping on it. Forensic engineering concluded that the main reason of the collapse was that the load of a jumping crowd exceeded the design load of 4 kN/m2, as specified in the Eurocode. The actual collapse was due to the combination of dynamic non-linear response of the structure and a dynamic load of the jumping crowd. Both the response and the load were investigated, and ultimately combined in an equivalent non-linear single-degree-of-freedom (SDOF) system with a dynamic load. A geometrically and physically non-linear finite element (FEM) model was used to obtain the backbone curve of the response. This curve was obtained with a push-down analysis. The push-down analysis was force-controlled to allow the grandstand to rotate and deform freely, so that the failure mechanism could develop freely. The jumping load was modelled with a Fourier series based on how high the crowd was jumping and how well coordinated they jumped. Based on video footage during the col-lapse these parameters were estimated, and the number of people counted. It was conclud-ed that the mass of the people was roughly 3.5 kN/m2. An equivalent SDOF system was set up based on analytical formulae and the backbone curve obtained with the FEM model. With this model the non-linear dynamic behavior of the grandstand due to the jumping load of the crowd was assessed. Failure was found with the estimated jumping load after roughly the same number of jumps as was seen on video. Expressing this non-linear dynamic behavior as a quasi-static load leads to a value of 9.8 kN/m2, which implies a dynamic amplification factor of 2.7. This is an unusually high am-plification factor for crowds, which is the result of the combination of a dynamic load and non-linear behavior. For design purposes it is therefore advised to use only the linear-elastic capacity of the structure for controlling dynamic loads.

In this paper, the nonlinear aeroelastic system behaviour of a typical section airfoil with two degrees of freedom is investigated. The aim of this work is to characterize the nonlinear dynamics and mapping the regions of limit cycle oscillations (LCO). Firstly, a previous numerical model [1] is simulated and analysed the time histories and phase portrait as function of initial conditions. Also, the bifurcation diagram and section of Poincaré are used to determine the behavior of the attractors. The experimental device consists of a typical section airfoil mounted on elastic structure. The translational motion or plunge motion 'h' is obtained by means of four vertical flexible beams with a linear stiffness 'k_h' and the rotational motion or pitch motion α is obtained by a torsional spring implemented with a horizontal beam, assuming a quadratic representation of a nonlinear dynamic stiffness 'k_α (α)' to represent the structural nonlinearity and to evaluate its effect on flutter phenomena. Further work, the model parameters will be updated and the experimental apparatus will also be characterized. The governing equations of motion of the aeroelastic system is presented in [2] [3]. The influence of a nonlinear aerodynamic lift and moment on the airfoil is considered under free stream velocity 'V∞' and the model includes the angular position of the control surface β. The quasi steady aerodynamic lift and moment are evaluated as [4] including the control surface effect in aerodynamic. In this work, a bifurcation diagram associated with the pitch degree of freedom was plotted and the diagram was obtained with a free stream velocity range between 0 and 110 m/s. It indicates the existence of a limit cycles for free stream velocity around 15m/s. Additionally, this work investigates the effect the elastic axis position in the bifurcation diagram, using the same procedure as above. Also, it can be observed that the nonlinearity affects the ocurrency of limit cycle by observation of Jacobian matrix that is directly dependent of the angular displacement. Thus, the nonlinearitties lead to limit cycle oscillations that cause the flutter phenomenon is aeroelastic sytem as obtained in the phase planes. Therefore, a bifurcation diagram was obtained and it reveals an existence of LCO in which its amplitude depend upon the free stream velocity and also the initial conditions. [1] R. C. M. Barbosa, L. C. S. Góes, A. Nabarrete, J. M. Balthazar, D. F. C. Zúniga, Nonlinear identification using polynomial NARMAX model and a stability analysis of an aeroelastic system, Proceedings of the XVII DINAME, 2017. [2] T., O’Neil, H., Gilliatt, T. W., Strganac, Investigations of aeroelastic response for a system with continuous structural nonlinearities, AIAA Meeting Papers on Disc, Paper 96 1390, 1996. [3] S. L., Kukreja. Nonlinear system identification for aeroelastic systems with application to experimental data, NASA, 2008. [4] D. Li, S. Guo, J. Xiang, Aeroelastic dynamic response and control of an airfoil section with control surface nonlinearities, Journal of Sound and Vibration, Vol. 329, pp. 4756, 2010.