Magnetic negative stiffness damper and its application to stay cables

Abstract

Negative stiffness dampers (NSDs) are an emerging type of dampers with both negative stiffness and damping behavior. In contrast with common positive stiffness, negative stiffness behavior implies that the instantaneous direction of the external force without damping is opposite to that of the deformation. This thesis presents a novel design of passive NSDs that is based on a magnetism principle. A magnetic NSD (MNSD), as a passive device, efficiently integrates negative stiffness and eddy-current damping in a simple and compact design. When applied to structural vibration control, passive MNSDs may achieve a performance comparable with semi-active or active control in some applications, e.g., in stay cable vibration mitigation. Two configurations of MNSDs, namely MNSD-A and MNSD-B, were presented, both of which were composed of numerous permanent magnets arranged in a conductive pipe. The design principles of both MNSD-A and MNSD-B were illustrated. The negative stiffness was contributed by the interaction forces among several co-axially placed magnets, and the eddy-current damping was contributed by the conductive pipe influenced by the magnetic field change as a result of magnet motions. The numerical modeling of the MNSDs has been performed for both negative stiffness and eddy-current damping. The numerical model adopted the Coulombian Model, in which the permanent magnets were simulated by surfaces with uniformly distributed magnetic monopoles. The experimental results of the MNSDs validated the accuracy of the model. The parametric study and optimization of two types of MNSDs was then conducted based on the numerical modeling. The effects of the magnet arrangement and dimensions on the negative stiffness and eddy-current damping characteristics were systematically investigated in the parametric study. The MNSDs were also optimized individually to maximize the negative stiffness and eddy-current damping coefficients. Based on the optimization results, some optimal design formulas were obtained to facilitate the quick design of MNSDs for different vibration mitigation applications in the future. The dynamic behavior of a taut cable with a linear passive NSD installed close to one cable end was systematically investigated. A passive NSD was represented by a combination of a negative stiffness spring and a viscous damper. By both analytical and numerical approaches, parametric analysis of negative stiffness and viscous damping were conducted to evaluate the vibration control performance of the linear passive NSD installed on a taut cable. Furthermore, the boundary of passive negative stiffness to maintain the stability of a taut cable was also derived. Results revealed that the asymptotic approach is only applicable to passive dampers with positive or moderate negative stiffness, and loses its accuracy when a passive NSD possesses significant negative stiffness. It has been found that the performance of a passive NSD could considerably exceed those of conventional viscous dampers. The influence of two practical factors, namely, the flexural rigidity of stay cables and the nonlinearity of MNSDs, on the vibration mitigation performance were further evaluated numerically. As a parameter often being overlooked in the dynamics of stay cables, cable flexural rigidity becomes more important after the installation of MNSDs. The actual influence of flexural rigidity depends on the negative stiffness coefficient and installation of the damper, and the boundary condition of a stay cable. Compared with cable flexural rigidity, the stiffness nonlinearity of MNSDs imposes relatively limited impact on the stay cable vibration. The vibration control performance of a passive MNSD was verified experimentally through a scaled stay cable model in the laboratory. The passive MNSD used in the experiment was composed of a viscous damper and a magnetic negative stiffness device with an adjustable stiffness coefficient. As the negative stiffness strength became stronger, the cable responses under various dynamic loadings became smaller, which indicated that high damping was added to the stay cable. The highest damping ratio achieved during the tests was around 10%, four times as large as the optimal damping ratio achievable by conventional viscous dampers. The numerical simulation results in consideration of the flexural rigidity and boundary conditions of the stay cable agreed well with the experimental results. Active control methods, like linear quadratic regulator (LQR), may also produce a force-deformation relationship with a negative-stiffness feature that benefits vibration mitigation of stay cables. Therefore, the vibration mitigation performances of a passive NSD and action LQR control were compared. The comparison indicated that a passive NSD can offer a stay cable with a high damping level comparable to active LQR control. However, the passive NSD also decreases the modal frequencies of a stay cable; whereas the active LQR increases the frequencies slightly. The dynamic response results also indicated that the active LQR control offers slightly better control performance than the passive NSD in various loading cases. The superiority of the LQR control over the passive NSD was explained through an output feedback control approach. Through a combination of theoretical, numerical and experimental studies, this thesis work clearly demonstrated the salient features of MNSDs and their superiority in stay cable vibration mitigation. Some challenges are also discussed based on the outcome of this work.