Energy Pumping with various Nonlinear Structures: Numerical Evidences

Numerical investigations are carried out on a linear structure, weakly coupled to a small non-linear attachment. The energy pumping phenomenon corresponds to a controlled one-way channeling of the vibrational energy to a passive non linear structure where it localizes and diminishes in time due to damping dissipation ([1], [2]). So the essential non-linearity of the attachment enables it to resonate with any of the linearized modes of the structure leading to energy pumping, i.e. passive, one-way, irreversible transfer of energy from the structure to the attachment ([3]). Thus, through energy pumping, vibrations of a linear structure (subjected to an external excitation) can be attenuated thanks to a weak coupling to a non-linear structure. By designing the nonlinear structure skilfully (the mass of the non linear structure will be significantly smaller than the corresponding mass of the linear structure), it is possible to modify the existing structural features slightly and creating “nonlinear modes of vibration” which causes energy pumping where vibrational energy initially localized in the linear oscillator is irreversibly transferred (“pumped”) to the nonlinear oscillator ([4]). The existing studies on this subject are mainly focus on the addition of regular nonlinearities (for instance a cubic stiffness non-linearity in [1] and [2]) and addition of non linear structures with the about size of the linear mass ([3]). However, it is necessary to increase the effectiveness of pumping and to use very small non linear structures to be able to apply it to real structures. This is why we analize more precisely this nonlinear additional structure which is coupled to the initial structure by giving numerical evidences. Indeed, different non linear structures (piecewise linear system, chaotic system) and efficiency of energy pumping are studied in each case in order to be able to apply it to Civil Engineering. Hence, we are interested in the addition of structures with non-smooth nonlinearity like a piecewice linear system because the practical design of such system is easier (change of stiffness) and it will be thus possible to consider a complete analytical study later on. Moreover, the study of nonlinear structures being able to answer in a chaotic way in the ignition area of energy pumping is planned to try to solve the problem of minimal energy level necessary to reach the stable orbit responsible for energy pumping. The aim is always to improve the efficiency. As a specific application, attenuation of vibrations of a building (as shown in Figure 1) is studied with two building models. The building can be modelised by different linear systems: a building model with floors (as show in Figure 2 where 8 floors are considered) and a flexible continuous building are considered. To attenuate vibrations, different nonlinearities like a piecewise linear system can be coupled to the linear one. In the case of flexible continuous building, a stochastic excitation can be analized, in particular to be able to consider sollicitations like seisms. An indicator of efficiency is found with the standard deviation μ of the linear response as shown in Figure 3. This figure highlights the fact that energy pumping allows better attenuation of vibrations (40%) than coupling a linear absorber (10%). The numerical studies show a certain robustness of energy pumping when the parameters are slightly modified.