Asynchronous modes of vibrations in linear conservative systems: an illustrative discussion of plane framed structures

The theme of asynchronous modes of vibration is recast in this paper to address the case of linear conservative discrete systems. In fact, it is known that linear non-conservative discrete systems such as those characterized by a non-symmetric stiffness matrix may undergo asynchronous free vibrations. Yet, non-conservativeness is not mandatory for the occurrence of asynchronous modes, as it is illustrated in this investigation of simple plane frames, provided the system parameters are properly chosen. Firstly, a one-storey-frame model, with consistent-mass and stiffness matrices, is studied and conditions for the occurrence of asynchronous modes are established. Next, a three-storey shear building is modelled, with lumped-mass matrix, and isolated asynchronous modes are not observed. However, non-isolated asynchronous modes are seen to exist, including a case in which the first floor is at rest. In this scenario, the upper floors play the role of vibration controllers, sparing the lower columns from oscillating. This might be welcome from the architectural viewpoint, so that pilotis would not be a priori ruled out in case of earthquake-borne vibration. Usually the participating masses of the asynchronous modes are low (and even null). To maximize the asynchronous-mode participating mass, a final model of the three-storey frame is discussed, this time taking into account the flexural stiffness of both the columns and the beams, although still using lumped-mass matrix. In this case, it is seen that much larger participating masses can be obtained for the asynchronous mode. Since asynchronous modes are associated with the vibration localization phenomenon, they can be potentially explored in the design of vibration controllers or, alternatively, energy harvesting systems.