The variety of important rigid-block-like elements that can be modelled as rigid blocks explains why several studies have been focused on protection strategies and devices able to reduce the amplitude of the oscillations during a rocking motion and prevent the overturning of these elements. In most of the cases, such studies proposed the use of external passive devices such as base isolation, while only a few studies proposed active protection systems. This study proposes the use of a device that combines two dynamic mass absorbers to enhance the protection of rigid-block-like elements against overturning. Such a device is obtained by connecting a dynamic mass absorber in the shape of a pendulum to a moving mass running on the top of the block that works as a second dynamic mass absorber. In turn, the moving mass is connected to the block by a visco-elastic device. The rigid block is assumed to be placed on a deformable surface modelled as a continuous layer of vertical visco-elastic elements that only react to compression. Additionally, the slide motion is assumed to be prevented. The equations of motion are obtained via a Lagrangian approach. The effectiveness of this protection device is analysed by comparing the overturning spectra of three similar systems: the stand-alone block, the block with only the pendulum dynamic mass absorber (i.e., connected at the top of the block in a fixed point), and the block with the combined dynamic mass absorber. Such overturning spectra provide the amplitude of the one-sine impulsive excitation able to overturn the block versus its circular frequency. The seismic effectiveness of the two combined mass dampers is investigated through rocking maps. These maps show contour plots of the maximum rocking angle in the parameter plane base length-slenderness of the block. The results show that the combination of the dynamic mass absorbers reduces the maximum rocking angle and the size of the overturning regions on the seismic maps with respect to those obtained with the stand-alone block and the block with the pendulum dynamic mass absorber.
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