Modeling and design of bidirectional pendulum tuned mass dampers using axial or tangential homogeneous friction damping

As a development of the classical pendulum vibration absorber, bidirectional pendulum TMDs (BTMDs) have been recently proposed, capable to resonate with the main structure along both its horizontal directions by virtue of their optimally designed three-dimensional (3D) pendulum surface. To provide BTMDs with the required energy dissipation capability, two damping mechanisms based on respectively axial and tangential friction were invented as an alternative to ordinary viscous dashpots. The first one consists of a vertical axial-friction damper connecting the BTMD to the main structure. The second one consists of a tangential friction spatially variable along the pendulum surface in proportion to the modulus of the surface gradient vector. Both mechanisms are fundamentally characterized by a nonlinear but homogeneous first-order model which makes their effectiveness independent from the excitation level. This paper compares the two friction paradigms with the classical viscous one. To this purpose, first a unifying fully nonlinear 3D model is established through Lagrangian mechanics, then an optimal design method is proposed, based on either H∞ or H2 norm minimization criteria. Extensive numerical simulations are performed to show the pros and cons of the three damping options and of the two optimization approaches. Results demonstrate that the three types exhibit a similar performance against unidirectional excitation but that the axial-friction type loses most of its effectiveness under bidirectional excitation whenever the pendulum surface is axial- or nearly axial-symmetrical, because of the insurgence of a peculiar rotational motion which virtually deactivates the friction damper. Results also show that the H∞ design criterion is more robust than the H2 design criterion, and that both criteria outperform previous simplified approaches proposed in the literature. It is concluded that, once properly designed and until stroke demand does not exceed their intrinsic stroke limitations, BTMDs are an effective vibration control strategy, which can be implemented through a variety of damping options, and that the two homogeneous friction mechanisms, and particularly the tangential one, are promising paradigms to provide amplitude-independent damping to engineering pendular systems.