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We experimentally and theoretically study the dynamics of a one-dimensional array of pendula with a mild spatial gradient in their self-frequency and where neighboring pendula are connected with weak and alternating coupling. We map their dynamics to the topological Su-Schrieffer-Heeger (SSH) model of charged quantum particles on a lattice with alternating hopping rates in an external electric field. By directly tracking the dynamics of a wavepacket in the bulk of the lattice, we observe Bloch oscillations, Landau-Zener transitions, and coupling between the isospin (i.e. the inner wave function distribution within the unit cell) and the spatial degrees of freedom (the distribution between unit cells). We then use Bloch oscillations in the bulk to directly measure the non-trivial global topological phase winding and local geometric phase of the band. We measure an overall evolution of 3.1 $\pm$ 0.2 radians for the geometrical phase during the Bloch period, consistent with the expected Zak phase of $\pi$. Our results demonstrate the power of classical analogs of quantum models to directly observe the topological properties of the band structure, and sheds light on the similarities and the differences between quantum and classical topological effects., Comment: 17 pages, 4 figures

We introduce a method to design topological mechanical metamaterials that are not constrained by Newtonian dynamics. The unit cells in a mechanical lattice are subjected to active feedback forces that are processed through autonomous controllers, pre-programmed to generate the desired local response in real-time. As an example, we focus on the quantum Haldane model, which is a two-band system with directional complex coupling terms, violating Newton's third law. We demonstrate that the required topological phase, characterized by chiral edge modes, can be achieved in an analogous mechanical system only with closed-loop control. We then show that our approach enables to realize, for the first time, a modified version of the Haldane model in a mechanical metamaterial. Here, the complex-valued couplings are polarized in a way that modes on opposite edges of a lattice propagate in the same direction, and are balanced by counter-propagating diffusive bulk modes. The proposed method is general and flexible, and could be used to realize arbitrary lattice parameters, such as non-local or nonlinear couplings, time dependent potentials, non-Hermitian dynamics, and more, on a single platform., Comment: 4+ pages, plus supplemental material