Elastic waveguides with time-modulated stiffness feature a frequency-periodic dispersion spectrum, where branches merge at multiple integers of half the modulation frequency and over a finite wave number range. In this range, frequency becomes complex, with its real part remaining constant. The vanishing group velocity associated with these flat bands leads to frequency-selective reflection at an interface between a nonmodulated medium and a time-modulated one, which converts a broadband input into a narrow-band output centered at the half modulation frequency. This behavior is illustrated in an elastic waveguide in transverse motion, where modulation is implemented experimentally by an array of piezoelectric patches shunted through a negative electrical capacitance controlled by a switching circuit. The switching schedule defines the modulation frequency and allows the selection of the output frequency. This implementation is suitable for the investigation of numerous properties of time-space modulated elastic metamaterials, such as nonreciprocity and one-way propagation, and can lead to the implementation of novel functionalities for acoustic wave devices operating on piezoelectric substrates.

In this work, we experimentally achieve 1-kHz-wide directional band gaps for elastic waves spanning a frequency range from approximately 8 to 11 kHz. One-way propagation is induced by way of a periodic waveguide consisting of an aluminum beam partially covered by a tightly packed array of piezoelectric patches. The patches are connected to shunt circuits, where switches allow a periodic modulation in time of the beam properties. A traveling stiffness profile is imposed by phasing the modulation of each active element, inducing the propagation of a plane wave along the material and establishing unidirectional wave propagation at band-gap frequencies.

We present an enhanced version of the parametric nonlinear reduced-order model for shape imperfections in structural dynamics we studied in a previous work. In this model, the total displacement is split between the one due to the presence of a shape defect and the one due to the motion of the structure. This allows to expand the two fields independently using different bases. The defected geometry is described by some user-defined displacement fields which can be embedded in the strain formulation. This way, a polynomial function of both the defect field and actual displacement field provides the nonlinear internal elastic forces. The latter can be thus expressed using tensors, and owning the reduction in size of the model given by a Galerkin projection, high simulation speedups can be achieved. We show that the adopted deformation framework, exploiting Neumann expansion in the definition of the strains, leads to better accuracy as compared to the previous work. Two numerical examples of a clamped beam and a MEMS gyroscope finally demonstrate the benefits of the method in terms of speed and increased accuracy.

The computation of the steady-stateresponse of large finite element discretized systems subject to periodic excitations is unfeasible because of excessive run time and memory requirements. One could in principle resort to reduced order models stemming from the high fidelity counterparts, which typically require a solution time orders of magnitude smaller. However, when many simulations are required, as in the case of parametric studies, the overall effort could be still significant and the analysis process could be severely hindered. In this work, we propose a sensitivity approach to assess the influence of model parameters on the nonlinear dynamic response. As opposed to the costly evaluation of reduced order solutions over a range of excitation frequencies and model parameters, the sensitivities of a nominal response allow one to approximate the dynamic response by a simple evaluation of an expansion in the directions spanning the parameter space. Special care must be taken on the closure equation that needs to be appended to the system of equations stemming from the harmonic balance method. We discuss the limitations of the current constant frequency approach and propose an improvement. We demonstrate the merits of the proposed approach on a micro-electro-mechanical system affected by parameterized manufacturing defects. Leveraging from a previous contribution, the nonlinear response and the sensitivities are obtained from a reduced order model which is analytical in the defect parameters. Our procedure is able to deliver accurate probability density functions of quantities of interest (e.g. nonlinear resonance peaks, triple solution bandwidth, etc) against statistical distributions of manufacturing defects at negligible computational cost.