Abstract

Materials with time-varying properties enable direction-dependent vibration transmission, meaning that interchanging the source and receiver changes transmission characteristics such as amplitude, phase, or wave speed, resulting in nonreciprocal behaviour. While unidirectional transmission in long, weakly modulated systems has been widely studied, the transmission characteristics of short, strongly modulated systems remain underexplored. This thesis addresses this gap, aiming to expand the application of materials and devices with time-varying mechanical properties. The focus is on discrete models of spatiotemporally modulated systems, where effective elasticity changes harmonically in time and space.
A methodology is developed to accurately predict the steady-state response of spatiotemporally modulated systems in response to external harmonic drive. The formulation is valid for strongly modulated systems of an arbitrary number of units. Using this methodology, vibration transmission characteristics of both weakly and strongly modulated systems are investigated. Contributions of primary and sideband resonances, and their overlaps, to nonreciprocity are clarified. The effects of modulation amplitude and wavenumber on the resonance frequencies are discussed.
The contribution of phase to nonreciprocity is highlighted, a feature that is often overlooked in the literature. It is shown that the difference between the transmitted phases is the primary contributor to nonreciprocity in short systems. To further emphasize the significant role of phase, a nonreciprocal response regime is introduced which is characterized by equal transmitted amplitudes in opposite directions. A nonreciprocal phase shift is the sole contributor to nonreciprocity in this case. A methodology is developed for achieving nonreciprocal phase shifts in short, weakly modulated systems based on the envelope of the response. A formulation is also presented that ensures the shapes of the transmitted response envelopes have the same shape but different phases.
Parametric stability is analyzed using Floquet theory, revealing the influence of key system parameters, including modulation phase, wavenumber, amplitude, frequency, damping, and system size. Perturbation theory shows that parametric instability occurs at specific frequency combinations of the unmodulated system. Instability is more likely at higher modulation frequencies, whereas lower modulation frequencies provide wide stable amplitude ranges. These insights enhance the design and safe operation of spatiotemporally modulated systems, potentially broadening their applications.