Frequency-domain design for non-conventional Active Vibration Control

Abstract

Vibrations deteriorate the performance of machines and instruments, especially when high precision and efficiency are required. Multiple approaches for vibration mitigation exist, mostly constituting separate bodies of research. This thesis establishes a connection between the active vibration control practice, advances in other control fields, and metamaterials research. To this end, three research gaps are addressed.

First, in Chapter 2, the design requirements of active vibration control are expressed in the frequency domain, using the loop-shaping approach commonly used in motion control. The use of the proposed approach is shown in the experimental evaluation of a vibration isolation system based on piezoelectric stack actuators.

Second, the loop-shaping approach is related to the design for bandgap in active metastructures. Chapter 3 adopts a modal analysis approach for finite metamaterial beams, relating the underlying control problem to the active damping of a single-degree-of-freedom system by assuming an infinite number of infinitesimally small transducer pairs distributed along a beam. This allows the application of design methods developed in the preceding chapter. The experiments demonstrate that controllers initially developed for damping resonance peaks can effectively induce bandgaps, even in structures featuring a small number of sparsely placed transducer pairs. Chapter 4 studies when the obtained models and approximations are accurate, highlighting the correlation between the minimal number of transducers required for model accuracy and the dominant vibration mode within the controller's targeted frequency range.

Third, the frequency-domain approach is applied for the design of fractional order and reset controllers for vibration mitigation to relax the limitations imposed using low-order linear controllers. In Chapter 5, a design for a fractional-order resonant element tailored for AVC, which preserves the characteristics of its integer-order counterpart but provides greater design freedom, is presented and evaluated in a simplified vibration isolation system. In Chapter 6, the same element is implemented within a unit cell of a granular metamaterial. For such a fractional-order metamaterial, both the dispersion characteristics of the infinite structure and the transmissibility of a finite chain are presented.

The use of nonlinear elements, like reset systems, poses additional challenges in vibration control. Since an exact frequency-domain representation of such elements does not exist, their behaviour is approximated using the describing functions. While this enables the loop-shaping design, the describing function approximation does not represent the system well in the presence of wide-band excitations and multiple resonance peaks in the plant. Chapter 7 explores how such conditions influence the reset elements and how to ensure that the use of reset is still beneficial. Additionally, assessing the stability of a reset system solely based on controller dynamics and experimentally measured plant frequency response is an open problem. To address this, the Negative Imaginary systems approach for stability analysis, originally developed for AVC of flexible systems with uncertain dynamics, is extended to reset systems in Chapter 8.