Date of Award

5-2011

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Legacy Department

Mechanical Engineering

Advisor

Daqaq, Mohammed M.

Committee Member

Li , Gang

Committee Member

Schiff , Scott

Committee Member

Vahidi , Ardalan

Abstract

During the last three decades, a considerable amount of research has been directed toward understanding the influence of time delays on the stability and stabilization
of dynamical systems. From a control perspective, these delays can either have a compounding and destabilizing effect, or can actually improve controllers'
performance. In the latter case, additional time delay is carefully and deliberately introduced into the feedback loop so as to augment inherent system delays and
produce larger damping for smaller control efforts. While delayed-feedback algorithms have been successfully implemented on discrete dynamical systems with
limited degrees of freedom, a critical issue appears in their implementation on systems consisting of a large number of degrees of freedom or on infinite-dimensional structures. The reason being that the presence of delay in the control loop renders the characteristic polynomial of the transcendental type which produces infinite number of eigenvalues for every discrete controller's gain and time delay. As a result, choosing a gain-delay combination that stabilizes the lower vibration modes can easily destabilize the higher modes. To address this problem, this dissertation introduces the concept of filter-augmented delayed-feedback control algorithms and applies it to mitigate vibrations of various structural systems both theoretically and experimentally. In specific, it explores the prospect of augmenting proper filters in the feedback loop to enhance the robustness of delayed-feedback controllers allowing them to simultaneously mitigate the response of different vibration modes using a single sensor and a single gain-delay actuator combination. The dissertation goes into delineating the influence of filter's dynamics (order and cut-off frequency) on the stability maps and damping contours clearly demonstrating the possibility of effectively reducing multi-modal oscillations of infinite-dimensional structures when proper filters are augmented in the feedback loop. Additionally, this research illustrates that filters may actually enhance the robustness of the controller to parameter's uncertainties at the expense of reducing the controller's effective damping.
To assess the performance of the proposed control algorithm, the dissertation presents three experimental case studies; two of which are on structures whose dynamics can be discretized into a system of linearly-uncoupled ordinary differential equations (ODEs); and the third on a structure whose dynamics can only be reduced into a set of linearly-coupled ODEs. The first case study utilizes a filter-augmented delayed-position feedback algorithm for flexural vibration mitigation and external disturbances rejection on a macro-cantilever Euler-Bernoulli beam. The second deals with implementing a filter-augmented delayed-velocity feedback algorithm for vibration mitigation and external disturbances rejection on a micro-cantilever sensor. The third implements a filter-augmented delayed-position feedback algorithm to suppress the coupled flexural-torsional oscillations of a cantilever beam with an asymmetric tip rigid body; a problem commonly seen in the vibrations of large wind turbine blades.
This research also fills an important gap in the open literature presented in the lack of studies addressing the response of delay systems to external resonant
excitations; a critical issue toward implementing delayed-feedback controllers to reduce oscillations resulting from persistent harmonic excitations. To that end,
this dissertation presents a modified multiple scaling approach to investigate primary resonances of a weakly-nonlinear second-order delay system with cubic
nonlinearities. In contrast to previous studies where the implementation is confined to the assumption of linear feedback with small control gains; this effort proposes
an approach which alleviates that assumption and permits treating a problem with arbitrarily large gains. The modified procedure lumps the delay state into unknown
linear damping and stiffness terms that are function of the gain and delay. These unknown functions are determined by enforcing the linear part of the steady-state
solution acquired via the Method of Multiple Scales to match that obtained directly by solving the forced linear problem. Through several examples, this research
examines the validity of the modified procedure by comparing its results to solutions obtained via a Harmonic Balance approach demonstrating the ability of the
proposed methodology to predict the amplitude, softening-hardening characteristics, and stability of the resulting steady-state responses.

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