On the Complex Non-Minimum Phase Zeros in Flexure Mechanisms

flexure mechanism;complex non-minimum phase zeros;dynamics;Mechanical Engineering;Engineering;Mechanical Engineering

Cui, LeqingWang, Kon-Well ;

University of Michigan

关键词: flexure mechanism;    complex non-minimum phase zeros;    dynamics;    Mechanical Engineering;    Engineering;    Mechanical Engineering;   

Others  :  https://deepblue.lib.umich.edu/bitstream/handle/2027.42/140940/leqing_1.pdf?sequence=1&isAllowed=y

瑞士|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

The objective of this thesis is to investigate the genesis and consequences of complex non-minimum phase (CNMP) zeros observed experimentally in the dynamics of large displacement flexure mechanisms. The knowledge gained through this research may be used to overcome traditional tradeoffs between large displacement and good dynamic performance in various applications that employ flexure mechanisms. Applications include high precision positioning stages for wafer inspection, motion stages for atomic force microscopy, and micro electro-mechanical systems for high speed scanning.In the first part of this study, a lumped parameter approach is proposed to model the CNMP behavior. It is shown that CNMP zeros appear due to the inherent geometric nonlinearities in the mechanics of a simple beam (i.e., arc length conservation) as well as due to small structural asymmetry (e.g., manufacturing tolerances) at certain operating points in the mechanism’s workspace. In addition, a hypothesis of a correlation between CNMP zeros and a phenomenon known as curve veering is proposed. This hypothesis is verified in later part of the thesis.Second, to understand the physics of CNMP zeros, a systematic investigation is presented, assuming a broader scope of lightly damped flexible systems. The analysis of modal residues based on modal theory is conducted to find the conditions for the existence of each possible type of zeros (i.e., marginally minimum phase, real non-minimum phase, and complex non-minimum phase). To study the transition between these types of zeros, a graphical technique is proposed that utilizes the sequences of the dominant modes and the sequences of each decomposed mode’s sign. It is shown that CNMP zeros appear in a system that is decomposed as two closely spaced modes and a remainder mode with alternating signs, and these modes dominate at their corresponding frequency regions. Furthermore, the impact of CNMP zeros on closed loop control, as well as the initial properties (e.g., initial undershoot) of the step response for a system with CNMP zeros are visualized via this technique. Third, the correlation between CNMP zeros and curve veering is examined. It is shown under certain conditions that CNMP zeros appear in the presence of curve veering. The physical insights of these conditions are presented. This finding reveals a conditional connection between the two fields (i.e., CNMP and curve veering) that have been separately studied in the literature.Next, an experimental setup is designed to validate the above findings. To conduct the validation, a design procedure is presented to minimize the extraneous factors. The frequency responses are then measured and match well with the model prediction.Lastly, a mechatronic system (including mechanical system and control system) design approach is presented based on the knowledge developed above, in order to achieve better closed loop performance in large motion range flexure stages compared to the state-of-the-art stages. The study is based on a simplified system considered in this thesis. It is shown that a system with a sequence of alternating signs is easier to stabilize compared to a system with non-alternating signs, even when the former one has CNMP zeros. Therefore, it is recommended to design the mechanical system by studying the sequence of decomposed modes’ signs (and the sequence of dominant modes) to achieve the best possible dynamic characteristics for good control performance. An example is presented with both simulation and experiment.