Ultra-precision manufacturing (UPM) machines are designed to fabricate and measure complex parts having micrometer-level features and nanometer-level tolerances/surface finishes. Therefore, they must be isolated from deleterious effects of vibration to enable them to meet stringent precision requirements. UPM machine builders often prefer passive isolators for reducing vibration because they are easy to use, cost effective, energy neutral and reliable. A long-standing rule-of-thumb in passive isolation system design, recommended in academic literature and industrial practice, is to decouple all the vibration modes of an isolated machine by aligning the isolator mounting locations with the center of gravity (CG) of the machine. However, there is anecdotal evidence scattered in the literature that suggests that locating isolators such that vibration modes are coupled could help reduce vibrations of passively-isolated machines. What is lacking, however, is a proper understanding of when and how to use mode coupling to achieve superior vibration reduction.xiv This doctoral dissertation research seeks to provide a theoretical foundation as well as analysis-based design guidelines and tools for reducing unwanted vibration in passively-isolated UPM machines using mode coupling. Its primary contributions are threefold. Firstly, it uses eigenvalue and perturbation analyses on a single-variable, proportionally (or modally) damped, planar isolation system to demonstrate that the drastic reduction of vibration caused by mode coupling is primarily linked to so-called “critical configurations” induced by curve veering and mode localization. It therefore clears the misconception purported in academic literature and industrial practice that the vibration-reduction effects of mode coupling on UPM machines are simply linked to damping. Secondly, it proves analytically that mode coupling (with or without damping effects included, and provided that it is properly carried out) is almost always better than the recommended practice of modal decoupling with regard to vibration reduction in passively-isolated UPM machines. The dissertation therefore provides design guidelines for properly exploiting weak mode coupling for vibration reduction in UPM machines. Thirdly, it proposes a framework for reformulating the generalized (multivariable, 3-D) UPM isolator placement problem as a linear feedback controller design problem whose gains represent isolator locations. It thus provides a powerful engineering tool for using linear control theory, in all its wealth and elegance, for the optimization and analyses of passive isolator placement.

The theoretical work presented in this dissertation is backed up by simulations and experiments conducted on prototypes of UPM machines. The results demonstrate that, when properly exploited, mode coupling could bring about huge reductions in UPM machine vibration compared to modal decoupling;​ for example, up to 40% reduction in residual vibration and 50% reduction in transmissibility are demonstrated experimentally in Chapter 3. Even though this dissertation is presented in the context of UPM machines, the reader will discover that its methods and findings are applicable to the placement of passive isolators/suspensions/dampers in automotive, aerospace, civil, and other applications.