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The book consists of two main parts: structural synthesis methods for a precision elastic system, including effective approximations; and the application of precision functional elastic systems at reference and operating conditions. Each part provides theoretical basics and a large variety of examples of application and recommendations for parametric and structural optimization. A handbook as well as a textbook, it gives theoretical and practical tools to researchers, instrument system designers, engineers, metrologists, and also to students of college engineering courses. Special consideration is dedicated to the theory and applications of flexible helicoids, notch flexure hinges, and perforated plates whose methods of structural synthesis need development.This book comprehensively reflects the summary of theoretical and experimental studies whose conclusions are effective for optimized structural synthesis in precision elasticity, as well as demonstrates a large experience and options in the synthesis and application of precision elastic guides, mechanisms, correctors, transducers, elastic helicoids, and sensors in instruments and machines including their miniaturization for atomic force microscopy and other advanced solutions. The main focus of this book is in the possible simplification of the corresponding analytical apparatus. This tractable and digestible approach allows for increase of accuracy, sensitivity, responsiveness, and functional ranges in the synthesized precision elastic systems.