This book presents a rigorous yet practical framework for analyzing and optimizing the nonlinear dynamic behavior of mechanical systems, with applications spanning microelectromechanical systems (MEMS), robotics, and advanced structural components. Centered on Spectral Submanifold (SSM) theory, it shows how modern optimization techniques can be integrated to design systems that operate reliably under complex nonlinear dynamic conditions. The book begins by introducing the fundamentals of SSM theory and its role in modeling and understanding nonlinear vibrations. It then reviews key structural and topology optimization methods, such as density-based formulations and size optimization, highlighting their relevance for engineering design. Building on this foundation, the authors develop an SSM-based optimization framework for tailoring the nonlinear dynamic response of mechanical systems. Dedicated chapters address adjoint-based sensitivity analysis for frequency response and backbone curves, enabling efficient gradient-based optimization in large-scale design spaces, including topology optimization problems involving thousands of design variables. The proposed framework is illustrated through a series of parametric and topology optimization examples, including oscillator chains, resonators, and MEMS gyroscopes, demonstrating its scalability, versatility, and industrial relevance. Each chapter balances theoretical insight with practical implementation, guiding readers from foundational concepts to real-world applications. Aimed at graduate students, researchers, and practicing engineers in applied mechanics, nonlinear dynamics, and structural optimization, this book offers both a deep theoretical foundation and practical tools for modern mechanical system optimization.
