This chapter presents a comprehensive foundation for the mathematical modeling of mass–stiffness–damping systemsMass–stiffness–damping system (MKC systemsMKC system), which form the backbone of many real-world dynamic applications across mechanical, electrical, civil, and aerospace systemsAerospace system. Focusing on systems described by second-order ordinary differential equationsOrdinary Differential Equation (ODE), the chapter introduces the canonical MKC model and elaborates on its physical interpretation, structural propertiesStructural property, and practical relevance. Core modeling assumptions—such as symmetrySymmetry, definiteness, and energy conservationEnergy conservation—are used to align mathematical representationMathematical representation with real-world physical behavior. Emphasis is placed on lumped-parameter multibody systemsMultibody system, enabling intuitive yet rigorous modeling of structures like vehicle suspensions, robotic manipulators, rotating machineryRotating machinery, and large-scale civil infrastructures. A variety of modeling case studies are explored, including active vehicle suspension systems (quarter-, half-, and full-vehicleFull-vehicle models), single- and multi-link manipulators, and complex rotor-bearing systems. The chapter also addresses modeling techniques for distributed parameterDistributed parameter systems through finite-element discretizationDiscretization, highlighting their role in control design for large-scale flexible structures such as bridgesBridge and wind turbine towersWind turbine tower. The importance of simplification through model reductionModel reduction and validation is underscored, establishing this chapter as a crucial stepping stone for the control strategies introduced in subsequent chapters. It provides both theoretical rigor and practical relevance for engineers and researchers alike.