A single-degree-of-freedom (SDOF) system is an idealized dynamic system with one independent coordinate of motion, characterized by a single mass, spring, and damper. It is the fundamental building block of structural dynamics and the basis for the response spectrum concept.
A single-degree-of-freedom (SDOF) system is the simplest idealization of a dynamic structural system, consisting of a single mass (m) connected to a fixed base by a spring (stiffness k) and a damper (damping coefficient c). Its motion is described by a single coordinate — the displacement (u) of the mass relative to the base. The equation of motion for an SDOF system subjected to ground acceleration (ü_g) is: m·ü + c·u̇ + k·u = -m·ü_g, where ü and u̇ are the acceleration and velocity of the mass. Dividing by m yields the canonical form: ü + 2ζω·u̇ + ω²·u = -ü_g, where ω = √(k/m) is the natural frequency and ζ = c/(2√(km)) is the damping ratio. These two parameters — ω and ζ — completely define the dynamic behavior of an SDOF system.
The SDOF system is the foundation of earthquake engineering theory. Every response spectrum — whether acceleration, velocity, or displacement — is computed by subjecting a family of SDOF systems with varying periods and a fixed damping ratio to a ground motion and recording their peak responses. This is possible because any linear multi-degree-of-freedom (MDOF) structure can be decomposed into a set of independent SDOF systems through modal analysis: each mode behaves like an SDOF oscillator with its own natural frequency, damping, and effective mass. The total structural response is then obtained by superposing the modal responses. This decomposition is the basis of modal response spectrum analysis and is one of the most powerful tools in earthquake engineering.
Despite its simplicity, the SDOF system captures the essential physics of structural response to earthquakes. It demonstrates resonance (amplified response when the excitation frequency matches ω), the role of damping in limiting response, and the effects of period on the frequency content of structural response. It is used pedagogically to teach structural dynamics, computationally as a benchmark for validating more complex models, and practically in preliminary design and in code-based simplified procedures. However, the SDOF idealization has important limitations: it cannot represent higher-mode effects, torsional response, or the spatial distribution of mass and stiffness in real structures. For these reasons, MDOF models are used in detailed analysis, with the SDOF framework retained as the conceptual foundation. Nonlinear SDOF systems — with bilinear, hysteretic, or degrading stiffness — are also used extensively to study inelastic structural behavior and to derive inelastic response spectra, which form the basis for modern performance-based design.