4.1 Single-degree-of-freedom systems

Earthquakes shake buildings, but how do we predict their effects? Single-degree-of-freedom (SDOF) systems simplify complex structures into manageable models. These systems use mass, stiffness, and damping to represent a building's key properties and analyze its response to ground motion.

SDOF analysis forms the foundation of earthquake engineering. By solving equations of motion and applying various solution methods, engineers can estimate a structure's behavior during seismic events. This knowledge helps design safer buildings and assess existing ones for earthquake resistance.

Single-Degree-of-Freedom Systems in Earthquake Engineering

Components of SDOF systems

• SDOF system characterizes structural behavior with single coordinate represents primary motion direction (lateral displacement)
• Mass (m) models inertial properties resists acceleration (building floors, bridge deck)
• Stiffness (k) represents system's ability to resist deformation returns to equilibrium (columns, beams)
• Damping (c) dissipates energy reduces vibration amplitude over time (structural connections, material properties)
• Simple pendulum exemplifies SDOF motion single point mass swings in arc
• Mass-spring-damper system demonstrates interplay of SDOF components
• Single-story building simplified as SDOF assumes rigid floor diaphragm

Equation of motion for ground motion

• Newton's Second Law $F = ma$ forms basis for deriving equation
• Inertial force $m\ddot{u}$ opposes acceleration
• Damping force $c\dot{u}$ opposes velocity
• Elastic restoring force $ku$ opposes displacement
• Ground motion force $-m\ddot{u}_g$ external excitation
• Equation balances forces $m\ddot{u} + c\dot{u} + ku = -m\ddot{u}_g$
• Relative motion $u = u_t - u_g$ separates total displacement from ground movement
• $u_t$ represents absolute displacement of mass
• $u_g$ denotes ground displacement caused by earthquake

Solution methods for SDOF systems

• Duhamel integral method solves linear systems using convolution
• $u(t) = \frac{1}{m\omega_d}\int_0^t p(\tau)e^{-\zeta\omega_n(t-\tau)}\sin[\omega_d(t-\tau)]d\tau$ gives displacement response
• Numerical integration techniques solve complex nonlinear systems
• Central Difference Method uses finite difference approximations
• Newmark's Method assumes acceleration variation within time step
• Runge-Kutta Method provides high-accuracy solution for ODEs
• Free vibration solution characterizes system's inherent dynamic properties
• Natural frequency $\omega_n = \sqrt{\frac{k}{m}}$ determines oscillation rate
• Damped natural frequency $\omega_d = \omega_n\sqrt{1-\zeta^2}$ accounts for damping effect
• Damping ratio $\zeta = \frac{c}{2m\omega_n}$ quantifies energy dissipation

SDOF response to ground excitations

• Harmonic excitation produces steady-state response amplified at resonance
• Earthquake excitation analyzed through time history or response spectrum methods
• Displacement response tracks structural deformation
• Velocity response indicates energy content
• Acceleration response relates to inertial forces
• Natural period influences response magnitude duration
• Damping ratio affects peak response energy dissipation
• Ground motion frequency content determines excitation effectiveness
• Peak response values assess maximum demands (drift, base shear)
• Ductility demand evaluates inelastic deformation capacity
• Energy dissipation quantifies system's ability to absorb seismic input