Multi-degree-of-freedom systems are crucial in earthquake engineering, offering a more accurate representation of complex structures. These systems feature multiple masses, stiffness elements, and damping components, allowing for the analysis of coupled motion and higher mode effects in buildings.

Matrix notation simplifies the representation of MDOF systems, using mass, stiffness, and damping matrices. The general equation of motion for MDOF systems incorporates displacement, velocity, and acceleration vectors, providing a comprehensive framework for analyzing structural responses to external forces.

Fundamentals of Multi-Degree-of-Freedom Systems

Extension to MDOF systems

• SDOF vs MDOF systems differ in degrees of freedom and motion complexity
• MDOF systems feature multiple masses, stiffness elements, and damping components
• Coupled motion in MDOF systems results in interdependent movement of different parts
• MDOF analysis in earthquake engineering provides more accurate representation of complex structures (high-rise buildings) and captures higher mode effects (torsional responses)

Matrix notation for equations of motion

• MDOF systems represented by mass [M], stiffness [K], and damping [C] matrices
• General MDOF equation: $[M]{\ddot{u}} + [C]{\dot{u}} + [K]{u} = {F(t)}$
• Equation components include displacement ${u}$, velocity ${\dot{u}}$, acceleration ${\ddot{u}}$, and external force ${F(t)}$ vectors
• Global matrices assembled from element matrices consider structural connectivity
• Boundary conditions incorporated into matrices reflect support conditions (fixed, pinned)

Analysis and Response of MDOF Systems

Natural frequencies and mode shapes

• Undamped free vibration eigenvalue problem: $([K] - \omega^2[M]){\phi} = {0}$
• Characteristic equation roots yield natural frequencies (eigenvalues)
• Natural frequencies relate to stiffness and mass distribution in structure
• Mode shapes (eigenvectors) represent vibration patterns at each natural frequency
• Orthogonality of mode shapes enables mathematical simplification
• Modal matrix compiles mode shapes as column vectors
• Eigenvalue problem solving methods:
  1. Determinant search iteratively finds roots
  2. Inverse iteration refines initial estimates
  3. Subspace iteration efficient for large systems

Response calculation using modal superposition

• Modal superposition decouples equations of motion for simplified analysis
• Modal participation factors quantify each mode's contribution to overall response
• Effective modal mass indicates significance of each mode in total structural mass
• Modal superposition analysis steps:
  1. Transform to modal coordinates
  2. Solve uncoupled modal equations
  3. Transform back to physical coordinates
• Modal combination methods (SRSS, CQC) combine individual mode responses
• Response spectrum analysis applies modal superposition to earthquake ground motion
• Modal superposition assumes linear elastic behavior and neglects some higher-order effects
• Direct integration methods provide alternative for nonlinear or highly damped systems