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Structural Dynamics

Interactive, step-by-step calculators for a graduate-level structural dynamics course: SDOF and MDOF vibration theory, response spectra, numerical integration and modal analysis. Every formula is cited to its source textbook equation.

📚 References: Chopra (2012) Dynamics of Structures 4th ed. · Bathe (2014) Finite Element Procedures 2nd ed. · Cook (2001) Concepts and Applications of FEA 4th ed. · Clough & Penzien · Paz (2004) Structural Dynamics
Related course This course pairs with Finite Element Analysis — the numerical method frequently applied to model dynamic systems. The Finite Element Method is a distinct academic subject with its own theoretical foundations, not a sub-branch of Structural Dynamics.  Finite Element Analysis →
SDOF Systems Single Degree of Freedom
SDOF · Free Vibration ✓

Free Vibration

ωn, Tn, ccr, ζ · Underdamped / critical / overdamped · Response time history plot

Chopra §2.1–2.2 · Eq.(2.2)(2.9)(2.12)
SDOF · Forced Vibration ✓

Forced Harmonic Vibration

F(t)=F₀sinΩt · DMF = f(β,ζ) · Resonance detection · Phase angle · Response plot

Chopra §3.2 · Eq.(3.9)
SDOF · Numerical Integration ✓

Newmark-β Integration

Average acceleration (β=¼,γ=½) · Step-by-step time history · Unconditionally stable · Any loading

Chopra §5.2 · Newmark (1959)
SDOF · Duhamel ✓

Duhamel Integral

General loading · Convolution integral · Numerical evaluation · Comparison with Newmark

Chopra §4.1 · Eq.(4.4)
SDOF · Spectrum ✓

Response Spectrum

Sd, Sv, Sa from TBDY 2018 design spectrum · Sa(T,ζ) → SDOF peak response · Deformation spectrum

Chopra §6.1 · TBDY 2018 Md.2.3
MDOF Systems Multi Degree of Freedom — Modal Analysis
MDOF · 2DOF ✓

2-DOF Free Vibration

Eigenvalue problem [K−ω²M]{φ}=0 · Natural frequencies ω₁,ω₂ · Mode shapes φ₁,φ₂ · T₁,T₂

Chopra §10.2 · Eq.(10.5)
MDOF · Modal Superposition ✓

Modal Analysis + SRSS/CQC

n-story shear frame · Modal participation · Peak modal responses · SRSS and CQC combination

Chopra §12–13 · TBDY 2018 Md.4.8
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