Home / Structural Dynamics

Structural Dynamics

Interactive, step-by-step calculators for graduate-level structural dynamics and FEM courses. Every formula is cited to its source textbook equation. Inputs are editable — change a value and the engine recomputes instantly.

📚 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
This page covers two separate university courses: Structural Dynamics (SDOF/MDOF vibration theory — top section) and the Finite Element Method (direct stiffness method, element formulations — bottom section, marked with FEM). They are distinct academic subjects with different theoretical foundations; they share this page only because FEM tools are frequently applied to dynamics problems.
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
Finite Element Method Sub-branch of Dynamics — element matrices · assembly · solution
FEM · 1D Bar ✓

1D Bar Element

Local stiffness k=EA/L · 2-element assembly · Boundary conditions · Displacement solution u=K⁻¹f

Cook §2.2 · Eq.(2.2-1)
FEM · 2D Truss ✓

2D Truss Element

Transformation matrix T · Global stiffness · Multi-element assembly · Nodal displacements & member forces

Cook §2.4 · Eq.(2.4-4)
FEM · Beam ✓

Euler-Bernoulli Beam Element

4×4 stiffness matrix · Hermite shape functions · Consistent mass matrix · Moment & shear diagram

Bathe §4.2 · Cook §4.2
FEM · 2D Plane ✓

CST Plane Element

Constant Strain Triangle · B matrix · Constitutive matrix D · Ke=t·A·BᵀDB · Plane stress/strain

Cook §6.2 · Eq.(6.2-5)(6.2-11)
FEM · Frame ✓

Plane Frame Analysis

Combined beam+bar DOF · Global assembly · Support conditions · Displacements, reactions, M/V diagrams

Cook §4.3 · Bathe §4.2
FEM · Dynamics ✓

FEM + Free Vibration

Beam K & consistent M → eigenvalue → ω, T, mode shapes · Comparison with analytical solution

Bathe §9.4 · Chopra + Cook
← All categories Steel Examples →