Structural dynamics covers natural frequency, vibration response, and simple seismic checks before detailed earthquake analysis.

Scope (structural dynamics)

This page covers time-varying response in three linked ideas: natural frequency (SDOF), vibration response (harmonic magnification with damping), and seismic response (equivalent static base shear as a simple code-style check)—basics only for earthquake work. The handbook splits Chapter 3 (general dynamics) from Section IV (Ch. 17–21 earthquake engineering). Full treatment of Ch. 18–21 (damage, building and bridge seismic design, performance-based design) is on the Earthquake Engineering page. In South Africa use SANS 10160 for actions and hazard data from the brief.

Handbook alignment. Handbook of Structural Engineering (2nd ed., Chen & Lui, CRC Press, 2004)—Chapter 3 (structural dynamics: SDOF, damping, resonance) and Section IV earthquake context. This page delivers mechanics and simple seismic hand checks; Ch. 18–21 (damage, buildings, bridges, performance-based seismic design) are developed on Earthquake Engineering. Licensed reference; real seismic design uses site spectra and software—not these illustrations alone.

SDOF period check (concept)

  1. Idealise a mode or a dominant mass–spring path with stiffness k and mass m in consistent units.
  2. Compute natural frequency fn and period T = 1/fn; compare T to the spectrum of the loading (e.g. machine speed, gusts, or code period).
  3. If forcing frequency is close to fn, assess amplification with the harmonic magnification calculator (damping ζ).
  4. For earthquake, obtain Cs from code/spectrum and use the base shear calculator; use software for multi-modal and time-history work (Ch. 17–21).

Structural Engineering — Structural Dynamics

Natural frequency, vibration response (harmonic magnification), and seismic response (base shear)—Handbook Ch. 3; for Ch. 18–21 see Earthquake Engineering.

Introduction

Structural dynamics extends static equilibrium with inertia and, usually, damping. Chapter 3 of the handbook develops free and forced vibration, resonance, and SDOF idealisations. Section IV starts with Ch. 17 (earthquake fundamentals); Ch. 18–21 cover damage, building and bridge seismic design, and performance-based seismic design on the Earthquake Engineering page.

Free vibration is sinusoidal at the natural frequency; forced vibration (harmonic, transient, or random) produces amplification near resonance. Real structures use modal analysis; SDOF here is a teaching approximation.

Displacement versus time for simple harmonic motion
Undamped harmonic motion (displacement vs time). See licence on Wikimedia Commons
Displacement versus time with damping decay
Damped response: oscillations decay with time (concept). See licence on Wikimedia Commons

Handbook Chapter 3 and Section IV — scope (not a summary)

Chapter 3 is a full dynamics text (SDOF, MDOF, damping, spectra introduction). Section IV (Ch. 17–21) develops earthquake engineering for buildings and bridges. This page covers basic SDOF and a hand base shear illustration only.

  • Chapter 3 — Equations of motion, free and forced vibration, damping models, resonance, introduction to modal analysis and seismic response concepts.
  • Chapters 17–21 — Hazard, spectra, ductility, building and bridge seismic design, performance-based methods—see Earthquake Engineering.

What structural dynamics analysis produces

  • Natural frequencies and mode shapes (or equivalent period estimates) for comparison with forcing frequencies.
  • Time-history or spectral results for seismic or wind design when required by code or client.
  • Peak accelerations, displacements, and member forces for fatigue or comfort checks (floors, footbridges).
  • Documentation of damping assumptions and modelling choices (mass participation, boundary conditions).

Code and actions (South Africa)

SANS 10160 structures actions for buildings and industrial structures; seismic and other dynamic parts apply when relevant. Combine with material design standards (e.g. concrete, steel) for resistance checks. This page’s SDOF formula is illustrative; seismic design uses site-specific spectra and often multi-modal or time-history analysis.

Undamped SDOF:   ωn = √(k/m) ,   fn = ωn/(2π) ,   T = 1/fn

Notation (common)

  • m — mass (kg).
  • k — stiffness (N/m) for the SDOF idealisation.
  • ωn — natural circular frequency (rad/s).
  • fn — natural frequency (Hz); T — period (s).
  • ζ (zeta) — damping ratio (fraction of critical damping) in damped models.

Examples in practice

  • Floor vibration: walking or rhythmic activity near a natural frequency of the floor bay.
  • Earthquake: spectral acceleration vs period; modal combination or time history.
  • Wind: along-wind and cross-wind effects on slender structures or masts.
  • Machinery: rotating imbalance at a fixed forcing frequency.

Handbook-linked calculators

Three blocks aligned with the usual teaching path (Ch. 3 + seismic overview in Ch. 17–21):

  • Natural frequency — undamped SDOF fn and T from k and m.
  • Vibration response — steady-state harmonic magnification MF (damped) vs frequency ratio r = f/fn.
  • Seismic response — equivalent static base shear V = CsW (you supply Cs from code/spectrum).

Response spectra, modal combination, and performance-based procedures (Ch. 19–21) require software and project-specific hazard data.

Calculator — natural frequency (SDOF)

Handbook — Chapter 3

Undamped single-degree-of-freedom oscillator: enter k in N/m and m in kg. For distributed systems, equivalent k and m must come from a structural model.

fn = (1/2π) √(k/m) ,   T = 1/fn

Natural frequency

Undamped SDOF natural frequency fn and period T from stiffness k and mass m.

Key terms

Natural frequency
Cycles per second (Hz) at which an undamped system oscillates freely.
Period (T)
Time for one full oscillation; T = 1/f.

Calculator — vibration response (harmonic magnification)

Handbook — Chapter 3 (forced vibration / resonance)

Frequency ratio r = f / fn (forcing vs natural). Damped steady-state magnification MF = 1 / √[(1−r²)² + (2ζr)²]. Near r = 1 with low ζ, response grows—compare with the message in the output. Use for machinery or harmonic loading; earthquake design uses response spectra (Ch. 17–21), not this alone.

MF at resonance avoidance: compare f and fn from Calculator above (enter fn or compute from k, m first).

Magnification factor

Steady-state harmonic displacement magnification vs frequency ratio r = f/fn for a given damping ratio ζ.

Calculator — seismic response (equivalent static base shear)

Handbook — Section IV, Chapters 17–21 (conceptual link to lateral-force procedures)

V = Cs W: seismic coefficient × seismic weight (concept used in many codes). Enter Cs from your spectrum or code procedure—this block only multiplies. Full seismic design uses period-dependent spectra, ductility, and combinations per SANS 10160 and project geotechnical input.

V = Cs × W

Base shear

Equivalent static seismic base shear V = Cs × W from supplied coefficient and seismic weight.

Software and dynamic analysis

Multi-storey frames, bridges, and equipment supports need mass and stiffness models with appropriate boundary conditions and damping. Examples of ecosystems used for dynamic and seismic work (official sites):

No product endorsement—verify that analysis settings, damping, and combination rules match your jurisdiction and peer-review requirements. The SDOF calculator is not a substitute for code-compliant seismic or wind design.

Diagram sources

Educational schematics. Files in Images/structural-dynamics/ were downloaded from Wikimedia Commons into this repo (not copied from other topic folders). Confirm licence on each Commons file page before reuse.