Structural Dynamics

Introduction

Structural dynamics studies how structures respond to time-varying loads—earthquakes, wind gusts, machinery, footsteps, and impacts. Unlike static design, the dynamic problem includes mass and damping, so forces and displacements depend on frequency content and system stiffness. Whether you are sizing a tuned mass damper in a tower, checking floor vibrations in an office, or determining seismic drifts in a hospital, dynamics links structural loads, analysis, wind design, and seismic design with a continuous load path to the foundation.

Dynamic performance = mass + stiffness + damping + input. Match the structure’s natural behavior to the hazard and occupancy needs.

Structural Dynamics Basics & Performance Goals

The goals of dynamic design are life safety and collapse prevention during extreme events, and comfort and functionality during everyday excitations. We quantify:

  • Natural frequencies & modes: The “preferred motions” of the system; resonance occurs when input frequencies align with these modes.
  • Damping: Mechanisms that dissipate energy (material, friction, devices), reducing peak response and settling time.
  • Demand characterization: Ground motion time histories, response spectra, wind spectra, impulse loads, or machinery forcing functions.
  • Acceptability criteria: Drift, acceleration, and strain limits for strength and serviceability; comfort criteria for floors and tall buildings.

When Dynamics Matter Most

Slender/tall buildings, long-span floors and roofs, equipment platforms, pedestrian bridges, nonstructural components (facades, ceilings), and structures on soft soils.

SDOF Fundamentals: The Building Block

Many dynamic problems can be understood with a single-degree-of-freedom (SDOF) model—a mass on a spring with a dashpot. The equation of motion under a forcing function \(f(t)\) is:

SDOF Equation of Motion

\( m\ddot{u}(t) + c\dot{u}(t) + k\,u(t) = f(t) \)
\(m\)Generalized mass
\(c\)Damping coefficient
\(k\)Stiffness

The natural circular frequency is \( \omega_n=\sqrt{k/m} \) and the period is \( T=2\pi/\omega_n \). Under ground motion \( \ddot{u}_g(t) \), the effective forcing is inertial: \( m\ddot{u}(t)+c\dot{u}(t)+k\,u(t)=-m\ddot{u}_g(t) \).

Damping, Resonance & Frequency Response

Damping ratio \( \zeta = c/(2\sqrt{km}) \) controls peak response and bandwidth. Small damping can lead to large resonant amplification when the forcing frequency \( \omega \) approaches \( \omega_n \).

Harmonic Amplification (Concept)

\( \text{Amp}(\omega) \propto \dfrac{1}{\sqrt{\left(1-(\omega/\omega_n)^2\right)^2 + \left(2\zeta\,\omega/\omega_n\right)^2}} \)
\(\zeta\)Damping ratio
\(\omega/\omega_n\)Frequency ratio

Did you know?

Doubling damping can reduce resonant accelerations dramatically with modest impact on strength design—but make sure the device provides energy dissipation in the right modes.

MDOF & Modal Analysis: Real Buildings

Real structures have many degrees of freedom. We compute mode shapes \( \phi_i \) and natural frequencies \( \omega_i \), then express response as a combination of modal contributions. This decouples the equations and clarifies which modes dominate (e.g., first sway, second sway, torsion).

Modal Superposition (Concept)

\( \mathbf{M}\ddot{\mathbf{u}} + \mathbf{C}\dot{\mathbf{u}} + \mathbf{K}\mathbf{u} = \mathbf{p}(t) \Rightarrow \sum q_i(t)\,\boldsymbol{\phi}_i \)
\(\mathbf{M},\mathbf{K}\)Mass & stiffness matrices
\(q_i(t)\)Generalized modal coordinates

Modal participation factors indicate how strongly each mode participates for a given excitation. For design, we often combine modal results using square-root-of-sum-of-squares (SRSS) or complete quadratic combination (CQC) when modes are closely spaced.

Response Spectra & Seismic Design

In seismic design, ground motions are represented by elastic response spectra that plot maximum SDOF response (acceleration, velocity, displacement) versus period for a reference damping (often 5%). Structures with period \( T \) “read off” spectral accelerations \( S_a(T) \) to estimate base shear and story forces before reductions for ductility.

Conceptual Base Shear

\( V \propto \dfrac{S_a(T)\,W}{R/I_e} \)
\(W\)Seismic weight (mass)
\(R\)Response modification (ductility)
\(I_e\)Importance factor

For authoritative seismic hazard data and spectra resources, start with USGS Earthquake Hazards and code entry points at ASCE and ICC. Then implement system-specific detailing covered in our seismic design guide.

Wind-Induced Response & Occupant Comfort

Wind loads are dynamic and can excite along-wind buffeting and across-wind vortex shedding. For tall, slender buildings, accelerations—not strength—often govern. Comfort criteria are given as RMS accelerations or peak values for various occupancies. Aerodynamic shaping, structural stiffness, and supplemental damping reduce motion.

Practical Levers

Add stiffness with cores/braces, tweak mass distribution, introduce viscous dampers or tuned mass dampers, and coordinate facade flexibility to accommodate drift and interstory movement. See our wind design overview.

Floor Vibration & Serviceability

Human perception and equipment sensitivity can limit floor spans more than strength. Rhythmic walking, fitness areas, labs, and offices have different vibration limits. We evaluate frequency, damping, and response to footfall or machine forcing; acceptable ranges may be defined by acceleration thresholds or response factors.

Footfall & Resonance (Concept)

\( a_\text{peak} \propto \dfrac{F_0}{m}\, \text{Amp}(\omega/\omega_n,\zeta) \)
\(F_0\)Footfall forcing amplitude
\(m,\zeta\)Modal mass & damping

For long-span floors, consider composite action, tuned stiffness, and damping measures (nonstructural partitions, ceiling systems). Coordinate with steel, concrete, and timber choices during early layout.

Soil–Structure Interaction (SSI)

Foundations and surrounding soils modify dynamic response by adding flexibility and damping. Ignoring SSI can over- or under-predict drifts and forces. Use springs and dashpots calibrated to geotechnical data for rocking, sliding, and radiation damping. Coordinate with foundation design to ensure anchors, piles, and mats accommodate dynamic demands.

Design Tips

Include rotational and translational springs at bases; check uplift and sliding under dynamic combinations; and evaluate kinematic interaction for embedded foundations where appropriate.

Modeling Workflow: From Concept to Verification

  1. Define inputs: Select seismic and wind demands consistent with adopted standards (see ASCE, ICC, and USGS).
  2. Establish mass model: Include superimposed dead loads and nonstructural masses that move with the structure; define torsional mass distribution.
  3. Pick analysis method: ELF/response spectrum for regular buildings; modal time history or nonlinear procedures for irregular/performance targets (see seismic design).
  4. Model stiffness & damping: Choose cracked section properties where appropriate; use realistic damping ratios and consider modal damping.
  5. Check results: Base shears vs. code minima, mode shapes vs. expectations, energy balance, and reaction sums vs. applied demands.
  6. Detail to match model: Provide collectors, chords, and connection ductility to realize the assumed load path; coordinate with load path analysis.
  7. Document & inspect: Put drift/acceleration criteria, diaphragm assumptions, and damping devices on drawings; plan special inspections.

Important

Model and detail the same reality. If diaphragms are assumed semi-rigid in the model, provide nailing/attachments that deliver that stiffness; if bases are flexible, show it in anchorage and foundation springs.

Response Control: Tuned Mass & Damping Devices

When stiffness or mass changes are impractical, supplemental damping reduces motion efficiently. Tuned mass dampers (TMDs) target a specific mode; viscous/hysteretic dampers dissipate energy across a broader band. For seismic applications, buckling-restrained braces and added damping/steel links provide stable hysteresis.

TMD Tuning (Concept)

\( \mu = \dfrac{m_d}{m_\text{eff}},\ \ \omega_d \approx \omega_\text{target},\ \ \zeta_d \rightarrow \text{maximize reduction in } a_\text{rms} \)
\(m_d\)Damper mass
\(m_\text{eff}\)Modal effective mass

Always verify maintenance access, operability, and fail-safe behavior of devices. Document acceptance criteria for drift and acceleration improvements.

Testing, Commissioning & Structural Health Monitoring

Field testing validates assumptions and informs retrofits. Ambient vibration testing (operational modal analysis) extracts frequencies and mode shapes without controlled excitation. Long-term monitoring with accelerometers and strain gauges supports performance-based operation and post-event assessments.

Commissioning Checklist

Measure as-built frequencies and damping; confirm device tuning; benchmark floor accelerations; and create a baseline for future comparisons after windstorms or earthquakes.

Codes, Standards & Trusted References

Dynamic design draws on building codes and consensus standards. While editions vary by jurisdiction, these stable homepages are authoritative starting points:

  • ASCE: Minimum design loads and hazard criteria. Visit asce.org.
  • ICC: International Building Code resources. Visit iccsafe.org.
  • USGS: Seismic hazard data & maps. Visit usgs.gov.
  • NIST: Structural research and guidance on performance and resilience. Visit nist.gov.
  • FEMA Building Science: Seismic and wind mitigation resources. Visit fema.gov.

For topic integration, see our guides on wind design, seismic design, and foundational structural analysis.

Frequently Asked Questions

When should I move beyond response spectra?

Use modal or nonlinear response history when the building is irregular, higher-mode/torsional effects are pronounced, or when performance-based objectives (e.g., immediate occupancy) govern.

What damping ratio should I assume?

5% is a common baseline for elastic analysis, but actual damping depends on materials, nonstructural elements, and devices. For serviceability, measured or literature-based values provide better predictions.

Do partitions and ceilings help floor vibrations?

Yes—nonstructural components add damping and sometimes stiffness. Include reasonable allowances when supported by testing or accepted guidance.

How do I treat equipment-induced vibrations?

Identify forcing frequencies, avoid resonance with support modes, and add isolation mounts or tuned devices as needed. Verify anchor and inspection details.

Key Takeaways & Next Steps

Structural dynamics connects how a structure is shaped and detailed to how it moves under time-varying loads. Start with credible hazard definitions, build mass–stiffness–damping models that reflect reality, and verify against hand checks and field data. Use modal insight to target the right levers—stiffness, mass, and damping—and ensure the load path with collectors, chords, and anchorage is continuous to the foundations.

Continue with our focused pages on wind design, seismic design, and material-specific behavior in steel, concrete, and timber. With thoughtful modeling, damping strategies, and inspections, your projects can be both safe and comfortable—performing as intended from everyday vibrations to rare earthquakes.

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