Structural Dynamics: Natural Frequency, Resonance, and Damping
A structure under a static load deforms, reaches equilibrium, and stays there. A structure under a dynamic load — one that varies in magnitude or direction with time — vibrates. The amplitude of that vibration depends not only on the magnitude of the applied force but on the ratio of the forcing frequency to the natural frequency of the structure itself. When these frequencies align, resonance occurs, and the structural response can be many times larger than what a static analysis of the same peak force would predict. Understanding this relationship is the foundation of seismic engineering, wind engineering, and the vibration serviceability of floors and footbridges.
Natural Frequency and Period
Every structure has one or more natural frequencies at which it vibrates freely after an initial disturbance, without continuing external force. The lowest of these is the fundamental (first-mode) natural frequency. For a single-degree-of-freedom (SDOF) system — a lumped mass m attached to a spring of stiffness k and a dashpot providing damping — the natural angular frequency is:
ωn = √(k/m) (radians per second)
The natural frequency in hertz and the fundamental period are:
fn = ωn / (2π) T = 1 / fn = 2π√(m/k)
These relationships identify the two levers available to the engineer. Increasing stiffness k raises the natural frequency and shortens the period. Increasing mass m lowers the natural frequency and lengthens the period. A stiff concrete shear wall building 10 stories tall might have a fundamental period near 0.3 seconds; a flexible 40-story steel moment frame might reach 4 seconds or more. ASCE 7 provides approximate period formulas, such as Ta = Ct hnx, calibrated to measured building periods, for use in preliminary seismic design.
Resonance and Dynamic Amplification
Resonance occurs when the forcing frequency of an applied load matches the natural frequency of the structure. Under this condition, energy is added to the vibrating system every cycle at exactly the rate most effective for building amplitude. In the absence of damping, the response amplitude would grow without bound. In real structures, damping limits the amplitude, but resonance still produces response many times larger than the equivalent static case.
The dynamic amplification factor (DAF) — also called the magnification factor — relates the steady-state dynamic displacement amplitude to the static deflection produced by the same peak force:
DAF = 1 / √[(1 − β2)2 + (2ζβ)2]
where β = ω / ωn is the frequency ratio and ζ is the damping ratio. At resonance (β = 1) with 5% damping (ζ = 0.05, typical for reinforced concrete): DAF = 1/(2 × 0.05) = 10. The structure sees ten times the displacement it would experience under the static application of the same peak force. This is why periodic excitation sources — rotating machinery, rhythmic crowd motion, passing vehicles — must always be checked against the building natural frequency before installation.
Damping
Damping is the mechanism by which vibration energy is dissipated rather than stored in the system. In civil structures, damping arises from several sources acting simultaneously: internal material damping (hysteretic behavior of concrete and yielding steel), friction at connections and between non-structural elements, and aerodynamic damping from the surrounding air. The aggregate effect is expressed as a fraction of critical damping ζ.
| Structure Type | Typical Damping Ratio ζ |
|---|---|
| Welded steel frame | 0.01 – 0.02 |
| Reinforced concrete frame | 0.03 – 0.05 |
| Masonry structures | 0.05 – 0.07 |
| Timber structures | 0.05 – 0.10 |
When inherent damping is insufficient to control resonant response — as in long-span footbridges and tall slender towers — supplemental damping can be added. Tuned mass dampers (TMDs) attach a secondary mass-spring system to the structure, tuned to the building’s fundamental frequency and designed to vibrate 90 degrees out of phase with the primary structure. The TMD absorbs energy from the main structure, substantially reducing resonant amplitudes without adding stiffness. The Taipei 101 skyscraper employs a 660-metric-ton pendulum TMD suspended from the 92nd floor; without it, wind-induced accelerations at occupied floors would exceed human comfort thresholds during typhoon events.
Multi-Degree-of-Freedom Systems and Mode Shapes
Real buildings have multiple degrees of freedom — each floor can move independently in translation and rotation. An N-story building has N natural frequencies and N corresponding mode shapes. The first mode (lowest frequency, longest period) typically shows all floors displacing in the same direction with amplitude that grows toward the top, similar to a cantilever vibrating in its first bending mode. Higher modes show floors moving in opposing directions; they contribute less to base shear because their effective mass participation is smaller.
Seismic codes use response spectrum analysis to determine peak modal forces. The response spectrum relates peak spectral acceleration to period for a given site and hazard level. Peak forces in each mode are combined using SRSS (square root of sum of squares) for well-separated modes or CQC (complete quadratic combination) for closely spaced modes. The result is the probable maximum total response without requiring a full time-history analysis.
Floor vibration from walking is a serviceability problem, not a strength problem. Human perception threshold for vertical floor acceleration is roughly 0.5% g for office environments and up to 1.5% g for walking paths. Long-span composite floors with low mass and low inherent damping can fall below these thresholds without any structural inadequacy. AISC Design Guide 11 provides a practical procedure for checking walking-induced vibration using natural frequency and modal mass.
Understanding natural frequency and damping is not an academic exercise reserved for high-rise work. It is the basis for checking whether a footbridge will resonate with jogging pedestrians, whether a gymnasium floor will bounce under rhythmic exercise, whether a mechanical equipment pad will transmit vibration to occupied spaces, and whether wind will drive a slender tower into cross-wind resonance — conditions where static analysis alone gives no warning of trouble.