Structural Dynamics: Vibration Analysis and Dynamic Response
Every structure vibrates. Wind gusts make skyscrapers sway gently. Footsteps cause floors to bounce. Earthquakes shake buildings violently. Structural dynamics is the field that studies how structures respond to time-varying loads, ensuring that vibrations stay within acceptable limits and do not lead to failure.
The Tacoma Narrows Bridge collapse in 1940 is the most famous example of what happens when structural dynamics is misunderstood. The bridge’s torsional vibrations grew until the deck twisted apart. The lesson was clear: static analysis alone is insufficient for structures that must resist dynamic loads.
Single-Degree-of-Freedom Systems
The simplest dynamic model is a single-degree-of-freedom (SDOF) system: a mass supported by a spring and a damper. This model captures the essential physics of structural vibration.
Equation of Motion
The equation of motion for an SDOF system under forced vibration is:
m × u-double-dot + c × u-dot + k × u = F(t)
where m is mass, u-double-dot is acceleration, c is damping coefficient, u-dot is velocity, k is stiffness, u is displacement, and F(t) is the time-varying force.
The natural circular frequency is ωn = (k/m)^0.5. The natural frequency in cycles per second is fn = ωn/2π. The natural period is Tn = 1/fn. A typical 10-story office building has a natural period of approximately 1.0 to 1.5 seconds. A 50-story skyscraper has a natural period of 4 to 6 seconds. The period increases with building height approximately as Tn = 0.1N for steel frames, where N is the number of stories.
Damping
Damping is the process by which vibrational energy is dissipated. In structures, damping comes from various sources: friction at connections, cracking of non-structural elements, soil-structure interaction, and material hysteretic behavior.
The critical damping ratio ccr = 2(mk)^0.5. Actual damping in structures is expressed as a fraction of critical damping. Typical values are 0.5 to 1 percent for steel buildings in the elastic range, 2 to 5 percent for concrete buildings, and 5 to 10 percent for base-isolated structures.
The damping ratio has a dramatic effect on resonant response. At resonance, the dynamic amplification factor is approximately 1/(2ζ). With 1 percent damping, the amplification is 50 — a small harmonic force can produce large displacements.
Resonance
Resonance occurs when the forcing frequency matches the natural frequency of the structure. The response grows cycle by cycle until limited by damping. A pedestrian walking at 2 Hz can excite a footbridge with a natural frequency near 2 Hz, potentially causing unsafe vibrations.
The London Millennium Bridge famously experienced significant lateral vibration on opening day because pedestrians walking in step excited the bridge’s lateral natural frequency. Retrofit with dampers solved the problem.
Multi-Degree-of-Freedom Systems
Real structures require multi-degree-of-freedom (MDOF) models. A frame building with N stories can be modeled as N masses connected by interstory stiffness.
Modal Analysis
Modal analysis decomposes the response of an MDOF system into a set of independent SDOF responses called modes. Each mode has a natural frequency, a damping ratio, and a mode shape that describes the deformed pattern of the structure at that frequency.
The fundamental mode of a shear building has all masses moving in the same direction, with displacement increasing from bottom to top. The second mode has the lower half moving in one direction and the upper half in the opposite direction, with a node point at mid-height.
The number of modes required for accurate response analysis depends on the excitation and the structure. For earthquake response, the first three to five modes typically capture 90 percent of the building’s response. For wind response of tall buildings, the fundamental mode often dominates.
Modal Superposition
The total response is the sum of contributions from each mode. The modal participation factor measures how strongly each mode is excited by the applied loading. The effective modal mass — the fraction of total mass participating in a mode — indicates which modes matter.
The sum of effective modal masses from all modes equals the total mass of the structure. Typically, the fundamental mode captures 60 to 80 percent of the total mass participation. Higher modes capture the remainder.
Response Spectrum Analysis
Response spectrum analysis is the standard method for seismic design of buildings. The response spectrum shows the maximum acceleration, velocity, or displacement of an SDOF oscillator as a function of its natural period for a given ground motion.
For each significant mode, the engineer reads the spectral acceleration from the design spectrum, computes the modal force, and combines modal contributions using the square root of the sum of squares (SRSS) or complete quadratic combination (CQC) methods for closely spaced modes.
Response spectrum analysis is efficient — it provides the maximum response without requiring time-history analysis. However, it loses phase information and does not directly compute response time histories.
Time-History Analysis
Time-history analysis directly integrates the equations of motion in the time domain. The structure is subjected to ground acceleration records from actual earthquakes or synthetic motions. The analysis produces displacement, velocity, and acceleration time histories at each degree of freedom.
Nonlinear time-history analysis accounts for material yielding, cracking, and geometric nonlinearity. It is required for performance-based design and for verification of structures with damping devices, base isolation, or other energy dissipation systems.
The choice of ground motion records significantly affects the results. Building codes require a minimum of three to seven pairs of horizontal ground motion components scaled to match the design spectrum.
Dynamic Soil-Structure Interaction
Soil flexibility affects the dynamic response of structures. The soil allows the foundation to rock and sway, increasing the natural period compared to a fixed-base structure. This period lengthening reduces spectral accelerations in most cases.
However, soil-structure interaction also increases damping through radiation damping — energy that radiates away from the foundation into the soil. The net effect is usually beneficial for seismic response, but the analysis is complex and requires coupled soil-structure models.
Vibration Control
When structural vibrations exceed acceptable limits, control measures are required. Tuned mass dampers are devices consisting of a mass, spring, and damper attached to the structure. The damper is tuned to the natural frequency of the primary structure, absorbing vibrational energy. Taipei 101 uses a 660-metric-ton tuned mass damper suspended from the 87th to the 92nd floor to control wind-induced motion.
Viscoelastic dampers dissipate energy through shear deformation of viscoelastic material. They are installed in building frames as braces or wall panels. Buckling-restrained braces yield in both tension and compression, providing energy dissipation while maintaining lateral stiffness.
Human-Induced Vibrations
Floor vibrations from human activity — walking, dancing, aerobic exercise — must be evaluated for serviceability. The AISC Design Guide 11 provides acceptance criteria based on peak acceleration. Floors with natural frequencies below 3 Hz are most susceptible to walking excitation.
Heel-drop tests and walking path analyses evaluate existing floors. Remedies include stiffening the floor, adding damping, or modifying the activity causing the vibration.
Wind-Induced Vibrations
Tall buildings respond to wind dynamically. Vortex shedding occurs when wind passing around a building creates alternating low-pressure zones on the leeward side. The shedding frequency depends on wind speed and building width. When the shedding frequency matches the building natural frequency, resonance can occur.
Aerodynamic shaping — chamfered corners, tapered profiles, slots — disrupts vortex formation. The Burj Khalifa’s stepped, asymmetrical profile was designed specifically to avoid organized vortex shedding.
Dynamic Testing of Structures
Forced vibration testing uses shakers or eccentric mass vibrators to excite structures at controlled frequencies. The response is measured with accelerometers to identify natural frequencies, mode shapes, and damping ratios. Ambient vibration testing uses wind, traffic, or microtremors as excitation and extracts modal properties using output-only techniques like frequency domain decomposition.
Modal testing is standard for validating analytical models. Measured natural frequencies from a new building are typically within 5 percent of analytical predictions for the fundamental mode but can deviate by 10 to 20 percent for higher modes. Model updating adjusts analytical parameters to match test results, improving the accuracy of response predictions.
Health monitoring uses continuous vibration measurements to detect structural damage. Changes in natural frequency indicate stiffness loss from cracking or yielding. Changes in mode shapes indicate localized damage. The Z24 Bridge in Switzerland was monitored for eight months, and controlled damage at the pier was detectable through modal parameter changes.
Frequently Asked Questions
What is the difference between static and dynamic analysis? Static analysis assumes loads are applied slowly and remain constant. Dynamic analysis accounts for time-varying loads and the resulting inertial forces, which can amplify the response significantly.
How is damping measured in existing structures? Damping is measured through ambient vibration testing, forced vibration testing, or free vibration decay tests. The logarithmic decrement method uses the decay rate of free vibrations to calculate the damping ratio.
What is a mode shape? A mode shape is the pattern of displacement that a structure assumes when vibrating at a particular natural frequency. Each mode has a characteristic shape independent of the excitation magnitude.
Can structural dynamics be ignored for low-rise buildings? Low-rise buildings with high natural frequencies (short periods) may have low dynamic amplification from earthquakes or wind. However, dynamic effects from rotating machinery, human activity, or blast loading may still be significant.
Earthquake Engineering — Structural Analysis Basics — Bridge Engineering Guide