Beginner — Fundamentals of Vibration Theory
Start here if you are new to structural dynamics.
Single Degree of Freedom (SDOF) Systems
The SDOF system is the fundamental building block of structural dynamics. It consists of a mass, a spring, and a damper — representing any structure whose motion can be described by a single displacement coordinate. The equation of motion is m·ü + c·u̇ + k·u = F(t), where m is mass, c is damping coefficient, k is stiffness, and F(t) is the applied force. Understanding SDOF behavior is essential for predicting how structures respond to dynamic loads such as wind, earthquakes, and machinery vibration.
The natural frequency of an undamped SDOF system is ω_n = √(k/m), and the natural period is T_n = 2π/ω_n. These intrinsic properties determine how the system responds to dynamic excitation. When the forcing frequency approaches the natural frequency, resonance occurs — causing large amplitude oscillations that can lead to structural failure. The damping ratio ζ = c/(2√(km)) quantifies energy dissipation, with ζ < 1 indicating underdamped systems (most civil structures), ζ = 1 critically damped, and ζ > 1 overdamped.
Free and Forced Vibration
Free vibration occurs when a structure is displaced from equilibrium and released, oscillating at its natural frequency while amplitude decays due to damping. The logarithmic decrement δ = ln(u_n/u_{n+1}) = 2πζ/√(1-ζ²) measures the rate of amplitude decay and provides an experimental method for determining the damping ratio from a free vibration test. Forced vibration occurs when a time-varying load is applied — harmonic, periodic, impulsive, or general dynamic loading.
Under harmonic excitation F(t) = F₀ sin(ωt), the steady-state response amplitude depends on the frequency ratio r = ω/ω_n and the damping ratio. The dynamic amplification factor D = 1/√((1-r²)² + (2ζr)²) quantifies how much the dynamic displacement exceeds the static displacement. At resonance (r ≈ 1), the amplification is limited only by damping — D_max ≈ 1/(2ζ). This explains why damping is critical for controlling vibration amplitudes in structures near resonance.
Damping Mechanisms and Models
Damping in structures arises from multiple mechanisms: viscous damping (material velocity-dependent), hysteretic damping (internal material friction), Coulomb friction (sliding at connections and interfaces), and radiation damping (energy propagating away through the ground). Viscous damping is the most commonly assumed model because it yields linear equations of motion. Equivalent viscous damping approximates all damping mechanisms as an equivalent viscous damping ratio.
Typical damping ratios for civil structures: welded steel buildings 1-2%, bolted steel buildings 2-3%, reinforced concrete buildings 3-5%, prestressed concrete 2-3%, and bridges 1-3%. Damping increases with response amplitude (amplitude-dependent damping). In practice, Rayleigh damping (proportional damping) C = αM + βK is often used for multi-degree of freedom systems, where mass-proportional damping α dominates low-frequency response and stiffness-proportional damping β dominates high-frequency response.
Intermediate — MDOF Systems and Modal Analysis
Build on fundamentals with multi-degree of freedom analysis.
Multi-Degree of Freedom (MDOF) Systems
Real structures have many degrees of freedom. An MDOF system is described by the matrix equation of motion: [M]{ü} + [C]{u̇} + [K]{u} = {F(t)}. The mass matrix [M], damping matrix [C], and stiffness matrix [K] capture the spatial distribution of properties. For a shear building model (lumped masses at floor levels), the matrices are tri-diagonal, making them computationally efficient to solve. The number of degrees of freedom equals the number of independent displacement coordinates needed to describe the structural configuration.
MDOF systems have multiple natural frequencies and mode shapes. The eigenvalue problem [K - ω²M]{φ} = {0} yields n natural frequencies ω_j and n mode shape vectors {φ}_j. Each mode shape describes the relative displacement pattern when the structure vibrates at that natural frequency. The fundamental (lowest) mode typically dominates the dynamic response for most loading conditions. Higher modes become significant for structures with irregular geometry or for loads with high-frequency content.
Modal Superposition Method
Modal superposition is the most powerful technique for linear dynamic analysis. The physical displacements {u} are expressed as a linear combination of mode shapes: {u} = Σ_{j=1}^{n} {φ}_j q_j(t), where q_j(t) are generalized (modal) coordinates. Substituting into the MDOF equation and using orthogonality properties uncouples the system into n independent SDOF equations: M_j·q̈_j + C_j·q̇_j + K_j·q_j = P_j(t). Each modal equation can be solved independently using Duhamel's integral or numerical integration.
The modal participation factor Γ_j = {φ}_jᵀ[M]{1} / ({φ}_jᵀ[M]{φ}_j) measures how strongly a particular mode is excited by the loading. Modes with small participation factors contribute little to the total response. In practice, only the first few modes (typically those with cumulative mass participation exceeding 90%) need to be considered. The effective modal mass M_eff,j = Γ_j² · M_j represents the portion of total structural mass participating in each mode — a key concept in seismic design.
Response Spectrum Analysis
A response spectrum plots the maximum response (acceleration, velocity, or displacement) of SDOF systems with varying natural periods for a given ground motion. The design response spectrum, specified by seismic codes (ASCE 7, Eurocode 8, IS 1893), envelopes the response spectra of multiple earthquake records with appropriate amplification factors. For elastic design, the spectral acceleration S_a(T) at each period T is read from the design spectrum.
In response spectrum analysis for MDOF systems, the maximum modal responses are combined using the square root of sum of squares (SRSS) for well-separated modes, or the complete quadratic combination (CQC) method for closely spaced modes. The resulting base shear V is compared to the minimum base shear required by code. Modal mass participation must be checked — codes typically require at least 90% mass participation in each orthogonal direction. Use the Seismic Load Calculator to apply response spectrum methods per code provisions.
Advanced — Nonlinear Dynamics and Earthquake Response
For senior students and practicing engineers.
Nonlinear Dynamic Analysis
When structures undergo large deformations or material yielding, the linear assumptions of modal analysis break down. Nonlinear dynamic analysis accounts for material nonlinearity (steel yielding, concrete crushing, gap opening), geometric nonlinearity (P-Δ effects, large displacements), and boundary nonlinearity (uplift, sliding at supports). The equation of motion becomes [M]{ü} + {F_D(u̇)} + {F_S(u)} = {F(t)}, where the damping and stiffness forces depend nonlinearly on the response.
Direct integration methods solve the nonlinear equations step by step. The Newmark-β method (constant average acceleration: β = 1/4, or linear acceleration: β = 1/6) is the most widely used implicit scheme. For nonlinear problems, iteration (Newton-Raphson) within each time step ensures equilibrium at the end of the step. The Hilber-Hughes-Taylor (HHT-α) method introduces numerical damping to filter out spurious high-frequency content while maintaining accuracy for low-frequency response. Time step selection must resolve the highest frequency of interest — typically Δt = T_min/20.
Earthquake Response of Structures
Earthquake ground motion subjects structures to base acceleration ü_g(t), which appears as an equivalent force P_eff(t) = -M·{1}·ü_g(t) on the structure. The response depends on the frequency content of the ground motion relative to the structure's natural frequencies. Near-fault ground motions with velocity pulses can cause disproportionately large demands on long-period structures. Soil-structure interaction (SSI) modifies the effective period and damping — soft soils amplify long-period motions as described in site classification systems (Site Classes A through F per ASCE 7).
Inelastic response during earthquakes is characterized by the ductility factor μ = u_max/u_yield and the strength reduction factor R = V_elastic/V_yield. The equal displacement principle states that for medium-to-long period structures, the inelastic displacement approximately equals the elastic displacement, implying R ≈ μ. For short-period structures, the equal energy principle applies: R ≈ √(2μ - 1). Modern performance-based design uses nonlinear response history analysis to verify drift limits and collapse prevention under design-level and maximum considered earthquakes.
Vibration Control and Isolation
Vibration control technologies protect structures from excessive dynamic response. Passive control includes base isolation (lead rubber bearings, friction pendulum bearings) which shifts the fundamental period away from dominant earthquake energy, and dampers (viscous, viscoelastic, metallic yielding, friction) which dissipate energy. Base isolation reduces floor accelerations by 50-80% compared to fixed-base buildings, making it ideal for hospitals, data centers, and critical facilities.
Active control systems use sensors and actuators to apply counteracting forces in real time — used in tall buildings for wind response control (e.g., the active mass damper in the Taipei 101 tower). Semi-active systems (magnetorheological dampers, variable orifice dampers) combine the reliability of passive systems with the adaptability of active systems, requiring minimal power. Tuned mass dampers (TMDs) absorb energy through a secondary mass tuned to the structure's fundamental frequency — effective for wind-induced vibrations but less robust for seismic loading due to frequency shift during yielding.
Practice Exercises
Exercise 1: SDOF Natural Frequency
A water tower has a mass of 50,000 kg and a lateral stiffness of 8,000 kN/m. Calculate the natural frequency, natural period, and critical damping coefficient. If the damping ratio is 2%, determine the damped natural frequency.
Exercise 2: Harmonic Excitation Response
An SDOF system with m = 2,000 kg, k = 500 kN/m, and ζ = 3% is subjected to a harmonic force F(t) = 10 sin(25t) kN. Calculate the frequency ratio, dynamic amplification factor, and steady-state displacement amplitude. Determine the resonant amplitude if the forcing frequency equals the natural frequency.
Exercise 3: Modal Analysis of a Two-Story Frame
A two-story shear building has floor masses m_1 = 10,000 kg and m_2 = 8,000 kg, with story stiffnesses k_1 = 20,000 kN/m and k_2 = 15,000 kN/m. Determine the natural frequencies and mode shapes. Calculate the modal participation factors and effective modal masses. Verify that the sum of effective modal masses equals the total mass.
Exercise 4: Response Spectrum Base Shear
A 5-story building located in Seismic Design Category D has a fundamental period T = 0.6 s, total weight W = 20,000 kN, and S_DS = 1.0 g, S_D1 = 0.6 g. Using ASCE 7 equivalent lateral force procedure, calculate the design base shear, story forces, and overturning moment. Verify the minimum base shear requirement. Use the Seismic Load Calculator to verify your results.
Related Calculators
Seismic Load Calculator
Calculate seismic base shear and story forces per ASCE 7, IS 1893, and Eurocode 8.
Moment of Inertia Calculator
Calculate section properties including moment of inertia for dynamic analysis.
Wind Load Calculator
Compute wind loads per ASCE 7 for dynamic wind response analysis.
Shear Force Diagram Calculator
Analyze shear and moment for determinate structures under dynamic loads.
Live Load & Dead Load Calculator
Calculate mass participation for dynamic analysis.
Concrete Shear Wall Calculator
Design shear walls for lateral load resistance in seismic regions.
References
- Chopra, A.K. Dynamics of Structures: Theory and Applications to Earthquake Engineering. 5th ed., Pearson, 2020.
- Clough, R.W. and Penzien, J. Dynamics of Structures. 3rd ed., Computers & Structures Inc., 2003.
- Chopra, A.K. Earthquake Dynamics of Structures: A Primer. 2nd ed., EERI, 2005.
- Paz, M. and Leigh, W. Structural Dynamics: Theory and Computation. 5th ed., Springer, 2004.
- ASCE 7-22. Minimum Design Loads and Associated Criteria for Buildings and Other Structures. American Society of Civil Engineers, 2022.
- Civil Engineering Handbook — Structural dynamics chapter with vibration analysis and seismic design guidance.
- Engineering Formula Library — Dynamic response and vibration formulas.
- Engineering Standards Reference — ASCE 7, Eurocode 8, IS 1893 seismic provisions.
- Engineering Glossary — Definitions of structural dynamics terms.