Structural Dynamics

Reference

Every symbol, term and formula the course uses, in one searchable place.

This is a companion to the lessons, not a substitute for them. Every entry names the module that teaches it, so you can go back to where the idea is explained rather than take a formula on trust.

Each formula carries its assumptions, and in dynamics that matters more than usual. Almost everything here assumes the structure stays linear, and the expressions that do not — the hysteretic ones — say so. A magnification factor applied to a structure that has yielded, or modal superposition applied where the damping is not classical, will both give you a confident and wrong answer.

Search filters all three sections at once — the tab counts update as you type, so you can see where else a term appears.

Natural periodModule 4: The time a structure takes to complete one cycle of free vibration. It follows from the mass and the stiffness alone, and it is the first quantity to check against a hand estimate on any dynamic model.Not to be confused: Forcing period — a property of the load, not of the structure
Damping ratioModule 5: Damping expressed as a fraction of the critical value. Below 1 the structure oscillates and the amplitude decays; at 1 or above it returns without oscillating.Not to be confused: Damping coefficient c, which carries units and cannot be compared between structures
Critical dampingModule 5: The least damping that prevents oscillation altogether. It is a reference value rather than a design target — no building is anywhere near it.
Logarithmic decrementModule 5: The natural logarithm of the ratio of two successive peaks in a free-decay record. It is how damping is measured from a test rather than assumed from a table.
ResonanceModule 6: Forcing at or near the natural frequency, where the response builds cycle on cycle and is limited only by damping. The magnification approaches 1/(2ζ).Not to be confused: Dynamic amplification, which occurs across a band of frequencies and not only at resonance
TransmissibilityModule 6: The fraction of an applied force that reaches the supports, or of a support motion that reaches the structure. It falls below 1 only when the forcing frequency exceeds √2 times the natural frequency.
Duhamel integralModule 7: The response to any load, built by treating it as a sequence of impulses and superposing their individual responses. Valid only for a linear system, because it is superposition made explicit.
Dynamic load factorModule 7: Peak dynamic response divided by the static response to the same peak load. It is 2 for a suddenly applied constant load on an undamped system.
Newmark methodModule 8: A family of step-by-step integration schemes for the equation of motion, parameterised by γ and β. Constant average acceleration (γ = ½, β = ¼) is unconditionally stable and the usual default.Not to be confused: Central difference, which is explicit and only conditionally stable
Period elongationModule 8: The apparent lengthening of the period introduced by a numerical integration scheme. For constant average acceleration it is (π²/3)(Δt/T)², about 3.3% at Δt/T = 0.1.
Degree of freedomModule 2: An independent coordinate needed to describe the structure's displaced position. A shear building has one per floor; a rigid-diaphragm model has three.
Mode shapeModule 11: The pattern in which a structure moves when vibrating at one of its natural frequencies. Every point reaches its maximum at the same instant; only the shape is fixed, not the scale.Not to be confused: Deflected shape under a static load, which depends on where the load is applied
Effective modal massModule 11: The share of the total mass that a mode mobilises in a given direction. It is independent of how the mode was scaled, sums to the total mass over all modes, and is the correct measure of a mode's importance.Not to be confused: Participation factor, which depends on the scaling and is not a measure of importance
OrthogonalityModule 11: The property that φₘᵀMφₙ and φₘᵀKφₙ vanish for m ≠ n. It is what allows the coupled equations to be separated into independent single-degree-of-freedom equations.
Modal decouplingModule 12: Transforming the coupled equations of motion into one independent equation per mode. It requires the damping to be classical; without that, the modes stay coupled.
Classical dampingModule 12: Damping whose matrix the undamped mode shapes diagonalise. Rayleigh damping is classical by construction; concentrated dampers and base isolation generally are not.Not to be confused: Non-classical damping, which requires complex modal analysis or direct integration
Rayleigh dampingModule 12: A damping matrix formed as αM + βK. It delivers the target damping ratio at exactly two chosen frequencies and something different at every other.
Base excitationModule 13: Loading applied by moving the supports rather than by applying a force. The effective load is −Mιü_g, so it is proportional to the mass.
Response spectrumModule 14: The peak response of every possible single-degree-of-freedom oscillator to one ground motion, plotted against period. It discards all timing information.Not to be confused: A time history, which retains when each peak occurred and in what order
Pseudo-accelerationModule 14: ωₙ²S_d. Defined so that the peak elastic force is exactly m·PSa. It is not the true peak acceleration, and the gap widens with damping.
SRSS and CQCModule 14: Rules for combining modal peaks that do not occur at the same instant. SRSS assumes the modes are independent; CQC accounts for correlation and is required when periods are close.
HysteresisModule 16: Force–displacement behaviour in which the path taken depends on the history. The loop encloses an area, and that area is energy the structure absorbed and did not return.Not to be confused: Elastic behaviour, where loading and unloading retrace the same line and no energy is lost
Ductility demandModule 16: Peak displacement divided by yield displacement. A demand is what the earthquake asks of the structure; a capacity is what the detailing can supply, and the two must be quoted together.
Tangent stiffnessModule 16: The slope of the force–displacement curve at the current point. It is what a Newton iteration linearises about, and it drops from k to αk the moment a member yields.Not to be confused: Secant stiffness, the slope from the origin, used for equivalent linearisation only
Equal-displacement ruleModule 16: The observation that a yielding structure of moderate to long period reaches roughly the same peak displacement as an elastic one would. It gives R = μ, and it means a behaviour factor reduces strength without reducing drift.Not to be confused: Equal-energy rule, R = √(2μ − 1), which applies below the corner period
Residual displacementModule 16: The permanent offset a yielded structure is left with when the shaking stops. It is a separate demand from peak displacement, is badly conditioned, and has governed real demolition decisions.
Base isolationModule 17: Deliberately lengthening a structure's period by placing a flexible layer beneath it, so it sits lower on the acceleration branch of the spectrum. It converts force demand into displacement demand.Not to be confused: Supplemental damping, which reduces response without changing the period
Period shiftModule 17: The ratio of isolated to fixed-base period. Along the velocity branch of a spectrum it is roughly the force reduction factor and roughly the displacement amplification factor at the same time.
MoatModule 17: The clear gap around an isolated building that lets it move. Its width is much greater than the analysis displacement, after bidirectional resultant, torsion, bounding bearing properties and any residual offset.
Velocity exponentModule 18: The exponent α in a viscous damper's force law F = C|v|^α. Values below 1 flatten the force–velocity curve so a rare, fast event does not impose an extreme force on the frame.
Added damping ratioModule 18: The damping a device contributes to a particular mode. It goes as the SQUARE of the relative modal displacement across the device, which is why placement matters more than size.
Tuned mass damperModule 19: A secondary mass on a spring, tuned near the structure's own frequency, which moves out of phase with it so its inertia force opposes the motion. One resonant peak becomes two.Not to be confused: A viscous damper, which acts on velocity and is broad-band rather than tuned
Den Hartog optimumModule 19: The tuning and damper damping that minimise a tuned mass damper's peak response, found by making two fixed points on the response curve equal in height. It deliberately gives two equal peaks, not a deep trough.
MistuningModule 19: A tuned mass damper's frequency differing from the structure's. A 10% error nearly doubles the peak response, which is why the tuning target must be measured on the completed structure.
Semi-active controlModule 19: A device whose properties can be changed in real time but which cannot add energy to the structure. It cannot destabilise, and it fails to a passive damper.Not to be confused: Active control, which supplies force from an external power source and can add energy
Seismic massModule 20: The mass that actually participates in the inertia: full dead load, the quasi-permanent fraction of imposed load, cladding and permanent equipment. It is a physical quantity and carries no partial factor.Not to be confused: A load combination, which is a force and does carry factors
Base shear coefficientModule 20: Base shear divided by seismic weight. It tests mass, spectrum, units and combination in a single number, and a value outside roughly 0.02 to 0.5 is a unit error until proved otherwise.

Notation used throughout the course
Symbol	Meaning	Unit	Introduced
m	MassSeismic mass is a physical quantity and carries no partial factor	kg	Module 3
c	Viscous damping coefficient	N·s/m	Module 3
k	StiffnessForce per unit displacement in the direction being modelled	N/m	Module 3
c_cr	Critical damping coefficient2√(km) = 2mωₙ — the least damping that prevents oscillation	N·s/m	Module 5
ζ	Damping ratioc/c_cr. 2–5% for an ordinary building; 15–25% with supplemental dampers	—	Module 5
ωₙ	Undamped natural circular frequency√(k/m)	rad/s	Module 4
ω_D	Damped natural circular frequencyωₙ√(1 − ζ²); within 0.5% of ωₙ below ζ = 0.1	rad/s	Module 5
fₙ	Natural frequencyωₙ/2π	Hz	Module 4
Tₙ	Natural period1/fₙ. The number to check any modal output against first	s	Module 4
ω	Forcing circular frequency	rad/s	Module 6
β	Frequency ratioω/ωₙ. Resonance is β = 1	—	Module 6
R_d	Displacement magnification factorDynamic amplitude ÷ static deflection under the same peak force	—	Module 6
δ	Logarithmic decrementln of the ratio of successive peaks	—	Module 5
u(t)	Displacement relative to the groundRelative displacement is what strains the structure	m	Module 3
u_g(t)	Ground displacement	m	Module 13
u_t	Total (absolute) displacementu + u_g. Absolute acceleration is what people and equipment feel	m	Module 13
h(t)	Unit impulse response functionResponse to a unit impulse; the kernel of the Duhamel integral	1/(kg·s)	Module 7
Δt	Integration time stepSet by the HIGHEST mode retained, never by the first	s	Module 8
γ, β	Newmark parametersγ = 1/2, β = 1/4 is constant average acceleration — unconditionally stable	—	Module 8
M	Mass matrix	kg	Module 10
C	Damping matrix	N·s/m	Module 12
K	Stiffness matrix	N/m	Module 10
φₙ	Mode shape of mode nA shape, not a displacement — its scale is arbitrary	—	Module 11
Φ	Modal matrixMode shapes as columns	—	Module 11
Mₙ	Generalised (modal) mass of mode nφₙᵀMφₙ. Depends on how the mode was scaled	kg	Module 11
Γₙ	Modal participation factorScale-dependent — not a measure of importance	—	Module 11
M_eff,n	Effective modal massScale-INDEPENDENT, sums to the total mass, and is the one to judge modes by	kg	Module 11
qₙ(t)	Modal coordinateHow much of mode n is present at time t	m	Module 12
α, β	Rayleigh damping coefficientsC = αM + βK — exact at two frequencies and wrong at every other	1/s, s	Module 12
S_d, PSv, PSa	Spectral displacement, pseudo-velocity, pseudo-accelerationPSa = ωₙ²S_d by definition, not by approximation	m, m/s, m/s²	Module 14
PGA	Peak ground accelerationThe zero-period ordinate of any spectrum	m/s²	Module 13
V/W	Base shear coefficient0.02–0.5 for an ordinary building; outside that, suspect a unit error	—	Module 15
f_y	Yield force	N	Module 16
u_y	Yield displacementf_y/k	m	Module 16
α	Post-yield stiffness ratio0 is elastic–perfectly plastic; 0.02–0.05 is realistic and numerically kinder	—	Module 16
μ	Displacement ductility demand, or TMD mass ratiou_max/u_y in Module 16; m₂/m₁ in Module 19	—	Module 16
R	Strength reduction (behaviour) factorReduces strength, NOT displacement	—	Module 16
k_t	Tangent stiffnessThe slope at the current point — what the solver linearises about	N/m	Module 16
k_sec	Secant stiffnessOrigin to current point — for equivalent linearisation, never for the solver	N/m	Module 16
K_iso	Total horizontal isolator stiffness	N/m	Module 17
T_iso	Isolated periodSet by the bearings and the mass; the superstructure changes it by a per cent or two	s	Module 17
C, α	Damper coefficient and velocity exponentC has different units for different α, so two dampers' C values are not comparable	N·(s/m)^α, —	Module 18
θ	Damper brace angle to the horizontalEffectiveness goes as cos²θ	deg	Module 18
f	TMD tuning ratioω₂/ω₁. Optimum is below 1, so the damper's PERIOD is longer	—	Module 19

Equation of motion, single degree of freedom

Foundations · Module 3

m u ¨ + c u ˙ + k u = p (t)

Assumes: Linear stiffness and linear viscous damping; small displacements; mass constant.

Natural circular frequency

Foundations · Module 4

ω_{n} = \frac{k}{m}, T_{n} = \frac{2 π}{ω _{n}}

Assumes: Undamped; a single degree of freedom, or one mode of a larger system with its generalised mass and stiffness.

Critical damping and damping ratio

Free vibration · Module 5

c_{cr} = 2 k m = 2 m ω_{n}, ζ = \frac{c}{c _{cr}}

Assumes: Viscous damping. Friction and hysteretic damping are not viscous and only approximate this form.

Damped free vibration

Free vibration · Module 5

u (t) = e^{- ζ ω_{n} t} [u_{0} cos ω_{D} t + \frac{u ˙ _{0} + ζ ω _{n} u _{0}}{ω _{D}} sin ω_{D} t]

Assumes: Underdamped, ζ < 1; ω_D = ωₙ√(1 − ζ²); no applied load after t = 0.

Logarithmic decrement

Free vibration · Module 5

δ = ln \frac{u _{i}}{u _{i + 1}} = \frac{2 π ζ}{1 - ζ ^{2}} \approx 2 π ζ

Assumes: Successive peaks one full cycle apart; the approximation is within 1% for ζ below about 0.15.

Harmonic magnification factor

Forced vibration · Module 6

R_{d} = \frac{1}{( 1 - β ^{2} ) ^{2} + ( 2 ζ β ) ^{2}}

Assumes: Steady state only — the transient has died away; harmonic force of constant amplitude; β = ω/ωₙ.

Magnification at resonance

Forced vibration · Module 6

R_{d}_{β = 1} = \frac{1}{2 ζ}

Assumes: Steady state at exactly β = 1. The true maximum occurs at β = √(1 − 2ζ²) and is marginally higher.

Force transmissibility

Forced vibration · Module 6

T R = \frac{1 + ( 2 ζ β ) ^{2}}{( 1 - β ^{2} ) ^{2} + ( 2 ζ β ) ^{2}}

Assumes: Harmonic steady state. Isolation is only achieved for β > √2, and more damping HURTS above that.

Duhamel integral

Arbitrary loading · Module 7

u (t) = \frac{1}{m ω _{D}} \int_{0}^{t} p (τ) e^{- ζ ω_{n} (t - τ)} sin ω_{D} (t - τ) d τ

Assumes: Linear system — it is superposition written out. At rest at t = 0, or the free-vibration terms must be added.

Newmark integration

Numerical methods · Module 8

u ˙_{i + 1} = u ˙_{i} + [(1 - γ) Δ t] u ¨_{i} + (γ Δ t) u ¨_{i + 1}

Assumes: γ = ½ avoids artificial damping; β = ¼ gives unconditional stability. γ > ½ introduces numerical damping deliberately.

Period elongation, constant average acceleration

Numerical methods · Module 8

\frac{Δ T}{T} \approx \frac{π ^{2}}{3} (\frac{Δ t}{T})^{2}

Assumes: γ = ½, β = ¼; small Δt/T. About 3.3% at Δt/T = 0.1 — the error is in the period, not the amplitude.

Matrix equation of motion

Multi-degree of freedom · Module 10

M u ¨ + C u ˙ + Ku = p (t)

Assumes: Linear; M and K symmetric and constant; degrees of freedom consistent between the three matrices.

Generalised eigenproblem

Multi-degree of freedom · Module 11

(K - ω_{n}^{2} M) ϕ_{n} = 0

Assumes: Undamped free vibration. M positive definite — a zero on the mass diagonal makes the problem singular.

Effective modal mass

Multi-degree of freedom · Module 11

M_{eff, n} = \frac{( ϕ _{n}^{T} M ι ) ^{2}}{ϕ _{n}^{T} M ϕ _{n}}

Assumes: Independent of mode scaling, unlike the participation factor. Sums to the total mass in the direction of ι.

Modal equation

Modal analysis · Module 12

q ¨_{n} + 2 ζ_{n} ω_{n} q ˙_{n} + ω_{n}^{2} q_{n} = \frac{ϕ _{n}^{T} p ( t )}{M _{n}}

Assumes: Classical damping — the mode shapes must diagonalise C. Test it; do not assume it.

Rayleigh damping

Modal analysis · Module 12

C = α M + β K, ζ_{n} = \frac{α}{2 ω _{n}} + \frac{β ω _{n}}{2}

Assumes: Delivers the target at exactly two frequencies. Below the first anchor and above the second the damping is higher, often much higher.

Base excitation

Earthquake response · Module 13

m u ¨ + c u ˙ + k u = - m u ¨_{g} (t)

Assumes: u is RELATIVE to the ground. The absolute acceleration is ü + ü_g, and that is what equipment feels.

Spectral relations

Response spectra · Module 14

P S v = ω_{n} S_{d}, P S a = ω_{n}^{2} S_{d}

Assumes: Definitions, not approximations. PSa gives the exact peak elastic force m·PSa; it is not the true peak acceleration.

CQC combination

Response spectra · Module 14

r \approx i \sum j \sum ρ_{ij} r_{i} r_{j}

Assumes: Required when modal periods are within about 10% of one another; SRSS is the special case ρ = δ_ij.

Bilinear restoring force

Nonlinear response · Module 16

f_{s} = k (u - u_{p}) + α k u_{p}

Assumes: Rate-independent, kinematic hardening, no strength or stiffness degradation and no pinching.

Hysteretic energy per cycle, elastic–perfectly plastic

Nonlinear response · Module 16

E_{H} = 4 f_{y} u_{y} (μ - 1)

Assumes: α = 0, symmetric full cycle at constant amplitude. A positive α makes the loop slightly fatter.

Strength reduction factors

Nonlinear response · Module 16

R = μ (equal displacement), R = 2 μ - 1 (equal energy)

Assumes: Empirical observations, not derivations. Equal displacement above the corner period, equal energy below it.

Secant stiffness and effective period

Nonlinear response · Module 16

k_{sec} = \frac{k [ 1 + α ( μ - 1 )]}{μ}, T_{eff} = T_{n} \frac{k}{k _{sec}}

Assumes: Equivalent linearisation at the peak. Not valid inside a time-stepping solver, which needs the tangent stiffness.

Isolated period

Base isolation · Module 17

T_{iso} = 2 π \frac{M}{K _{iso}}, K_{iso} = \frac{4 π ^{2} M}{T _{iso}^{2}}

Assumes: Superstructure much stiffer than the isolation layer; M is the mass ABOVE the bearings, including the base slab.

The isolation trade

Base isolation · Module 17

\frac{V _{iso}}{V _{fixed}} \approx \frac{T _{fixed}}{T _{iso}} \approx \frac{S _{d, fixed}}{S _{d, iso}}

Assumes: The spectrum is on its velocity branch, where S_a falls as 1/T; equal damping in both cases.

Viscous damper force

Supplemental damping · Module 18

F = C ∣ v ∣^{α} sign (v)

Assumes: v is the velocity ACROSS the damper. C has units N·(s/m)^α, so C values for different α are not comparable.

Damper energy per cycle, linear

Supplemental damping · Module 18

E_{D} = π C ω u_{0}^{2}

Assumes: α = 1 and harmonic motion of amplitude u₀. Nonlinear dampers carry a gamma-function coefficient instead of π.

Added modal damping ratio

Supplemental damping · Module 18

ζ_{added} = \frac{\sum _{j} C _{j} cos ^{2} θ _{j} ϕ _{rel, j}^{2}}{2 ω _{n} M _{n}}

Assumes: Linear dampers only. φ_rel is the relative modal displacement across the damper, and it enters SQUARED.

Den Hartog optimum tuning

Structural control · Module 19

f_{opt} = \frac{1}{1 + μ}, ζ_{2, opt} = \frac{3 μ}{8 ( 1 + μ )}

Assumes: Harmonic forcing of an UNDAMPED main structure. Real structural damping and earthquake excitation both shift the optimum.

Achievable TMD peak

Structural control · Module 19

R_{peak} = 1 + \frac{2}{μ}

Assumes: Optimum tuning and damping, undamped main structure. Once the mass ratio is chosen, this is the best available.

Empirical first period

Model checking · Module 20

T_{1} \approx 0.075 H^{3/4} or T_{1} \approx 0.1 N

Assumes: H in metres, N storeys; moment-frame buildings of ordinary proportions. A check, not a design value.

Base shear coefficient

Model checking · Module 20

\frac{V}{W} = \frac{V}{m g}

Assumes: W is the seismic weight, unfactored. Outside roughly 0.02 to 0.5, suspect a unit error before anything else.

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