Structural Dynamics & Response Spectrum Notes

Last reviewed 16 Sept 2026 · 7 min read

Dynamic loads

A load is dynamic when its magnitude, direction or position varies with time fast enough to produce significant inertia forces. Examples: earthquakes, wind gusts, machine vibrations, moving vehicles, blasts, waves.

In dynamic analysis, inertia ( $m \overset{u}{¨}$ ) and damping ( $c \overset{u}{˙}$ ) forces are considered with elastic forces.
Degrees of freedom (DOF) — number of independent displacements needed to define the displaced position of all masses.
Lumped mass idealisation — mass concentrated at floor levels (e.g. shear building with rigid floors and axially inextensible columns → one lateral DOF per floor).

SDOF system

FormulaEquation of motion

m \overset{u}{¨} + c \overset{u}{˙} + k u = F (t)

$m$ = mass; $c$ = viscous damping coefficient; $k$ = stiffness; $u$ = displacement.

Undamped natural circular frequency $ω_{n} = \frac{k}{m}$ (rad/s) Natural period $T_{n} = \frac{2 π}{ω _{n}} = 2 π \frac{m}{k}$ Natural frequency $f_{n} = \frac{1}{T _{n}}$ (Hz)

Stiffer structures → shorter periods; heavier structures → longer periods.
For a cantilever column with a lumped top mass: $k = 3 E I / L^{3}$ ; for a column fixed at both ends (sway): $k = 12 E I / L^{3}$ ; storey stiffness = sum of column stiffnesses.

Damping

FormulaDamping relations

Critical damping $c_{cr} = 2 m ω_{n} = 2 k m$ Damping ratio $ζ = \frac{c}{c _{cr}}$ Damped natural frequency $ω_{D} = ω_{n} 1 - ζ^{2}$ (≈ $ω_{n}$ for small ζ)

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

Logarithmic decrement:

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

Case	Behaviour
$ζ < 1$ — underdamped	Oscillates with decaying amplitude (all practical structures, ζ ≈ 2–7%)
$ζ = 1$ — critically damped	Returns to rest fastest without oscillation
$ζ > 1$ — overdamped	Returns slowly without oscillation

Forced harmonic vibration

For $F (t) = F_{0} sin ω t$ , the steady-state amplitude is $u_{0} = (u_{s t})_{0} \times R_{d}$ where $(u_{s t})_{0} = F_{0} / k$ .

FormulaDynamic amplification factor

R_{d} = \frac{1}{( 1 - r ^{2} ) ^{2} + ( 2 ζ r ) ^{2}}, r = \frac{ω}{ω _{n}}

At resonance ( $r = 1$ ): $R_{d} = \frac{1}{2 ζ}$
$r ≪ 1$ : $R_{d} \to 1$ (quasi-static); $r ≫ 1$ : $R_{d} \to 0$ (mass hardly moves)

Transmissibility (force transmitted to support / applied force):

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

Vibration isolation ( $T R < 1$ ) requires $r > 2$ .

Earthquake excitation

With ground acceleration $\overset{u}{¨}_{g} (t)$ and relative displacement $u$ :

m \overset{u}{¨} + c \overset{u}{˙} + k u = - m \overset{u}{¨}_{g} (t) \Rightarrow \overset{u}{¨} + 2 ζ ω_{n} \overset{u}{˙} + ω_{n}^{2} u = - \overset{u}{¨}_{g} (t)

The response depends only on $ω_{n}$ (or $T_{n}$ ) and $ζ$ for a given ground motion.
Base shear $V_{b} = k u_{ma x}$ .

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