Civil engineering mechanics & structural foundations of Shake Table Resonance.
During severe earthquake events, seismic ground motions transmit harmonic and transient shock waves through building foundations, inducing base excitations that travel upward into the superstructure. When external ground shaking frequency ω matches the building's fundamental natural frequency ω_n (frequency ratio β = ω / ω_n ≈ 1.0), the structure enters catastrophic resonance, amplifying roof accelerations and inter-story drifts to many times the ground input. Evaluated on a computer-controlled single-axis earthquake shake table, a structural frame is subjected to harmonic sinusoidal base acceleration: u_g(t) = a_base · sin(ω·t). The steady-state relative displacement response is governed by the Dynamic Amplification Factor (DAF): DAF = 1 / √[(1 - β²)² + (2·ζ·β)²], where ζ is the viscous damping ratio. At exact resonance (β = 1.0), the theoretical amplification factor reaches its maximum value: DAF_max = 1 / (2·ζ). For a typical civil engineering steel frame with damping ratio ζ ≈ 5.88% (0.0588), the peak resonant amplification factor is exactly DAF = 1 / (2 · 0.0588) = 8.5000. In addition, the phase angle between base excitation and structural response shifts from 0° (quasi-static in-phase tracking at low frequencies β << 1), to exactly 90° at resonance (β = 1.0), and finally to 180° (complete out-of-phase motion at high frequencies β >> 1). In this laboratory shake table experiment, base acceleration amplitudes from 0.2 to 1.2 m/s² at resonant frequency (f = 2.80 Hz) are delivered to the model to calibrate the resonant amplification slope = 8.5000, verify the DAF spectrum, and evaluate seismic inertia forces.
Governing Equations of Harmonic Base Excitation
When an SDOF structural model of mass $m$, stiffness $k$, and damping $c$ is subjected to ground acceleration $\ddot{u}_g(t) = a_{base} \sin(\omega t)$, the equation of relative motion $u(t) = x(t) - u_g(t)$ is:
\[ m \ddot{u} + c \dot{u} + k u = -m \ddot{u}_g(t) = -m a_{base} \sin(\omega t) \]The steady-state relative displacement amplitude is $u_0 = \frac{m a_{base}}{k} \cdot \text{DAF}$, where the Dynamic Amplification Factor is:
\[ \text{DAF} = \frac{1}{\sqrt{(1 - \beta^2)^2 + (2 \zeta \beta)^2}} \]where $\beta = \omega / \omega_n$ is the frequency tuning ratio. Total absolute roof acceleration amplitude $a_{roof}$ at resonance ($\beta = 1.0$) is related to base acceleration by:
\[ a_{roof} = \text{DAF}_{res} \cdot a_{base} = \frac{1}{2\zeta} \cdot a_{base} \]For damping ratio $\zeta = 0.05882$ ($5.88\%$), the theoretical resonant amplification slope is:
\[ \text{DAF} = \frac{1}{2 \times 0.05882} = 8.5000 \]Phase Angle Shift Mechanics
The phase lag angle $\phi$ between base acceleration and structural response is:
\[ \tan \phi = \frac{2 \zeta \beta}{1 - \beta^2} \]At low frequencies ($\beta \to 0$), $\phi \to 0^\circ$; at resonance ($\beta = 1$), $1 - \beta^2 = 0$, giving $\phi = 90^\circ$; at high frequencies ($\beta \to \infty$), $\phi \to 180^\circ$.
Viva voce questions & analytical theory
What is the physical definition of the Dynamic Amplification Factor (DAF)?
DAF is the dimensionless ratio of the maximum dynamic response amplitude (displacement or acceleration) of a structure to the static response that would be produced by a constant force equal to the dynamic force amplitude: DAF = u_dyn / u_stat.
Why is the resonant amplification factor bounded in real structures?
In an idealized undamped system (ζ = 0), resonant amplitude would grow infinitely; in real structures, internal material hysteresis, joint friction, and aerodynamic drag dissipate energy, bounding maximum resonant amplification to DAF_max = 1 / (2ζ).
What is the phase angle relationship between base excitation and structural response across frequency ratios?
At low excitation frequencies (β << 1), response is in-phase with base acceleration (φ ≈ 0°); at resonance (β = 1), response lags the excitation by exactly 90° (φ = 90°); at high frequencies (β >> 1), response is completely out of phase (φ = 180°).
How does base isolation protect buildings from resonant earthquake ground motions?
Base isolation pads (elastomeric lead-rubber bearings) decouple the building from ground shaking, shifting the building's fundamental period from T = 0.5 s (in the peak earthquake energy band) to T = 2.5 to 3.0 s (where earthquake ground accelerations are minimal).
What is a sine-sweep test and why is it used in modal analysis?
A sine-sweep excites the structure continuously across a broad frequency band, measuring the transfer function (Frequency Response Function, FRF) to identify all resonant modal frequencies, mode shapes, and modal damping ratios.
Why do flexible skyscrapers experience lower seismic base shears than stiff low-rise masonry buildings?
Flexible tall towers have long fundamental periods (T > 2.0 s) far above dominant earthquake ground motion frequencies, meaning they act as natural low-pass filters that avoid resonance and reduce spectral acceleration.
Experimental precautions and laboratory safety
- Securely torque all foundation bolts connecting the model to the shake table platen to prevent rattling and parasitic slip.
- Ensure maximum roof drift does not exceed column yield limits during resonant sine-dwell excitation.
- Set shake table software emergency displacement interlocks to trigger automatic shutdown if stroke limits are approached.
- Verify that accelerometer cable leads have generous strain-relief loops so they do not exert pulling forces on the model.
Theoretical assumptions & boundary condition limits
- The civil engineering physical models apply within the specified calibrated operational window (0.2 to 1.2 m/s²).
- Assumes homogeneous, isotropic continuum material behavior without unmodeled micro-fracture anomalies.
- Continuous sensor readings reflect ordinary least-squares regression across successive observations.
- Extreme inelastic yielding or cyclic degradation requires non-linear constitutive continuum damage mechanics.