Response Spectrum Value⚠ unverified
Mechanical / Vibration Analysis · Return the pseudo-spectral acceleration for a given period and damping
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| Sa | Sa | m/s**2 | 1.0 | Pseudo-spectral acceleration |
| T | T | s | 1.0 | Oscillator period |
| zeta | ζ | — | 1.0 | Damping ratio (dimensionless) |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | Sa | m/s**2 | Pseudo-spectral acceleration, in metres per second squared (m/s**2), returned unchanged |
The science & history
Understanding the Parameters
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Oscillator period $T$ — the horizontal axis of a response spectrum. Structures are reduced to their fundamental period and $S_a(T)$ read off; short, stiff buildings and tall, flexible ones sit at different points on the spectrum and see very different accelerations.
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Damping ratio $\zeta$ — each spectrum is drawn for a fixed damping (commonly 5 % for buildings). More damping lowers the whole spectrum, especially near resonant peaks.
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Pseudo-spectral acceleration $S_a$ — "pseudo" because $S_a \approx \omega_n^2 S_d$ (spectral displacement times frequency squared); it gives the peak base shear as $F = m\,S_a$ directly, which is why designers use it. It bundles the entire dynamic analysis into one number per period.
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Why it is a lookup, not a formula — $S_a$ depends on the entire ground-motion time history, not just $T$ and $\zeta$; it is computed by integrating the SDOF equation of motion for that record, or taken from a smoothed design spectrum in a building code (ASCE 7, Eurocode 8).
Derivation (Approaching a Proof)
For a given ground acceleration history $\ddot{u}_g(t)$, each SDOF oscillator obeys
$$\ddot{x} + 2\zeta\omega_n\dot{x} + \omega_n^2 x = -\ddot{u}_g(t), \qquad \omega_n = \frac{2\pi}{T}.$$
Integrate this (numerically, e.g. by Duhamel's integral or a time-stepping scheme) for the record, and record the peak relative displacement $S_d = \max_t |x(t)|$. The spectral displacement, pseudo-velocity, and pseudo-acceleration are
$$S_d = \max|x|, \qquad S_v = \omega_n S_d, \qquad S_a = \omega_n^2 S_d.$$
Repeating for a range of periods $T$ (at fixed $\zeta$) traces the response spectrum $S_a(T)$. There is no algebraic shortcut — the peak response is an outcome of the simulation — which is exactly why the registry's $S_a = S_a$ can only return a pre-computed value.
Dimensional check. $S_a = \omega_n^2 S_d = (\text{rad/s})^2\cdot\text{m} = \text{m}/\text{s}^2$ — an acceleration, as the output label states. The pass-through returns whatever $S_a$ (in m/s²) is supplied.
History and Development
The response spectrum was introduced by Maurice Biot (1930s–40s) and made the foundation of earthquake-resistant design by George Housner and others: rather than analyse a structure against many ground-motion records, engineers characterise the hazard once as a smoothed spectrum and read each structure's demand from its period. Modern building codes (ASCE 7, Eurocode 8) specify design response spectra, and the same shock-response spectrum (SRS) concept qualifies equipment against pyroshock and transient events.
Related Concepts: Natural Frequency Mass Spring, Damping Ratio, Magnification Factor, Random Vibration Rms, Rayleigh Damping Alpha, Quality Factor Q
Notes: Registry is a pass-through ($S_a=S_a$; $T$, $\zeta$ unused). A response spectrum is simulated from a ground-motion record (or read from a code design spectrum), not a formula. Pseudo-acceleration $S_a=\omega_n^2 S_d$ → base shear $F = m S_a$. Usually plotted at 5 % damping.