ktnsrm

Spectral power density models and Spectral Representation Method

Introduction

Stochastic process models are responsible for characterising ground motions, representing the stochastic excitations applied upon engineering structures¹. To this end, a series of power spectral density models are developed and employed in stochastic dynamic analysis². Notably, Kanai Tajimi model plays a foundational role³. Beyond which, nonstationary model attract more attention in recent years.

functionality

• Define a base Kanai Tajimi model;
• Define both separable and non-separable evolutionary power spectral density models;
• Generate sample realizations via the Spectral Representation Method;
• A general framework enabling easy addition of more nonstationary models via subclassing.

Examples

1. Kanai Tajimi PSD model

$$S(\omega) = S_{0} \frac{1+[2 \zeta (\omega/\omega_{g})]^2}{[1-(\omega/\omega_{g})^2]^2+[2 \zeta (\omega/\omega_{g})]^2}$$

where $w_{g}=5 \pi$ rad/s; $\zeta$ = 0.63; $S_{0}$ = 0.011;

2. separable EPSD

Define an evolutionary spectrum in the form $$S(\omega, t)=g(t)^2S(\omega)$$

with an example of modulating function: $$g(t)=b(e^{-ct} - e^{-2ct})$$ where $b$=4, $c$=0.8

3. non-separable EPSD

An evolutionary spectrum with fully coupled time and frequency nonstationarity. Define an example EPS: $$S(\omega, t) =\frac{\omega^2}{5 \pi} e^{-0.15t} t^{2} e^{-(\frac{\omega}{5 \pi})^2 t}$$

4. Spectral Representation Method

Verification of the Spectral Representation Method with a stationary process with known PSD.

License

ktnsrm was created by Y. Chen. It is licensed under the terms of the MIT license.

Footnotes

1. Kiureghian etc. Nonlinear stochastic dynamic analysis for performance-basedearthquake engineering. ↩

2. Conte and Peng. Fully nonstationary analytical earthquake ground-motion model. ↩

3. Lai etc. Statistical characterization of strong ground motions using power spectral density function. ↩