If the response PSD (RPSD) curve after a random vibration analysis describes the RMS (1-sigma) response, how can I get the RPSD that describes the 3-sigma response?

It is the integral of the RPSD curve wrt frequency that gives you the *square* of the RMS value (where RMS = 1 sigma). To get 1 sigma, you integrate RPSD and take the square root, so for 3 sigma:

int RPSD dFreq = 1 sigma^2

3 sqrt (int RPSD dFreq) = 3 sigma

sqrt (int 9 RPSD dFreq) = 3 sigma

In summary, to get the 3-sigma response, you would need 9 times the RPSD.

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