peteroznewman
Subscriber

Yes, you have the right idea.

The time history of the earthquake can be run into the base of a SDOF mass-spring-damper oscillator with a specific natural frequency and the peak response S is recorded. That is repeated for a large number of SDOF oscillators of different natural frequencies to create the table. Note that damping has been included in the generation of the spectrum table already, so damping is not added in the Response Spectrum analysis as that would be double-counting. Only the spectrum table values that match one of the modal frequencies are used or the table is log-log interpolated to generate a spectrum value that matches each modal frequency. If the spectrum table doesn’t go as high in frequency as the modes, then the last value in the table is used. Note that the peak response put in the spectrum table could be displacement, velocity or acceleration.

Modal analysis provides a list of natural frequencies, mode shapes and participation factors. Recall that the modal participation factor (PFi) measures the amount of mass moving in a specific direction for a unit displacement and i goes from 1 to n for the number of modes. The Mode Coefficient (Ai) depends on whether the input spectrum table is displacement, velocity or acceleration. The Mode Coefficient is defined as the amplification factor that is multiplied by the eigenvector to give the actual displacement in each mode. If you had a spectrum table of peak displacements, then the Mode Coefficient is simply the Spectrum value times the Participation Factor.  Ai = Si*PFi

Since you have a spectrum table of peak acceleration, you have to divide by the frequency (w) squared. Ai = Si*PFi/w^2

Since you have modes that go higher than your spectrum table, all higher modes will be multiplied by the last value in the table.

The modal displacement response is {Ri} = Ai{EVi} where EV is the Eigenvector for the ith mode.

The modal acceleration response is {Ri} = Ai/wi^2{EVi} where wi is the Natural Frequency of the ith mode.

Lastly, the modes must be combined to get the total response. The default method is Square Root of the Sum of the Squares (SRSS) Method.

So to answer your original first question, the acceleration response is calculated from the acceleration spectrum value times the participation factor then combine all modes in a SRSS calculation. So in your imagined case with four modes, you must also say what the participation factor is to complete the calculation.