2016-04-14

Matrix Iteration, Modal Contributions and Static Correction

Returning on our last topic, we see one example of the Rayleigh-Ritz procedure, we discuss the Subspace Iteration Method and finally we do an exercise on subspace iteration.

-oOo-

The second part was devoted to the analysis of the relative importance of the different modal contributions to the response.

Modal contributions

We analyze the common case in which the dynamic load can be expressed by a constant load vector \(\boldsymbol r\) modulated by an adimensional function of time, \(f(t)\) (e.g., the seismic excitation can be described in such terms).

  • We start our argument introducing the modal partecipation factor: the MPF measures the "tuning" between the loading vector and the modal shape.
  • Reasoning on the definition of the MPF, we find that it is possible to represent the load as a sum of modal load contributions, \(\boldsymbol r = \sum \boldsymbol r_i\).
  • Next, we recognize that it is possible, mode for mode, to represent the modal response in terms of a constant, pseudo-static response, due to the modal load contribution, modulated by a (nearly) adimensional response function.
  • Introducing a) the modal contribution factor, the ratio between the modal response contribution and the static response, b) the dynamic response amplification factor and c) the peak factors, we finally see that it is possible to single out the effects of the static response and the dynamic amplification.

Static correction

We have recognized that, for a given response quantity, we have a static response that, mode by mode, is modulated by a a dynamic amplification and a constant modal contribution factor.

For modal frequencies much larger than the excitation frequency, the dynamic amplification tends to unity and the response is essentially static.

From this consideration, the static correction procedure insert into the response the higher modes contributions in term of an enveloping of a static response that, surprise, can be computed also in terms of the lower modes characteristics!

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Giacomo Boffi