Verification with Simulated Data

Generalized Modal Identification Using Hysteretic Models

4.5 Verification with Simulated Data

needs to be evaluated to determine the direction of steepest descent and then a one-dimensional minimization would be performed in this direction. However, the present approach needs no such evaluations.

(2) The initial estimates of

b!

and

b2

may be taken from the

a!

and

a2

identified by the four-parameter "nonparametric" model which are generally very dose to the optimal parameters of the model. This saves considerable computational effort in finding appropriate initial estimates for

bJ:

and

b2.

As described in Section 2.6, the dominant modes are identified one at a time by performing a succession of single-mode identifications. The optimal estimate

13r

is determined by a simple and direct one-dimensional minimization scheme as outlined in Section 2.6.4. For given {3r, the backbone parameters

bi

^and

b2

^{of the}

generalized restoring force hr are estimated by a parametric identification technique described in Section 4.4.4. Note that the initial estimates of the modal parameters

b!, b2

and

13r

are taken from the optimal values of

a!, a2

and

13r

obtained in previous identification using the the four-parameter "nonparametric" model. Since the initial values of parameters estimated in this way are generally very close to the final optimal values, the optimization process used to refine the parameters is performed very efficiently. No convergence problems have been encountered.

Determined by minimizing the prediction error based on the acceleration peaks of the system response, the results for the optimal models are given in Table 4.1.

Only a two-mode model is identified for the same reason as in previous nonparamet- ric study. Comparing Table 4.1 with Table 3.1, it is clear that the model parameters

b!' b2

and {3r are indeed close to their counterparts,

a!' a2

^and

rr'

r

=

1, 2, obtained in Chapter 3. This supprots the point made above regarding the closeness of the values of these parameters.

The negative sign of b~ and b~ indicated the softening stiffness behavior of the hysteretic system. This is also illustrated by the. identified generalized modal restoring force diagrams depicted in Figure 4. 7.

The quality of the response match using two modes is shown in Figures 8-10.

In general, the model fits the actual time history response better than does the four- parameter nonparametric model. This improvement is considered the consequence that the hysteretic component of the response has been identified by employing the two-parameter distributed element model.

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element hysteretic model to scaled El Centro accelerogram.

(a) First mode.

(b) Second mode.

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It is next desirable to examine how well the identified model can predict the hysteretic response to other excitations and how much it improves the results pre- dicted by the four-parameter nonparam.etric model.

4.5.2 Model Validation - Response Prediction

To continue the verification study, the nonlinear model identified in section 4.4.2 is used to predict the roof response of the same system when subjected to other base excitations. The same scaled Taft and Parkfield earthquakes and corresponding response of the top mass are employed as in Chapter 3 and the prediction results are compared.

Figures 11-13 show the time histories predicted by the model and "measured"

from the system to the scaled Taft earthquake. A similar comparison for the re- sponse to the scaled Parkfield earthquake is presented in Figures 14-16. The predic- tion error P based on acceleration peaks is summarized in the last two columns of Table 4.1 for both cases. By comparing all these results with their counterparts in Chapter 3, it is dearly seen that the prediction of the time history of the response made by the optimal two-parameter distributed-element model is superior to that obtained using the four-parameter nonhysteretic model. Especially significant are the better reproduction of the hysteretic features of the response such as the drift displacement shown in Figures 10 and 13.

It is concluded that the two-parameter distributed-element model with a small number of modes is capable of predicting the hysteretic response, including the permanent displacement, under different base excitations. Noting that the hys- teretic behavior is more pronouned in the response used for prediction than for identification, this example emphasizes the importance of employing an appropri- ate hysteretic model in the identification study of a hysteretic system so that the

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Mode Frequency Band (Hz) t o.o-4.o 2 4.0--7.5

Identlflcatlon El Centrol1l b~ b2

pr

p (1/sec2 ) 1/(cm · sec)2 Initial Optimal Inltlal Optimal Initial Optimal 2.t3 X t02 2.t5 X t02 -6.74 x to-t -6.69 x to-t 1.6t0 0.992 t.38 x to-t t.44 X t03 1.62 X t03 -8.29 X 102 -3.25 X t03 -0.442 -0.383 8.66 x to-2 -------------- (1) Simulated response with the El Centro, t94.0, SOOE, scaled to a peak of 57% g (2) Simulated response with the Taft, 1952, S69E, scaled to a peak of 50% g (3) Simulated response with the Parkfield, 1966, N65E, scaled to a peak of 61% g Prediction Taft(2) Park.8eldl3l

.

p p 1.38 x to-t 1.22 x to-t I 9.24 x to-2 1.29 x to-t Table 4.1 Identification and Prediction Results of the Two-Parameter Distributed-Element Hysteretic Model Using Simulated Data

identified model is capable of prediCting the response to the input motions other than the identification signal.