Particular attention is paid to the identification and modeling of the behavior of structural dynamical systems in the nonlinear hysteretic response regime. For this purpose, special attention is paid to the identification and modeling of the response behavior of nonlinear hysteretic systems under the influence of base motion.
Chapter 2
- Introduction
- Nonlinear System
- Modal Representation
- Identification Problem
- Minimization Criterion
- Identification Methodology
- Summary
In the present formulation, the discrepancy between the model and system state is measured by the prediction error P. As a means of sharpening the result, the evaluation of the modal response is performed iteratively in the generalized modal identification method.
Chapter 3
Generalized Modal Identification Using Nonhysteretic Models
- Introduction
- Nonparametric Identification Techniques
- Nonhysteretic Restoring Force Models
- Four-Parameter Nonparametric Model
- Verification with Simulated Data
However, a method for the general form of the restoring force is reported to be available. A non-parametric model is used to obtain an initial non-hysteretic estimate of the generalized restoring force for each condition.
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Summary
Caughey, 'N-linear Analysis, Synthesis and Identification Theory', Proceedings of the Symposium on Testing and Identification of Nonlinear Systems, California Institute of Technology, March 1975. Caughey, 'A Non parametric Identification Technique for Nonlinear Dynamic Problems', ASME Journal of Applied Mechanics, Vol. Caughey, "Nonparametric Identification of Nearly Arbitrary Nonlinear Systems," ASME Journal of Applied Mechanics, Vol.
Iwan, "On the Steady-State Response of a One-Dimensional Yielding Continuum," ASME Journal of Applied Mechanics, September 1970. Iwan, "Response of Multi-Degree-of-Freedom Yielding Systems," ASCE Journal of Engineering Mechanics Division, vol. Gate, "Effective period and damping of a class of hysteretic structures", Earthquake Engineering and Structural Dynamics, Vol.
Chapter 4
Generalized Modal Identification Using Hysteretic Models
Introduction
The Backbone Curve
Hysteretic Restoring Force Models
As shown in Figure 4.4(a), this model approximates the spine curve by line segments with two different slopes, (k1 + k2 ) and k2 , which can be expressed as This system is constructed by adding a second linear spring to the elastoplastic system shown in Figure_4.3(b). However, in system modeling and identification, it is difficult to describe the detailed hysteretic behavior of real systems using this simplified model, especially when transient response is important.
Both the elastoplastic and bilinear models are too simplified to describe the actual hysteretic behavior observed in Figure 4-1. As shown in Figure 4.5, the hysteresis loops generated by the model are not always closed under cyclic displacement loading and the loops float continuously under certain types of cyclic force loading. Because the model's hysteretic behavior is based on the physics of a given mechanical system, no mathematical rules are needed to ensure physical hysteresis loops under complicated load histories.
Two-Parameter Distributed-Element Model
4.8) };.r is the estimate of the generalized modal restoring force hr given by the two-parameter distributed element model. It is observed that the "non-parametric" model with four parameters provides a good non-hysteretic estimate of the nonlinear stiffness behavior of the system. By fitting the backbone of the model to the expression fr(yr), one can obtain
In this study, r(yr) is chosen in the form (4.8), which has two parameters. We will see later that this two-parameter distributed element model can capture the essential features of the hysteretic behavior under consideration. Let ht be the estimate of the restoring force hr by the two-parameter distributed element model. and b2, which appear in the backbone relationship.
Verification with Simulated Data
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. This supports the point made above about the closeness of the values of these parameters. This improvement is considered to be the result of the hysteretic component of the response being identified using the two-parameter distributed element model.
By comparing all these results with their counterparts in Chapter 3, it becomes clear that the prediction of the response time history made by the optimal two-parameter distributed element model is superior to the prediction obtained using the non- hysteretic model with four parameters. Particularly significant is the better reproduction of the hysteretic characteristics 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 able to predict the hysteretic response. , including the permanent displacement, under different base excitations.
Summary
Masing, "Eigenspannungen und Verfestigung beim Messing (Self-stretching and hardening for copper)," Proceedings of the 2nd International Congress of Applied Mechanics, Zurich, Switzerland, 1926. Iwan, "The Steady-State Response of the Double Bilinear Hysteretic System," ASME Journal of Applied Mechanics, Vol. Iwan, "On a class of models for the yield behavior of continuous and composite systems," ASME Journal of Applied Mechanics, Vol.
Naghdi, "On the Characterization of Hardening in Plasticity", ASME Journal of Applied Mechanics, Vol. Lutes, "Response of Bilinear Hysteretic System to Stationary Case Excitation," Journal of the Acoustic Society of America, Vol. Jennings, “Hysteretic response of a nine-story reinforced concrete building,” International Journal of Earthquake Engineering and Structural Dynamics, Vol.
Chapter 5
Application to Pseudo-Dynamic Test Data
Introduction
Pseudo-Dynamic Testing Method
Based on the known mass distribution of the test structure and the assumption of mass accumulating at each degree of freedom, the mass matrix M is. The viscous damping matrix C is estimated from preliminary tests at low amplitudes assuming Rayleigh damping. Thus, after experimentally measuring ,.., f(i), equation (5.4) can be solved by an on-line computer, and the increments in the displacements at the nodal points can be determined.
Since in a pseudodynamic test the displacements to be imposed on the test structure are calculated from the structural restoring forces measured directly from the deformed structure, experimental errors related to displacement control and force measurement are inevitably introduced into the calculation process. Because of the large number of loading steps generally involved, the cumulative errors in the numerical results can be significant, even though the experimental feedback errors introduced at each step are small. Therefore, the higher frequency response is more sensitive to experimental errors and the cumulative error growth can be reduced by reducing the integration time interval At.
BRI Testing Program
In the elastic and inelastic tests, the test structure was subjected to the 821 W component of the Taft record of the 1952 Kern County, California earthquake scaled to a peak acceleration of 6.5% and 50%. The study of the elastic response data using system identification techniques, conducted by Jayakumar and Beck, revealed the cumulative effect of experimental errors inherent in the test [17-20]. They noted that negative damping present in the third mode and the gears calculated from system identification did not match well with the test gears.
The algorithm developed involves continuous alternation between the steepest descent and the modified Gauss-Newton methods for the simultaneous identification of the optimal parameter values in a (3N + I)-dimensional space where N is the number of floors. In the next two sections, an analysis of the inelastic test data is performed using the general modal identification method. In marked contrast to most nonlinear system identification approaches, only the input and roof response data are used in the analysis.
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Four-Parameter Nonhysteretic Model
The relative acceleration of the roof with respect to the ground is used as response data in the analysis. Only the two-mode model is specified here because the signal of the higher modes is very small, as can be seen in Figure 5.5. Note that the generalized restoring force for the second mode is comparable in amplitude to that for the first mode.
The quality of the acceleration match of the two-mode model is illustrated in Figure 5. Some high-frequency discrepancies are partly due to the control and measurement errors during the test. It is important that the model does not correctly estimate the peaks of the measured displacement, as shown in Figure 5.9.
Two-Parameter Hysteretic Model
The general characteristics of the hysteresis loops are very similar to those of the four-parameter non-parametric model shown in Figure 5.6. This is due to the fact that the hysteretic nature of the response can be identified by the hysteretic model used herein. Based on the identified generalized restoring force and linear mode shape of the first mode, an estimation of the inter-story restoring force behavior is attempted.
The mass distribution and the mode shape of the first mode from references l17,22) are used. All the results clearly show that the two-parameter distributed element model gives an improved representation of the non-linear response of the test structure and provides a way to estimate the hysteretic behavior of the inter-story restoring forces. Considering the fact that the two-parameter distributed element model has fewer parameters than the four-parameter non-parametric model, it is thought that the main reason for this improvement is that the hysteretic nature of the response has been identified.
Summary
Tanaka, "A simulation of the earthquake response of steel buildings," Proceedings of the 6th World Conference on Earth-. Tanaka, 'Pseudo-dynamic tests on a two-storey steel frame by a hybrid system with computer load test equipment', Proceedings of the 7th World Conference on Earthquake Engineering, Istanbul, Turkey, September 1980. Tanaka, 'Pseudo-dynamic tests on frames including bolted joints with high strength", Proceedings of the 7th World.
Lee and LW Lu, “Design Studies of the Six-Story Steel Test Building: American-Japanese Cooperative Earthquake Research Program,” Report No. Jayakumar, “Analysis of Elastic Pseudodynamic Test Data of a Whole Steel Structure Using System Identification,” Proceedings of the 6th Joint Technical Coordinating Committee Meeting, US-Japan Cooperative Research Program Utilizing Large-Scale Testing Facilities, Maui, Hawaii, June 1985. Jayakumar," System Identification Applied to Pseudo-Dynamic Test Data: A Treatment of Experimental Errors", Proceedings of the 9th ASCE Engineering Mechanics Specialty Conference on Dynamic Response of Structures, University of California, Los Angeles, California, April.
Chapter 6 Conclusions
The parameters of the model are determined by approximating the generalized modal restoring force in the sense of least squares. The second and third are used to study the predictive ability of the identified model. However, the identified model provides a good estimate of the system's nonlinear stiffness and energy dissipation.
The backbone relationship of the model is characterized by only two parameters that are based on insights obtained from previous non-parametric studies. This model uses the results of the previous non-parametric identification as an initial estimate for the model parameters. The model predictions for the hysteretic characteristics of the response time history, including the permanent displacement, are.
