Uncertainty Modeling in Vibration, Control and Fuzzy Analysis of Structural Systems

The following sections are included:

• STABILITY, VIBRATION AND CONTROL OF SYSTEMS

• Foreword

• Preface

• Contributors

• Contents

Methods for the analysis of structural systems under stochastic dynamic loading are presented. First a brief review of the theoretical background along with procedures to simulate stochastic processes is given. In a next step the methodology for calculating the response of MDOF-systems under stochastic loading is discussed. This section closes with a treatment of procedures of non-linear stochastic dynamics where it is focussed on larger systems as generally encountered in engineering practice. The efficiency of the various computational procedures is highlighted.

An approximate method to account for the effect of clumping of large peaks on first passage times and extreme values of narrow-band random vibrations, both Gaussian and non-Gaussian, is proposed. The method is based on the concept of joint crossing rates of a stochastic process. This makes it possible to introduce a correlation structure to the sequence of peak values, allowing the introduction of an approximate estimate of the effect of clumping on large excursions of the underlying narrow-band process. The advantage of the proposed method is that explicit, closed-form expressons for the clumping effect on first passage times and extreme values are obtained. The method is illustrated by application to specific examples.

A separatrix in the state space of a dynamical system is a boundary between regions of qualitatively dissimilar modes of dynamics. In the presence of weak external forcing the separatrix can be breached, allowing the system to change dynamical modes. Such breaches are a root cause of chaos in externally forced, continuous-time systems. The flux factor, a leading order expression of the state space flux across the separatrix, measures the “propensity” for chaotic dynamics. Two new uses of the flux factor are given. In the first of these, the flux factor is an objective function for a control against crossing a separatrix to maintain a system within a potential well. In the second, the flux factor is the basis of a measure of the relative stability of competing potential wells. This measure of relative stability is used to formulate and prove a “blowtorch” theorem for potential wells.

The method of equivalent statistical linearization, commonly used in frequency domain analyses of nonlinear stochastic systems, fails to capture the non-Gaussianity of the response of such systems in terms of its higher-order statistics. In addition, response energy in frequency ranges outside that of the input spectrum is not observed using this technique. Herein, methods of equivalent statistical quadratization and cubicization are proposed, whereby nonlinearities in the excitation of practical systems are cast in an equivalent quadratic or cubic form rather than in an equivalent linear form. The present techniques take advantage of the Gaussianity of the first order response (when the system input is Gaussian) to simplify the recasting of the nonlinearity in its approximate polynomial form. A Volterra series approach in the frequency domain leads to the development of transfer functions from which the response Spectrum as well as statistics of the response may be obtained. Response cumulants, computed up to the fourth order via direct integration or the Kac-Siegert technique, reveal the non-Gaussian character of the response and, when used in the framework of some available non-Gaussian probability density function models, indicate acceptable agreement with time-domain simulations of the original nonlinear differential equations. In addition, approximations for the response power spectral densities reveal spectral peaks outside the range of input frequencies which are brought about by nonlinear effects that are undetectable using linearization techniques. This, too, corroborates information gained from time-domain simulations.

The motivations for a non-standard form of structural control in civil engineering are illustrated with reference to environmental actions like wind and earthquake loading. Fuzzy control is robust, efficient and can be implemented as an adaptive strategy that can react to changes in system parameters. The theory underlying the fuzzy inference scheme is explained in light of the implementation of a control device using displacement and velocity feedback and producing a crisp control force. This paper summarizes the results of ongoing research aimed at the implementation of Fuzzy Logic Controllers (FLC) in active structural control. The numerical applications cover linear and nonlinear systems with one or several degrees of freedom.

The effect and handling of uncertainties is an important topic in the control of civil engineering structures. There exist uncertainties in (1) evaluating the condition and resistance of the structure and (2) predicting future loads. Because the consequence of any unsuitable control strategy for serviceability is not critical, applications of structural control in buildings have been implemented mainly for comfort purposes to date. On the other hand, active control of structures for safety purposes would involve much larger forces; the effect of uncertainties in such cases requires further investigation. The authors describe the results of some research that has been published in this area and outline open questions that need to be examined.

A general framework for structural fuzzy control is proposed. The proposed framework is organized in three consecutive stages system definition, state evaluation and fuzzy control. The first stage involves the development of a generic framework for system modeling and identification which should serve as a guide for structural system identification. The framework is capable of reflecting the same structural system in several forms based on its nature, the control scheme objectives and its behavior. The second stage defines a generic scheme for state evaluation. The proposed scheme provides a guide for several state evaluation techniques based on the system model and its properties. Because of the complexity and uncertainty involved in structural systems, fuzzy-based control is considered a potential candidate for structural control. The proposed system is capable of monitoring and controlling several control attributes. The main objective is to identify and improve variables responsible for any unsatisfactory state condition. The system also includes a self learning unit that is capable of defining and extracting new rules in order to improve the system's performance. The proposed concepts are applied to several structural systems with different control objectives in order to demonstrate the generality of the framework.

In this chapter, an attempt is made to apply the fuzzy logic to structural vibration control of earthquake-excited and wind-excited oscillations. The numerical examples of single-degree-of-freedom and two-degree-of-freedom models show that the fuzzy control technique is useful for reducing the amplitude and acceleration of structural oscillation due to earthquakes. Experiments on structural vibration control are conducted using a system of prestressing tendons with spring connected to a stepping motor system with fuzzy control rules. The model is subject to base motions produced by a small shaking table, and a fluctuating flow generated by a fan. These experimental results confirm the efficiency of fuzzy control for structural vibration problems.

Thus, it has been proven that fuzzy active control can provide practical and robust control of structural vibration with simple algorithms. However, in order to make the fuzzy active controller more practical, it is necessary to tune the parameters and shapes of membership functions used in the fuzzy inference rules. In this chapter, an attempt is also made to apply the technique of genetic algorithms (GA) for the self-tuning of the inference rules and their parameters. To demonstrate the applicability of GA in the self-tuning, an experiment is conducted on a single degree of freedom vibration model with a simple pendulum actuator.

Recently, probabilistic methods have been used extensively to model uncertainty in many design optimization problems. An alternative is to use fuzzy set based methods for modeling uncertainties. These methods use possibility theory for assessing the safety of a system. Fuzzy set based methods usually require much less information than probabilistic methods and they can utilize expert opinion. The first objective of the study described in this chapter is to understand how each approach maximizes safety. The second objective is to experimentally compare designs obtained using each method. A cantilevered truss structure is used as a test case. The truss is equipped with passive viscoelastic tuned dampers for vibration control. The structure is optimized by selecting locations for tuning masses added to the truss. The design requirement is that the acceleration at given points on the truss for a specified excitation be less than some limit. The properties of the dampers are the primary sources of uncertainty. They are described by their probability density functions in the probabilistic analysis and by their possibility functions in fuzzy sets based analysis.

A pair of alternate optimal designs are obtained from the probabilistic and the fuzzy set based optimizations, respectively. The probabilistic optimization minimizes the system probability of failure. Fuzzy set based optimization minimizes the system possibility of failure. Problem parameters (e.g., upper limits on the acceleration) are selected in a way that the probabilities of failure of the alternate designs differ significantly, so that the difference can be measured with a relatively small number of experiments in the laboratory.

The main difference in the way each method maximizes safety is that probabilistic optimization tries to mostly reduce the probabilities of failure of the modes that are easier to control, whereas fuzzy set based optimization tries to equalize the possibilities of failure of all failure modes.

The resulting optimum probabilistic and fuzzy set based designs are compared in the laboratory. The results confirm that, for the selected problem, probabilistic methods can provide designs that are significantly more reliable than designs obtained using fuzzy set based methods.

This chapter discusses the application of modern fuzzy logic control of mechanical structures. An overview of intelligent control with emphasis on fuzzy logic control design techniques is given. Then, a fuzzy logic controller was designed and employed to suppress the unwanted vibrations of a rotating elastic beam resulting after the impact with an external rigid body. The momentum balance method and an empirical coefficient of restitution was used in the collision of the two bodies. It is seen that the controller can be applied successfully to control the impact between two bodies.

The following sections are included:

• Author Index

• Subject Index

• About the Editors