The topic of Random Vibrations is the behavior of structural and mechanical systems when they are subjected to unpredictable, or random, vibrations. These vibrations may arise from natural phenomena such as earthquakes or wind, or from human-controlled causes such as the stresses placed on aircraft at takeoff and landing. Study and mastery of this topic enables engineers to design and maintain structures capable of withstanding random vibrations, thereby protecting human life. Random Vibrations will lead readers in a user-friendly fashion to a thorough understanding of vibrations of linear and nonlinear systems that undergo stochastic-random-excitation. Provides over 150 worked out example problems and, along with over 225 exercises, illustrates concepts with true-to-life engineering design problems Offers intuitive explanations of concepts within a context of mathematical rigor and relatively advanced analysis techniques. Essential for self-study by practicing engineers, and for instruction in the classroom.

As understanding of the engineering design and configuration processes grows, the recognition that these processes intrinsically involve imprecise information is also growing. This book collects some of the most recent work in the area of representation and manipulation of imprecise information during the syn thesis of new designs and selection of configurations. These authors all utilize the mathematics of fuzzy sets to represent information that has not-yet been reduced to precise descriptions, and in most cases also use the mathematics of probability to represent more traditional stochastic uncertainties such as un controlled manufacturing variations, etc. These advances form the nucleus of new formal methods to solve design, configuration, and concurrent engineering problems. Hans-Jurgen Sebastian Aachen, Germany Erik K. Antonsson Pasadena, California ACKNOWLEDGMENTS We wish to thank H.-J. Zimmermann for inviting us to write this book. We are also grateful to him for many discussions about this new field Fuzzy Engineering Design which have been very stimulating. We wish to thank our collaborators in particular: B. Funke, M. Tharigen, K. Miiller, S. Jarvinen, T. Goudarzi-Pour, and T. Kriese in Aachen who worked in the PROKON project and who elaborated some of the results presented in the book. We also wish to thank Michael J. Scott for providing invaluable editorial assis tance. Finally, the book would not have been possible without the many contributions and suggestions of Alex Greene of Kluwer Academic Publishers. 1 MODELING IMPRECISION IN ENGINEERING DESIGN Erik K. Antonsson, Ph.D., P.E.

This volume provides useful tools in Lie group analysis to solve nonlinear partial differential equations. Many of important issues in nonlinear wave dynamics and nonlinear fluid mechanics are presented: Homotopy techniques are used to obtain analytical solutions; fundamental problems and theories in classic and quantum dynamical systems are discussed; and numerous interesting results about dynamics and vibration in sensor and smart systems are presented. Interval computation and nonlinear modeling in dynamics and control are also briefly included.

ERMR 2006 included invited speakers, technical presentations, poster presentations, and a student paper competition. At the conference banquet, Dr. David Carlson of Lord Corporation addressed the conference attendees and gave a stirring speech on the history of ER and MR fluids, as well as current and future applications. A unique feature of the ERMR Conferences is that they comprehensively cover issues ranging from physics to chemistry to engineering applications of ER and MR materials held in a general session to enhance the interaction between the scientists and engineers. The sessions in ERMR 2006 were organized based into two Symposia: a) Materials and b) Applications. Topics covered in the Materials Symposium included: mechanisms, preparation, and characterization of ER and MR materials. Topics covered in the Applications Symposium included: ER and MR devices, control systems, system integration, and applications. This structure was implemented in order to enable interaction between attending scientists and engineers in both the Materials Symposium and the Applications Symposium, and to enhance the free flow of ideas, and the potential collaborative research opportunities. Sample Chapter(s). Chapter 1: Transient Behavior of Electrorheological Fluids in Shear Flow (471 KB). Contents: The Physical Mechanism to Reduce Viscosity of Liquid Suspensions (R Tao); Polar Molecular Type Electrorheological Fluids (K Lu et al.); Yield Stress in Ferrofluids? (H Shahnazian & S Odenbach); The Effect of Dwell Time on the Rheological Behavior of MR Fluids (M Ahmadian & F D Goncalves); The Methods of Measuring Shear Stress of Polar Molecule Dominated ER Fluids (R Shen et al.); Electrosensitive Lubricants (E V Korobko et al.); Study on Characteristics of an Electrorheological Fluid Coupling (Y Meng et al.); On the Control of a MR Torque Transfer Device (M H Elahinia et al.); Comparison of ERF Clutch Designs (D J Ellam et al.); Examination of Throughflow in a Radial ERF Clutch (S M Chen et al.); Two-Layered Magnetic Fluid Sloshing in a Rectangular Container (S Yoshida et al.); Design of the High-Performance MR Brake and Its Characteristics (T Kikuchi et al.); and other papers. Readership: Mechanical engineers, electrical engineers, physicists, chemists, chemical engineers and materials scientists.

7. 1. 1 Background Uncertainty can be considered as the lack of adequate information to make a decision. It is important to quantify uncertainties in mathematical models used for design and optimization of nondeterministic engineering systems. In general, - certainty can be broadly classi?ed into three types (Bae et al. 2004; Ha-Rok 2004; Klir and Wierman 1998; Oberkampf and Helton 2002; Sentz 2002). The ?rst one is aleatory uncertainty (also referred to as stochastic uncertainty or inherent - certainty) – it results from the fact that a system can behave in random ways. For example, the failure of an engine can be modeled as an aleatory uncertaintybecause the failure can occur at a random time. One cannot predict exactly when the engine will fail even if a large quantity of failure data is gathered (available). The second one is epistemic uncertainty (also known as subjective uncertainty or reducible - certainty) – it is the uncertainty of the outcome of some random event due to lack of knowledge or information in any phase or activity of the modeling process. By gaining information about the system or environmental factors, one can reduce the epistemic uncertainty. For example, a lack of experimental data to characterize new materials and processes leads to epistemic uncertainty.