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Random Vibration Mechanical, Structural and Earthquake Engineering Applications by Zach Liang and George C. Lee
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Random Vibration Mechanical, Structural and Earthquake Engineering Applications by Zach Liang and George C. Lee

PREFACE to Random Vibration Mechanical, Structural and Earthquake Engineering Applications

Understanding and modeling a vibration system and measuring and controlling its oscillation responses are important basic capacities for mechanical, structural, and earthquake engineers who deal with the dynamic responses of mechanical/structural systems.

Generally speaking, this ability requires three components: the basic theories of vibrations, experimental observations, and measurement of dynamic systems and analyses of the time-varying responses.

Among these three efforts, the former two are comparatively easily learned by engineering students. However, the third component often requires a mathematical background of random processes, which is rather abstract for students to grasp.

One course covering stochastic processes and random vibrations with engineering applications is already too much for students to absorb because it is mathematically intensive and requires students to follow an abstract thinking path through “pure” theories without practical examples.

To carry out a real-world modeling and analysis of specific types of vibration systems while following through the abstract pure thinking path of mathematical logic would require an additional course; however, there is no room in curriculums for such a follow-up course.

This has been the observation of the first author during many years of teaching random vibration. He frequently asked himself, How can one best teach the material of all three components in a one-semester course?

The authors, during the past 20 years, have engaged in an extensive research study to formulate bridge design limit states; first, for earthquake hazard and, subsequently,

expanded to multiple extreme natural hazards for which the time-varying issue of rare-occurring extreme hazard events (earthquakes, flood, vehicular and vessel collisions, etc.) had to be properly addressed.

This experience of formulating real-world failure probability–based engineering design criteria provided nice examples of using the important basic ideas and principles of random process

(e.g., correlation analysis, the basic relationship of the Wiener–Khinchine formula to transfer functions, the generality of orthogonal functions and vibration modes, and the principles and approaches of dealing with engineering random process).

We thus decided to emphasize the methodology of dealing with random vibration. In other words, we have concluded that it is possible to offer a meaningful course in random vibration to students of mechanical and structural engineering by changing the knowledge-based course approach into a methodology-based approach.

The course will guide them in understanding the essence of vibration systems, the fundamental differences in analyzing the deterministic and dynamic responses, the way to handle random variables, and the way to account for random process.

This is the basic approach that underlines the material developed in this Random Vibration Mechanical, Structural and Earthquake Engineering Applications book.

By doing so, we give up coverage of the rigorous mathematical logic aspect and greatly reduce the portion of random process.

Instead, many real-world examples and practical engineering issues are used immediately following the abstract concepts and theories.

As a result, students might gain the basic methodology to handle the generality of engineering projects and develop a certain capability to establish their own logic to systematically handle the issues facing the theory and application of random vibrations.

After such a course, students are not expected to be proficient in stochastic process and to model a random process, but they will be able to design the necessary measurement and observation,

to understand the basic steps and validate the accuracy of dynamic analyses, and to master and apply newly developed knowledge in random vibrations and corresponding system reliabilities.

With this approach, we believe it is possible to teach students the fundamental methodology accounting for random data and random process and apply them in engineering practice.

This is done in this Random Vibration Mechanical, Structural and Earthquake Engineering Applications book by embedding engineering examples wherever appropriate to illustrate the importance and approach to deal with randomness.

The materials are presented in four sections. The first is a discussion of the scope of random process, including engineering problems requiring the concept of probability to deal with.

The second is the overview of random process, including the time domain approach to define time-varying randomness, the frequency domain approach for the spectral analysis, and the statistical approach to account for the process.

The third section is dedicated specifically to random vibrations, a typical dynamic process with randomness in engineering practice.

The fourth section is the application of the methodology. In recent years, we used typical examples of developing fatigue design limit states for mechanical components and reliability-based extreme event design limit states for bridge components in teaching this course.

The nice performances and positive responses of the students have encouraged us to prepare this manuscript. Section I consists of two chapters.

Chapter 1 expresses the brief background and the objectives of this Random Vibration Mechanical, Structural and Earthquake Engineering Applications book, followed by a brief review of the theory of probability within the context at application to engineering.

The attempt is to only introduce basic concepts and formulas to prepare for discussions of random process. The review of the theory of probability is continued in Chapter 2,

with focus on treating random data measured as a function of certain basic random distributions for randomness in their actual applications.

This will also help engineers to gain a deeper understanding of the randomness in sequences. In this section, the essence of probability as the chance of occurrence in sample space,

the basic treatment to handle one-dimensional random variables by using two-dimensional deterministic probability distributions (PDF), and the tools to study the random variables of averaging (statistics) that changes quantities from random to deterministic are emphasized.

Two important issues in engineering practice, the uncertainty of data and the probability of failure, are introduced.

Section II begins with Chapter 3, where the random (also called stochastic) process is introduced in the time domain. The nature of time-varying variables is first explained by joint PDF through the Kolmogorov extension.

Because of the existence of the indices in both sample space and in the time domain, the averages should be well defined, in other words, the statistics must be used in rigorous conditions by identifying if the process is stationary as well as ergodic.

Although the averaged results of mean and variance are often easily understandable, the essence of correlation analysis is explained through the concept of function/variable orthogonality.

In Chapter 4, random process is further examined in the frequency domain. Based on the Wiener–Khinchine relations, the spectral analyses on the frequency components of the deterministic power spectrum density function of random process are carried out.

In these two chapters, several basic and useful models of random process are discussed. In Chapter 5, a new set of statistics for random processes that is different from averaging of the entire process is introduced, such as level crossing, peaks, and maxima.

To further understand the important engineering problems of cumulated damage, the Markov process is introduced, which is a continuous approach of introducing random processes based on engineering motivations.

Thus, due to the nature of random processes, which consists of a broad range of rather different types of mathematical models, to introduce each special process one by one is not an effective approach for students to learn.

This Random Vibration Mechanical, Structural and Earthquake Engineering Applications book employs an approach to present important processes within the context of practical engineering problems,

whereas the generality of dealing with randomness and the difference between random variables and processes are included.

Necessary mathematical logic, such as limits, differentiation, and integration on random variables is only considered for the purposes of understanding the nature of randomness.

Section III of this Random Vibration Mechanical, Structural and Earthquake Engineering Applications book focuses on the topic of vibration problems.

The basic concept is reviewed in Chapter 6, where the essence of vibration is emphasized based on energy exchange.

The basic parameters of the linear single-degree-of-freedom (SDOF) system are discussed, followed by the key issues of dynamic magnification factors, convolutions, and transfer functions.

The topic in Chapter 7 is on SDOF systems excited by random initial conditions and forcing functions. Together with the aforementioned correlation and spectral analyses, a new method of random process referred to as time series is also described.

In Chapter 8, the discussion is extended to linear multi-degree-of-freedom (MDOF) systems. The statistical analyses of direct approach based on model decoupling of proportionally and nonproportionally damped systems are discussed,

along with basic knowledge of eigenparameters, Rayleigh quotient, state variables, and equation and transfer function matrices.

Engineering examples of how to deal with random excitations, such as earthquake response spectrum and various types of white noises are considered for students to further gain insight into random processes and particular random vibrations.

Vibration is a special dynamic process and possesses time histories, whereas a random process is also a dynamic process. In the third section of this Random Vibration Mechanical, Structural and Earthquake Engineering Applications book,

we not only present the generality of these dynamic processes but also treat the vibration response as the output of a second-order linear system due to the input of a random process.

Section IV and the last part of the Random Vibration Mechanical, Structural and Earthquake Engineering Applications book provides more materials on the applications of random process and vibration.

Chapter 9 is especially dedicated to inverse problems, which are limited to system and excitation identifications.

In engineering practice, inverse problems can be much more difficult to solve, both due to possible dimension reductions and noise contaminations.

Measurement and testing, especially on vibration systems, should deal with uncertainties. Based on the methodology learned from previous chapters, statistical studies on random data and model identifications are discussed.

In Chapter 10, the failure of systems is further discussed in a more systematic fashion, followed by the concept of reliability.

For mechanical engineering applications, high cycle fatigue failure is further considered as a continuation of the topic in Chapter 5.

For structural engineering application, the example of load-and-resistance-factor-design under multiple hazard load effects are considered to explain how to deal with load combinations of several random processes, which is essentially different from the currently used bridge code based on bridge reliability.

In Chapter 11, nonlinear vibration with random excitation is briefly considered, along with an introduction of linearization procedure.

Again, the purpose of this chapter is not to systematically describe the system and response nonlinearity. Rather, it is intended to explain the nonlinear phenomena and the general approach of linearization.

In addition, a special method of Monte Carlo simulation is considered as a tool to study complex systems and their responses.

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