Keywords

1.1 Introduction

In the last 40 years, Ali H. Nayfeh has been the most influential, worldwide recognized, scholar and scientist in the area of nonlinear dynamics applied to mechanics and engineering. He embodied a rare mixture of scientific training and expertise/interests. Indeed, upon starting as a fluid dynamicist and an applied mathematician (60s and early 70s), he turned into a combined applied mathematician, dynamicist and physicist (70s and 80s) and then a comprehensive dynamicist, with resumed and enhanced attention to structural mechanics (90s and the new millennium). He worked in many scientific areas, dealing with perturbation techniques, nonlinear oscillations, aerodynamics, flight mechanics, acoustics, ship motion, hydrodynamic stability, nonlinear waves, structural dynamics, experimental dynamics, linear and nonlinear control, micromechanics, and providing outstanding contributions. During four decades he was also a brilliant educator for thousands of student, and wrote a meaningful number of important books.

Due to lack of space, it is impossible to dwell here on all aspects of Nayfeh’s scientific activity, thus choosing to solely focus on his contributions as a scholar and a successful books’ author, while leaving to a more extended paper [16] a general discussion on his many original contributions as a scientist.

Upon summarizing his life, publications and achievements [2, 4, 18] in Sect. 1.2, Nayfeh’s books are comprehensively discussed in Sect. 1.3, by analyzing specific features and novel aspects of a number of them, and by highlighting some under-lying common methodological features. Section 1.4 provides a summary portrait of Ali Nayfeh as an outstanding scholar and scientist in nonlinear dynamics, at the crossroad between applied mathematics and engineering.

1.2 Life, Publications, Achievements: A Short Summary

Ali H. Nayfeh was born on December 21, 1933 in Shuwaikah, Palestine, to an illiterate and poor family which, nonetheless, highly encouraged him to acquire education and maximum possible knowledge. Owed to the harsh conditions in his country and lack of higher-education institutes, Ali worked as a teacher of mathematics in small villages and towns for ten years. At the age of 26 he won a scholarship to study in the USA, and in only 5 years moved from a 1 year junior college up to getting Bachelor’s Degree in engineering science (1962), Master of Science (1963), and Ph.D. in Aeronautics and Astronautics (1964) at Stanford University.

For six years (1964–1970) he worked in the aerospace industry, upon which he was appointed at Virginia Tech as a Professor (1971) and then as a University Distinguished Professor (1976). Nayfeh:

  • Wrote: (i) 10 books in perturbation methods and nonlinear dynamics, several of them with thousands of citations, considered as the most valuable and fundamental references in their fields, translated into Russian, Chinese, and German, and used as textbooks in top schools; (ii) nearly 480 papers in refereed journals.

  • Was the Founder and Editor of two fundamental journals, Nonlinear Dynamics and Journal of Vibration and Control.

  • Advised 69 Ph.D. students: many of them became prominent scholars, department chairs and deans in top ranked institutes all over the world.

  • Established the highly successful series of 13 Conferences on Nonlinear Vibrations, Stability, and Dynamics of Structures at Virginia Tech (1986–2010).

  • Received honorary doctorates from Politechnika Szczecinska, Poland, Technical University of Munich, Germany, and Marine Technical University of St. Petersburg, Russia, along with an incredible number of Awards, including:

1995:

AIAA Pendray Aerospace Literature, “for seminal contributions to perturbation methods, nonlinear dynamics, acoustics, and boundary-layer transition.”

1996:

ASME Den Hartog, “for lifetime contributions to the teaching and practice of vibration engineering.”

2005:

ASME Lyapunov (first recipient), “for lifelong contributions to the field of nonlinear dynamics.”

2008:

ASME Thomas Caughey (first recipient), “for significant contributions to the field of nonlinear dynamics through practice, research, teaching, and outstanding leadership.”

2014:

Benjamin Franklin Medal in Mechanical Engineering, “for developing novel methods to model complex engineering systems in structural dynamics, acoustics, fluid mechanics, and electromechanical systems.”

Ali always aimed at contributing to the development of science in the Arab World. He established the college of engineering at King Abdel Aziz University in Jeddah, Saudi Arabia (1976), and an engineering college in Jordan where he served as Dean and Vice-president for engineering affairs for four years (1980). He also helped establishing a new graduate and internationally reputable program in mechanics in Tunisia (2002).

Upon retiring from Virginia Tech, he volunteered at the University of Jordan, under the condition of working without pay, helping scientists and researchers, and providing advice and consultations; he even established and funded a modern school in his birth village of Palestine, to offer the best education and produce a new generation of brilliant scientists.

Quoting from [18]: “Anyone who has met Dr. Nayfeh knows well his inexhaustible energy and deep desire for knowledge; and more importantly, his passion to share and spread his knowledge with others. … He was a brilliant scientist, a distinguished teacher, an inspiring motivator, a great community leader, and an amazing and lovely human. He will be truly and deeply missed.”

1.3 Nayfeh’s Books

Nayfeh wrote several books, generally clearly distinct from each other and of high impact. They are summarized in Fig. 1.1, by grouping them in three main areas, with well-identified and coherent features.

Fig. 1.1
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Ali Nayfeh’s books

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The first group is concerned with analytical methods, and includes all books on asymptotic techniques, with special emphasis on the method of multiple time scales (MMS) which was Nayfeh’s most originally addressed theme within the realm of applied mathematics. Within this overall context, the transition can be recognized from a theoretically-oriented approach [6] to a more introductory treatment of mathematical aspects [7], with also clear educational purposes [8], up to a modern popularization of the MMS [12] taking advantage of symbolic algebra, as well as the independent treatment of a method [9] playing an important role in nonlinear dynamics.

Books in the second group are more physically-oriented, and also more varied. Indeed, they range from the effective use of asymptotic techniques for the analysis of weakly nonlinear archetypal oscillators [13], to the presentation of computational and geometrical concepts, techniques and tools of the modern theory of dynamical systems also allowing to deal with strongly nonlinear and complex phenomena [11], up to the analysis of the involved interaction phenomena, regular or non-regular, which characterize the nonlinear dynamics of multi-mode models reliably representative of actual engineering systems [10].

The third group consists of a book [14] which, pushing forward the interest towards distributed parameter systems already apparent in [10], deals more generally with structural mechanics issues in terms of modelling, dynamic analyses, and exemplary phenomenological aspects.

In the following, the main features and novel aspects of some of these books with respect to their publication time are discussed, by citing a few author’s sentences from the corresponding prefaces (as per the writer’s choice) and by summarizing a few characterizing aspects.

1.3.1 Specific and Novel Features

Although being certainly linked with each other, Nayfeh’s two fundamental books on perturbation methods were driven by a different perspective.

  • Perturbation Methods (1973). “The different techniques are described using examples which start with model simple ordinary equations that can be solved exactly and progress toward complex partial differential equations”, with “examples drawn from different branches of physics and engineering”, and “the different techniques described as formal procedures without any attempt at justifying them rigorously.”

  • Introduction to Perturbation Techniques (1980). In contrast with the former book, where “coincise and advanced material” is dealt with, this second one presents the same material “in elementary way.” Indeed, “as a result of teaching perturbation methods for eight years to first-year and advanced graduate students at Virginia Polytechnic Institute and State University, I have selected a limited number of techniques and amplified their description considerably. Also I have attempted to answer the questions most frequently raised by my students. … A new chapter is devoted to the determination of the adjoints of homogeneous linear equations and the solvability conditions of linear inhomogeneous problems.”

However, Nayfeh’s research activity on mathematical aspects of ordinary differential equations was always paralleled by a deep interest towards physical aspects of the nonlinear oscillations described by those equations.

Nonlinear Oscillations (1979) was the comprehensive outcome of intense and extended applications of the multiple scale technique to obtain the solution of both archetypal oscillators (representing discrete systems, or simplified reduced models of continuous systems) and some multidegree-of-freedom models.

In the preface, referring to a comprehensive list of books on the same topic and generally with a similar title, it is stated that “the previously published books emphasized, and some exclusively treated, systems having a single degree of freedom”, whereas “the primary purpose of this book is to fill this void.” Specifically, Chap. 1 (Introduction) “attempts to abstract the entire book” and provides an useful and comprehensive overview of the addressed topics (according to a criterion later on adopted successfully in the introduction of other books), describing “only the physical phenomena, leaving all the algebra to the subsequent chapters.”

Seven chapters are devoted to Single-Degree-of-Freedom Systems (conservative, nonconservative, forced); Parametrically excited systems; Systems having finite degrees of freedom, with the treatment of internal resonance; Continuous systems (beams, string, plates); and Traveling waves. In particular, Chap. 7 “concentrates on the physical mechanisms and effects, restricting the attention to uniform systems with simple boundary conditions whose linear natural modes can be obtained analytically; then using the method of multiple scales to solve the equations describing the temporal functions”; Chap. 8 refers to “simple physical examples to explain nonlinear dispersive and nondispersive waves.”

Nonlinear Oscillations (1979) was translated in many languages, and became a fundamental classroom textbook in many academic institutions, summing up to 9402 citations (as of April 28, 2019, according to Google Scholar).

Between the 80s and the 90s, a substantial enlargement of research perspectives in nonlinear dynamics occurred within the community of applied mechanicians, with the attention being increasingly paid to strongly nonlinear dynamic phenomena. Nearly in parallel with such developments, the role played by experimental (physical) techniques for a reliable and exhaustive characterization of the dynamics of (mostly) multi/infinite-dimensional systems became fully apparent.

Although not providing specific foundational contributions to the development of these novel topics, Nayfeh swiftly grasped the importance of properly complementing the treatment of nonlinear oscillations via classical perturbation methods with the intensive use of advanced geometrical and computational techniques from the dynamical system theory also allowing to deal with strongly nonlinear and complex dynamics, along with the need to validate theoretical/numerical outcomes through experimental approaches.

Applied Nonlinear Dynamics (1995) testifies to Nayfeh’s great capability to grasp recent advancements and scientific trends, and timely make them available to people involved with research, and to practicing engineers working on challenging problems in applied mechanics.

Upon worthily summarizing the overall book contents in the introduction, seven chapters are dedicated to Equilibrium Solutions, Periodic Solutions, Quasiperiodic Solutions, Chaos, Numerical Methods, Tools to Analyze Motions, and Control. Specifically, quoting from ca.wiley.com website:

  • “Analytical approaches based on perturbation methods and dynamical systems theory are presented and illustrated through applications to a wide range of nonlinear systems.”

  • “Geometrical concepts, such as Poincaré maps, are treated at length, with a thorough discussion of stability and local and global bifurcation analyses for systems of differential equations and algebraic equations conducted with the aid of examples and illustrations. Continuation methods for fixed points and periodic solutions, and homotopy methods for determining fixed points, are detailed. Bifurcations of fixed points, limit cycles, tori, and chaos are discussed.”

  • Chaos is explored, with many routes treated at length, by also describing methods for “controlling bifurcations and chaos.”

  • “Numerical methods and tools (Poincaré sections, Fourier spectra, autocorrelation functions, Lyapunov exponents, dimension calculations)” for the analysis and characterization of motion in both the analytical and experimental context are presented.

Consistent with the awareness of being substantially a (though smart) user of techniques developed by others and with the willingness to provide an understandable framework to effectively implement them in applications, “proofs are not provided but references that provide them are included, some chapters (2, 3) are not written within a mathematically rigorous framework” [11], and many examples are used to explain the different concepts.

The last two books mark a progressive shift (or, indeed, an extension) of Nayfeh’s interests (already present in his earlier activity) towards applied dynamics problems in structural and mechanical engineering.

Nonlinear Interactions (2000) is built on the awareness that “an understanding of dynamic characteristics of a structural system is essential for its design and control.”

  1. (a)

    A variety of nonlinear interactions based on (2:1, 1:1, 3:1) autoparametric and combination resonances are addressed.

  2. (b)

    Another type of experimentally observed interaction between a directly excited high-frequency mode and a low-frequency mode, accompanied by a slow modulation of amplitude and phase of the former, is discussed. It is a mechanism of great practical importance, through which energy from high-frequency low-amplitude sources (such as rotating machineries, waves, propeller blades passing the rudder) can be transferred to large-amplitude low-frequency modes of supporting structures and foundations, resulting in their possible harmful oscillations, or may entail decrease of vibration of a main system and increase of its fatigue life to the expense of a sacrificial subsystem.

Upon the introduction, seven chapters include treatment of (5) Systems with widely spaced modes, (6) Multiple internal resonances, and (7) Nonlinear normal modes. Among meaningfully addressed topics, the following ones are mentioned.

  • Multidegree-of-freedom systems, with a marked interest to distributed parameter (structural) systems besides discrete (mechanical) ones.

  • Modulation equations: equilibrium solutions; stability and bifurcation analysis; dynamic solutions. This delineates a number of issues to be sequentially faced for a comprehensive nonlinear dynamic analysis of a generic system, according to a scheme later on adopted in hundreds of papers by other authors.

  • Confronting direct versus discretization approach in asymptotics.

  • Control strategies based on internal resonances and ensuing phenomena (e.g., saturation in 2:1 resonance).

  • Importance of higher-order approximations.

  • Comparing methods to obtain nonlinear normal modes for discrete and continuous systems (direct multiple scales, method of normal form, real- versus complex-valued invariant manifold approach).

Overall, the book is indeed a mine of information about the richness and variety of nonlinear phenomena, along with the theory behind them, and a valuable blend of author’s expertise on classical perturbation methods and nonlinear oscillations theory—revisited through symbolic algebra—with knowledge and tools from dynamical system theory and experimental nonlinear dynamics.

Twenty years after the publication of [13], the book can be considered as its continuation and update, and turns out to be “more in the genuine author’s attitude of mind than some other of his recent books” [15].

Linear and Nonlinear Structural Mechanics (2004) dwells on how the “nonlinear modeling and dynamic analysis of structures becomes a complex but important step in advancing the design and optimization of modern structural systems (with special attention to composites)” and aims “to close the gap between the practicing engineer and the applied mathematician in the modeling and analysis of geometrically nonlinear structures.”

  • “Mathematically consistent and systematic derivations of comprehensive and refined structural theories” of strings, cables, beams, plates and shells, both exact and approximate, are presented, also including laminates, integration with piezoelectric materials, thermoelasticity, and microbeams/plates.

  • “Physical meaning of linear and nonlinear structural mechanics” is detailed.

  • “Ready-to-use governing equations and boundary conditions” are provided, “ranging from simple linear to complex nonlinear.”

  • Exemplary nonlinear structural analyses of refined and/or reduced order models are summarized.

  • Some main treatments and outcomes of linear/nonlinear dynamics for beams, plates and shells are obtained, referring to refined and/or reduced order models.

1.3.2 Some Characterizing Methodological Aspects

Some main methodological aspects common to all Nayfeh’s books are highlighted and illustrated with examples in the following. His overall production exhibits four basic characteristics.

  1. 1.

    Topics are addressed and presented based on an incremental/additive (and overall inductive) perspective, via a series of case histories.

  2. 2.

    Specific outcomes (about analytical techniques, different methods and/or systems) are embedded into a unified, comprehensive, and comparative framework.

  3. 3.

    Concepts are illustrated with numerous examples and many exercises aimed to reinforce and assess progress in understanding.

  4. 4.

    Extended and updated bibliographies are provided.

The incremental/additive perspective can be recognized in the contents of anyone of Nayfeh’s books. Figure 1.2 shows two examples [13] concerned with the forced oscillations of systems with a single or finite number of degrees of freedom. Considering “simple systems that exhibit the essential ideas, instead of treating general systems for which the algebra is involved” produces lists of case studies of progressively increased difficulty. Basic nonlinearities (cubic, or quadratic and cubic) and self-sustained oscillations are addressed in the first example, considering different resonant (primary, subharmonic, superharmonic, combination, simultaneous) or non-resonant excitations. In turn, the second example refers to diverse physical systems and deals with various cases of resonance between excitation and system natural frequencies for those having quadratic nonlinearities. The adopted scheme is certainly repetitive. Yet, while being somehow encyclopedic, the underlying comprehensive perspective allows to easily point out differences and peculiarities of distinct nonlinear aspects of both the analytical treatment and the considered systems/excitations. This results in a meaningful amount of information about the richness and variety of nonlinear phenomena, and the theory behind them.

Fig. 1.2
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Incremental/additive approach in the contents of [13]

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A similar example (Fig. 1.3) is taken from [9], where the general idea of the method of normal forms to “use a ‘local’ (i.e., near-identity) coordinate transformation to ‘simplify’ the equations describing the dynamics of the system under consideration” is applied to parametrically excited systems.

Fig. 1.3
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Incremental/additive approach in the contents of [9]

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Two more incremental/additive examples (Fig. 1.4) refer to topics addressed by Nayfeh later on in the more engineering oriented stage of his activity, and also highlight the inductive character of his approach: the same form of modulation equations is recognized for systems belonging to a certain group of symmetry [10], and methods of linear/nonlinear structural mechanics can be readily extended to different and/or more complex structures [14].

Fig. 1.4
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Incremental/inductive approach in [10, 14]

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Overall, as explicitly declared in the prefaces of all Nayfeh’s books (starting with the more theoretical one [6]), the material is not presented “within a mathematically rigorous framework.” This originated criticisms from more dynamical system-oriented (and generally more rigorous) scientists. But this characterizing feature of Nayfeh’s scientific personality ensued from his being basically a scholar in engineering sciences capable of exploiting a strongly founded knowledge of fundamentals of applied mathematics to understand the nonlinear behavior of involved mechanical/structural systems; this being the actual main focus of his research interests. The inductive approach adopted by Nayfeh in his presentations of scientific material was likely brought to its extreme consequences. Yet, it was somehow in line with the perspective adopted by most of the earlier scientists active in nonlinear dynamics (including mathematicians), who “were not led to their discoveries by a process of deduction from general postulates or principles, but rather by a thorough examination of properly chosen particular cases”, with the generalization coming later “because it is far easier to generalize an established result than to discover a new line of argument” [5]. In this sense, Nayfeh’s approach was not too far from the purpose of studying concrete nonlinear systems with their “natural effects”, which also inspired earlier theoretical research in nonlinear dynamics at Moscow [1] and Kiev [3] schools.

It is anyway important to notice that such a ‘list-looking’ aspect of his presentations was constantly paralleled in all books by the successful attempt to achieve a unified, comprehensive, and comparative framework into which presenting the various case-studies and highlighting the relevant differences and peculiarities. Sentences from the prefaces of some Nayfeh’s books quoted in Fig. 1.5 witness the care he always paid to the matter.

Fig. 1.5
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A unified framework

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Two more aspects of Nayfeh’s fully effective educational and professional perspective as a book author are to be mentioned. Concepts were always illustrated through worked-out examples and exercises useful to reinforce understanding and assess progress of students. Moreover, very rich and (for their time) updated bibliographies were provided at the end of each book. They are still invaluable sources of information/knowledge about ‘classical’ and more recent books and journal/conference papers, for scholars in nonlinear dynamics. This is even more important in the current time in which scientific problems are sometime ‘rediscovered’, in a context of overall minor care paid to the historical evolution and advancement of science by some young scientist mostly aimed at attaining specific, and also possibly limited, goals.

1.4 Complementing Mathematical and Engineering Approaches to Nonlinear Systems

A fundamental part of Ali Nayfeh’s legacy consists of the conceptual, and mostly operational, contributions to the method of multiple time scales provided at the beginning of his scientific activity, and later on expanded in the direction of frequently needed higher-order expansions and in terms of popularization. As a matter of fact, multiple time scales are continuously being used by generations of scientists for solving nonlinear dynamics problems. This holds irrespective of mathematical warnings raised about pitfalls possibly occurring in the asymptotics of the MMS owed mostly to the anticipated choice of timescales, a feature which does not occur in the averaging method where they emerge by nonsecularity conditions without any a priori assumption [17]. But Nayfeh was most likely aware of those pitfalls. This is witnessed both by the reference constantly made to the need to properly select timescales (including fractional ones) depending on the dynamic problem at hand, and by his many performed comparisons of equivalent approximations provided by multiple scales and averaging. It is however to be noticed that Nayfeh’s enormous skill and experience about how properly selecting a priori the timescales necessary for a reliable asymptotic solution of a given problem ensue from exceptional personal features in terms of intuition and overall understanding which are not in the patrimony of all scientists.

As a matter of fact, looking at his outstanding scientific activity on a huge variety of themes [16], Nayfeh’s main nature of a scholar in engineering sciences could be further highlighted. While being well-acquainted with applied mathematics, he was fundamentally interested in using his skill and expertise to catch the nonlinear behavior of mechanical and structural systems. Indeed, moving from his earlier achievements on the nonlinear oscillations of simple models, in the second part of Nayfeh’s academic life further distinguishing and highly influential features in the realm of applied mechanics and engineering emerged clearly. Therein, with an unrivaled capability to grasp novel advancements and research trends, he succeeded in remarkably complementing a profound expertise on asymptotic methods and nonlinear oscillations theory, revisited through modern symbolic algebra, with the effective interpretation and organization of experimental outcomes, both in-house and from the literature, along with the smart use of knowledge and tools from modern dynamical system theory.