7R9. Nonlinear Interactions: Analytical, Computational, and Experimental Methods. - AH Nayfeh (VPI, Blacksburg VA). Wiley, New York. 2000. 782 pp. ISBN 0-471-17591-9. $110.00.

Reviewed by RA Ibrahim (Dept of Mech Eng, Wayne State Univ, 5050 Anthony Wayne Dr, Rm 2119 Engineering Bldg, Detroit MI 48202).

Professor Ali Nayfeh is a well-known authority in the area of nonlinear dynamics and has an impressive record of research monographs and hundreds of published papers. This book comes as a result of the growing research interest in the nonlinear modal interaction in dynamical systems under various types of excitations and resonance conditions. The book presents the-state-of-the-art in eight chapters supported by an extensive bibliography. Chapter 1 serves as an essential foundation and presents basic ingredients of analyzing nonlinear-coupled differential equations.

The analysis of coupled systems with two-to-one internal resonance (essentially dictated by quadratic nonlinearity) under external or parametric excitation is outlined in Chapter 1. It describes some physical systems that may possess 2-1 internal resonance such as a double pendulum, a planar elastic pendulum, autoparametric vibration absorber, ships constrained to pitch and roll interaction, parametric excitation of liquid sloshing modes, cylindrical shells, and other systems. The free oscillations of coupled undamped and damped nonlinear modes are described using multiple scales method. The concept of autoparametric vibration absorber, and the associated saturation phenomenon, is used to establish a specific control strategy. The second and first mode excitations, when the second mode frequency is close to twice the first mode frequency, are then examined for the solution and stability of fixed-point solution. The existence of multiple solutions in the form of jump and hysteresis is discussed in parallel to experimental results in terms of response amplitude dependence on excitation amplitude, and in the frequency domain. The chapter is extended to include simultaneous parametric and internal resonance conditions. Three cases of parametric excitation that include the second, first, and combination modes are considered. The last two sections of this chapter treat the influence of nonlinear damping and cubic nonlinearity on the saturation response characteristics of two-to-one internal resonance.

Chapter 3 presents the problem of one-to-one internal resonance encountered in nonlinear systems with symmetry such as the spherical pendulum, taut strings, axisymmetric shells, surface liquid sloshing in near-circular or near-square tanks, beams with near-circular or near square cross-sections, and near-circular or near-square plates and membranes. The analytical modeling of such systems is first developed then the work systematically proceeds by analyzing the response characteristics. The analysis includes primary resonance excitation in the time and frequency domains and the associated limit cycles represented in the phase plane. The fundamental parametric resonance in two- and three-degree-of-freedom, non-semi-simple, systems possessing quadratic and cubic nonlinearities is treated in the neighborhood of one-to-one internal resonance. Bifurcation diagrams indicating the dependence of different equilibrium solutions on the system quadratic and linear parameters are presented for different nonlinear systems. The chapter is closed by treating the case of combined parametric and external excitations of a taut string oscillating in the transverse planes for the periodic solutions which correspond to fixed-point solutions of the modulated equations. While nonlinear systems with cubic nonlinear coupling can experience one-to-one internal resonance, they also can exhibit three-to-one internal resonance. This case is considered in Chapter 4, which introduces some physical classical systems such as a double pendulum with a moving support, a hinged-fixed beam, rods subject to longitudinal excitation, and clamped-clamped buckled beams. The modulated equations using different techniques are derived for each system. The equilibrium solutions of the modulated equations are obtained and represented in the frequency domain for different values of system and external excitation parameters. The bifurcation diagrams demonstrating the relative locations of different fixed-points are plotted for specific system parameters. The corresponding phase portraits and time history records for different response regimes are presented and discussed in a coherent format. The studies include different cases of external and parametric excitations of different modes.

Chapter 5 deals with nonlinear modal interactions resulting from combination parametric, external, or internal resonances. Different classifications of combination resonances considered in the literature are reviewed. Experimental observations of selected two-, three-, and four-degree-of-freedom systems are then described in terms of frequency and excitation amplitude domains. The analyses of combination parametric, external sub-combination, combination internal, sub-combination internal resonances of selected systems are presented for the modulated equations, fixed points, and their stability.

The case of nonlinear modal interaction in the form of energy transfer from high-frequency to low-frequency modes is considered in Chapter 6. The mechanism responsible for this energy transfer is neither an autoparametric resonance, an external combination resonance, nor a parametric combination resonance. The mechanism is rather due to the interaction between slow dynamics of the high-frequency mode, represented by its amplitude and phase, with the dynamics, which is slow, of the low-frequency mode. This mechanism may be viewed as a one-to-zero internal resonance. The experimental evidence of this type of interaction is demonstrated for the cases of a parametrically excited cantilever beam, a transversely excited cantilever beam, frames, composite plates, and an externally excited circular rod. The analyses of these systems are then presented using different methods to derive the modulated equations, which are studied for fixed points and their stability. The response characteristics are presented in the frequency domain together with the boundaries between constant and oscillatory motions. The dynamics of the system in the neighborhood of unstable foci are considered for some cases. Typical and selected nonlinear phenomena, such as period-doubling bifurcations culminating in chaos, symmetry breaking bifurcations, the coexistence of multiple attractors, and the merging of attractors are included in this chapter.

Chapter 7 presents a more complicated form of nonlinear modal interaction which occurs when simultaneous internal resonances exist in the system dynamics. The treatment includes multiple two-to-one internal resonances in three-degree-of-freedom systems with quadratic nonlinearities, and four simultaneous internal resonances in suspended cables. The modulated equations of such systems are solved for fixed points and their stability. The bifurcation analysis of the fixed-point solutions is presented in terms of internal detuning parameters and excitation amplitude.

Chapter 8 addresses the problem of nonlinear normal modes and modal localization. The basic concepts of nonlinear normal modes according to RM Roenberg, and others, are introduced. Basically, nonlinear normal modes may be viewed as synchronous periodic solutions of the system nonlinear equations of motion. The estimation of nonlinear normal modes of different multi-degree-of-freedom and continuous systems constitutes the main theme of this chapter. Nonlinear normal modes of systems with geometric nonlinearities in the presence of internal resonance are analyzed using the complex-variable form of the invariant-manifold approach. The author documents those methods recently developed by others and reported in the literature.

Having read this review, this reviewer realized that the author has written not a book or a reference, but an encyclopedia on nonlinear modal interactions. Nonlinear Interactions: Analytical, Computational, and Experimental Methods serves both beginners and specialists in the area of nonlinear dynamics. The book is well written, self-contained, and coherent.