Nonlinear Interactions: Analytical, Computational, and Experimental Methods

Abstract

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 t