Vibration and stability. Advanced theory, analysis, and tools. 2nd ed.

*(English)*Zbl 1086.70001
Berlin: Springer (ISBN 3-540-40140-7/hbk). xxi, 404 p. (2003).

The introductory two chapters, “Vibration basics and eigenvalue problems of vibrations” and “Stability”, give surveys of fundamental concepts, methods and phenomena associated with vibrations (single degree-of-freedom systems, multiple degree-of-freedom systems, continuous systems and energy methods for setting up equations of motion), and present formulations and solution methods of eigenvalue problems (EVP) on the determination of critical buckling loads and modes using the theory of elastic stability (algebraic and differential EVP, stability-related EVPs, vibration-related EVPs, concepts of differential EVPs, properties of eigenvalues and eigenfunctions, methods of solution).

Ch. 3, “Nonlinear vibrations: Classical local theory”, provides classical tools for the local analysis of nonlinear vibrations. A nonlinear system typically has several states of static and dynamic equilibrium. Local theories concern solutions in an immediate vicinity of such states. Sources of nonlinearity are explained here by geometrical or material nonlinearities, or are associated with nonlinear forces or with physical configuration. Rich dynamic behaviour of vibrating systems is demonstrated on the main example: pendulum with oscillating support. Ch. 3 presents also qualitative analysis of the unforced response and the corresponding quantitative analysis methods (perturbation methods, straightforward expansion, multiple scales harmonic balance and averaging methods). On examples of pendulum equation with damping and forcing, and on externally excited Duffing systems, the author illustrates the construction of approximate nonlinear solutions for non-resonant and resonant cases together with their stability by perturbation and averaging techniques.

Ch. 4 is devoted to local analysis of nonlinear two-degree-of-freedom systems with nonlinear interaction of modes. This problem is discussed on the examples of autoparametric vibration absorber – a device purposefully designed to operate under conditions of combined internal and external resonance. The nonlinear dynamics of a non-shallow arch serves as an example of a structure for which internal resonance is more or less unavoidable. The follower-loaded double pendulum, displaying rich dynamics, allows to demonstrate how to employ multiple scales perturbation for systems written in the first-order matrix form. Here, the author investigates also nonlinear interactions associated with vibration-induced sliding of mass for the problems of pendulum with a sliding disc, string with a sliding point mass and vibration-induced fluid flow in pipes.

Ch. 5 contains a review of bifurcation analysis for general dynamic systems described by autonomous first-order differential equations. Codimension-one bifurcations of equilibriums for one- and \(n\)-dimensional systems (saddle-node, transcritical and pitchfork bifurcations), center manifold and normal form reductions in steady-state and dynamic bifurcation theory are presented here. Methods for the investigation of bifurcating periodic solutions and stability of bifurcation to perturbations are discussed with application to problems considered in Ch. 3 and 4.

Ch. 6 provides an introduction to chaotic vibrations with focusing on practical implications and applications. It considers tools for detecting chaotic vibrations, universal routes to chaos, tools for predicting the onset of chaos and examples of mechanical systems with chaos (Lorenz system, Duffing-type, pendulum-type, piecewise-linear, coupled autonomous, autoparametric and high-order systems). Elastostatic chaos, spatial and spatiotemporal chaos and controlling chaos problems are shortly discussed. In Ch. 7 on various examples the author discusses special effects of high-frequency excitation of mechanical systems. Finally, appendices present practical aspects of numerical simulations, the major exercises (tension control of rotating shafts, vibrations of a spring-tensioned beam, dynamics of microbeam), mathematical formulas used in perturbation and stability analysis, vibration modes and frequencies for structural elements, and properties of engineering materials. Every chapter is equipped by useful exercises.

The reviewed book will be very useful in engineering and scientific practice.

Ch. 3, “Nonlinear vibrations: Classical local theory”, provides classical tools for the local analysis of nonlinear vibrations. A nonlinear system typically has several states of static and dynamic equilibrium. Local theories concern solutions in an immediate vicinity of such states. Sources of nonlinearity are explained here by geometrical or material nonlinearities, or are associated with nonlinear forces or with physical configuration. Rich dynamic behaviour of vibrating systems is demonstrated on the main example: pendulum with oscillating support. Ch. 3 presents also qualitative analysis of the unforced response and the corresponding quantitative analysis methods (perturbation methods, straightforward expansion, multiple scales harmonic balance and averaging methods). On examples of pendulum equation with damping and forcing, and on externally excited Duffing systems, the author illustrates the construction of approximate nonlinear solutions for non-resonant and resonant cases together with their stability by perturbation and averaging techniques.

Ch. 4 is devoted to local analysis of nonlinear two-degree-of-freedom systems with nonlinear interaction of modes. This problem is discussed on the examples of autoparametric vibration absorber – a device purposefully designed to operate under conditions of combined internal and external resonance. The nonlinear dynamics of a non-shallow arch serves as an example of a structure for which internal resonance is more or less unavoidable. The follower-loaded double pendulum, displaying rich dynamics, allows to demonstrate how to employ multiple scales perturbation for systems written in the first-order matrix form. Here, the author investigates also nonlinear interactions associated with vibration-induced sliding of mass for the problems of pendulum with a sliding disc, string with a sliding point mass and vibration-induced fluid flow in pipes.

Ch. 5 contains a review of bifurcation analysis for general dynamic systems described by autonomous first-order differential equations. Codimension-one bifurcations of equilibriums for one- and \(n\)-dimensional systems (saddle-node, transcritical and pitchfork bifurcations), center manifold and normal form reductions in steady-state and dynamic bifurcation theory are presented here. Methods for the investigation of bifurcating periodic solutions and stability of bifurcation to perturbations are discussed with application to problems considered in Ch. 3 and 4.

Ch. 6 provides an introduction to chaotic vibrations with focusing on practical implications and applications. It considers tools for detecting chaotic vibrations, universal routes to chaos, tools for predicting the onset of chaos and examples of mechanical systems with chaos (Lorenz system, Duffing-type, pendulum-type, piecewise-linear, coupled autonomous, autoparametric and high-order systems). Elastostatic chaos, spatial and spatiotemporal chaos and controlling chaos problems are shortly discussed. In Ch. 7 on various examples the author discusses special effects of high-frequency excitation of mechanical systems. Finally, appendices present practical aspects of numerical simulations, the major exercises (tension control of rotating shafts, vibrations of a spring-tensioned beam, dynamics of microbeam), mathematical formulas used in perturbation and stability analysis, vibration modes and frequencies for structural elements, and properties of engineering materials. Every chapter is equipped by useful exercises.

The reviewed book will be very useful in engineering and scientific practice.

Reviewer: Boris V. Loginov (Ul’yanovsk)

##### MSC:

70-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of particles and systems |

74-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of deformable solids |

70Kxx | Nonlinear dynamics in mechanics |

74H45 | Vibrations in dynamical problems in solid mechanics |

74H55 | Stability of dynamical problems in solid mechanics |

74H65 | Chaotic behavior of solutions to dynamical problems in solid mechanics |