Chaotic Vibrations: An Introduction for Applied Scientists and Engineers

Abstract

BOOK REVIEWS principles, invariance under translation and fundamental solutions, integral representation theorem, finite difference methods, finite element methods, Petrov-Galerkin finite element methods and boundary element methods. Various special iterative methods for the finite difference equations are also discussed. Chapter 4: "Dissipative Systems". The numerical methods considered in Chapter 3 are revised to deal with the advection-diffusion type equations. Nonstandard methods such as upstream weighting, solution pathology at sharp fronts, and alternating direction methods are also introduced in this chapter. Chapter 5: "Nondissipative Systems" deals with transport equations such as the Euler equations, the Burger's equation, and the wave equation. Methods of characteristic, conservative finite-difference schemes, and nonstandard Galerkin finite element methods are considered. Chapter 6: "High-Order, Nonlinear, and Coupled Systems," considers special problems such as the biharmonic equation, nonlinear diffusion equations and compositional flows in porous media. The oil reservoir modeling is discussed at great length. The above list is not complete, merely representative of the topics covered in the book.

Nonlinear dynamics in flow through unsaturated fractured porous media: Status and perspectives

“…The attractor is a set of points in a phase space toward which nearly all trajectories converge, and the attractor describes an ensemble of states of the system. For a dynamic process, which is described by a system of evolution equations, the coordinates of a phase space are state variables or components of the state vector [Moon, 1987], so that the evolution of a system is described without direct time-dependent dynamic variables.…”

Belykh Attractor in Zaslavsky Map and Its Transformation Under Smoothing

“…In Section 1 we consider a concrete example of a dissipative system where this type of dynamics takes place, namely, a rotator under the action of periodic kicks. The description of the state change on a period of kicks leads to the Zaslavsky map [25,26], in which, when choosing the sawtooth form with a break for the function of intensity of the kicks on the rotation angle g(θ), the Belykh attractor occurs. Section 2 presents the corresponding analytic derivations reducing the problem to the standard form of the Belykh map [22].…”

Analysis of Bistable and Chaotic Piezoelectric Energy Harvesting Device Coupled With Diode Bridge Rectifier

“…where m is the equivalent mass of the beam, x is the displacement of the beam in the transverse direction, c is the damping, k l is the equivalent linear stiffness of the beam calculated by the finite element method (FEM), the term k nl provides a nonlinear restoring force at large x caused by the spacing of the magnets [8,9,10,17,18,19]. The signal of the parameters k l and k nl define the type of spring.…”

“…A lumped mass of 14 grams is attached to the free end of the ferromagnetic beam tip to improve the dynamic flexibility and to allow the motion caused by the magnets. Therefore, depending on the magnet spacing, the ferromagnetic beam may have five (with three stable), three (with two stable) or one (stable) equilibrium positions [7,8,16,17]. The equations that describe the device for the condition with three equilibrium positions for the fundamental vibration mode are [8,9,10]:…”

“…Among these harvesting devices, the use of nonlinear devices has been widely studied to extract energy from wideband non-harmonic vibration sources [3,4,5,6,7]. In this sense, the use of the structure investigated by Moon [17], composed by a nonlinear device that can present chaotic behavior, is one of the most used for energy harvesting purposes in recent years using only an electric resistance to dissipate the energy generated by the piezoelectric materials [6,7,8,9,10,18,19].…”