scholarly journals A Review on Invariant Manifolds and Targeted Energy Transfer

2018 ◽  
Vol 3 (2) ◽  
pp. 75-86
Author(s):  
Maaita Jamal Odysseas ◽  
Meletlidou Efthymia
Keyword(s):  
Energy Transfer ◽  
Dynamical Systems ◽  
Invariant Manifolds ◽  
Singular Perturbation Theory ◽  
Targeted Energy Transfer ◽  
Transfer Energy ◽  
Practical Applications ◽  
Nonlinear Targeted Energy Transfer ◽  
Self Tuning ◽  
Different Time Scales

We present a review on one of the latest developments in the field of dynamical systems, The nonlinear Targeted Energy Transfer (TET). The great significance of the phenomenon lies in the fact that the systems in which Nonlinear TET occurs present a form of self-tuning and can transfer energy over a wide variety of frequencies (resonances). This makes nonlinear TET particularly suitable in practical applications where it is necessary to extract energy from multiple ways of oscillation. Dynamical systems where nonlinear TET occurs are systems with different time scales and are singular. This property allows us to study such systems with the use of singular perturbation theory. It has been shown that Nonlinear TET is related to the bifurcation of the Slow Invariant Manifold of such systems and their slow flow.

scholarly journals SLOW INVARIANT MANIFOLDS AS CURVATURE OF THE FLOW OF DYNAMICAL SYSTEMS

2008 ◽  
Vol 18 (11) ◽  
pp. 3409-3430 ◽  
Author(s):  
JEAN-MARC GINOUX ◽  
BRUNO ROSSETTO ◽  
LEON O. CHUA
Keyword(s):  
Dynamical Systems ◽  
Invariant Manifolds ◽  
Piecewise Linear ◽  
Slow Manifold ◽  
Analytical Equation ◽  
Singular Perturbation Theory ◽  
High Dimensional ◽  
Geometric Singular Perturbation Theory ◽  
Flow Curvature ◽  
Dimensional Dynamical System

Considering trajectory curves, integral of n-dimensional dynamical systems, within the framework of Differential Geometry as curves in Euclidean n-space, it will be established in this article that the curvature of the flow, i.e. the curvature of the trajectory curves of any n-dimensional dynamical system directly provides its slow manifold analytical equation the invariance of which will be then proved according to Darboux theory. Thus, it will be stated that the flow curvature method, which uses neither eigenvectors nor asymptotic expansions but only involves time derivatives of the velocity vector field, constitutes a general method simplifying and improving the slow invariant manifold analytical equation determination of high-dimensional dynamical systems. Moreover, it will be shown that this method generalizes the Tangent Linear System Approximation and encompasses the so-called Geometric Singular Perturbation Theory. Then, slow invariant manifolds analytical equation of paradigmatic Chua's piecewise linear and cubic models of dimensions three, four and five will be provided as tutorial examples exemplifying this method as well as those of high-dimensional dynamical systems.

Using passive nonlinear targeted energy transfer to stabilize drill-string systems

2009 ◽  
Vol 23 (1) ◽  
pp. 148-169 ◽  
Author(s):  
R. Viguié ◽  
G. Kerschen ◽  
J.-C. Golinval ◽  
D.M. McFarland ◽  
L.A. Bergman ◽  
...  
Keyword(s):  
Energy Transfer ◽  
Drill String ◽  
Targeted Energy Transfer ◽  
Nonlinear Targeted Energy Transfer
Sign in / Sign up

Export Citation Format

Share Document