Slow High-Frequency Effects in Mechanics: Problems, Solutions, Potentials

Abstract

Strong high-frequency excitation (HFE) may change the "slow" (i.e. effective or average) properties of mechanical systems, e.g. their stiffness, natural frequencies, equilibriums, equilibrium stability, and bifurcation paths. This tutorial describes three general HFE effects: Stiffening — an apparent change in the stiffness associated with an equilibrium; Biasing — a tendency for a system to move towards a particular state which does not exist or is unstable without HFE; and Smoothening — a tendency for discontinuities to be apparently smeared out by HFE. The effects and a method for analyzing them are introduced, first in terms of simple physical examples, and then in generalized form for mathematical models covering broad classes of discrete and continuous mechanical systems. Several application examples are summarized. Three mathematical tools for analyzing HFE effects are described and compared: The Method of Direct Separation of Motions, the Method of Averaging, and the Method of Multiple Scales. The tutorial concludes with a suggestion that more vibration experts, researchers and students should be aware of HFE effects, for the benefit of general vibration troubleshooting, and also for furthering the creation of innovative technical devices and processes utilizing HFE effects.

Response of a magneto‐rheological fluid damper subjected to periodic forcing in a high frequency limit

“…In these cases the time-scales are introduced to account for nonlinear terms in the examined systems. On the other hand, the fast time scale can be introduced through excitation terms in the limit of very high excitation frequency [7,11,12,13]. In various nonlinear systems [7], subjected to additional excitation fast and slow time-scales, such splitting is realized physically.…”

“…1,2) in the original treatment by Thomsen, k 1 = 0 k 2 = 0 and ∆k = 0 [7,11] to describe the effect of dry friction. On the other hand, in Ref.…”

“…Following Thomsen (2005) [11] we define two time scales fast given by T 0 and slow by T 1 , respectively…”

“…Following Thomsen [7,11] discussion on dry friction, we decided to plot the results (Eq. 13) for three values of x 0 ω and k 2 = 0 and ∆k = 0 in Fig.…”

Stability and chaotic behavior of a PID controlled inverted pendulum subjected to harmonic base excitations by using the normal form theory

“…It is known that a dissipative pendulum subjected to vertical and horizontal harmonic disturbances of high frequency can be driven to several equilibrium points apart from the stable pointing-down position (examples can be found in Refs [21][22][23][24][25][26]). …”