scholarly journals SLOW HIGH-FREQUENCY EFFECTS IN MECHANICS: PROBLEMS, SOLUTIONS, POTENTIALS

2005 ◽  
Vol 15 (09) ◽  
pp. 2799-2818 ◽  
Author(s):  
JON JUEL THOMSEN
Keyword(s):  
High Frequency ◽  
Natural Frequencies ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Mechanical Systems ◽  
Frequency Effects ◽  
Direct Separation ◽  
High Frequency Excitation ◽  
Frequency Excitation ◽  
High Frequency Effects

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.

Multi-Frequency Multi-Modal Excitation of a Heated Annular Plate

2006 ◽  
Author(s):  
Haider N. Arafat ◽  
Ali H. Nayfeh
Keyword(s):  
Nonlinear Dynamics ◽  
Internal Resonance ◽  
Radial Direction ◽  
Natural Frequencies ◽  
Annular Plate ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Plate Theory ◽  
Frequency Excitation ◽  
The Difference

The forced nonlinear dynamics of a pre-buckled thermally loaded annular plate are investigated. The plate is modeled using the von Ka´rma´n plate theory and the heat equation. The heat, which is generated by the difference between the uniformly distributed temperatures at the inner and outer boundaries, is assumed to symmetrically flow in the radial direction. The amount of heat affects the natural frequencies, which may give rise to different internal resonance conditions. The method of multiple scales is used to examine the system axisymmetric responses when it is driven by an external multi-frequency excitation. The plate responses could be very complex exhibiting Hopf and cyclic-fold bifurcations, quasi-periodicity, chaos, and multiplicity of attractors.

Nonplanar Motion of a String due to Two-Frequency Excitation

2001 ◽  
Author(s):  
Kimihiko Yasuda ◽  
Keisuke Kamiya
Keyword(s):  
Numerical Simulation ◽  
Theoretical Analysis ◽  
Perturbation Method ◽  
Natural Frequencies ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Harmonic Excitation ◽  
Chaotic Motions ◽  
Frequency Excitation

Abstract It is known that, under certain conditions, a stretched string subjected to a planar harmonic excitation executes nonplanar motions due to the instability of the palanar motion. In recent years, studies on bifurcations of such nonplanar motions to amplitude modulated quasiperiodic motions and chaotic motions have been reported. However no literatures on the problem of nonplanar motions due to a multi-frequency excitation are found. In this paper, the possibility of nonplanar motions in a string due to a two-frequency excitation is studied. For this purpose two cases are considered, i.e. one in which both components of the excitation are in a plane, and one in which they are perpendicular to each other. In both cases the sum of the frequencies of the components is supposed to near to twice one of the natural frequencies of the string. Theoretical analysis using the perturbation method of multiple scales and numerical simulation are carried out to show that nonplanar motions occur.

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