induced HF vibrations could be used to create materials whose effective properties
change instantly, by changing properties of the HF signal (Blekhman 2008;
Thomsen and Blekhman 2007, Abusoua and Daqaq 2018), and mechanical system
parameters can be estimated using HF excitation (Abusoua and Daqaq 2017).
Nature also utilizes slow effects of HF excitation, e.g., in enabling primitive
organisms to drift and swim. And then there are also harmful effects, e.g.,
self-unscrewing bolts and nuts on vibrating machinery, and needle pointer instruments displaying bias error when operating in strongly vibrating environments.
Some HF excitation phenomena appear well outside usual experience, e.g. fluid can
be made to float above air by vibrating the air-fluid system strongly at high frequency, and even with objects floating upside down on the levitated air-fluid
interface, as if gravity is reversed (Apffel et al. 2020, Sorokin and Blekhman 2020).
In this chapter we first present an outline of a method for conveniently analyzing
systems with – and effects of – HF excitation. Hereafter the main ideas are illustrated in terms of three simple examples, representative of three main effects of HF
excitation, which we shall call stiffening, biasing, and smoothening. Then various
levels of generalizations are described, and finally some more involved examples
are briefly presented to illustrate the variety and character of applications. The main
reference in this area is the monograph by Blekhman (2000), while (Blekhman
2004), Fidlin (2002), Jensen (1999b), and Thomsen (2003a), can be considered as
more specialized offspring.
7.2 The Method of Direct Separation of Motions
(MDSM)
Here we describe a mathematical tool for dealing conveniently with linear and
nonlinear systems subjected to high frequency excitation, the Method of Direct
Separation (or Partition) of Motions (MDSM). The MDSM transforms a set of
differential equations, generally nonlinear, into two subsets: one describing the
‘fast’ (rapidly oscillating) components of motion, and another one the ‘slow’ (in
comparison) components. Typically, in applications, the subset describing slow
motions is of primary interest, whereas the fast motions constitute a small and
trivial overlay that is only interesting by its effect on the slow motions. The subset
of ‘slow’ equations takes into account the fast forces through added terms, so-called
vibrational forces, which describe the averaged influence of the fast forces.
Traditional perturbation approaches, such as the multiple scaling and averaging
methods, also produce approximate equations describing slow solution components, known as modulation equations (e.g. Nayfeh and Mook 1979). These
methods can be used as well for problems involving rapidly oscillating terms. For
example, averaging is used for such problems in Chelomei (1981), Fidlin (1999,
2001, 2004a,b), Fidlin and Thomsen (2001), and Sethna (1967) – and the method of
multiple scales in Fidlin (2000), Hansen (2000), Tcherniak (1999), and Tcherniak
and Thomsen (1998). But the MDSM is particularly well suited, in terms of
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7 Special Effects of High-Frequency Excitation
change instantly, by changing properties of the HF signal (Blekhman 2008;
Thomsen and Blekhman 2007, Abusoua and Daqaq 2018), and mechanical system
parameters can be estimated using HF excitation (Abusoua and Daqaq 2017).
Nature also utilizes slow effects of HF excitation, e.g., in enabling primitive
organisms to drift and swim. And then there are also harmful effects, e.g.,
self-unscrewing bolts and nuts on vibrating machinery, and needle pointer instruments displaying bias error when operating in strongly vibrating environments.
Some HF excitation phenomena appear well outside usual experience, e.g. fluid can
be made to float above air by vibrating the air-fluid system strongly at high frequency, and even with objects floating upside down on the levitated air-fluid
interface, as if gravity is reversed (Apffel et al. 2020, Sorokin and Blekhman 2020).
In this chapter we first present an outline of a method for conveniently analyzing
systems with – and effects of – HF excitation. Hereafter the main ideas are illustrated in terms of three simple examples, representative of three main effects of HF
excitation, which we shall call stiffening, biasing, and smoothening. Then various
levels of generalizations are described, and finally some more involved examples
are briefly presented to illustrate the variety and character of applications. The main
reference in this area is the monograph by Blekhman (2000), while (Blekhman
2004), Fidlin (2002), Jensen (1999b), and Thomsen (2003a), can be considered as
more specialized offspring.
7.2 The Method of Direct Separation of Motions
(MDSM)
Here we describe a mathematical tool for dealing conveniently with linear and
nonlinear systems subjected to high frequency excitation, the Method of Direct
Separation (or Partition) of Motions (MDSM). The MDSM transforms a set of
differential equations, generally nonlinear, into two subsets: one describing the
‘fast’ (rapidly oscillating) components of motion, and another one the ‘slow’ (in
comparison) components. Typically, in applications, the subset describing slow
motions is of primary interest, whereas the fast motions constitute a small and
trivial overlay that is only interesting by its effect on the slow motions. The subset
of ‘slow’ equations takes into account the fast forces through added terms, so-called
vibrational forces, which describe the averaged influence of the fast forces.
Traditional perturbation approaches, such as the multiple scaling and averaging
methods, also produce approximate equations describing slow solution components, known as modulation equations (e.g. Nayfeh and Mook 1979). These
methods can be used as well for problems involving rapidly oscillating terms. For
example, averaging is used for such problems in Chelomei (1981), Fidlin (1999,
2001, 2004a,b), Fidlin and Thomsen (2001), and Sethna (1967) – and the method of
multiple scales in Fidlin (2000), Hansen (2000), Tcherniak (1999), and Tcherniak
and Thomsen (1998). But the MDSM is particularly well suited, in terms of
388
7 Special Effects of High-Frequency Excitation