The changes in stiffness in turn imply changes in natural frequencies and
equilibrium stability, which can also be calculated in similar general terms.
However, we shall postpone this to the following section dealing with a far more
general class of systems.
7.5 A Fairly General Class of Discrete Systems
The general system (7.54) analyzed in the previous section covers many, but far from
all relevant systems with HF excitation. For example, in the equation of motion
(7.46) for Brumberg’s pipe, a velocity _
x appears in the HF excitation term, thus
excluding this system from the class (7.54). Also, to claim generality we need to
include systems with several degrees of freedom, and HF excitations more general
than a mono-frequency time harmonic. On the other hand, the system (7.1), used to
outline the MDSM, is so general that it is very difficult to extract meaningful
information regarding the HF effects of concern here (though see Kremer 2016).
So, as a useful compromise, we consider a general class of systems with HF
excitation that is broad enough to cover a large variety of systems of scientific and
industrial interest, but also sufficiently specific to provide results that are physically
interpretable. The purpose of the analysis is to set up expressions governing the
slow motions, and to quantify the stiffening, the biasing, and the smoothening
effects for this general system. As should be evident from the examples in Sect. 7.3,
these effects are understood to characterize the averaged behavior of the system, i.e.
what can be observed when ignoring the details of a small overlay of HF oscillations. This corresponds to lowpass filtering the observed response – either digitally,
if the response consists of computer simulated or real sampled data, or analogously,
as with response perceived by band limited devices such as physical measurement
instruments and human senses.
Of course there are many ways to study HF effects in a general sense, differing in
the level of generalization, and the particular effects at focus; the following is based
on Thomsen (2002, 2003a, 2005); for alternative attempts with a similar goal see
e.g. Blekhman and Sorokin (2016), Fidlin (2006), Kremer (2016).
7.5.1 The System
We consider dynamical systems that can be modeled by a finite number of second
order ordinary differential equations, generally nonlinear, and with time-explicit HF
excitation (Thomsen 2002):
7.4 A Slight but Useful Generalization
405
equilibrium stability, which can also be calculated in similar general terms.
However, we shall postpone this to the following section dealing with a far more
general class of systems.
7.5 A Fairly General Class of Discrete Systems
The general system (7.54) analyzed in the previous section covers many, but far from
all relevant systems with HF excitation. For example, in the equation of motion
(7.46) for Brumberg’s pipe, a velocity _
x appears in the HF excitation term, thus
excluding this system from the class (7.54). Also, to claim generality we need to
include systems with several degrees of freedom, and HF excitations more general
than a mono-frequency time harmonic. On the other hand, the system (7.1), used to
outline the MDSM, is so general that it is very difficult to extract meaningful
information regarding the HF effects of concern here (though see Kremer 2016).
So, as a useful compromise, we consider a general class of systems with HF
excitation that is broad enough to cover a large variety of systems of scientific and
industrial interest, but also sufficiently specific to provide results that are physically
interpretable. The purpose of the analysis is to set up expressions governing the
slow motions, and to quantify the stiffening, the biasing, and the smoothening
effects for this general system. As should be evident from the examples in Sect. 7.3,
these effects are understood to characterize the averaged behavior of the system, i.e.
what can be observed when ignoring the details of a small overlay of HF oscillations. This corresponds to lowpass filtering the observed response – either digitally,
if the response consists of computer simulated or real sampled data, or analogously,
as with response perceived by band limited devices such as physical measurement
instruments and human senses.
Of course there are many ways to study HF effects in a general sense, differing in
the level of generalization, and the particular effects at focus; the following is based
on Thomsen (2002, 2003a, 2005); for alternative attempts with a similar goal see
e.g. Blekhman and Sorokin (2016), Fidlin (2006), Kremer (2016).
7.5.1 The System
We consider dynamical systems that can be modeled by a finite number of second
order ordinary differential equations, generally nonlinear, and with time-explicit HF
excitation (Thomsen 2002):
7.4 A Slight but Useful Generalization
405