i.e. s 0 is a point set. Then the second integral in (7.99) vanishes, because s is
bounded and the integration intervals are infinitely short. Consequently, continuity is assured if the two other integrals are continuous. Their integrands are
continuous, because they are to be evaluated for the intervals s − and s + , where
_
z þ u
0
ðsÞ has constant sign and thus only sweeps the continuous part of s. But
the boundaries of the integration intervals should be continuous as well, that is,
s j should be continuous functions of _
z. By the implicit function theorem this is
true when φʺ(s j ) 6 ¼ 0, that is, if s j are simple zeroes of _
z þ u
0
ðsÞ.
4. In particular, it follows from the above results that ~ s is continuous at _
z ¼ 0 if φ′(s)
has a finite number of simple zeroes during one period of fast motions.
As described the smoothening effect on the discontinuity at zero occurs only if
the fast motions actually sweeps the discontinuity, and does not ‘rest’ there. An
example function that fails to satisfy this is φ′ = − sin(2s) for s 2 [0;p]; φ′ = 0 for
s 2 [p;2p], which has simple zeroes at s = 0, p/2, but infinitely many non-simple
zeroes for s 2 [p;2p]; hence this function does not smooth discontinuities.
An example function that does satisfy the requirements is a simple harmonic, e.g.
φ′ = − acos(s), which has simple zeroes at p/2 and 3p/2 and thus is smoothening. Its
effect on the discontinuous signum function appears in Table 7.2(a); it could represent the smoothening of the dry friction characteristic for a physical system subjected to HF excitation (see also the mass on a vibrated plane example in Sect. 7.3.2,
and Fig. 7.5).
Table 7.2(b) shows how fast motions smoothens the discontinuity of a function
that may more realistically model dry friction. Here the function s has negative slope
at the origin and thus may cause unstable oscillations of a corresponding physical
system, whereas the smoothed image has positive slope everywhere (For applications of this see Thomsen 1999; Feeny and Moon 2000; Chatterjee et al. 2004).
Discontinuities in derivatives may also be smoothened, as seen when replacing
s and ~ s with ds=d _
u and d~ s=d_ z (or higher order derivatives) in the above expressions
and arguments. Table 7.2(c,d) shows two such examples, which might represent
idealized physical processes involving saturation and ‘barrier’ behavior, respectively, in both cases with discontinuous changes in slope HF-smoothened.
Even if s has no discontinuity, the smoothed image ~ s may still differ from
s. Table 7.2(e–h) shows smoothed images of functions that are typically encountered in applications. The essential properties of such functions are captured by a
third order polynomial with zero constant term and coefficients s 1,2,3 :
sð_ zÞ ¼ s 1 _
z þ s 2 _
z
2
þ s 3 _
z
3
;
ð7:100Þ
whose smoothed image is, by (7.98) and recalling that u
0
h i ¼ 0:
~ sð_ zÞ ¼ s 2 ðu
0
Þ
2
D
E
þ s 3 ðu
0
Þ
3
D
E
þ s 1 þ 3s 3 ðu
0
Þ
2
D
E
_
z
þ s 2 _
z
2
þ s 3 _
z
3
:
ð7:101Þ
7.5 A Fairly General Class of Discrete Systems
421
bounded and the integration intervals are infinitely short. Consequently, continuity is assured if the two other integrals are continuous. Their integrands are
continuous, because they are to be evaluated for the intervals s − and s + , where
_
z þ u
0
ðsÞ has constant sign and thus only sweeps the continuous part of s. But
the boundaries of the integration intervals should be continuous as well, that is,
s j should be continuous functions of _
z. By the implicit function theorem this is
true when φʺ(s j ) 6 ¼ 0, that is, if s j are simple zeroes of _
z þ u
0
ðsÞ.
4. In particular, it follows from the above results that ~ s is continuous at _
z ¼ 0 if φ′(s)
has a finite number of simple zeroes during one period of fast motions.
As described the smoothening effect on the discontinuity at zero occurs only if
the fast motions actually sweeps the discontinuity, and does not ‘rest’ there. An
example function that fails to satisfy this is φ′ = − sin(2s) for s 2 [0;p]; φ′ = 0 for
s 2 [p;2p], which has simple zeroes at s = 0, p/2, but infinitely many non-simple
zeroes for s 2 [p;2p]; hence this function does not smooth discontinuities.
An example function that does satisfy the requirements is a simple harmonic, e.g.
φ′ = − acos(s), which has simple zeroes at p/2 and 3p/2 and thus is smoothening. Its
effect on the discontinuous signum function appears in Table 7.2(a); it could represent the smoothening of the dry friction characteristic for a physical system subjected to HF excitation (see also the mass on a vibrated plane example in Sect. 7.3.2,
and Fig. 7.5).
Table 7.2(b) shows how fast motions smoothens the discontinuity of a function
that may more realistically model dry friction. Here the function s has negative slope
at the origin and thus may cause unstable oscillations of a corresponding physical
system, whereas the smoothed image has positive slope everywhere (For applications of this see Thomsen 1999; Feeny and Moon 2000; Chatterjee et al. 2004).
Discontinuities in derivatives may also be smoothened, as seen when replacing
s and ~ s with ds=d _
u and d~ s=d_ z (or higher order derivatives) in the above expressions
and arguments. Table 7.2(c,d) shows two such examples, which might represent
idealized physical processes involving saturation and ‘barrier’ behavior, respectively, in both cases with discontinuous changes in slope HF-smoothened.
Even if s has no discontinuity, the smoothed image ~ s may still differ from
s. Table 7.2(e–h) shows smoothed images of functions that are typically encountered in applications. The essential properties of such functions are captured by a
third order polynomial with zero constant term and coefficients s 1,2,3 :
sð_ zÞ ¼ s 1 _
z þ s 2 _
z
2
þ s 3 _
z
3
;
ð7:100Þ
whose smoothed image is, by (7.98) and recalling that u
0
h i ¼ 0:
~ sð_ zÞ ¼ s 2 ðu
0
Þ
2
D
E
þ s 3 ðu
0
Þ
3
D
E
þ s 1 þ 3s 3 ðu
0
Þ
2
D
E
_
z
þ s 2 _
z
2
þ s 3 _
z
3
:
ð7:101Þ
7.5 A Fairly General Class of Discrete Systems
421