independent on u. Non-zero averages of this kind of terms are quite common where
nontrivial effects of HF excitation appear, e.g. they occur with the example systems
of the pendulum on a vibrating support (Sect. 7.3.1) and Fig. 7.7(a–g).
It is important to note that the vibrational force is generally a nonlinear function
of n j , since ^
u depends on n j (cf. (7.67)). Consequently, if v 1 is the vibrational force
corresponding to a specific time variation of the excitation n 1 (t, Xt), and v 2 is the
force corresponding to another excitation n 2 (t, Xt), then the force v corresponding
to the simultaneous action of both excitations does not, in general, equal v 1 + v 2 .
Thus one cannot readily predict the effects of combined excitations based on
knowledge of isolated effects (Thomsen 2008a considers effects of multifrequency
HF excitation). However, consideration to (7.69) shows that when M and f j are
independent of u, and h j is independent of du/dt, and s is linear in or independent of
du/dt – then one can indeed calculate the vibrational force for a sum of excitations
by adding up the vibrational forces for each excitation. None of the examples in
Sect. 7.3 or Fig. 7.7 fulfills this necessary condition for the validity of
superposition.
7.5.5 The Effects
Next we make general predictions for the stiffening, biasing, and smoothening
effects, based on the averaged system with vibrational forces as given by (7.69).
Note that (7.68) is valid on the quite general assumptions stated below the original
system (7.59), whereas various additional assumptions may be stated during the
following analysis. However, it will not be explicitly stated whether in each case the
system functions M, s, h, and f are allowed to depend on the slow time t, since this
is obvious from the context; thus the t in e.g. f(u, t) will be kept even when f is
actually not allowed to vary explicitly with t.
We shall base several results on a linearization of the averaged system (7.68)–
(7.69) near a possible static equilibrium ðz; _
zÞ ¼ ð~ z; 0Þ, where ~ z is defined by:
sð~ z; 0; tÞ þ vð~ z; 0; tÞ ¼ 0;
ð7:71Þ
where for consistency in notation we keep t in the function arguments, as explained
above, even though the existence of static equilibriums generally requires s and v to
be independent of t. The linearization is readily obtained using Taylor-expansion at
the equilibrium, and discarding nonlinear terms:
Mð~ z; tÞ€ z þ Cð~ zÞ þ DCð~ zÞ
ð
Þ _
z þ Kð~ zÞ þ DKð~ zÞ
ð
Þ ð z À ~ zÞ ¼ 0;
ð7:72Þ
7.5 A Fairly General Class of Discrete Systems
411
nontrivial effects of HF excitation appear, e.g. they occur with the example systems
of the pendulum on a vibrating support (Sect. 7.3.1) and Fig. 7.7(a–g).
It is important to note that the vibrational force is generally a nonlinear function
of n j , since ^
u depends on n j (cf. (7.67)). Consequently, if v 1 is the vibrational force
corresponding to a specific time variation of the excitation n 1 (t, Xt), and v 2 is the
force corresponding to another excitation n 2 (t, Xt), then the force v corresponding
to the simultaneous action of both excitations does not, in general, equal v 1 + v 2 .
Thus one cannot readily predict the effects of combined excitations based on
knowledge of isolated effects (Thomsen 2008a considers effects of multifrequency
HF excitation). However, consideration to (7.69) shows that when M and f j are
independent of u, and h j is independent of du/dt, and s is linear in or independent of
du/dt – then one can indeed calculate the vibrational force for a sum of excitations
by adding up the vibrational forces for each excitation. None of the examples in
Sect. 7.3 or Fig. 7.7 fulfills this necessary condition for the validity of
superposition.
7.5.5 The Effects
Next we make general predictions for the stiffening, biasing, and smoothening
effects, based on the averaged system with vibrational forces as given by (7.69).
Note that (7.68) is valid on the quite general assumptions stated below the original
system (7.59), whereas various additional assumptions may be stated during the
following analysis. However, it will not be explicitly stated whether in each case the
system functions M, s, h, and f are allowed to depend on the slow time t, since this
is obvious from the context; thus the t in e.g. f(u, t) will be kept even when f is
actually not allowed to vary explicitly with t.
We shall base several results on a linearization of the averaged system (7.68)–
(7.69) near a possible static equilibrium ðz; _
zÞ ¼ ð~ z; 0Þ, where ~ z is defined by:
sð~ z; 0; tÞ þ vð~ z; 0; tÞ ¼ 0;
ð7:71Þ
where for consistency in notation we keep t in the function arguments, as explained
above, even though the existence of static equilibriums generally requires s and v to
be independent of t. The linearization is readily obtained using Taylor-expansion at
the equilibrium, and discarding nonlinear terms:
Mð~ z; tÞ€ z þ Cð~ zÞ þ DCð~ zÞ
ð
Þ _
z þ Kð~ zÞ þ DKð~ zÞ
ð
Þ ð z À ~ zÞ ¼ 0;
ð7:72Þ
7.5 A Fairly General Class of Discrete Systems
411