Thus the natural frequencies and stability associated with the equilibrium are
determined by the eigenvalues k = k r , r = 1, …, 2n, which are roots of the characteristic polynomial:
k
2
Mð0; tÞ þ k Cð0Þ þ DCð0Þ
ð
Þ þ Kð0Þ þ DKð0Þ
¼ 0:
ð7:83Þ
The natural frequencies are given by |Im(k r )|, while the equilibrium in question is
stable to small disturbances only if Re(k r ) < 0 for all r. Both quantities are likely to
differ from the values obtained when there is no HF excitation, i.e. when DC
(0) = DK(0) = 0 in (7.83). For assessing stability in specific cases, it might be
convenient to use Ziegler’s system of classification (Sect. 1.9), by which statements
on stability are inferred from properties such as symmetry and definiteness of the
system matrices. Also, since effective structural properties (stiffness, damping,
stability etc.) can be controlled by changing the structure or the HF excitation or
both, some interesting inverse problems could be posed on using HF excitation as a
design variable in designing structures with specific or optimal low-frequency
properties.
7.5.7 Biasing
We define biasing as a change in (average) static equilibrium position as a consequence of HF excitation. Two important kinds will be considered: positional bias
and velocity bias:
Positional bias refers to a fixed translation of a static equilibrium. Examples of
positional bias appeared in the examples of the vibrated pendulum and the mass on
a vibrated plane (Sect. 7.3.2), and in Fig. 7.7(b,d,e,h). Assuming, as in the previous
section, that the general system (7.59) has a static equilibrium at u = 0 when
unexcited, then HF excitation may change this state of affairs, so that the equilibrium for the slow components of motion changes from ðz; _
zÞ ¼ ð0; 0Þ to ð~ z; 0Þ,
where ~ z 6 ¼ 0 is constant-valued. We term ~ z a quasi-equilibrium for the full motions
u, because these will actually oscillate at small amplitude and high frequency about
~ z, cf. (7.60). To calculate ~ z we use (7.68), inserting ðz; _
z; € zÞ ¼ ð~ z; 0; 0Þ, and find that
~ z is the solution of the generally nonlinear set of algebraic equations (7.71), i.e.
sð~ z; 0; tÞ þ vð~ z; 0; tÞ ¼ 0;
ð7:84Þ
with v given by (7.69). The equilibrium ~ z is stable when the real parts of the roots k
of the following characteristic polynomial are all negative:
k
2
Mð~ z; tÞ þ k Cð~ zÞ þ DCð~ zÞ
ð
Þ þ Kð~ zÞ þ DKð~ zÞ
¼ 0;
ð7:85Þ
where the matrices C and K are given by (7.73). More than one solution for ~ z may
exist, since s and v are generally nonlinear functions of position, or there may be no
416
7 Special Effects of High-Frequency Excitation
determined by the eigenvalues k = k r , r = 1, …, 2n, which are roots of the characteristic polynomial:
k
2
Mð0; tÞ þ k Cð0Þ þ DCð0Þ
ð
Þ þ Kð0Þ þ DKð0Þ
¼ 0:
ð7:83Þ
The natural frequencies are given by |Im(k r )|, while the equilibrium in question is
stable to small disturbances only if Re(k r ) < 0 for all r. Both quantities are likely to
differ from the values obtained when there is no HF excitation, i.e. when DC
(0) = DK(0) = 0 in (7.83). For assessing stability in specific cases, it might be
convenient to use Ziegler’s system of classification (Sect. 1.9), by which statements
on stability are inferred from properties such as symmetry and definiteness of the
system matrices. Also, since effective structural properties (stiffness, damping,
stability etc.) can be controlled by changing the structure or the HF excitation or
both, some interesting inverse problems could be posed on using HF excitation as a
design variable in designing structures with specific or optimal low-frequency
properties.
7.5.7 Biasing
We define biasing as a change in (average) static equilibrium position as a consequence of HF excitation. Two important kinds will be considered: positional bias
and velocity bias:
Positional bias refers to a fixed translation of a static equilibrium. Examples of
positional bias appeared in the examples of the vibrated pendulum and the mass on
a vibrated plane (Sect. 7.3.2), and in Fig. 7.7(b,d,e,h). Assuming, as in the previous
section, that the general system (7.59) has a static equilibrium at u = 0 when
unexcited, then HF excitation may change this state of affairs, so that the equilibrium for the slow components of motion changes from ðz; _
zÞ ¼ ð0; 0Þ to ð~ z; 0Þ,
where ~ z 6 ¼ 0 is constant-valued. We term ~ z a quasi-equilibrium for the full motions
u, because these will actually oscillate at small amplitude and high frequency about
~ z, cf. (7.60). To calculate ~ z we use (7.68), inserting ðz; _
z; € zÞ ¼ ð~ z; 0; 0Þ, and find that
~ z is the solution of the generally nonlinear set of algebraic equations (7.71), i.e.
sð~ z; 0; tÞ þ vð~ z; 0; tÞ ¼ 0;
ð7:84Þ
with v given by (7.69). The equilibrium ~ z is stable when the real parts of the roots k
of the following characteristic polynomial are all negative:
k
2
Mð~ z; tÞ þ k Cð~ zÞ þ DCð~ zÞ
ð
Þ þ Kð~ zÞ þ DKð~ zÞ
¼ 0;
ð7:85Þ
where the matrices C and K are given by (7.73). More than one solution for ~ z may
exist, since s and v are generally nonlinear functions of position, or there may be no
416
7 Special Effects of High-Frequency Excitation