Interestingly, the nonlinear parts of s (with coefficients s 2 and s 3 ) change the
linear properties of ~ s (i.e. the first two groups of terms), whereas the nonlinear
properties themselves are unaffected. For example, the slope of ~ s at _
z ¼ 0 is seen to
increase, as compared to s if the cubic nonlinearity of s is progressive (s 3 > 0), and
to decrease if it is recessive (s 3 < 0). Thus, the effective linear damping of a
physical system might be partially controlled by using HF excitation, or be created
out of nothing, as when s 1 = 0, s 3 6 ¼ 0. Also, a constant drifting term is seen to
appear, so that ~ sð0Þ 6 ¼ 0 when there are nonlinear velocity terms in s.
Finally we recall that the above results for s and φ hold as well for the corresponding vector functions s and ^
u of the general system (7.59).
7.5.9 Effects of Multiple HF Excitation Frequencies
When m > 1 in (7.59) the HF excitation may have more than a single excitation
frequency. The consequence of this can be calculated using the above general
results, with the following main conclusions (Thomsen 2008a):
• The stiffening effect is similar to that of mono-frequency excitation, provided
the excitation frequencies are well separated.
• For non-close and non-resonant excitation frequencies, the change in effective
static stiffness is proportional to the sum of squared excitation velocities, i.e. to
the input energy level.
• With two or more close excitation frequencies, there is an additional contribution of slowly oscillating stiffness, having magnitude order similar to the change
in static stiffness, and frequencies equal to the differences in non-close
frequencies.
• With close excitation frequencies, strong parametrical resonance can occur at
conditions that might not appear obvious, that is: when the difference in any two
excitation frequencies comes near 2 ~
x=k; where k is an integer (with k = 1
corresponding to primary parametric resonance), and ~
x is an effective natural
frequency of the system – which due to the HF-excitation is actually shifted
away from the natural frequency x without HF-excitation.
• Strong multifrequency HF-excitation can stabilize unstable quasistatic equilibriums, just as can monofrequency HF-excitation. Generally this holds only if the
frequencies are well separated, i.e. with differences much larger than the lowest
natural frequencies of the system. With two or more close high frequencies, the
effect of HFE may be stabilizing or destabilizing, dependent on particular
parameter values. Thus continuous broadband and random excitation does not
have a uniquely stabilizing effect paralleling that of monofrequency
HF-excitation, or multifrequency HF-excitation with non-close frequencies.
(See also Kremer 2018.)
7.5 A Fairly General Class of Discrete Systems
425
linear properties of ~ s (i.e. the first two groups of terms), whereas the nonlinear
properties themselves are unaffected. For example, the slope of ~ s at _
z ¼ 0 is seen to
increase, as compared to s if the cubic nonlinearity of s is progressive (s 3 > 0), and
to decrease if it is recessive (s 3 < 0). Thus, the effective linear damping of a
physical system might be partially controlled by using HF excitation, or be created
out of nothing, as when s 1 = 0, s 3 6 ¼ 0. Also, a constant drifting term is seen to
appear, so that ~ sð0Þ 6 ¼ 0 when there are nonlinear velocity terms in s.
Finally we recall that the above results for s and φ hold as well for the corresponding vector functions s and ^
u of the general system (7.59).
7.5.9 Effects of Multiple HF Excitation Frequencies
When m > 1 in (7.59) the HF excitation may have more than a single excitation
frequency. The consequence of this can be calculated using the above general
results, with the following main conclusions (Thomsen 2008a):
• The stiffening effect is similar to that of mono-frequency excitation, provided
the excitation frequencies are well separated.
• For non-close and non-resonant excitation frequencies, the change in effective
static stiffness is proportional to the sum of squared excitation velocities, i.e. to
the input energy level.
• With two or more close excitation frequencies, there is an additional contribution of slowly oscillating stiffness, having magnitude order similar to the change
in static stiffness, and frequencies equal to the differences in non-close
frequencies.
• With close excitation frequencies, strong parametrical resonance can occur at
conditions that might not appear obvious, that is: when the difference in any two
excitation frequencies comes near 2 ~
x=k; where k is an integer (with k = 1
corresponding to primary parametric resonance), and ~
x is an effective natural
frequency of the system – which due to the HF-excitation is actually shifted
away from the natural frequency x without HF-excitation.
• Strong multifrequency HF-excitation can stabilize unstable quasistatic equilibriums, just as can monofrequency HF-excitation. Generally this holds only if the
frequencies are well separated, i.e. with differences much larger than the lowest
natural frequencies of the system. With two or more close high frequencies, the
effect of HFE may be stabilizing or destabilizing, dependent on particular
parameter values. Thus continuous broadband and random excitation does not
have a uniquely stabilizing effect paralleling that of monofrequency
HF-excitation, or multifrequency HF-excitation with non-close frequencies.
(See also Kremer 2018.)
7.5 A Fairly General Class of Discrete Systems
425