a
Ã
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4ðx à À 2Þ
3c
s
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
8
ffiffiffiffiffiffi ffi
b/q
p
À 1
À
Á
3c
s
; c 6 ¼ 0:
ð3:151Þ
Here the radical must be non-negative, so a peak point so defined only exists
when either c > 0 and b ! q, or when c < 0 and b
q. In the linear case the
response under parametric resonance is unlimited, i.e. a
*
! ∞, even with damping.
With the example parameters given in the legend of Fig. 3.10 inserted into (3.150)
and (3.151), one finds the peak response frequency x
* = 1, and the corresponding
peak response amplitude a
*
% 2.8, agreeing with the values that can be read from
the axis values.
Influence of System Parameters Fig. 3.11 shows a set of frequency-responses
obtained for varying levels of excitation q. It appears that stronger excitation
implies larger response amplitudes, and a wider resonant region. Note too that even
small levels of excitation, if resonant, may feed off large amplitudes, even though
the level of damping is significant (here 5%). This is also a characteristic associated
with parametric excitation: small inputs may create large outputs, even in the
presence of damping
5
.
Fig. 3.12 depicts the influence of the level of (small) damping. Increased
damping tends to lower the response-peak, whereas the width of the resonant region
is virtually unaffected.
Fig. 3.13 shows the frequency responses for three values of the coefficient of
nonlinearity. The response-peaks bend to the left when c < 0 (softening), to the
right when c > 0 (hardening), and straight up when c = 0 (linear case). Note that
the linear, near-resonant response is unbounded. Generally, for a system in
Fig. 3.11. Frequency responses a(x) for three different levels of system excitation q. (c = –1/6,
b = 0.05)
5
This also means parametric excitation can be used for amplifyingor ‘pumping’ motions or signals
(Rhoads et al. 2008; Rhoads and Shaw 2010; Thomas et al. 2013; Neumeyer et al. 2017, 2019),
and even for attenuatingor damping vibrations (Dohnal 2008).
142
3 Nonlinear Vibrations: Classical Local Theory
Ã
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4ðx à À 2Þ
3c
s
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
8
ffiffiffiffiffiffi ffi
b/q
p
À 1
À
Á
3c
s
; c 6 ¼ 0:
ð3:151Þ
Here the radical must be non-negative, so a peak point so defined only exists
when either c > 0 and b ! q, or when c < 0 and b
q. In the linear case the
response under parametric resonance is unlimited, i.e. a
*
! ∞, even with damping.
With the example parameters given in the legend of Fig. 3.10 inserted into (3.150)
and (3.151), one finds the peak response frequency x
* = 1, and the corresponding
peak response amplitude a
*
% 2.8, agreeing with the values that can be read from
the axis values.
Influence of System Parameters Fig. 3.11 shows a set of frequency-responses
obtained for varying levels of excitation q. It appears that stronger excitation
implies larger response amplitudes, and a wider resonant region. Note too that even
small levels of excitation, if resonant, may feed off large amplitudes, even though
the level of damping is significant (here 5%). This is also a characteristic associated
with parametric excitation: small inputs may create large outputs, even in the
presence of damping
5
.
Fig. 3.12 depicts the influence of the level of (small) damping. Increased
damping tends to lower the response-peak, whereas the width of the resonant region
is virtually unaffected.
Fig. 3.13 shows the frequency responses for three values of the coefficient of
nonlinearity. The response-peaks bend to the left when c < 0 (softening), to the
right when c > 0 (hardening), and straight up when c = 0 (linear case). Note that
the linear, near-resonant response is unbounded. Generally, for a system in
Fig. 3.11. Frequency responses a(x) for three different levels of system excitation q. (c = –1/6,
b = 0.05)
5
This also means parametric excitation can be used for amplifyingor ‘pumping’ motions or signals
(Rhoads et al. 2008; Rhoads and Shaw 2010; Thomas et al. 2013; Neumeyer et al. 2017, 2019),
and even for attenuatingor damping vibrations (Dohnal 2008).
142
3 Nonlinear Vibrations: Classical Local Theory