Response Backbone Fig. 3.10 also shows, in dash-dotted line, the backbone (or
skeleton ) curve of the frequency response; it is obtained by zeroing the forcing and
damping parameters in the frequency response equation. For the present example
this means letting q = b = 0 in (3.129)–(3.130), and inserting r = x – 2 from
(3.120), which gives C 1 = 0 and, considering only the positive branch of the
backbone:
a ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4ðx À 2Þ
3c
s
ðon backbone):
ð3:149Þ
The radical must be non-negative, which means that for c > 0 the backbone
curve is defined only for x ! 0 (i.e. it bends to the right, towards higher frequencies), while for c < 0 the backbone curve is defined only for x
0 (i.e. it
bends left, towards lower frequencies). In the linear case c ! 0 which gives a !
± ∞, i.e. the backbone points straight up, reflecting that for linear systems the free
oscillation frequency does not depend on amplitude. The backbone is an important
curve, and more easy to calculate than the forced frequency response. It organizes
all frequency responses of the system under study, independent of input forcing and
damping, and tells which way a nonlinear frequency response bends and how much.
It also describes the relation between oscillation amplitude and frequency, a very
characteristic feature for nonlinear systems. In the present example, with the
parametrically excited pendulum, we have c = –1/6 and (3.149) gives x ¼
2ð1 À
1
16 a
2
Þ; which is just twice (since the excitation is parametric) the nonlinear
natural frequency of the freely oscillating undamped pendulum, cf. (3.81).
Response backbones can also be determined from numerical models, using, e.g.,
free-decay (“ring down”) simulation or pseudo-arclength continuation, and from
experimental measurements by, e.g., free-decay measurement or control-based
continuation techniques; see, e.g., Denis et al. (2017), Givois et al. (2020), Londoño
et al. (2015), Peter et al. (2016), Renson et al. (2016).
Peak Response The maximum response is often of interest in applications. It
can often be rather easily calculating by noting that the peak point of the response
curve is located where the backbone intersects the response curve. This point can be
found by first solving (3.130) with C 1 = 0 to give the peak response frequency, x
* :
x
Ã
¼ 2
ffiffiffi
b
q
s
;
ð3:150Þ
which is independent on the nonlinearity parameter c – and then substitute this
frequency into in (3.129), along with r = x – 2 from (3.120), to give the corresponding peak response amplitude, a
* :
3.6 The Forced Response – Multiple Scales Analysis
141
skeleton ) curve of the frequency response; it is obtained by zeroing the forcing and
damping parameters in the frequency response equation. For the present example
this means letting q = b = 0 in (3.129)–(3.130), and inserting r = x – 2 from
(3.120), which gives C 1 = 0 and, considering only the positive branch of the
backbone:
a ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4ðx À 2Þ
3c
s
ðon backbone):
ð3:149Þ
The radical must be non-negative, which means that for c > 0 the backbone
curve is defined only for x ! 0 (i.e. it bends to the right, towards higher frequencies), while for c < 0 the backbone curve is defined only for x
0 (i.e. it
bends left, towards lower frequencies). In the linear case c ! 0 which gives a !
± ∞, i.e. the backbone points straight up, reflecting that for linear systems the free
oscillation frequency does not depend on amplitude. The backbone is an important
curve, and more easy to calculate than the forced frequency response. It organizes
all frequency responses of the system under study, independent of input forcing and
damping, and tells which way a nonlinear frequency response bends and how much.
It also describes the relation between oscillation amplitude and frequency, a very
characteristic feature for nonlinear systems. In the present example, with the
parametrically excited pendulum, we have c = –1/6 and (3.149) gives x ¼
2ð1 À
1
16 a
2
Þ; which is just twice (since the excitation is parametric) the nonlinear
natural frequency of the freely oscillating undamped pendulum, cf. (3.81).
Response backbones can also be determined from numerical models, using, e.g.,
free-decay (“ring down”) simulation or pseudo-arclength continuation, and from
experimental measurements by, e.g., free-decay measurement or control-based
continuation techniques; see, e.g., Denis et al. (2017), Givois et al. (2020), Londoño
et al. (2015), Peter et al. (2016), Renson et al. (2016).
Peak Response The maximum response is often of interest in applications. It
can often be rather easily calculating by noting that the peak point of the response
curve is located where the backbone intersects the response curve. This point can be
found by first solving (3.130) with C 1 = 0 to give the peak response frequency, x
* :
x
Ã
¼ 2
ffiffiffi
b
q
s
;
ð3:150Þ
which is independent on the nonlinearity parameter c – and then substitute this
frequency into in (3.129), along with r = x – 2 from (3.120), to give the corresponding peak response amplitude, a
* :
3.6 The Forced Response – Multiple Scales Analysis
141