are close to linear, today’s preferred excitation is typically either impulsive (e.g. by
modal impact hammers) or broadband pseudo-random noise (e.g. by vibration
shakers) (Brandt 2011; Ewins 2000). Frequency sweeps can also be used, though
less efficiently. But whatever the excitation, as long as the systems behaves
approximately linearly there should be no difference (except for the noise) in the
resulting measured frequency responses. And since linear systems cannot bifurcate,
there will certainly be no turning points or branching bifurcations.
With nonlinear systems this is all different. In particular the frequency response
will look differently for different response levels. This appears clearly from, e.g.,
Fig. 3.18, depicting for a cubically hardening nonlinearity the straight-up trivial
resonance peak for low levels of excitation, and a strongly bent-towards-higherfrequencies peak for higher excitation levels. If lab measurements were to be made
for such a system (could be a slender beam clamped between unmovable supports),
only sweep testing could give a meaningful picture of the frequency response. But
even this would be misleading, or at least only partially informing:
First, for excitation levels provoking nonlinearity, the frequency response
obtained from sweeping up in frequency would be quite different from that obtained
from sweeping down, i.e. there would be hysteresis: The resonance peak for the
upsweep would have a vertical drop on the right side, and would be wider than the
downsweep peak – which would have a vertical increase on the right side, but be
less wide that the upsweep peak. To a trained analysist this would be a clear sign of
nonlinearity (along with the presence of higher harmonics in the spectrum).
Secondly, the top of the peak would probably not be observed. Certainly not so
during downsweeps, since the stationary state only jumps up to the high-level
response branch at a some frequency below the peak top frequency. But also not
during upsweeps; dependent on the sweep rate, and the presence of other disturbances, the response in such experiments typically drops down to the low-level
response branch well before the peak top frequency.
5
And thirdly, the unstable branch of the frequency response will not be observed
at all. This is more important that one might think: A theoretically predicted frequency response such as, e.g., the one Fig. 3.18 needs experimental testing to be
trusted. As for the way we have been doing science for hundreds of years, since
Galileo, a physical theory can only be considered scientific if (1) it can be falsified
(physical theories cannot be ‘proved’), and (2) it can be tested experimentally. So,
the prediction of the unstable branches of nonlinear frequency responses are not
really ‘scientific’ – unless they can be experimentally observed.
5
The same happens with numerical simulation, where a frequency response for a nonlinear system
is obtained by sweeping up/incrementing frequency (rather than by numerical continuation;
Sect. 5.11.2): The response typically drops down on the low-level branch well before the peak top
frequency. To come closer to the peak top, one needs to use very small frequency increments, wait
until all transients have surely decayed, and start simulation with initial conditions corresponding
to the solution obtained for the previous frequency.
5.12 Bifurcation Analysis and Continuation in Lab Experiments
305
modal impact hammers) or broadband pseudo-random noise (e.g. by vibration
shakers) (Brandt 2011; Ewins 2000). Frequency sweeps can also be used, though
less efficiently. But whatever the excitation, as long as the systems behaves
approximately linearly there should be no difference (except for the noise) in the
resulting measured frequency responses. And since linear systems cannot bifurcate,
there will certainly be no turning points or branching bifurcations.
With nonlinear systems this is all different. In particular the frequency response
will look differently for different response levels. This appears clearly from, e.g.,
Fig. 3.18, depicting for a cubically hardening nonlinearity the straight-up trivial
resonance peak for low levels of excitation, and a strongly bent-towards-higherfrequencies peak for higher excitation levels. If lab measurements were to be made
for such a system (could be a slender beam clamped between unmovable supports),
only sweep testing could give a meaningful picture of the frequency response. But
even this would be misleading, or at least only partially informing:
First, for excitation levels provoking nonlinearity, the frequency response
obtained from sweeping up in frequency would be quite different from that obtained
from sweeping down, i.e. there would be hysteresis: The resonance peak for the
upsweep would have a vertical drop on the right side, and would be wider than the
downsweep peak – which would have a vertical increase on the right side, but be
less wide that the upsweep peak. To a trained analysist this would be a clear sign of
nonlinearity (along with the presence of higher harmonics in the spectrum).
Secondly, the top of the peak would probably not be observed. Certainly not so
during downsweeps, since the stationary state only jumps up to the high-level
response branch at a some frequency below the peak top frequency. But also not
during upsweeps; dependent on the sweep rate, and the presence of other disturbances, the response in such experiments typically drops down to the low-level
response branch well before the peak top frequency.
5
And thirdly, the unstable branch of the frequency response will not be observed
at all. This is more important that one might think: A theoretically predicted frequency response such as, e.g., the one Fig. 3.18 needs experimental testing to be
trusted. As for the way we have been doing science for hundreds of years, since
Galileo, a physical theory can only be considered scientific if (1) it can be falsified
(physical theories cannot be ‘proved’), and (2) it can be tested experimentally. So,
the prediction of the unstable branches of nonlinear frequency responses are not
really ‘scientific’ – unless they can be experimentally observed.
5
The same happens with numerical simulation, where a frequency response for a nonlinear system
is obtained by sweeping up/incrementing frequency (rather than by numerical continuation;
Sect. 5.11.2): The response typically drops down on the low-level branch well before the peak top
frequency. To come closer to the peak top, one needs to use very small frequency increments, wait
until all transients have surely decayed, and start simulation with initial conditions corresponding
to the solution obtained for the previous frequency.
5.12 Bifurcation Analysis and Continuation in Lab Experiments
305