a) For p = 0 and initial conditions y(0) = _
y(0) = 0.10, examine the numerically
computed transient response of the unloaded system when, respectively, x
2 = 1
and x
2 = –
1
2 . Is it as could be expected? Compare it with the answers of
Questions B.2.4a–c.
b) Check the quantitative predictions of Question B.2.5a when x
2 = 1, X % x and
p ( 1. Do the stationary amplitudes and dominating frequencies of the
numerical solution agree with the theoretical predictions? What are the deviations between predicted and simulated amplitudes when, respectively, p = 0.01
and p = 0.1?
c) Check, as in B.2.6(b) above, the quantitative predictions of Question B.2.5b
when x
2 = −
1
2 , X % 1 and p ( 1.
d) Simulate the system for x
2 = 1, X = 1.2 and increasing values of p. What is
observed?
e) Simulate the system for x
2 = 1, p = 0.2 and increasing values of X. This corresponds to tracing an amplitude curve of the frequency response obtained in
Question B.2.5a. Do the simulation results agree with the theoretical
predictions?
f) When x
2 < 0 many of the solutions will be chaotic. For x
2 = –
1
2 , choose a
frequency of excitation X < 2 and an amplitude of excitation p < 0.5 for which
the phase plane orbits appear chaotic. Make sure that the corresponding solutions are chaotic by examining frequency spectrum, Poincaré map and largest
Lyapunov exponent (remember to cut off transients). Investigate then the routes
to chaos involved when, respectively, X is varied while p is held constant, and
vice versa. Sketch the associated bifurcation diagrams for X and p. Do you
observe period-doublings, intermittent chaos, transient chaos, or quasiperiodic
responses?
g) Test the predictive chaos criterion obtained in Question B.2.5c, and discuss its
applicability for the present system.
B.2.7 Conclusions
Summarize your findings in brief form, discussing implications in terms of the
physical system, and bring up a few suggestions for a possible extended analysis.
B.3 Dynamics of a Microbeam
Background. The simply supported microbeam system in Fig. B.4 has been proposed as an alternative to the typical cantilever design for microbeams used in
atomic force microscopes (AFM). Your task is to analyze possible responses of the
new proposal, and to provide theoretical results that can aid in designing a first
experimental model. When reporting you should have two readerships in mind: The
468
Appendix B: Major Exercises
y(0) = 0.10, examine the numerically
computed transient response of the unloaded system when, respectively, x
2 = 1
and x
2 = –
1
2 . Is it as could be expected? Compare it with the answers of
Questions B.2.4a–c.
b) Check the quantitative predictions of Question B.2.5a when x
2 = 1, X % x and
p ( 1. Do the stationary amplitudes and dominating frequencies of the
numerical solution agree with the theoretical predictions? What are the deviations between predicted and simulated amplitudes when, respectively, p = 0.01
and p = 0.1?
c) Check, as in B.2.6(b) above, the quantitative predictions of Question B.2.5b
when x
2 = −
1
2 , X % 1 and p ( 1.
d) Simulate the system for x
2 = 1, X = 1.2 and increasing values of p. What is
observed?
e) Simulate the system for x
2 = 1, p = 0.2 and increasing values of X. This corresponds to tracing an amplitude curve of the frequency response obtained in
Question B.2.5a. Do the simulation results agree with the theoretical
predictions?
f) When x
2 < 0 many of the solutions will be chaotic. For x
2 = –
1
2 , choose a
frequency of excitation X < 2 and an amplitude of excitation p < 0.5 for which
the phase plane orbits appear chaotic. Make sure that the corresponding solutions are chaotic by examining frequency spectrum, Poincaré map and largest
Lyapunov exponent (remember to cut off transients). Investigate then the routes
to chaos involved when, respectively, X is varied while p is held constant, and
vice versa. Sketch the associated bifurcation diagrams for X and p. Do you
observe period-doublings, intermittent chaos, transient chaos, or quasiperiodic
responses?
g) Test the predictive chaos criterion obtained in Question B.2.5c, and discuss its
applicability for the present system.
B.2.7 Conclusions
Summarize your findings in brief form, discussing implications in terms of the
physical system, and bring up a few suggestions for a possible extended analysis.
B.3 Dynamics of a Microbeam
Background. The simply supported microbeam system in Fig. B.4 has been proposed as an alternative to the typical cantilever design for microbeams used in
atomic force microscopes (AFM). Your task is to analyze possible responses of the
new proposal, and to provide theoretical results that can aid in designing a first
experimental model. When reporting you should have two readerships in mind: The
468
Appendix B: Major Exercises