about a buckled state of equilibrium u(x, t) 6 ¼ 0, and that the nonlinear damping
is negligible (η = 0). For the single-mode approximation (B.25), determine a
first-order multiple scales approximation y = y 0 + O(e) for the beam response
near the primary external resonance X %
ffiffiffiffiffiffiffiffiffiffiffi ffi
À2x 2
p
when (b, q, c) = O(e), e ( 1.
(Introduce ~
x
2 = –2x
2 > 0 and f(t) = y (t) − ~ y, where ~ y denotes the buckled
state of equilibrium). Set up the frequency response equation governing stationary oscillations.
c) Set up a predictive criterion for chaotic oscillations when x
2 < 0 and η = 0. The
criterion can be based on Moon’s Multiwell Potential Criterion and results from
Question b above, but feel free to suggest and examine alternatives. (If Question
b was not answered you may assume stationary oscillations of the form fðtÞ
yðtÞ À ~ y ¼ a cosðXt À wÞ þ OðeÞ where a is a known function of X.)
B.3.7 Numerical Analysis
This part assumes you to use a computer program capable of simulating nonlinear
systems of the form (B.25). In addition, the program should provide plots of time
series, phase planes, frequency spectra, Poincaré maps, and Lyapunov exponents.
Unless otherwise stated you can assume b = 0.02, c = 0.9, η = 0.1, f vdW = 0, and
use y(0) = _
yð0Þ = 0.01 for the initial conditions.
a) For q = 0 and initial conditions y(0) = _
yð0Þ = 0.1, examine the numerically
computed transient response for the purely statically loaded system when,
x
2 = 1 and x
2 = –
1
2 , respectively. Is it as could be expected? Compare to the
predictions obtained with Question 4.
b) Check the quantitative predictions of Question 5a for x
2 = 1, X % 2x, and
small q. Do the stationary amplitudes and dominating frequencies of the
numerical solution agree with the theoretical predictions? For example, what are
the deviations between predicted and simulated amplitudes when, respectively,
q = 0.1 and q = 0.5?
c) Check, as in Question b above, the quantitative predictions obtained in Question
5b for x
2 = –
1
2 , X % ~
x, η = 0, small q, and initial conditions y(0) = 1.01,
_
yð0Þ = 0.01.
d) Simulate the system for x
2 = 1, q = 0.3 and increasing/decreasing values of X.
This corresponds to tracing an amplitude curve of the frequency response
obtained in Question 5a. Do the simulation results agree with the theoretical
predictions?
e) When x
2 < 0 chaotic solutions may exist. For x
2 = –
1
2 , choose a frequency of
excitation X < 4 and an amplitude of excitation q < 1 for which the phase plane
orbits appear to be chaotic. Make sure that the corresponding solutions really are
chaotic by examining frequency spectrum, Poincaré map, and largest Lyapunov
exponent (remember to cut off transients).
474
Appendix B: Major Exercises
is negligible (η = 0). For the single-mode approximation (B.25), determine a
first-order multiple scales approximation y = y 0 + O(e) for the beam response
near the primary external resonance X %
ffiffiffiffiffiffiffiffiffiffiffi ffi
À2x 2
p
when (b, q, c) = O(e), e ( 1.
(Introduce ~
x
2 = –2x
2 > 0 and f(t) = y (t) − ~ y, where ~ y denotes the buckled
state of equilibrium). Set up the frequency response equation governing stationary oscillations.
c) Set up a predictive criterion for chaotic oscillations when x
2 < 0 and η = 0. The
criterion can be based on Moon’s Multiwell Potential Criterion and results from
Question b above, but feel free to suggest and examine alternatives. (If Question
b was not answered you may assume stationary oscillations of the form fðtÞ
yðtÞ À ~ y ¼ a cosðXt À wÞ þ OðeÞ where a is a known function of X.)
B.3.7 Numerical Analysis
This part assumes you to use a computer program capable of simulating nonlinear
systems of the form (B.25). In addition, the program should provide plots of time
series, phase planes, frequency spectra, Poincaré maps, and Lyapunov exponents.
Unless otherwise stated you can assume b = 0.02, c = 0.9, η = 0.1, f vdW = 0, and
use y(0) = _
yð0Þ = 0.01 for the initial conditions.
a) For q = 0 and initial conditions y(0) = _
yð0Þ = 0.1, examine the numerically
computed transient response for the purely statically loaded system when,
x
2 = 1 and x
2 = –
1
2 , respectively. Is it as could be expected? Compare to the
predictions obtained with Question 4.
b) Check the quantitative predictions of Question 5a for x
2 = 1, X % 2x, and
small q. Do the stationary amplitudes and dominating frequencies of the
numerical solution agree with the theoretical predictions? For example, what are
the deviations between predicted and simulated amplitudes when, respectively,
q = 0.1 and q = 0.5?
c) Check, as in Question b above, the quantitative predictions obtained in Question
5b for x
2 = –
1
2 , X % ~
x, η = 0, small q, and initial conditions y(0) = 1.01,
_
yð0Þ = 0.01.
d) Simulate the system for x
2 = 1, q = 0.3 and increasing/decreasing values of X.
This corresponds to tracing an amplitude curve of the frequency response
obtained in Question 5a. Do the simulation results agree with the theoretical
predictions?
e) When x
2 < 0 chaotic solutions may exist. For x
2 = –
1
2 , choose a frequency of
excitation X < 4 and an amplitude of excitation q < 1 for which the phase plane
orbits appear to be chaotic. Make sure that the corresponding solutions really are
chaotic by examining frequency spectrum, Poincaré map, and largest Lyapunov
exponent (remember to cut off transients).
474
Appendix B: Major Exercises