B.2.5 Quantitative Analysis of the Loaded System
a) Assume that x
2 > 0, so that the pre-tensioned spring either causes the beam to
be tensioned when u(x, t) = 0, or causes compressive forces that are less than the
critical buckling load. Assume further that the load p from the unbalanced
engine is sufficiently small that the beam performs small (but finite) oscillations
about the statical equilibrium position u(x, t) = 0. For the single-mode
approximation (B.17), determine a first-order multiple scales approximation
y = y 0 + ey 1 + O(e
2 ) for the beam response near primary external resonance X
% x, when b, c, p = O(e), e ( 1. Compute the frequency response equation
governing stationary oscillations, and sketch stationary amplitudes as a function
of excitation frequency X. Discuss briefly the changes in beam response that
occur when X is varied slowly across x, as compared to the linear case (c = 0).
b) Assume now that x
2 < 0, so that when u(x, t) = 0 the supporting spring loads
the beam by a compressive force exceeding the critical buckling load. Assume
further that the load p is sufficiently small for the beam to perform small but
finite oscillations about a buckled state of equilibrium u(x, t) 6 ¼ 0. For the
single-mode approximation (B.17), determine a first-order multiple scales
approximation y = y 0 + ey 1 + O(e
2 ) for the beam response near primary external
resonance X % ~
x when b, c, p = O(e), e ( 1. (Introduce ~
x
2 –2x
2 > 0 and
η(t) y(t) − ~ y, where ~ y denotes the buckled state of statical equilibrium).
Compute the frequency response equation governing stationary oscillations.
Discuss briefly the dynamic response as compared to the unbuckled case treated
in Question B.2.5a above.
c) Set up a predictive criterion for chaotic oscillations to appear when x
2 < 0. The
criterion should be based on Moon’s Multiwell Potential Criterion and results
from Question B.2.5b above, but feel free to examine alternatives. (If this
question was not answered you may assume stationary oscillations having the
form η(t) = y(t) − ~ y = acos(Xt − w) + O(e), where a is a known function of X.)
B.2.6 Numerical Analysis
This part assumes you have access to a computer with a program simulating
nonlinear systems. The program should be capable of computing solutions for the
single-mode approximation (B.17), that is, for:
€ y þ 2b_ y þ x
2 y þ cy
3
¼ pX
2 cosðXtÞ:
ðB:18Þ
Further, the program should provide plots of time-series, phase planes, frequency
spectra, Poincaré maps and Lyapunov exponents. Unless otherwise stated you can
assume b = 0.05 and c = 0.5 for the damping and the nonlinearity, respectively,
and y(0) = _
y (0) = 0.01 for the initial conditions.
Appendix B: Major Exercises
467
a) Assume that x
2 > 0, so that the pre-tensioned spring either causes the beam to
be tensioned when u(x, t) = 0, or causes compressive forces that are less than the
critical buckling load. Assume further that the load p from the unbalanced
engine is sufficiently small that the beam performs small (but finite) oscillations
about the statical equilibrium position u(x, t) = 0. For the single-mode
approximation (B.17), determine a first-order multiple scales approximation
y = y 0 + ey 1 + O(e
2 ) for the beam response near primary external resonance X
% x, when b, c, p = O(e), e ( 1. Compute the frequency response equation
governing stationary oscillations, and sketch stationary amplitudes as a function
of excitation frequency X. Discuss briefly the changes in beam response that
occur when X is varied slowly across x, as compared to the linear case (c = 0).
b) Assume now that x
2 < 0, so that when u(x, t) = 0 the supporting spring loads
the beam by a compressive force exceeding the critical buckling load. Assume
further that the load p is sufficiently small for the beam to perform small but
finite oscillations about a buckled state of equilibrium u(x, t) 6 ¼ 0. For the
single-mode approximation (B.17), determine a first-order multiple scales
approximation y = y 0 + ey 1 + O(e
2 ) for the beam response near primary external
resonance X % ~
x when b, c, p = O(e), e ( 1. (Introduce ~
x
2 –2x
2 > 0 and
η(t) y(t) − ~ y, where ~ y denotes the buckled state of statical equilibrium).
Compute the frequency response equation governing stationary oscillations.
Discuss briefly the dynamic response as compared to the unbuckled case treated
in Question B.2.5a above.
c) Set up a predictive criterion for chaotic oscillations to appear when x
2 < 0. The
criterion should be based on Moon’s Multiwell Potential Criterion and results
from Question B.2.5b above, but feel free to examine alternatives. (If this
question was not answered you may assume stationary oscillations having the
form η(t) = y(t) − ~ y = acos(Xt − w) + O(e), where a is a known function of X.)
B.2.6 Numerical Analysis
This part assumes you have access to a computer with a program simulating
nonlinear systems. The program should be capable of computing solutions for the
single-mode approximation (B.17), that is, for:
€ y þ 2b_ y þ x
2 y þ cy
3
¼ pX
2 cosðXtÞ:
ðB:18Þ
Further, the program should provide plots of time-series, phase planes, frequency
spectra, Poincaré maps and Lyapunov exponents. Unless otherwise stated you can
assume b = 0.05 and c = 0.5 for the damping and the nonlinearity, respectively,
and y(0) = _
y (0) = 0.01 for the initial conditions.
Appendix B: Major Exercises
467