fluids. Also, there are heat exchangers where gases flow around tubes, with water
flowing inside. Flow-induced vibrations of tubes or fuel rods can lead to fatigue.
For periodic vibrations this fatigue is ‘predictable’, so that services can be planned
in time. The author is unaware of any studies related to the fatigue-time associated
with chaotic vibrations.
Machine noise is often associated with chaotic processes. Considering typical
machinery driven by electrical motors, we put in a periodic power signal and
observe a symphony of different sounds in a broadband frequency range. This is
deterministic chaos (e.g., Moon and Broschart 1991).
Plasticity and creep may be responsible for chaotic vibrations. Poddar et al.
(1988) describe a case in which nine different finite-element codes totally disagreed
upon the transient response of an elastic-plastic beam. Reducing the model of the
elastic-plastic beam to a fourth-order system, they provided evidence of chaotic
vibrations. Thus, small differences in numerical procedures were blown up to yield
highly different results after a number of time steps.
6.7 Elastostatical Chaos
Chaos is usually associated with dynamics, that is, with initial value problems
defined on the infinite domain of a time variable. Boundary value problems, on the
other hand, seem to preclude chaos, because boundaries are defined on the finite
domain of a physical object.
Nevertheless, it has been demonstrated that certain complicated phenomena
encountered in elastostatics become explainable when considered as asymptotically
chaotic processes (El Naschie and Al Athel 1989; El Naschie 1990a, b). This is
pertinent, e.g., to localized buckling of shells and the formation of soliton-like
homoclinic loops at seemingly random locations along very long steel tapes. The
term elastostatical chaos was introduced to describe the phenomenon.
As an example we consider the differential equation governing static deformations of a simply supported Euler elastica having length l and bending stiffness EI:
u
00
þ k
2 sin u ¼ 0; k
2
P
EI
; u
0
ð0Þ ¼ u
0
ðlÞ ¼ 0;
ð6:62Þ
where u = u(s) is the angle of cross-sectional rotation along an arch-measuring
variable s, P is the compressive axial force and u
0
¼ du=ds. Suppose the elastica
possesses an initial imperfection in form of a sinusoidal crookedness η(s) = asin
(xs). Then, replacing u by u + η, (6.62) becomes:
u þ a sinðxsÞ
ð
Þ
00 þ k
2 sin u þ a sinðxsÞ
ð
Þ¼0;
ð6:63Þ
or, when for a small crookedness one can assume a ( 1, to first order:
378
6 Chaotic Vibrations
flowing inside. Flow-induced vibrations of tubes or fuel rods can lead to fatigue.
For periodic vibrations this fatigue is ‘predictable’, so that services can be planned
in time. The author is unaware of any studies related to the fatigue-time associated
with chaotic vibrations.
Machine noise is often associated with chaotic processes. Considering typical
machinery driven by electrical motors, we put in a periodic power signal and
observe a symphony of different sounds in a broadband frequency range. This is
deterministic chaos (e.g., Moon and Broschart 1991).
Plasticity and creep may be responsible for chaotic vibrations. Poddar et al.
(1988) describe a case in which nine different finite-element codes totally disagreed
upon the transient response of an elastic-plastic beam. Reducing the model of the
elastic-plastic beam to a fourth-order system, they provided evidence of chaotic
vibrations. Thus, small differences in numerical procedures were blown up to yield
highly different results after a number of time steps.
6.7 Elastostatical Chaos
Chaos is usually associated with dynamics, that is, with initial value problems
defined on the infinite domain of a time variable. Boundary value problems, on the
other hand, seem to preclude chaos, because boundaries are defined on the finite
domain of a physical object.
Nevertheless, it has been demonstrated that certain complicated phenomena
encountered in elastostatics become explainable when considered as asymptotically
chaotic processes (El Naschie and Al Athel 1989; El Naschie 1990a, b). This is
pertinent, e.g., to localized buckling of shells and the formation of soliton-like
homoclinic loops at seemingly random locations along very long steel tapes. The
term elastostatical chaos was introduced to describe the phenomenon.
As an example we consider the differential equation governing static deformations of a simply supported Euler elastica having length l and bending stiffness EI:
u
00
þ k
2 sin u ¼ 0; k
2
P
EI
; u
0
ð0Þ ¼ u
0
ðlÞ ¼ 0;
ð6:62Þ
where u = u(s) is the angle of cross-sectional rotation along an arch-measuring
variable s, P is the compressive axial force and u
0
¼ du=ds. Suppose the elastica
possesses an initial imperfection in form of a sinusoidal crookedness η(s) = asin
(xs). Then, replacing u by u + η, (6.62) becomes:
u þ a sinðxsÞ
ð
Þ
00 þ k
2 sin u þ a sinðxsÞ
ð
Þ¼0;
ð6:63Þ
or, when for a small crookedness one can assume a ( 1, to first order:
378
6 Chaotic Vibrations