The force and frequency-responses obtained for the autoparametric vibration
absorber are common to a range of mechanical systems subjected to conditions of
internal resonance. The following section provides yet an example.
4.3 Nonlinear Mode-Coupling of Non-shallow Arches
Non-shallow arches are predisposed to a condition of internal resonance. Hence,
excitation energy imparted into one mode may spill over into other modes, as
illustrated above for the autoparametric vibration absorber.
Bolotin, in the 1950s, studied the parametrically excited vibrations of
non-shallow arches (Bolotin 1964). Through laboratory experiments he succeeded
in validating many of his own theoretical predictions concerning regions of
dynamic instability and magnitudes of vibration amplitude. However, he also
reported the interesting experimental observation, that ‘… with the approach to
resonance, the picture of vibration is more complex and difficult to understand’
(Bolotin 1964, p. 331). Bolotin hypothesized that these irregularities were consequences of non-idealities in the physical model. At certain conditions, he believed,
the mathematical model ceased to be a valid descriptor of reality.
This took place in the pre-chaos era. Thomsen (1992) later reconsidered this
same system, under the hypothesis that the model of Bolotin was perfectly adequate, but that the strange experimental findings were signs of chaotic dynamics.
Indeed, it was found that the simple mathematical arch-model of Bolotin displayed
chaotic vibrations under conditions similar to those of the laboratory experiments.
We summarize below a multiple scales analysis of Bolotin’s arch-model. This
analysis will shed further light on the phenomenon of modal interaction, and will
provide clues for explaining the strange experimental findings. However, local
perturbation analysis only predicts those of the possible responses that are smooth
and non-chaotic. To study chaotic responses we need to rely on numerical simulation; for this we return to the non-shallow arch in Chap. 6.
4.3.1 The Model
Fig. 4.6(a) shows a hinged-hinged circular arch of radius R, opening span 2a,
bending stiffness EI and mass per unit length qA. A time-harmonic force having
magnitude P(t) = P 0 + P t cos(ht) is applied vertically at the crown. The arch is nonshallow, that is, the angle 2a is so large that the first mode of linear vibration is
antisymmetric whereas the second is symmetric (Fig. 4.6(b)).
The linearized equation of motion is a fifth-order partial differential equation in
the radial and tangential displacements u(u, t) and v(u, t), respectively.
Approximate solutions to this equation are sought in terms of the fundamental mode
of antisymmetric vibration, described by:
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4 Nonlinear Multiple-DOF Systems: Local Analysis
absorber are common to a range of mechanical systems subjected to conditions of
internal resonance. The following section provides yet an example.
4.3 Nonlinear Mode-Coupling of Non-shallow Arches
Non-shallow arches are predisposed to a condition of internal resonance. Hence,
excitation energy imparted into one mode may spill over into other modes, as
illustrated above for the autoparametric vibration absorber.
Bolotin, in the 1950s, studied the parametrically excited vibrations of
non-shallow arches (Bolotin 1964). Through laboratory experiments he succeeded
in validating many of his own theoretical predictions concerning regions of
dynamic instability and magnitudes of vibration amplitude. However, he also
reported the interesting experimental observation, that ‘… with the approach to
resonance, the picture of vibration is more complex and difficult to understand’
(Bolotin 1964, p. 331). Bolotin hypothesized that these irregularities were consequences of non-idealities in the physical model. At certain conditions, he believed,
the mathematical model ceased to be a valid descriptor of reality.
This took place in the pre-chaos era. Thomsen (1992) later reconsidered this
same system, under the hypothesis that the model of Bolotin was perfectly adequate, but that the strange experimental findings were signs of chaotic dynamics.
Indeed, it was found that the simple mathematical arch-model of Bolotin displayed
chaotic vibrations under conditions similar to those of the laboratory experiments.
We summarize below a multiple scales analysis of Bolotin’s arch-model. This
analysis will shed further light on the phenomenon of modal interaction, and will
provide clues for explaining the strange experimental findings. However, local
perturbation analysis only predicts those of the possible responses that are smooth
and non-chaotic. To study chaotic responses we need to rely on numerical simulation; for this we return to the non-shallow arch in Chap. 6.
4.3.1 The Model
Fig. 4.6(a) shows a hinged-hinged circular arch of radius R, opening span 2a,
bending stiffness EI and mass per unit length qA. A time-harmonic force having
magnitude P(t) = P 0 + P t cos(ht) is applied vertically at the crown. The arch is nonshallow, that is, the angle 2a is so large that the first mode of linear vibration is
antisymmetric whereas the second is symmetric (Fig. 4.6(b)).
The linearized equation of motion is a fifth-order partial differential equation in
the radial and tangential displacements u(u, t) and v(u, t), respectively.
Approximate solutions to this equation are sought in terms of the fundamental mode
of antisymmetric vibration, described by:
220
4 Nonlinear Multiple-DOF Systems: Local Analysis