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Relaxation oscillations and buckling prognosis for shallow thin shells


This paper reports the possibilities of predicting buckling of thin shells with nondestructive techniques during operation. It examines shallow shells fabricated from high-strength materials. Such structures are known to exhibit surface displacements exceeding the thickness of the elements. In the explored shells, relaxation oscillations of significant amplitude could be generated even under relatively low internal stresses. The problem of cylindrical shell oscillation is mechanically and mathematically modeled in a simplified form by conversion into an ordinary differential equation. To create the model, works by many authors who studied the geometry of the surface formed after buckling (postbuckling behavior) were used. The nonlinear ordinary differential equation for the oscillating shell matches the well-known Duffing equation. It is important that there is a small parameter before the second time derivative in the Duffing equation. This circumstance enables a detailed analysis of the obtained equation to be performed and a description to be given of the physical phenomena—relaxation oscillations—that are unique to thin high-strength shells. It is shown that harmonic oscillations of the shell around the equilibrium position and stable relaxation oscillations are defined by the bifurcation point of the solutions to the Duffing equation. This is the first point in the Feigenbaum sequence at which stable periodic motions are converted into dynamic chaos. The amplitude and period of the relaxation oscillations are calculated based on the physical properties and the level of internal stresses within the shell. Two cases of loading are reviewed: compression along generating elements and external pressure. It is emphasized that if external forces vary in time according to the harmonic law, the periodic oscillation of the shell (nonlinear resonance) is a combination of slow and stick-slip movements. Because the amplitude and the frequency of the oscillations are known, this fact enables an experimental facility to be proposed for prediction of shell buckling with nondestructive techniques. The following requirement is set as a safety factor: maximum load combinations must not cause displacements exceeding the specified limits. Based on the results of the experimental measurements, a formula is obtained to estimate the safety against buckling (safety factor) of the structure.

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Correspondence to Yu. S. Neustadt.

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Neustadt, Y.S., Grachev, V.A. Relaxation oscillations and buckling prognosis for shallow thin shells. Z. Angew. Math. Phys. 71, 142 (2020).

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Mathematics Subject Classification

  • 74B20
  • 74K25
  • 74J35
  • 62K99


  • Elastic shells
  • Buckling
  • Relaxation oscillations
  • Duffing oscillator
  • Safety factor
  • Experimental buckling prediction