Characteristic spectra of the curvature functional: a numerical study in bifurcation

ABSTRACT

A method is described for the eignevalues of piecewise smooth C2 extremum-energy curves. Typical interpolants are investigated within the framework of their eigensystems, and conclusions are presented concerning their natural modes of vibration, stability state, and limits of existence.In the present discussion the word “spline” means exclusively an interpolating elastica.

REFERENCES

• 1: COOPE, I. D. 1993. Curve interpolation with nonlinear spiral splines. IMA J. Num. Anal. 13, 327-341.
• 2: John A. Edwards, Exact equations of the nonlinear spline, ACM Transactions on Mathematical Software (TOMS), v.18 n.2, p.174-192, June 1992 [doi>10.1145/146847.146925]
• 3: GILMORE, R. 1981. Catastrophe Theory for Scientists and Engineers. John Wiley and Sons, Inc., New York, NY.
• 4: B. K. P. Horn, The Curve of Least Energy, ACM Transactions on Mathematical Software (TOMS), v.9 n.4, p.441-460, Dec. 1983 [doi>10.1145/356056.356061]
• 5: LANCZOS, C. 1970. The Variational Principles of Mechanics. 4th ed. University of Toronto Press, Toronto, Canada.
• 6: LARKIN, F. M. 1966. An interpolation procedure based on fitting elasticas to given data points. Rep. CLM-R284. Culham Laboratory, Abingdon, UK.
• 7: RAYLEIGH, J. W. S. 1894. Theory of Sound. 2nd ed. Macmillan Press Ltd., London, UK.
• 8: TRUESDELL, C. 1983. The influence of elasticity on analysis: The classic heritage. Bull. AMS 9, 3, 293-310.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.1 Interpolation
          Subjects: Spline and piecewise polynomial interpolation

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.3 Numerical Linear Algebra
          Subjects: Eigenvalues and eigenvectors (direct and iterative methods)
      G.1.6 Optimization
          Subjects: Constrained optimization
      G.1.7 Ordinary Differential Equations
          Subjects: Boundary value problems

  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.2 Algorithms
          Subjects: Algebraic algorithms

  J. Computer Applications
  J.2 PHYSICAL SCIENCES AND ENGINEERING
      Subjects: Physics; Engineering
  J.6 COMPUTER-AIDED ENGINEERING
      Subjects: Computer-aided design (CAD); Computer-aided manufacturing (CAM)

General Terms:
Algorithms, Theory

Keywords:
Morse theory, acoustics, buckling, catastrophes, degenerate critical points, deterministic chaos, dynamical systems, eigenanalysis, elastica, elasticity, energy extrema, generalized coordinates, modal analysis, structural stability, variational methods, vibrations