On the lift-off constant for elastically supported plates

Bass, R.; Horák, J.; McKenna, P.

doi:10.1090/S0002-9939-04-07428-3

On the lift-off constant for elastically supported plates
HTML articles powered by AMS MathViewer

by R. F. Bass, J. Horák and P. J. McKenna

Proc. Amer. Math. Soc. 132 (2004), 2951-2958

DOI: https://doi.org/10.1090/S0002-9939-04-07428-3

Published electronically: June 2, 2004

PDF | Request permission

Abstract:

In this paper we continue the study begun by Kawohl and Sweers of the precise constant at which the elastic foundation supporting a bending plate can allow lift-off in the case of downward loading. We provide a number of numerical results and a rigorous result on a different counterexample than the one suggested in Kawohl and Sweers (2002). Important open problems are summarized at the conclusion.

References

Richard F. Bass, Probabilistic techniques in analysis, Probability and its Applications (New York), Springer-Verlag, New York, 1995. MR 1329542
Richard F. Bass and Krzysztof Burdzy, Conditioned Brownian motion in planar domains, Probab. Theory Related Fields 101 (1995), no. 4, 479–493. MR 1327222, DOI 10.1007/BF01202781
Boggio, T., Sull’equilibrio delle piastre elastiche incastrate, Rend. Acc. Lincei 10 (1901), 197- 205.
Boggio, T., Sulle funzioni di Green d’ordine $m$, Rend. Circ. Mat. Palermo 20 (1905), 97-135.
M. Cranston, E. Fabes, and Z. Zhao, Potential theory for the Schrödinger equation, Bull. Amer. Math. Soc. (N.S.) 15 (1986), no. 2, 213–216. MR 854557, DOI 10.1090/S0273-0979-1986-15478-9
M. Cranston and T. R. McConnell, The lifetime of conditioned Brownian motion, Z. Wahrsch. Verw. Gebiete 65 (1983), no. 1, 1–11. MR 717928, DOI 10.1007/BF00534989
M. Cranston, Lifetime of conditioned Brownian motion in Lipschitz domains, Z. Wahrsch. Verw. Gebiete 70 (1985), no. 3, 335–340. MR 803674, DOI 10.1007/BF00534865
Charles V. Coffman, On the structure of solutions $\Delta ^{2}u=\lambda u$ which satisfy the clamped plate conditions on a right angle, SIAM J. Math. Anal. 13 (1982), no. 5, 746–757. MR 668318, DOI 10.1137/0513051
C. V. Coffman and R. J. Duffin, On the structure of biharmonic functions satisfying the clamped plate conditions on a right angle, Adv. in Appl. Math. 1 (1980), no. 4, 373–389. MR 603137, DOI 10.1016/0196-8858(80)90018-4
Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
Hans-Christoph Grunau and Guido Sweers, Positivity for perturbations of polyharmonic operators with Dirichlet boundary conditions in two dimensions, Math. Nachr. 179 (1996), 89–102. MR 1389451, DOI 10.1002/mana.19961790106
Hadamard, J., Mémoire sur le problème d’analyse relatif à l’équilibre des plaques élastiques incastrées, Mémoires presentes par divers savants à l’Académie des Sciences, Vol. 33 (1908), 1-128.
Hadamard, J., Sur certains cas intéressants du problème biharmonique, Atti IVe Congr. Intern. Mat. Rome (1909), 12-14.
Bernd Kawohl and Guido Sweers, On ‘anti’-eigenvalues for elliptic systems and a question of McKenna and Walter, Indiana Univ. Math. J. 51 (2002), no. 5, 1023–1040. MR 1947867, DOI 10.1512/iumj.2002.51.2275
Lin, Li, and Adams, G. G., Beam on tensionless elastic foundation J. Eng. Mech., 113 (1986), 542-553.
P. J. McKenna and W. Walter, Nonlinear oscillations in a suspension bridge, Arch. Rational Mech. Anal. 98 (1987), no. 2, 167–177. MR 866720, DOI 10.1007/BF00251232
Patil, S. P., Response of infinite railroad track to vibrating mass, J. Eng. Mech., 114 (1988) 688-703.
Johann Schröder, Operator inequalities, Mathematics in Science and Engineering, vol. 147, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 578001