A comparison of certain elastic dissipation mechanisms via decoupling and projection techniques
Author:
David L. Russell
Journal:
Quart. Appl. Math. 49 (1991), 373-396
MSC:
Primary 73K05; Secondary 35Q72, 73B05, 73D35
DOI:
https://doi.org/10.1090/qam/1106398
MathSciNet review:
MR1106398
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Abstract: In this paper we study the Euler-Bernoulli elastic beam model, modified in a variety of ways to achieve an asymptotically linear relationship between damping rate and frequency. We review the so-called spatial hysteresis model and then introduce the thermoelastic/shear diffusion model, which is obtained by coupling the originally conservative elastic equations to two different diffusion processes. We then use a decoupling/triangulation process to project the coupled system onto the subspace corresponding to the lateral displacements and velocities and show that the projected system agrees in many significant respects with the spatial hysteresis model. The procedure also indicates some possibly desirable modifications in the elastic term of the spatial hysteresis model.
J. D. Achenbach, Wave Propagation in Elastic Solids, North-Holland/American Elsevier, Amsterdam-New York-Oxford, 1973, pp. 391 ff.
C. W. Bert, Material damping: an introductory review of mathematical models, measures and experimental technique, J. Sound. Vibration 29, 272–292 (1973)
- G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping, Quart. Appl. Math. 39 (1981/82), no. 4, 433–454. MR 644099, DOI https://doi.org/10.1090/S0033-569X-1982-0644099-3
- Scott Walter Hansen, Frequency-proportional damping models for the Euler-Bernoulli beam equation, ProQuest LLC, Ann Arbor, MI, 1988. Thesis (Ph.D.)–The University of Wisconsin - Madison. MR 2637169
- Fa Lun Huang, On the holomorphic property of the semigroup associated with linear elastic systems with structural damping, Acta Math. Sci. (English Ed.) 5 (1985), no. 3, 271–277. MR 856278, DOI https://doi.org/10.1016/S0252-9602%2818%2930548-4
- Isaak A. Kunin, Elastic media with microstructure. I, Springer Series in Solid-State Sciences, vol. 26, Springer-Verlag, Berlin-New York, 1982. One-dimensional models; Translated from the Russian. MR 664203
- James E. Potter, Matrix quadratic solutions, SIAM J. Appl. Math. 14 (1966), 496–501. MR 201457, DOI https://doi.org/10.1137/0114044
- Robert C. Rogers, Some remarks on nonlocal interactions and hysteresis in phase transitions, Contin. Mech. Thermodyn. 8 (1996), no. 1, 65–73. MR 1388608, DOI https://doi.org/10.1007/s001610050029
D. L. Russell, Mathematical models for the elastic beam with frequency-proportional damping, Frontiers in Applied Mathematics (H. T. Banks, ed.), SIAM, to appear
- D. L. Russell, On the positive square root of the fourth derivative operator, Quart. Appl. Math. 46 (1988), no. 4, 751–773. MR 973388, DOI https://doi.org/10.1090/S0033-569X-1988-0973388-X
- David L. Russell, Mathematics of finite-dimensional control systems, Lecture Notes in Pure and Applied Mathematics, vol. 43, Marcel Dekker, Inc., New York, 1979. Theory and design. MR 531035
- David L. Russell, A general framework for the study of indirect damping mechanisms in elastic systems, J. Math. Anal. Appl. 173 (1993), no. 2, 339–358. MR 1209323, DOI https://doi.org/10.1006/jmaa.1993.1071
S. P. Timoshenko, Vibration Problems in Engineering, 2nd ed., Van Nostrand, Princeton, New Jersey, 1955
J. D. Achenbach, Wave Propagation in Elastic Solids, North-Holland/American Elsevier, Amsterdam-New York-Oxford, 1973, pp. 391 ff.
C. W. Bert, Material damping: an introductory review of mathematical models, measures and experimental technique, J. Sound. Vibration 29, 272–292 (1973)
G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping, Quart. Appl. Math. 39, 433–454 (1982)
S. W. Hansen, Frequency-proportional damping models for the Euler-Bernoulli beam equation, Thesis, University of Wisconsin-Madison, December, 1988
F.-L. Huang, Proof of the holomorphic semigroup property for an Euler-Bernoulli beam structural damping model, to appear
I. A. Kunin, Elastic Media with Microstructure I; One-dimensional Models, Springer Series in Solid State Sciences, vol. 26, Springer-Verlag, New York, 1982
J. E. Potter, Matrix quadratic solutions, SIAM J. Appl. Math. 14, 496–501 (1966)
R. C. Rogers and D. L. Russell, to appear
D. L. Russell, Mathematical models for the elastic beam with frequency-proportional damping, Frontiers in Applied Mathematics (H. T. Banks, ed.), SIAM, to appear
D. L. Russell, On the positive square root of the fourth derivative operator, Quart. Appl. Math. 46, 751–773 (1988)
D. L. Russell, Mathematics of Finite-Dimensional Control Systems: Theory and Design, Lecture Notes in Pure and Appl. Math., vol. 43, Marcel Dekker, New York, 1979 (cf. p. 225, ff.)
D. L. Russell, A general framework for the study of indirect damping mechanisms in elastic systems, submitted to J. Math. Anal. Appl.
S. P. Timoshenko, Vibration Problems in Engineering, 2nd ed., Van Nostrand, Princeton, New Jersey, 1955
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Article copyright:
© Copyright 1991
American Mathematical Society