A mathematical model for linear elastic systems with structural damping
Authors:
G. Chen and D. L. Russell
Journal:
Quart. Appl. Math. 39 (1982), 433-454
MSC:
Primary 70J20; Secondary 34C28, 34C35
DOI:
https://doi.org/10.1090/qam/644099
MathSciNet review:
644099
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Abstract: We present a mathematical model exhibiting the empirically observed damping rates in elastic systems. The models studied are of the form ($A$ the relevant elasticity operator) \[ \ddot x + B\dot x + Ax = 0\] with $B$ related in various ways to the positive square root, ${A^{1/2}}$, of $A$. Comparison with existing “ad hoc” models is made.
R. P. Boas, A trigonometric moment problem, J. London Math. Soc. 14, 242–244 (1939)
N. Dunford and J. T. Schwartz, Linear operators, Interscience, New York, 1957
Y. C. Fung, Foundations of solid mechanics, Prentice-Hall, Englewood Cliffs, 1965
- Walter C. Hurty and Moshe F. Rubinstein, Dynamics of structures, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0177532
- Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
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- Jerzy Zabczyk, On decomposition of generators, SIAM J. Control Optim. 16 (1978), no. 4, 523–534. MR 512915, DOI https://doi.org/10.1137/0316035
R. P. Boas, A trigonometric moment problem, J. London Math. Soc. 14, 242–244 (1939)
N. Dunford and J. T. Schwartz, Linear operators, Interscience, New York, 1957
Y. C. Fung, Foundations of solid mechanics, Prentice-Hall, Englewood Cliffs, 1965
W. C. Hurty and M. F. Rubinstein, Dynamics of structures, Prentice-Hall, Englewood Cliffs, 1964
T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, 1966
J. L. Lions, Optimal control of systems governed by partial differential equations, Springer-Verlag, New York, 1971
R. S. Phillips, Dissipative operators and hyperbolic systems of partial differential equations, Trans. Amer. Math. Soc. 90, 123–254 (1959)
D. L. Russell, Controllability and stabilizability theory for linear partial differential equations, SIAM Review 20, 639–739 (1978)
R. H. Scanlon and R. Rosenbaum, Introduction to the theory of aircraft vibration and flutter, Macmillan, New York, 1951
J. Zabczyk, On decomposition of generators, SIAM J. Cont. Opt. 16, 523–534 (1978)
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Article copyright:
© Copyright 1982
American Mathematical Society
