On an elastic dissipation model for a cantilevered beam
Authors:
W. T. van Horssen and M. A. Zarubinskaya
Journal:
Quart. Appl. Math. 61 (2003), 565-573
MSC:
Primary 74K10; Secondary 35B35, 35Q72, 74H40
DOI:
https://doi.org/10.1090/qam/1999837
MathSciNet review:
MR1999837
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Abstract: In this paper we will study an elastic dissipation model for a cantilevered beam. This problem for a cantilevered beam has been formulated by D. L. Russell as an open problem in [1, 2]. To determine the relationship between the damping rates and the frequencies we will use a recently developed, adapted form of the method of separation of variables. It will be shown that the dissipation model for the cantilevered beam will not always generate damping. Moreover, it will be shown that some solutions can become unbounded.
D. L. Russell, On the positive root of the fourth derivative operator, Quarterly of Applied Mathematics, vol. XVI, No. 4, (1988), p. 751–773.
- David L. Russell, A comparison of certain elastic dissipation mechanisms via decoupling and projection techniques, Quart. Appl. Math. 49 (1991), no. 2, 373–396. MR 1106398, DOI https://doi.org/10.1090/qam/1106398
- 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
- Peng-Fei Yao, On the nonnegative square root of the fourth derivative operators, J. Differential Equations 148 (1998), no. 2, 318–333. MR 1643179, DOI https://doi.org/10.1006/jdeq.1998.3464
- C. R. MacCluer, Boundary value problems and orthogonal expansions, IEEE Press, Piscataway, NJ, 1994. Physical problems from a Sobolev viewpoint. MR 1285355
- W. T. Van Horssen, On the applicability of the method of separation of variables for partial difference equations, J. Difference Equ. Appl. 8 (2002), no. 1, 53–60. MR 1884591, DOI https://doi.org/10.1080/10236190211942
- Shuping Chen, Kangsheng Liu, and Zhuangyi Liu, Spectrum and stability for elastic systems with global or local Kelvin-Voigt damping, SIAM J. Appl. Math. 59 (1999), no. 2, 651–668. MR 1654395, DOI https://doi.org/10.1137/S0036139996292015
- Shu Ping Chen and Roberto Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math. 136 (1989), no. 1, 15–55. MR 971932
- Shu Ping Chen and Roberto Triggiani, Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications, J. Differential Equations 88 (1990), no. 2, 279–293. MR 1081250, DOI https://doi.org/10.1016/0022-0396%2890%2990100-4
- Shu Ping Chen and Roberto Triggiani, Gevrey class semigroups arising from elastic systems with gentle dissipation: the case $0<\alpha <\frac 12$, Proc. Amer. Math. Soc. 110 (1990), no. 2, 401–415. MR 1021208, DOI https://doi.org/10.1090/S0002-9939-1990-1021208-4
D. L. Russell, On mathematical models for the elastic beam with frequency-proportional damping, Chapter 4 in Control and Estimation in Disturbed Parameter Systems, edited by H. T. Banks, vol. 11 of Frontiers in Applied Mathematics, SIAM Pubs., 1992.
D. L. Russell, On the positive root of the fourth derivative operator, Quarterly of Applied Mathematics, vol. XVI, No. 4, (1988), p. 751–773.
D. L. Russell, A comparison of certain elastic dissipation mechanisms via decoupling and projection techniques, Quarterly of Applied Mathematics, Vol. XIX, No. 2, (1991), p. 373–396.
G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping, Quarterly of Applied Mathematics, Vol. XVI, No. 1, (1982), p. 433–454.
P. F. Yao, On the negative square root of the fourth derivative operators, J. of Differential Equations 148, (1998), p. 318–333.
C. R. MacCluer, Boundary value problems and orthogonal expansions, New York, IEEE Press, (1994).
W. T. van Horssen, On the applicability of the method of separation of variables for partial difference equations, Journal of Difference Equations and Applications, vol. 8, No. 1, (2002), p. 53–60.
S. Chen, K. Liu, and Z. Liu, Spectrum and stability for elastic systems with global or local Kelvin-Voigt damping, SIAM J. on Applied Mathematics, vol. 59, No. 2, (1998), p. 651–668.
S. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific Journal of Mathematics, vol. 136, No. 1, (1989), p. 15–55.
S. Chen and R. Triggiani, Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications, J. of Differential Equations, 88 (1990), p. 279–293.
S. Chen and R. Triggiani, Gevrey class semigroups arising from elastic systems with gentle dissipation: the case $0 < \alpha < \frac {1}{2}$, Proceedings of the American Mathematical Society, Vol. 110, No. 2 (1990), p. 401–415.
D. L. Russell, On mathematical models for the elastic beam with frequency-proportional damping, Chapter 4 in Control and Estimation in Disturbed Parameter Systems, edited by H. T. Banks, vol. 11 of Frontiers in Applied Mathematics, SIAM Pubs., 1992.
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© Copyright 2003
American Mathematical Society