or u0 (ξ) to u0 (1)) for rapidly decaying components).
Using solution [1.61] and boundary conditions [1.57], we obtain the frequency of oscillations:
Formula [1.62] at φ1 = φ2 = 1 coincides with Bolotin’s formula [1.18]. Thus, the WKBJ method generalizes the DEE method to the problems with variable coefficients.
As it is mentioned in Chen et al. (1991, 1992), the short-wave (high-frequency) asymptotics gives the same results as the DEE approach for domains of simple geometry. At the same time, DEE does not cover the cases of different geometries (circular, elliptical, etc.) or non-self-adjoint problems. The short-wave asymptotics in the form of the Keller–Rubinow approach (Keller and Rubinow 1960) in Chen et al. (1991) allows more ready extension to other geometries and is more aptly generalizable to dissipative boundary conditions. In other words, it gives the possibility to overcome degeneration of the DEE case.
1.8. Conclusion: DEEM, highly recommended
The importance and usefulness of a particular calculation method is determined by its wide application when studying practically important systems and phenomena. From this point of view, the importance of DEEM is not in doubt.
From the very beginning, DEEM was originated by Bolotin for the analysis of overhead power lines (Bolotin et al. 1958).
DEEM was used to obtain estimates for the density of natural frequencies of shallow shell vibrations (Gavrilov 1961b; Bolotin 1963; Stearn 1970; Zhinzher and Khromatov 1971, 1972a, 1972b; Moskalenko 1972, 1975). This is very important during the study of random vibrations of elastic structures (Bolotin 1966; Birger and Panovko 1968; Moskalenko 1968; Bolotin 1984; Elishakoff et al. 1994).
The influence of the magnetic field on the distribution of plate and shell vibration frequencies was studied in Bagdasaryan (1986), Koreshkova and Khromatov (2009), Golubeva et al. (2013) and Khromatov and Golubeva (2013).
A lot of research devoted to the problems of aeroelastic stability and supersonic flutter can be mentioned (Zhinzher 1983; Zhinzher and Kadarmetov 1984, 1986; Dubovskikh et al. 1996).
We also mention the optimal control problem for continuous systems (Andrianov and Iskra 1991).
DEEM and its generalizations are important particular cases of high-frequency asymptotics. The effectiveness of this method for analyzing the main types of plates and shells used in engineering practices has been proven through experience. The main advantage of DEEM consists of its simplicity and good compatibility with variational approaches.
Naturally, DEEM is not a panacea. For example, when considering a mixed boundary value problem with many points of change in the boundary conditions, the method based on the homotopy parameter (Andrianov et al. 2014) seems more suitable.
Nevertheless, in general, we hope that our review has convinced researchers that DEEM and its generalizations occupied an honorable place in the arsenal of analytical methods for solving the dynamics and stability problems of thin-walled structures.
1.9. Acknowledgments
Several years ago, Professor I. Elishakoff pointed out that it would be useful to prepare a new review of Bolotin’s method, since his previous review on this topic was written in 1976. We are grateful to him for this idea.
CONFLICTS OF INTEREST. – The authors declare no conflict of interest.
1.10. Appendix
Professor Elishakoff enjoys historiography of science and his historical research is read with great interest. Bubnov or Galerkin? Timoshenko or Ehrenfest? The chicken or the egg?
We also provided a little historical research. Bolotin wrote about the possibility of extending the range of applicability of DEEM to PDEs with variable coefficients (Bolotin et al. 1961, Chapter II): “If the coefficients of the equation change slowly, then it is advisable to combine this method with the Wentzel–Brillouin–Kramers method or its related Blumenthal-Shtaerman approach” (translated by us). After reading the relevant papers, we were convinced that Blumenthal used the “WKB method”, created to solve problems of quantum mechanics in 1926, already in 1912 (Blumenthal 1912, 1914). He created this asymptotic method for solving problems of the shell theory. H. Reissner used Blumenthal’s approach in 1912 (Reissner 1912), as did Shtaerman in 1924 (Shtaerman 1924).
With these remarks, we are certainly not going to interfere with the complex priority history of the WKB approach (Wikipedia 2020). We recall Nayfeh’s remark concerning one well-known asymptotic method (Nayfeh 2000, p. 232): “The method of multiple scales is so popular that it is being rediscovered just about every 6 months”. A lot of phenomena in completely different fields of science are described using similar or directly identical equations. Researchers, as a rule, do not search for methods of their solution in areas far from them, but simply rediscover them. The corresponding methods are naturally given different names in different fields of science. Surprisingly, this does not lead to the “Tower of Babel effect”.
1.11. References
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Andrianov, I.V. (2008). Asymptotic construction of nonlinear normal modes for continuous systems. Nonl. Dyn., 51(1–2), 99–109.
Andrianov, I.V. and Iskra, V.S. (1991). Use of Bolotin’s asymptotic method in the optimal control problem. Probl. Mashinostr., 36, 79–82.
Andrianov, I.V. and Kholod, E.G. (1985). Natural nonlinear oscillations of shallow shells. Struct. Mech. Theory Struct., 4, 51–54.
Andrianov, I.V. and Kholod, E.G. (1993a). Intermediate asymptotical forms in nonlinear dynamics of shells. Mech. Solids, 28(2), 160–165.
Andrianov, I.V. and Kholod, E.G. (1993b). Non-linear free vibration of shallow cylindrical shell by Bolotin’s asymptotic method. J. Sound Vib., 165(1), 9–17.
Andrianov, I.V. and Kholod, E.G. (1995). Bolotin’s asymptotic method for nonlinear free vibration of shells. SAMS, 18–19, 211–213.
Andrianov, I.V. and Krizhevskiy, G.A. (1987). Modified asymptotic method for the problems of stiffened constructions dynamics. Struct. Mech. Theory Struct., 2, 66–68.
Andrianov, I.V. and Krizhevskiy, G.A. (1988). Calculation of skew plate natural oscillation by approximate method. Izv. VUZov. Civil Eng. Archit., 12, 46–49.
Andrianov, I.V. and Krizhevskiy, G.A. (1989). Analytical investigation of geometrically nonlinear oscillation of sector plates, reinforced by radial ribs. Dokl. AN Ukr. SSR, ser. A, 11, 30–33.
Andrianov,
