Modern Trends in Structural and Solid Mechanics 2

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”.

Abramowitz, M. and Stegun, I.A. (1965). Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables . Dover Publications, New York.

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, I.V. and Krizhevskiy, G.A. (1991). Investigation of natural oscillation of circle and sector plates with consideration of geometrical nonlinearity. Mech. Solids , 26(2), 143–148.

Andrianov, I.V. and Krizhevsky, G.A. (1993). Free vibration analysis of rectangular plates with structural inhomogeneity. J. Sound Vib ., 162(2), 231–241.

Andrianov, I.V., Manevitch, L.I., Kholod, E.G. (1979). On the nonlinear oscillation of rectangular plates. Struct. Mech. Theory Struct ., 5, 48–51.

Andrianov, I.V., Awrejcewicz, J., Manevitch, L.I. (2004). Asymptotical Mechanics of Thin-Walled Structures: A Handbook . Springer-Verlag, Heidelberg, Berlin.

Andrianov, I.V., Awrejcewicz, J., Danishevs’kyy, V.V., Ivankov, A.O. (2014). Asymptotic Methods in the Theory of Plates with Mixed Boundary Conditions . John Wiley & Sons, Chichester.

Avramov, K.V. and Mikhlin, Y.V. (2013). Review of applications of nonlinear normal modes for vibrating mechanical systems. Appl. Mech. Rev ., 65(2), 020801-20.

Awrejcewicz, J., Andrianov, I.V., Manevitch, L.I. (1998). Asymptotic Approaches in Nonlinear Dynamics: New Trends and Applications . Springer-Verlag, Heidelberg, Berlin, New York.

Babich, V.M. and Buldyrev, V.S. (1991). Asymptotic Methods in Short-Wavelength Diffraction Theory . Springer, Berlin.

Babich, V.M., Buldyrev, V.S., Molotkov, I.A. (1985). A Space-Time Ray Method . Leningrad University, Leningrad.

Bagdasaryan, G.E. (1986). Application of V.V. Bolotin’s asymptotic methods for investigation of magnetoelastic vibration of rectangular plates. Probl. Mashinostr ., 25, 63–68.

Bauer, S.M., Filippov, S.B., Smirnov, A.L., Tovstik, P.E., Vaillancourt, R. (2015). Asymptotic Methods in Mechanics of Solids . Birkhäuser, Basel.

Birger, I.A. and Panovko, Y.G. (1968). Prochnost. Ustoichivost. Kolebaniya (Strength. Stability. Oscillations Handbook) 3 . Mashinostroyenie, Moscow.

Blumenthal, O. (1912). Über asymptotische Integration von Differentialgleichungen mit Anwendung auf eine asymptotische Theorie der Kugelfunctionen. Archiv Math. Physik, ser. 3 , 19, 136–174.

Blumenthal, O. (1914). Über asymptotische Integration von Differentialgleichungen mit Anwendung auf die Berechnung von Spannungen in Kugelschalen. Z. Math. Physik , 62, 343–358. Extract previously appeared in Proc. Fifth Intern. Cong. Math ., Cambridge (1913), II, 319–327.

Bolotin, V.V. (1960a). Dynamic edge effect in the elastic vibrations of plates. Inzh. Sb ., 31, 3–14.

Bolotin, V.V. (1960b). The edge effect in the oscillations of elastic shells. J. Appl. Math. Mech ., 24(5), 1257–1272.

Bolotin, V.V. (1961a). A generalization of the asymptotic method of the eigenvalue problems for rectangular regions. Inzh. Zh ., 1(3), 86–92.

Bolotin, V.V. (1961b). An asymptotic method for the study of the problem of eigenvalues of rectangular regions. Problems of Continuum Mechanics , SIAM, 56–68.

Bolotin, V.V. (1961c). Asymptotic method in the theory of oscillations of elastic plates and shells. Tr. Konf. po Teorii Plastin i Obolochek , Kazan State University, 21–26.

Bolotin, V.V. (1961d). The natural oscillations of a rectangular elastic parallelepiped. J. Appl. Math. Mech ., 25(1), 220–227.

Bolotin, V.V. (1963). On the density of the distribution of natural frequencies of thin elastic shells. J. Appl. Math. Mech ., 27(2), 538–543.

Bolotin, V.V. (1966). Broadband random vibrations of elastic systems. Int. J. Solids Struct ., 2(1), 105–124.

Bolotin, V.V. (1970). Application of edge effect theory to forced vibration analysis of elastic systems. Trudy Moscow Energet. Inst. Dyn. Soprot. Mater ., 74, 180–192.

Bolotin, V.V. (1984). Random Vibrations of Elastic Systems . Springer, Dordrecht.

Bolotin, V.V. (2006). 80th birthday tribute. J. Appl. Math. Mech ., 70(2), 161–175.

Bolotin, V.V., Marein N.S., Vinokurov A.I., Poznyak E.L., Ivovich V.A. (1958). Vibration and vibrational strength of overhead power lines . Nauch. Dokl. Vish. Shkoly. Energetika , 2, 55–62.

Bolotin, V.V., Makarov, V.P., Mishenkov, G.V., Shveiko, Yu.Yu. (1960). Asymptotic method of investigating the eigenfrequency spectrum of elastic plates. Rasch. Prochn ., 6, 231–253.

Bolotin, V.V., Gol’denblat, I.I., Smirnov, A.F. (1961). Modern Problems of Structural Mechanics . Stroyizdat, Moscow.