An algorithm of solving a quasi-regular infinite system of linear algebraic equations following from a boundary-value problem describing the stationary forced vibrations of an isotropic rectangular prism in the plane linear elastic case is outlined. The algorithm employs Koyalovich’s limitants, which makes it possible to estimate the upper and lower bounds for the entire infinite set of unknowns and the natural frequencies of the prism. Additionally, the sums of all the functional series in the representation of the solution of the boundary-value problem are found in the rectangular domain
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P. S. Bondarenko, “On uniqueness for infinite systems of linear equations,” Mat. Sb., 29, No. 2, 403–418 (1951).
V. T. Grinchenko, Equilibrium and Steady-State Vibrations of Elastic Finite-Size Bodies [in Russian], Naukova Dumka, Kyiv (1978).
V. T. Grinchenko and V. V. Meleshko, Harmonic Vibrations and Waves in Elastic Bodies [in Russian], Naukova Dumka, Kyiv (1981).
A. N. Guz and V. T. Golovchan, Diffraction of Elastic Waves in Multiply Connected Bodies [in Russian], Naukova Dumka, Kyiv (1972).
L. V. Kantorovich and V. I. Krylov, Approximate Methods of Higher Analysis, Fizmatgiz, Moscow–Leningrad (1962).
B. M. Koyalovich, “Study on infinite systems of linear algebraic equations,” Izv. Fiz.-Mat. Inst., No. 3, 41–167 (1930).
S. O. Papkov, “Dynamic problem for a rectangular prism,” Visn. Sevastopol. Derzh. Tekhn. Univ., Ser. Mekh. Energ. Ekol., No. 67, 5–18 (2005).
S. O. Papkov, “Stationary forced vibrations of a prism with given boundary displacements,” Akust. Visn., 11, No. 4, 36–43 (2008).
S. O. Papkov and V. V. Meleshko, ”Flexural vibrations of a rectangular plate with free edges,” Teor. Prikl. Mekh., 46, 104–111 (2009).
S. O. Papkov and V. N. Chekhov, “Regular infinite systems of algebraic equations describing long-period deformation of a prism,” Uchen. Zap. Tavrich. Nats. Univ., No. 1, 81–86 (2001).
S. O. Papkov and V. N. Chekhov, “Determining the resonant frequencies of stationary forced vibrations of a rectangular prism,” Vestn. Sevastopol Nats. Tehkn. Univ., Ser. Fiz. Mat., No. 43, 149–158 (2003).
S. O. Papkov and V. N. Chekhov, “On study of the stationary forced vibrations of a rectangular prism,” in: Trans. Acust. Symp. “Konsonans-2003,” Inst. Gidromekh. NAN Ukrainy (2003), pp. 181–189.
S. O. Papkov and V. N. Chekhov, “Localization of the natural frequencies of a rectangular prism by eliminating unknowns in the quasiregular infinite system,” Dop. NAN Ukrainy, No. 10, 57–62 (2004).
S. O. Papkov and V. N. Chekhov, “Nonstationary deformation of a rectangular prism,” in: Trans. Acust. Symp. “Konsonans-2005,” Inst. Gidromekh. NAN Ukrainy (2005), pp. 255–260.
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Elementary Functions [in Russian], Nauka, Moscow (1981).
V. N. Chekhov and A. V. Pan, ”On limiting expressions for Koyalovich’s limitants,” Dop. NAN Ukrainy, No. 3, 31–36 (2007).
V. N. Chekhov and A. V. Pan, ”Improving the convergence of series for a biharmonic problem in a rectangle,” Dinam. Sist., No. 3, 135–144 (2008).
Val. N. Chekhov, “Stress state of a cross-base prism under torsion,” Int. Appl. Mech., 44, No. 11, 1265–1278 (2008).
V. T. Grinchenko and A. F. Ulitko, “Dynamic problem of elastic theory for a rectangular prism,” Int. Appl. Mech., 7, No. 9, 979–984 (1971).
T. V. Karnaukhova and E. V. Pyatetskaya, “Resonant vibrations of a clamped viscoelastic rectangular plate,” Int. Appl. Mech., 45, No. 8, 762–771 (2009).
A. S. Kosmodamianskii, “Three-dimensional problems of the theory of elasticity for multiply connected plates: Survey,” Int. Appl. Mech., 19, No. 12, 1045–1061 (1983).
V. A. Shaldyrvan and G. S. Bulanov, “Method of homogeneous solutions in problems with mixed boundary conditions,” Int. Appl. Mech., 25, No. 9, 57–61 (1989).
T. A. Vasil’ev and V. A. Shaldyrvan, “Local stress singularities in mixed axisymmetric problems of the bending of circular cylinders,” Int. Appl. Mech., 48, No. 2, 176–187 (2012).
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Translated from Prikladnaya Mekhanika, Vol. 49, No. 5, pp. 62–76, September–October 2013.
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Papkov, S.O., Chekhov, V.N. Limiting Limitants in Dynamic Problems for a Rectangular Prism. Int Appl Mech 49, 555–569 (2013). https://doi.org/10.1007/s10778-013-0589-3
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DOI: https://doi.org/10.1007/s10778-013-0589-3