7de20a28-d9dc-4723-96ff-44cf2fab7d4f20211206091614268naun:naunmdt@crossref.orgMDT DepositInternational Journal of Mechanics1998-444810.46300/9104http://www.naun.org/cms.action?id=2828128202112820211510.46300/9104.2021.15https://www.naun.org/cms.action?id=23280Nikolay V.Banichukshlinsky Institute for Problems in Mechanics RAS, Moscow, 119526, RussiaAlexander A.BarsukState University of Moldova, Chișinău, MoldovaSvetlana Yu.Ivanovashlinsky Institute for Problems in Mechanics RAS, Moscow, 119526, RussiaTero T.TuovinenJAMK University of Applied Sciences Jyväskylä, Jyväskylä, FinlandThe paper considers the problems and the methods of spectral analysis of elastic structural systems. The presented consideration focuses on the translation-invariant spectral formulations. Some periodic representations and the spectral decomposition are derived. In the context of general analysis of translation-invariant systems, the particular problems of structural vibration and stability are solved in analytical form.1262021126202124625328https://www.naun.org/main/NAUN/mechanics/2021/a562003-028(2021).pdf10.46300/9104.2021.15.28https://www.naun.org/main/NAUN/mechanics/2021/a562003-028(2021).pdf10.1007/978-3-319-01745-7Banichuk, N., Jeronen, J., Neittaanmäki, P., Saksa, T., Tuovinen, T. 2014. Mechanics of Moving Materials, Berlin/Heidelberg: Springer. 10.1007/978-3-030-23803-2_6Banichuk, N., Barsuk, A., Jeronen, J., Tuovinen, T. Neittaanmäki, P. 2020. Stability of Axially Moving Materials, Berlin/Heidelberg: Springer. Yakubovich, V.A., Starjinsky, V.M. 1972. Linear Differential Equations with Periodic Coefficients and Their Applications, Moscow: Nauka (in Russian). Bolotin, V.V. 1963. Nonconservative Problem of the Theory of the Elastic Stability, New York: Pergamon Press. 10.1007/978-3-030-49882-5_7Banichuk, N., Ivanova, S., Jeronen, J. “Moving web and dynamical problem of aerothermoelastic vibrations and instability,” D.A. Indeitsev and A.M Krivtsov (Eds.): APM 2019., LNME, pp.66–71, Springer Nature Switzerland AG. 2020. 10.1016/j.compstruc.2006.09.007Kulachenko, A., Gradin, P., Koivurova, H. “Modiling the dynamical behaviour of a paper web. Parts I,II,” Computers and Structures, vol. 85, pp. 151–157, 2007. Perkins, N.C. “Linear dynamics of a translating string on an elastic foundation,” Journal of Vibration and Acoustic, vol. 112, no. 2, pp. 2–7, 1990. 10.1115/1.2789016Parker, R.G. “On the eigenvalues and critical speed stability of gyroscopic continua,” ASME Journal of Applied Mechanics, vol. 65, pp. 134–140, 1998. 10.1243/jmes_jour_1973_015_031_02Simpson, A. “Transverse modes and frequencies of beams translating between fixed end supports,” Journal of Mechanical Engineering Science, vol. 15, pp. 159–164, 1973. 10.1007/s10409-005-0066-2Wang, Y., Huang, L., and Liu, X. “Eigenvalue and stability analysis for transverse vibrations of axially moving strings based on Haniltonian dynamics,” Acta Mechanica Sinica, vol. 21, pp. 485–494, 2005. 10.1016/j.jfluidstructs.2009.09.006Banichuk, N., Jeronen, J., Neittaanmäki, P., and Tuovinen, T. “Static instability analysis for travelling membranes and plates interacting with axially moving ideal fluid,” Journal of Fluids and Structures, vol. 26, no. 2, pp. 274–291, 2010. Kronig, R. de L., Penney, W.G. 1931. Quantum mechanics of electrons in crystal lattices, London: Proc. Roy-Soc., vol. 130 A.