Dispersion characteristics and frequency-dependent attenuation of the bending waves propagating in a beam lying on the viscoelastic base

The paper considers an infinite beam lying on the deformable foundation and performing the bending vibrations. Such an idealization is acceptable, if the optimal damping devices are positioned at the beam boundaries, i.e. the boundary fixing parameters are those that do not reflect the disturbances incident on it. This makes it possible to consider the beam model without taking into account the boundary conditions, and vibrations propagating along it could be considered as the traveling bending waves. The paper assumes that the deformable foundation is formed from the Voigt-Kelvin rheological material containing parallel connection of the elastic (spring) and viscous (damper) elements. This material total stress is equal to the stresses sum in the viscous and elastic elements exposed to the identical deformations. The beam midline is assumed to be inextensible. Solution to the problem could be obtained in a form of the traveling harmonic wave with real frequency and the complex wave number. The wave number real part characterizes the propagation constant, using which the wave phase and group velocities are computed, and the imaginary part is the exponential law index attenuating the wave. The paper determines the flexural wave dispersion characteristics and regularities in its frequency-dependent attenuation for different values of the dimensionless parameter specified as the damping coefficient ratio to the deformable foundation rigidity.
EDN: BHLXDW, https://elibrary/bhlxdw

References

[1] Filippov A.P., Kokhmanyuk S.S., Vorobyev Yu.S. Vozdeystvie dinamicheskikh nagruzok na elementy konstruktsiy [Impact of dynamic loads on structural elements]. Kiev, Naukova dumka Publ., 1974. 176 p. (In Russ.).

[2] Kokhmanyuk S.S., Yanyutin V.G., Romanenko L.G. Kolebaniya deformiruemykh sistem pri impulsnykh i podvizhnykh nagruzkakh [Fluctuations of deformable systems under impulse and mobile loads]. Kiev, Naukova dumka Publ., 1980. 231 p. (In Russ.).

[3] Fryba L. Vibration of solids and structures under moving loads. London, Thomas Telford, 1999. 494 p.

[4] Vesnitskiy A.I. Volny v sistemakh s dvizhushchimisya granitsami i nagruzkami [Waves in systems with moving boundaries and loads]. Moscow, Fizmatlit Publ., 2001. 320 p. (In Russ.).

[5] Anisimov V.N., Litvinov V.L. Rezonansnye svoystva mekhanicheskikh obektov s dvizhushchimisya granitsami [Resonance properties of mechanical objects with moving boundaries]. Samara, Izd-vo SamGTU Publ., 2009. 139 p. (In Russ.).

[6] Ivanchenko I.I. Dinamika transportnykh sooruzheniy [Dynamics of transport facilities]. Moscow, Nauka Publ., 2011. 574 p. (In Russ.).

[7] Metrikin A.V., Verichev S.N., Vostrukhov A.V. Fundamentalnye zadachi vysokoskorostnogo nazemnogo transporta [Fundamental problems of high-speed land transport]. LAP Lambert Academic Publ., 2014. 200 p. (In Russ.).

[8] Litvinov V.L., Anisimov V.N. Matematicheskoe modelirovanie i issledovanie kolebaniy odnomernykh mekhanicheskikh sistem s dvizhushchimisya granitsami [Mathematical modelling and study of vibrations of one-dimensional mechanical systems with moving boundaries.]. Samara, Izd-vo SamGTU Publ., 2017. 150 p. (In Russ.).

[9] Ellington J.P. The beam discrete elastic supports. Bulletin of the International Railway Congress Association, 1957, vol. 34, no. 12, pp. 933–941.

[10] Fuss N.I. Opyt teorii o soprotivlenii, prichinyaemom dorogami vsyakogo roda chetyrekhkolesnym i dvukhkolesnym povozkam s opredeleniem obstoyatelstv, pri kotorykh odna iz sikh povozok poleznee drugikh [Experience of the theory of resistance caused by roads to all kinds of four-wheeled and two-wheeled carriages with determination of circumstances in which one of these carriages is more useful than others]. V: Akademicheskie sochineniya, vybrannye iz pervogo toma Deyaniy Imperatorskoy akademii nauk. Ch. 1 [In: Academic works selected from the first volume of the Acts of the Imperial Academy of Sciences.]]. Sankt–Petersburg, 1801, pp. 373–422. (In Russ.).

[11] Winkler E. Die Lehre von der Elastizität und Festigkeit mit besonderer Rücksicht auf ihre Anwendung in der Technik. Prague, Dominicius, 1868. 388 p.

[12] Pasternak P.L. Osnovy novogo metoda rascheta fundamentov na uprugom osnovanii pri pomoshchi dvukh koeffitsientov posteli [Basics of a new method for calculating foundations on an elastic base using two bedding factors]. Moscow, Stroyizdat Publ., 1954. 56 p. (In Russ.).

[13] Erofeev V.I., Kolesov D.A., Lisenkova E.E. Features of wave generation by a source moving along a one-dimensional flexible guide lying on an elastic-inertial foundation. Akusticheskiy zhurnal, 2016, vol. 62, no. 6, pp. 639–647, doi: https://doi.org/10.7868/S0320791916060058 (in Russ.). (Eng. version: Acoust. Phys., 2016 vol. 62, no. 6, pp. 643–650, doi: https://doi.org/10.1134/S1063771016060051)

[14] Frolov K.V., ed. Vibratsii v tekhnike. T. 1. Kolebaniya lineynykh system [Vibrations in engineering. Vol. 1. Vibrations of linear systems]. Moscow, Mashinostroenie Publ., 1999. 504 p. (In Russ.).

[15] Erofeev V.I., Kazhaev V.V., Semerikova N.P. Volny v sterzhnyakh. Dispersiya. Dissipatsiya. Nelineynost [Waves in rods. Dispersion. Dissipation. Nonlinearity]. Moscow, Fizmatlit Publ., 2002. 208 p. (In Russ.).

[16] Vesnitskiy A.I., Miloserdova I.V. Matched end damper of flexural vibrations of a beam. Akusticheskiy zhurnal, 1995, vol. 41, no. 4, pp. 572–575. (In Russ.).

[17] Vesnitskiy A.I. Izbrannye trudy po mekhanike [Selected works on mehancis]. Nizhniy Novgorod, Nash dom Publ., 2010. 248 p. (In Russ.).

[18] Raevskiy A.S., Raevskiy S.B. Kompleksnye volny [Complex waves]. Moscow, Radiotekhnika Publ., 2010. 223 p. (In Russ.).

[19] Gerasimov S.I., Erofeev V.I., Soldatov I.N. Volnovye protsessy v sploshnykh sredakh [Wave processes in solid medium]. Sarov, RFYaTs-VNIIEF Publ., 2012. 258 p. (In Russ.).

[20] Tikhonov A.N., Samarskiy A.A. Uravneniya matematicheskoy fiziki [Equations of mathematic physics]. Moscow, Nauka Publ., 1977. 736 p. (In Russ.).

[21] Gerasimov S.I., Erofeev V.I. Calculation of flexural-and-torsional vibrations of a rocket track rail. Problemy mashinostroeniya i nadezhnosti mashin, 2016, no. 3, pp. 25–27. (In Russ.). (Russ. ed.: J. Mach. Manuf. Reliab., 2016, vol. 45, no. 3, pp. 211–213, doi: https://doi.org/10.3103/S1052618816030055)

[22] Radin V.P., Chirkov V.P., Poznyak E.V. et al. Stability of a rod with the elastic joint exposed to distributed non-conservative load. Izvestiya vysshikh uchebnykh zavedeniy. Mashinostroenie [BMSTU Journal of Mechanical Engineering], 2023, no. 5, pp. 3–13, doi: http://dx.doi.org/10.18698/0536-1044-2023-5-3-13 (in Russ.).

[23] Ovchinnikov V.F., Kapitanov D.V., Glazova E.G. Influence of external cubic viscosity on the dynamics of a cantilever rod loaded with a tracking force. Problemy prochnosti i plastichnosti [Problems of Strength and Plasticity], 2024, vol. 86, no. 1, pp. 83–93, doi: https://doi.org/10.32326/1814-9146-2024-86-1-83-93 (in Russ.).