Damping Properties of Vibrations of Three-Layer VIscoelastic Plate

Safarov Ismail Ibrahimovich; Teshayev Muhsin Khudoyberdiyevich; Boltayev Zafar Ixtiyorovich; Akhmedov Maqsud Sharipovich

doi:doi:10.11648/j.ijtam.20170306.13

Damping Properties of Vibrations of Three-Layer VIscoelastic Plate

Safarov Ismail Ibrahimovich, Teshayev Muhsin Khudoyberdiyevich, Boltayev Zafar Ixtiyorovich, Akhmedov Maqsud Sharipovich

Published in International Journal of Theoretical and Applied Mathematics (Volume 3, Issue 6)

Received: 28 September 2017 Accepted: 3 November 2017 Published: 30 November 2017

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Abstract

The work is devoted to the study of harmonic waves in a hereditarily elastic plate with two viscoelastic coatings, the properties of the material, which are described by the equations of state in integral form. The fractional exponential function of Rabotnov and Koltunov-Rzhanitsyn was chosen as the kernel of the integral operator. Two cases are considered: the case of a stress-strain state symmetric and antisymmetric in the normal coordinate (VAT). In the study of natural oscillations, the properties of those modes that are time-dependent by harmonic law are investigated. For both cases, dispersion equations are derived, which are solved numerically. Asymptotics of the roots of dispersion equations for small and large frequencies are also obtained. The analysis of the obtained solutions made it possible to draw conclusions about the influence of hereditary factors on the behavior of dispersion curves. A comparative analysis of numerical solutions and their asymptotics is carried out.

Published in	International Journal of Theoretical and Applied Mathematics (Volume 3, Issue 6)
DOI	10.11648/j.ijtam.20170306.13
Page(s)	191-198
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2017. Published by Science Publishing Group

Keywords

Dispersion Equations, Stress-Strain State, Hereditarily Elastic Layer, Asymptotics

References

[1]	Kulesh M. A., Shardakov I. N. Wave dynamics of elastic media. PSU, Perm, 2007.- 60p.
[2]	Kayumov S. S., Safarov I. I. Propagation and diffraction of waves in dissipative - inhomogeneous cylindrical deformable mechanical systems. Tashkent, 2002, 214p.
[3]	Davies R. M. Stress waves in solids. Мoscow, 1961, -104 p.
[4]	Miker T., Meitzler A. Waveguide propagation in extended cylinders and plates. - Phys. Acoustics. Principles and methods. Trans. from English, 1966, 1 A, p. 140-203
[5]	Natif A., Jones D., Henderson J. Dampening of vibrations: Persanal-M., 1968, 448 p.
[6]	Safarov II, Teshaev M. Kh., Boltaev Z. I. Wave processes in a mechanical waveguide. LAP LAMBERT Academic publishing (Germany). 2012., 217 pp.
[7]	Safarov I. I., Rashidov M., Kayumov S. S. Propagation of waves in dissipatively irregular planar bodies. International Conference "Aviation and Cosmonautics", 2006. Moscow p. 112-113.
[8]	Seymov V. M., Trofimchuk O. A., Savitsky O. A. Vibrations and waves in layered media. -Kyiv: Science. Dumka, 1990.-224 p.
[9]	Timoshenko S. P. Plates and shells. - Moscow: Nauka, 1966, 597 p.
[10]	Achenbach J. D., Keshava S/P/ Free waves in a plate supted by a sani-infinite continuum –Trans. ASME. Ser.E J.Mech, 1967, V.34. #2. p.398-404
[11]	Safarov I. I., Boltaev Z. I., Akhmedov M. Distribution of the natural waves. LAP LAMBERT Academic Publishing Saarbrucren Dentschland /Germanu/-2015. -110p.
[12]	Safarov I. I., Akhmedov M., Rajabov O. Vibrations of plates and shells with attached concentrated mass. LAP LAMBERT Academic Publishing Saarbrucren Dentschland /Germanu/-2015 - 92p.
[13]	Mirsaidov M. M., Troyanovsky I. E. Dynamics of inhomogeneous systems with allowance for internal dissipation and wave entrainment of energy. Tashkent: 1990. -170 p.
[14]	Safarov I. I., Akhmedov M. S., Boltaev Z. I. DUCTING IN EXTENDED PLATES OF VARIABLE THICKNES. Sciences of Europe (Praha, Czech Republic). Vol 2, No 1 (1) (2016). р.21-39.
[15]	Safarov I. I., Boltaev Z. I., Akhmedov M.Sh., Radzhabov O. I. Influence of the density of a liquid on the wave process in a viscoelastic fluid system. "The first independent scientific bulletin." Kiev, Monthly Scientific Journal, HF No. 20489-10289RR, No. 6 / 2016. P. 109-121.
[16]	Safarov I. I., Teshaev M. Kh., Boltaev Z. I., Axmedov M.Sh.Coommon natural in dissipative inhomogeous plane Bodies /Discovery, 2016,52,(251) 2108-2126.
[17]	Hileo Saito, Kinchi Sato. Propagation of flexural waves and oscillations of multilayer rods and beams. Applied Mechanics. №2,1962.P.78-87.
[18]	Teshaev M. Kh., Boltaev Z. I., Nuriddinov B. Z. Of Own and Forced Vibrations of Dissipative Inhomogeneous Mechanical Systems. Applied Mathematics, 2017, 8. P.1001-1015
[19]	Rabotnov Yu. N. Elements of hereditary mechanics of solids. M.: Nauka press.1977. 383 p.
[20]	Rzhanitsyn A. R. Creep theory. Moscow: Stroyizdat press, 1968.16 p.
[21]	Koltunov M. A. To the problem of the choice of kernels in solving problems with allowance for creep and relaxation // Mechanics of polymers. 1966. №4. P. 483-497.
[22]	Koltunov M. A., Mayboroda V. P., Zubchaninov V. G. Strength calculations of products made of polymer materials. M.: “Mashinastroyenie” press. 1983. 239 p.
[23]	Korn G., Korn T. Handbook of Mathematics (for scientists and engineers). M.: Nauka press, 1984.- 832 p.
[24]	Koltunov M. A. Creep and relaxation.-М: “Wisshaya shkola” press, 1976.- 276 p.
[25]	Adamov A. A., Matveenko V. P., Trufanov N. A., Shardakov I. N. Methods of applied viscoelasticity. Ekaterinburg. 2003. -411p.

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Safarov Ismail Ibrahimovich, Teshayev Muhsin Khudoyberdiyevich, Boltayev Zafar Ixtiyorovich, Akhmedov Maqsud Sharipovich. (2017). Damping Properties of Vibrations of Three-Layer VIscoelastic Plate. International Journal of Theoretical and Applied Mathematics, 3(6), 191-198. https://doi.org/10.11648/j.ijtam.20170306.13

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ACS Style

Safarov Ismail Ibrahimovich; Teshayev Muhsin Khudoyberdiyevich; Boltayev Zafar Ixtiyorovich; Akhmedov Maqsud Sharipovich. Damping Properties of Vibrations of Three-Layer VIscoelastic Plate. Int. J. Theor. Appl. Math. 2017, 3(6), 191-198. doi: 10.11648/j.ijtam.20170306.13

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AMA Style

Safarov Ismail Ibrahimovich, Teshayev Muhsin Khudoyberdiyevich, Boltayev Zafar Ixtiyorovich, Akhmedov Maqsud Sharipovich. Damping Properties of Vibrations of Three-Layer VIscoelastic Plate. Int J Theor Appl Math. 2017;3(6):191-198. doi: 10.11648/j.ijtam.20170306.13

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@article{10.11648/j.ijtam.20170306.13,
  author = {Safarov Ismail Ibrahimovich and Teshayev Muhsin Khudoyberdiyevich and Boltayev Zafar Ixtiyorovich and Akhmedov Maqsud Sharipovich},
  title = {Damping Properties of Vibrations of Three-Layer VIscoelastic Plate},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {3},
  number = {6},
  pages = {191-198},
  doi = {10.11648/j.ijtam.20170306.13},
  url = {https://doi.org/10.11648/j.ijtam.20170306.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20170306.13},
  abstract = {The work is devoted to the study of harmonic waves in a hereditarily elastic plate with two viscoelastic coatings, the properties of the material, which are described by the equations of state in integral form. The fractional exponential function of Rabotnov and Koltunov-Rzhanitsyn was chosen as the kernel of the integral operator. Two cases are considered: the case of a stress-strain state symmetric and antisymmetric in the normal coordinate (VAT). In the study of natural oscillations, the properties of those modes that are time-dependent by harmonic law are investigated. For both cases, dispersion equations are derived, which are solved numerically. Asymptotics of the roots of dispersion equations for small and large frequencies are also obtained. The analysis of the obtained solutions made it possible to draw conclusions about the influence of hereditary factors on the behavior of dispersion curves. A comparative analysis of numerical solutions and their asymptotics is carried out.},
 year = {2017}
}

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TY  - JOUR
T1  - Damping Properties of Vibrations of Three-Layer VIscoelastic Plate
AU  - Safarov Ismail Ibrahimovich
AU  - Teshayev Muhsin Khudoyberdiyevich
AU  - Boltayev Zafar Ixtiyorovich
AU  - Akhmedov Maqsud Sharipovich
Y1  - 2017/11/30
PY  - 2017
N1  - https://doi.org/10.11648/j.ijtam.20170306.13
DO  - 10.11648/j.ijtam.20170306.13
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 191
EP  - 198
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20170306.13
AB  - The work is devoted to the study of harmonic waves in a hereditarily elastic plate with two viscoelastic coatings, the properties of the material, which are described by the equations of state in integral form. The fractional exponential function of Rabotnov and Koltunov-Rzhanitsyn was chosen as the kernel of the integral operator. Two cases are considered: the case of a stress-strain state symmetric and antisymmetric in the normal coordinate (VAT). In the study of natural oscillations, the properties of those modes that are time-dependent by harmonic law are investigated. For both cases, dispersion equations are derived, which are solved numerically. Asymptotics of the roots of dispersion equations for small and large frequencies are also obtained. The analysis of the obtained solutions made it possible to draw conclusions about the influence of hereditary factors on the behavior of dispersion curves. A comparative analysis of numerical solutions and their asymptotics is carried out.
VL  - 3
IS  - 6
ER  -

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Author Information

Safarov Ismail Ibrahimovich

Department of “Mathematics”, Tashkent Khimical-Technological Institute, Tashkent, Uzbekistan
Teshayev Muhsin Khudoyberdiyevich

Department of “Mathematics”, Bukhara Engineering-Technological Institute, Bukhara, Uzbekistan
Boltayev Zafar Ixtiyorovich

Department of “Mathematics”, Bukhara Engineering-Technological Institute, Bukhara, Uzbekistan
Akhmedov Maqsud Sharipovich

Department of “Mathematics”, Bukhara Engineering-Technological Institute, Bukhara, Uzbekistan

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Plain Text BibTeX RIS

APA Style

Safarov Ismail Ibrahimovich, Teshayev Muhsin Khudoyberdiyevich, Boltayev Zafar Ixtiyorovich, Akhmedov Maqsud Sharipovich. (2017). Damping Properties of Vibrations of Three-Layer VIscoelastic Plate. International Journal of Theoretical and Applied Mathematics, 3(6), 191-198. https://doi.org/10.11648/j.ijtam.20170306.13

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ACS Style

Safarov Ismail Ibrahimovich; Teshayev Muhsin Khudoyberdiyevich; Boltayev Zafar Ixtiyorovich; Akhmedov Maqsud Sharipovich. Damping Properties of Vibrations of Three-Layer VIscoelastic Plate. Int. J. Theor. Appl. Math. 2017, 3(6), 191-198. doi: 10.11648/j.ijtam.20170306.13

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AMA Style

Safarov Ismail Ibrahimovich, Teshayev Muhsin Khudoyberdiyevich, Boltayev Zafar Ixtiyorovich, Akhmedov Maqsud Sharipovich. Damping Properties of Vibrations of Three-Layer VIscoelastic Plate. Int J Theor Appl Math. 2017;3(6):191-198. doi: 10.11648/j.ijtam.20170306.13

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@article{10.11648/j.ijtam.20170306.13,
  author = {Safarov Ismail Ibrahimovich and Teshayev Muhsin Khudoyberdiyevich and Boltayev Zafar Ixtiyorovich and Akhmedov Maqsud Sharipovich},
  title = {Damping Properties of Vibrations of Three-Layer VIscoelastic Plate},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {3},
  number = {6},
  pages = {191-198},
  doi = {10.11648/j.ijtam.20170306.13},
  url = {https://doi.org/10.11648/j.ijtam.20170306.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20170306.13},
  abstract = {The work is devoted to the study of harmonic waves in a hereditarily elastic plate with two viscoelastic coatings, the properties of the material, which are described by the equations of state in integral form. The fractional exponential function of Rabotnov and Koltunov-Rzhanitsyn was chosen as the kernel of the integral operator. Two cases are considered: the case of a stress-strain state symmetric and antisymmetric in the normal coordinate (VAT). In the study of natural oscillations, the properties of those modes that are time-dependent by harmonic law are investigated. For both cases, dispersion equations are derived, which are solved numerically. Asymptotics of the roots of dispersion equations for small and large frequencies are also obtained. The analysis of the obtained solutions made it possible to draw conclusions about the influence of hereditary factors on the behavior of dispersion curves. A comparative analysis of numerical solutions and their asymptotics is carried out.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Damping Properties of Vibrations of Three-Layer VIscoelastic Plate
AU  - Safarov Ismail Ibrahimovich
AU  - Teshayev Muhsin Khudoyberdiyevich
AU  - Boltayev Zafar Ixtiyorovich
AU  - Akhmedov Maqsud Sharipovich
Y1  - 2017/11/30
PY  - 2017
N1  - https://doi.org/10.11648/j.ijtam.20170306.13
DO  - 10.11648/j.ijtam.20170306.13
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 191
EP  - 198
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20170306.13
AB  - The work is devoted to the study of harmonic waves in a hereditarily elastic plate with two viscoelastic coatings, the properties of the material, which are described by the equations of state in integral form. The fractional exponential function of Rabotnov and Koltunov-Rzhanitsyn was chosen as the kernel of the integral operator. Two cases are considered: the case of a stress-strain state symmetric and antisymmetric in the normal coordinate (VAT). In the study of natural oscillations, the properties of those modes that are time-dependent by harmonic law are investigated. For both cases, dispersion equations are derived, which are solved numerically. Asymptotics of the roots of dispersion equations for small and large frequencies are also obtained. The analysis of the obtained solutions made it possible to draw conclusions about the influence of hereditary factors on the behavior of dispersion curves. A comparative analysis of numerical solutions and their asymptotics is carried out.
VL  - 3
IS  - 6
ER  -

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