Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

For citation:

Kaplunov J. D., Prikazchikova L. A. Low-Frequency Vibration Modes of Strongly Inhomogeneous Elastic Laminates. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2018, vol. 18, iss. 4, pp. 447-457. DOI: 10.18500/1816-9791-2018-18-4-447-457, EDN: VQLCNR

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
30.11.2018
Full text:
download
(downloads: 151)
Language: 
English
Heading: 
Mechanics
Article type: 
Article
UDC: 
539.3
DOI: 
10.18500/1816-9791-2018-18-4-447-457
EDN: 
VQLCNR

Low-Frequency Vibration Modes of Strongly Inhomogeneous Elastic Laminates

Autors: 
Kaplunov Julius Davidovich, Keele University
Prikazchikova Ludmila A., Keele University
Abstract: 

The dynamic behaviour of thin multi-layered structures, composed of contrasting “strong” and “weak” layers, is considered. An asymptotic procedure for analysing the lowest cutoffs is developed. A polynomial frequency equation is derived, along with the linear equations for the associated eigenforms corresponding to displacement variation across the thickness. For a five-layered laminate with clamped faces two term expansions for eigenfrequencies and eigenforms are compared with those obtained from the exact solution of the original problem for thickness resonances.

Key words: 
asymptotic
high-contrast
thin elastic structure
thickness fluctuations
low frequency
cutoff
References: 
  1. Kossovich L. Yu. Nestacionarnye zadachi teorii uprugih tonkih obolochek [Nonstationary problems in the theory of elastic thin shells]. Saratov, Saratov Univ. Press, 1986 (in Russian).
  2. Kaplunov J. D., Kossovich L. Y., Nolde E. V. Dynamics of thin walled elastic bodies. Academic Press, 1998.
  3. Beresin V. L., Kossovich L. Y., Kaplunov J. D. Synthesis of the dispersion curves for a cylindrical shell on the basis of approximate theories. Journal of sound and vibration, 1995, vol. 186, no. 1, pp. 37–53. DOI: https://doi.org/10.1006/jsvi.1995.0432
  4. Le K. C. Vibrations of shells and rods. Berlin, Springer, 1999.
  5. Berdichevsky V. Л. Variacionnye principy mekhaniki sploshnoj sredy [Variational Principles of Continuum Mechanics]. Moscow, Nauka, Glav. red. fiz.-mat. lit., 1983 (in Russian).
  6. Kaplunov J., Prikazchikov D. A., Prikazchikova L. A. Dispersion of elastic waves in a strongly inhomogeneous three-layered plate. International Journal of Solids and Structures, 2017, vol. 113–114, pp. 169–179. DOI: https://doi.org/10.1016/j.ijsolstr.2017.01.042
  7. Kossovich L. Yu., Shevtsova Yu. V. Asymptotic approximations of three-dimensional dynamic equations of elasticity theory in case of two-layered plates. Problems of strenght and plasticity, 2005, vol. 76, pp. 102–111 (in Russian). DOI: https://doi.org/10.32326/1814-9146-2005-67-1-102-110
  8. Prikazchikova L., Ece Aydın Y., Erba¸ s B., Kaplunov J. Asymptotic analysis of an anti-plane dynamic problem for a three-layered strongly inhomogeneous laminate. Mathematics and Mechanics of Solids, 2018. DOI: https://doi.org/10.1177/1081286518790804
  9. Kaplunov J., Prikazchikov D., Sergushova O. Multi-parametric analysis of the lowest natural frequencies of strongly inhomogeneous elastic rods. Journal of Sound and Vibration, 2016, vol. 366, pp. 264–276. DOI: https://doi.org/10.1016/j.jsv.2015.12.008
  10. Vinson J. R. The behavior of sandwich structures of isotropic and composite materials. CRC Press, 1999.
  11. Ivanov I. V. Analysis, modelling, and optimization of laminated glasses as plane beam. International Journal of Solids and Structures, 2006, vol. 43, no. 22–23, pp. 6887–6907. DOI: https://doi.org/10.1016/j.ijsolstr.2006.02.014
  12. Schulze S.-H., Pander M., Naumenko K., Altenbach H. Analysis of laminated glass beams for photovoltaic applications. International Journal of Solids and Structures, 2012, vol. 49, pp. 2027–2036. DOI: https://doi.org/10.1016/j.ijsolstr.2012.03.028
  13. Lee P., Chang N. Harmonic waves in elastic sandwich plates. Journal of Elasticity, 1979, vol. 9, pp. 51–69. DOI: https://doi.org/10.1007/BF00040980
  14. Kaplunov J. D. Long-wave vibrations of a thinwalled body with fixed faces. The Quarterly Journal of Mechanics and Applied Mathematics, 1995, vol. 48, no. 3, pp. 311–327. DOI: https://doi.org/10.1093/qjmam/48.3.311
  15. Kaplunov J. D., Nolde E. V. Long-wave vibrations of a nearly incompressible isotropicplate with fixed faces. The Quarterly Journal of Mechanics and Applied Mathematics, 2002, vol. 55, no. 3, pp. 345–356. DOI: https://doi.org/10.1093/qjmam/55.3.345
  16. Kaplunov J. D., Kossovich L. Yu., Rogerson G. A. Direct asymptotic integration of the equations of transversely isotropic elasticity for a plate near cut-off frequencies. Quarterly Journal of Mechanics and Applied Mathematics, 2000, vol. 53, no. 2, pp. 323–341.
  17. Nolde E. V., Rogerson G. A. Long wave asymptotic integration of the governing equations for a pre-stressed incompressible elastic layer with fixed faces. Wave Motion, 2002, vol. 36, no. 3, pp. 287–304. DOI: https://doi.org/10.1016/S0165-2125(02)00017-3
  18. Rogerson G. A., Sandiford K. J., Prikazchikova L. A. Abnormal long wave dispersion phenomena in a slightly compressible elastic plate with non-classical boundary conditions. International Journal of Non-Linear Mechanics, 2007, vol. 42, no. 2, pp. 298–309. DOI: https://doi.org/10.1016/j.ijnonlinmec.2007.01.005
Received: 
16.07.2018
Accepted: 
11.11.2018
Published: 
07.12.2018
Short text (in English):
download
(downloads: 128)
  • 1154 reads