Skip to main content
SpringerLink
Log in

LOW FREQUENCY VIBRATIONS OF A LONG ELASTIC STRIP

  • Published:
Mechanics of Solids Aims and scope Submit manuscript

Abstract—

The application of the iterative Saint-Venant-Picard-Banach method is described on the example of constructing the solution of the system for differential equations of motion of the theory of elasticity with a small parameter for a long strip at perturbation frequencies commensurate with the frequencies of transverse vibrations of the beam. The system is transformed in such a way that the equations are integrated sequentially and without increasing in order. The unknowns are calculated using the first-order Picard operators so that the previously calculated unknowns are input for the next equation, etc. The integrals of all unknowns for problem that make it possible to fulfill all the boundary conditions on the long and short sides are found. The convergence of the solution is ensured using a small thin-walled parameter in accordance with the Banach’s contraction principle. Satisfaction of the boundary conditions at the long edges leads to two equations for slowly and two singular ones for rapidly varying components of the solution, which depend only on the longitudinal coordinate. It is shown that when these equations are reduced to one with the loss of a rapidly changing component, the Timoshenko equation is obtained. The presentation is illustrated by two examples of loading a strip with a transverse distributed load and a concentrated force. A technique for establishing orders of magnitude by a small parameter relative to the load is described.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.

Similar content being viewed by others

Notes

  1. A slowly varying function is such a function that, being differentiated with respect to the argument x, does not change its asymptotic order in ε, while a rapidly changing function being differentiated with respect to x increases by ε–1 times.

REFERENCES

  1. A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity (Cambridge Univ. Press, Cambridge, 1927).

    MATH  Google Scholar 

  2. R. O. Friedrichs, “Asymptotic phenomena in mathematical physics,” Bull. Amer. Math. Soc. 61 (6), 485–504 (1955).

    Article  MathSciNet  Google Scholar 

  3. E. I. Grigolyuk and I. T. Selezov, Nonclassical Theories of Rod, Plate, and Shell Vibrations, in Results in Science and Technology. Mechanics of Deformable Solids, Vol. 5 (VINITI, Moscow, 1973) [in Russian].

  4. S. P. Timoshenko, “On the correction for shear of the differential equation for transverse vibrations of prismatic bar,” Phil. Mag. Ser. 6 (41), No. 245, 744–746 (1921).

  5. S. P. Timoshenko, “On the transverse vibrations of bars of uniform cross sections,” Phil. Mag. Ser. 6 (43), 125–131 (1922).

  6. Ya. S. Uflyand, “Wave propagation in transverse oscillations of rods and plates,” Prikl. Mat. Mekh. 12 (3), 287–300 (1948).

    MathSciNet  Google Scholar 

  7. R. D. Mindlin, “Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates,” J. Appl. Mech. 18 (1), 31–38 (1951).

    Article  ADS  Google Scholar 

  8. C. L. Dolph, “On the Timoshenko theory of transverse beam vibrations,” Quart. Appl. Math. 12 (2), 175–187 (1954).

    Article  MathSciNet  Google Scholar 

  9. J. R. Hutchinson, “On Timoshenko beams of rectangular cross–section,” J. Appl. Mech. 71, 359–367 (2004). https://doi.org/10.1115/1.1751186

    Article  MATH  ADS  Google Scholar 

  10. N. G. Stephen, “The second spectrum of Timoshenko beams theory– Further assessment,” J. Sound Vib. 292, 372–389 (2006). https://doi.org/10.1016/j.jsv.2005.08.003

    Article  MATH  ADS  Google Scholar 

  11. V. V. Nesterenko, “A theory for transverse vibrations of the Timoshenko beam,” J. Appl. Math. Mech. 57, 669–677 (1993).

    Article  MathSciNet  Google Scholar 

  12. A. K. Abramyan, D. A. Indeitsev, and V. A. Postnov, “Running and standing waves of Timoshenko beam,” Mech. Solids 53 (2), 203–210 (2018). https://doi.org/10.3103/S0025654418020115

    Article  ADS  Google Scholar 

  13. X. Q. Wanga, “Timoshenko beam theory: A perspective based on the wave–mechanics approach,” Wave Motion. 57, 64-87 (2015).

    Article  MathSciNet  Google Scholar 

  14. N. I. Demochkin, K. S. Morgachev, and L.I. Fridman, “Reliability domain of the Timoshenko model in dynamics of rods and plates,” Mech. Solids 43, 957–964 (2008). https://doi.org/10.3103/S0025654408060137

    Article  ADS  Google Scholar 

  15. E. M. Zveriaev, “Interpretation of semi-invers Saint-Venant method as iteration asymptotic method,” in Shell Structures: Theory and Application (Taylor & Francis Group, London, 2006), pp. 191–198.

  16. E. M. Zveriaev, “Saint-Venant–Picard–Banach method for integrating thin-walled systems equations of the theory of elasticity,” Mech. Solids 55, 1042–1050 (2020). https://doi.org/10.3103/S0025654420070225

    Article  ADS  Google Scholar 

  17. A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Dover Publ., New York, 1999).

    Google Scholar 

  18. E. Kamke, Handbook on Ordinary Differential Equations (Nauka, Moscow, 1971) [in Russian].

    Google Scholar 

  19. E. L. Lindelöf, “Sur l’application des méthodes d' approximation successives a l'étude des intégrales réeles des équations différentielles ordinaires,” J. Math. Pures Appl. Ser. 4. 10, 117–128 (1894).

    MATH  Google Scholar 

  20. E. Picard, “Mémoire sur la théorie des équations aux dérivées partielles et la méthode des approximations successives,” J. Math. Pures Appl. Ser. 4 6, 145-210 (1890).

    MATH  Google Scholar 

  21. A. Granas, Fixed Point Theory (Springer, New York, 2003).

    Book  Google Scholar 

  22. R. De Pascalis, M. Destrade, and G. Saccomandi, “The stress field in a pulled cork and some subtle points in the semi-inverse method of nonlinear elasticity,” Proc. R. Soc. A. Mat. Phys. Eng. Sci. 463 (2087, 2945–2959 (2007). https://doi.org/10.1098/rspa.2007.0010

  23. R. De Pascalis, K. R. Rajagopal, and G. Saccomandi, “Remarks on the use and misuse of the semi–inverse method in the nonlinear theory of elasticity,” Q. J. Mech. Appl. Math. 62 (4), 451–464 (2009).

    Article  MathSciNet  Google Scholar 

  24. E. Bulgariu, “On the Saint–Venant’s problem in microstretch elasticity,” Libertas Math. XXXI, 147–162 (2011).

    MathSciNet  MATH  Google Scholar 

  25. S. Chiriëta, “Saint–Venant’s problem and semi–inverse solutions in linear viscoelasticity,” Acta Mech. 94, 221–232 (1992).

    Article  MathSciNet  Google Scholar 

  26. L. Placidi and A.R. El Dhaba, “Semi-inverse method à la Saint-Venant for two-dimensional linear isotropic homogeneous second-gradient elasticity,” Math. Mech. Solids. 22 (5), 919–937 (2017). https://doi.org/10.1177/1081286515616043

    Article  MathSciNet  MATH  Google Scholar 

  27. Ye. M. Zveryayev, “Analysis of the hypotheses used when constructing the theory of beams and plates,” J. Appl. Math. Mech. 67 (3), 425–434 (2003). https://doi.org/10.1016/S0021-8928(03)90026-8

    Article  MathSciNet  MATH  Google Scholar 

  28. Ye. M. Zveryayev, “A consistent theory of thin elastic shells,” J. Appl. Math. Mech. 80 (5), 409–420 (2016). https://doi.org/10.1016/j.jappmathmech.2017.02.008

    Article  MathSciNet  MATH  Google Scholar 

  29. Ye. M. Zveryayev and G. I. Makarov, “A general method for constructing Timoshenko-type theories,” J. Appl. Math. Mech. 72 (2), 197–207 (2008). https://doi.org/10.1016/j.jappmathmech.2008.04.004

    Article  MathSciNet  Google Scholar 

  30. E. M. Zveryayev and L.V. Olekhova, “Reduction 3D equations of composite plate to 2D equations on base of mapping contraction principle,” KIAM Preprint № 95 (KIAM RAS, Moscow, 2014).

    Google Scholar 

  31. A. I. Likhoded and V. V. Sidorov, “Certain convergence features of the decomposition method by tones vibrations concerning continuum and finite-element models,” Kosmonaut. Raketostr., No. 2 (71), 20–27 (2013).

  32. V. V. Lalin and Le Tu Quang Trung, “Calculation of building structures for several dynamic effects with a static accounting of higher forms of oscillation,” Struct. Mech. Eng. Construct. Build. 16 (3), 171–178 (2020). https://doi.org/10.22363/1815-5235-2020-16-3-171-178

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Keldysh Institute of Applied Mathematics, 125047, Moscow, Russia

    E. M. Zveryaev

  2. Moscow Aviation Institute (National Research University), 125993, Moscow, Russia

    E. M. Zveryaev

Authors
  1. E. M. Zveryaev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. M. Zveryaev.

Additional information

Translated by A.A. Borimova

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zveryaev, E.M. LOW FREQUENCY VIBRATIONS OF A LONG ELASTIC STRIP. Mech. Solids 56, 980–995 (2021). https://doi.org/10.3103/S0025654421060261

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0025654421060261

Keywords:

Search

Navigation