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Stability of a flat arch subjected to deterministic and stochastic loads taking into account nonlocal damping


In the example of a flat arch, a problem of vibrational stability of systems subjected to deterministic and stochastic loads is considered taking into account the geometric nonlinearity and nonlocal damping of a material, which is characteristic of certain types of composites and nanomaterials. To study the stability of arch motion within the deterministic statement of the problem (stability in the sense of Lyapunov) and the stability in the almost certain stochastic statement, a method is used which is based on the calculation of the maximal Lyapunov’s exponent. The influence of damping and loading parameters on the degree of system stability is analyzed.

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  1. 1.

    Banks, H.T. and Inman, D.J., On damping mechanisms in beams, J. Appl. Mech., 1991, vol. 58, pp. 716–723.

    Article  MATH  Google Scholar 

  2. 2.

    Sears, A. and Batra, R., Macroscopic properties of carbon nanotubes from molecular-mechanics simulations, Phys. Rev. B, 2004, vol. 69, pp. 235–406.

    Google Scholar 

  3. 3.

    Ahmadi, G., Linear theory of nonlocal viscoelasticity, Int. J. Non-Linear Mech., 1975, vol. 10, pp. 253–258.

    Article  MATH  Google Scholar 

  4. 4.

    Lei, Y., Eriswell, M.I., and Adhikari, S., A Galerkin method for distributed systems with non-local damping, Int. J. Solids Struct., 2006, vol. 43, pp. 3381–3400.

    Article  MATH  Google Scholar 

  5. 5.

    Kumar, D., Heinrich, C., and Waas, A.M., Buckling analysis of carbon nanotubes modeled using nonlocal continuum theories, J. Appl. Phys., 2008, vol. 103, p. 073521.

    Article  Google Scholar 

  6. 6.

    Potapov, V.D., On the stability of columns under stochastic loading Taking into account nonlocal damping, J. Mach. Manuf. Reliab., 2012, vol. 41, no. 4, p. 284.

    Article  Google Scholar 

  7. 7.

    Potapov, V.D., Stability via nonlocal continuum mechanics, Int. J. Solids Struct., 2013, vol. 50, pp. 637–641.

    Article  Google Scholar 

  8. 8.

    Sudak, L.J., Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics, J. Appl. Phys., 2003, vol. 94, pp. 7281–7287.

    Article  Google Scholar 

  9. 9.

    Tylikowski, A., Dynamic stability of carbon nanotubes, Mech. Mech. Eng. Int. J., 2006, vol. 10, pp. 160–166.

    Google Scholar 

  10. 10.

    Zhang, Y.Q., Liu, G.R., and Wang, J.S., Smal-scale effects on buckling of multiwalled carbon nanotubes under axial compression, Phys. Rev. B, 2004, vol. 70, p. 205430.

    Article  Google Scholar 

  11. 11.

    Filippov, A.P., Kolebaniya deformiruemykh sistem (Oscillations of Deformed Systems), Moscow: Mashinostroenie, 1970.

    Google Scholar 

  12. 12.

    Benettin, G., Galgani, L., Giorgolly, A., and Strelcyn, J.M., Liapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them, Meccanica, 1980, vol. 15, pp. 9–20, 21–30.

    Article  MATH  Google Scholar 

  13. 13.

    Potapov, V.D., Stability of Stochastic Elastic and Viscoelastic Systems, Chichester: Wiley, 1999.

    Google Scholar 

  14. 14.

    Shalygin, A.P. and Palagin, Yu.I., Prikladnye metody staticheskogo modelirovaniya (Applied Methods for Statistical Simulation), Leningrad: Mashinostroenie, 1986.

    Google Scholar 

Download references


Additional information

Original Russian Text © V.D. Potapov, 2013, published in Problemy Mashinostroeniya i Nadezhnosti Mashin, 2013, No. 6, pp. 9–16.

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Potapov, V.D. Stability of a flat arch subjected to deterministic and stochastic loads taking into account nonlocal damping. J. Mach. Manuf. Reliab. 42, 450–456 (2013).

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  • Machinery Manufacture
  • Maximal Lyapunov Exponent
  • Viscoelastic System
  • Arch Length
  • Stochastic Load