Abstract
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.
Similar content being viewed by others
References
Banks, H.T. and Inman, D.J., On damping mechanisms in beams, J. Appl. Mech., 1991, vol. 58, pp. 716–723.
Sears, A. and Batra, R., Macroscopic properties of carbon nanotubes from molecular-mechanics simulations, Phys. Rev. B, 2004, vol. 69, pp. 235–406.
Ahmadi, G., Linear theory of nonlocal viscoelasticity, Int. J. Non-Linear Mech., 1975, vol. 10, pp. 253–258.
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.
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.
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.
Potapov, V.D., Stability via nonlocal continuum mechanics, Int. J. Solids Struct., 2013, vol. 50, pp. 637–641.
Sudak, L.J., Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics, J. Appl. Phys., 2003, vol. 94, pp. 7281–7287.
Tylikowski, A., Dynamic stability of carbon nanotubes, Mech. Mech. Eng. Int. J., 2006, vol. 10, pp. 160–166.
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.
Filippov, A.P., Kolebaniya deformiruemykh sistem (Oscillations of Deformed Systems), Moscow: Mashinostroenie, 1970.
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.
Potapov, V.D., Stability of Stochastic Elastic and Viscoelastic Systems, Chichester: Wiley, 1999.
Shalygin, A.P. and Palagin, Yu.I., Prikladnye metody staticheskogo modelirovaniya (Applied Methods for Statistical Simulation), Leningrad: Mashinostroenie, 1986.
Additional information
Original Russian Text © V.D. Potapov, 2013, published in Problemy Mashinostroeniya i Nadezhnosti Mashin, 2013, No. 6, pp. 9–16.
About this article
Cite this article
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). https://doi.org/10.3103/S1052618813060101
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1052618813060101