Abstract
A method of hypotheses has been developed to construct a mathematical model of micropolar elastic thin beams. The method is based on the asymptotic properties of the solution ofan initial boundary value problem in a thin rectangle within the micropolar theory of elasticity with independent displacement and rotation fields. An applied model of the dynamics of micropolar elastic thin beams was constructed in which transverse shear strains and related strains are taken into account. The constructed dynamics model was used to solve problems of free and forced vibrations of a micropolar beam. Free vibration frequencies and modes, forced vibration amplitudes, and resonance conditions were determined. The obtained numerical calculation results show the specific features of free vibrations of thin beams. Micropolar thin beams have a free vibration frequency which is almost independent of the thin beam size, but depends only on the physical and inertial properties of the micropolar material. It is shown for the micropolar material that the free vibration frequency values of beams can be readily adjusted and hence a large vibration frequency separation can be achieved, which is important for studying resonance.
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References
Physical Mesomechanics of Heterogeneous Media and Computer-Aided Design of Materials, Panin, V.E., Ed., Cambridge: Cambridge Interscience Publ., 1998.
Smolin, I.Yu., Makarov, P.V., and Bakeev, R.A., Generalized Model of Elastic-Plastic Medium with Independent Plastic Rotation, Fiz. Mezomekh., 2004, vol. 7, spec. iss., part 1, pp. 89–92.
Novatskii, V., Theory of Elasticity, Moscow: Mir, 1975.
Lyalin, A.E., Pirozhkov, V.A., and Stepanov, R.D., On Propagation of Surface Waves in the Cosserat Continuum, Akust. J., 1982, vol. 28, no. 6, pp. 838–840.
Erofeyev, V.I., Wave Processes in Solids with Microstructure, Moscow: Moscow State University, 1999.
Kulesh, M.A., Matveenko, V.P., and Shardakov, I.N., Propagation of Surface Elastic Waves in the Cosserat Medium, Acous. Phys., 2006, vol. 52, no. 2, pp. 186–193.
Korepanov, V.V., Kulesh, M.A., Matveenko, V.P., and Shardakov, I.N., Analytical and Numerical Solutions for Static and Dynamic Problems of the Asymmetric Theory of Elasticity, Phys. Mesomech., 2007, vol. 10, no. 5–6, pp. 281–293.
Kulesh, M.A., Matveenko, V.P., and Shardakov, I.N., Dispersion and Polarization of Surface Rayleigh Waves for the Cosserat Continuum, Mech. Solids, 2007, vol. 42, no.4, pp. 583–594.
Varygina, M.P., Sadovskaya, O.V., and Sadovskii, V.M., Numerical Simulation of Spatial Wave Movements in the Moment Elastic Medium, Problems of Mechanics and Acoustics of Media with Micro-andNanostructure: “Nanomech-2009”, Nizhnii Novgorod, 2009, pp. 1–13 (electronic source).
Morozov, N.F., Structural Mechanics of Materials and Structural Elements. Interaction of Nano-, Micro-, Mesoand Macroscales under Deformation and Fracture, Izv. RAN. Mekh. Tv. Tela, 2005, no. 4, pp. 188–189.
Ivanova, E.A., Krivtsov, A.M., and Morozov, N.F., Derivation of Macroscopic Relations of the Elasticity of Complex Crystal Lattices Taking into Account the Moment Interactions at the Microlevel, J. Appl. Math. Mech., 2007, vol. 41, no. 4, pp. 543–561.
Altenbach, H. and Eremeyev, V.A., On the Linear Theory of Micropolar Plates, Z. Angew. Math. Mech., 2009, vol. 89, no. 4, pp. 242–256.
Sargsyan, S.H., Mathematical Model of Micropolar Elastic Thin Plates and Their Strength and Stiffness Characteristics, J. Appl. Mech. Phys., 2012, vol. 53, no. 2, pp. 275–282.
Sargsyan, S.H., General Theory of Micropolar Elastic Thin Shells, Phys. Mesomech., 2012, vol. 15, no. 1–2, pp. 69–79.
Sargsyan, S.H., The General Dynamic Theory of Micropolar Elastic Thin Shells, Dokl Phys., 2011, vol. 56, no. 1, pp. 39–42.
Hassanpour, S. and Heppler, G.R., Uncomplicated Torsion and Bending Theories for Micropolar Elastic Beams, Proc. 11th World Congress on Computational Mechanics, 5th Euro. Conf on Computational Mechanics, 6th Euro. Conf on Computational Fluid Dynamics, Onate, E., Oliver, J., and Huerta, A., Eds., Barselona, 2014, vol. II, pp. 142–153.
Altenbach, J., Altenbach, H., and Eremeyev, V.A., On Generalized Cosserat-Type Theories of Plates and Shells: A Short Review and Bibliography, Arch. Mech., 2009, spec. iss. doi 10.1007/s00419-009-0365-3
Sargsyan, S.H. and Sargsyan, A.H., General Dynamic Theory of Micropolar Elastic Thin Plates with Free Rotation and Special Features of Their Natural Oscillations, Acous. Phys., 2011, vol. 57, no. 4, pp. 461–469.
Sargsyan, S.H. and Sargsyan, A.H., Vibration Model of Micropolar Thin Shell Oscillations, Acous. Phys., 2013, vol. 59, no. 2, pp. 148–158.
Sargsyan, A.H. and Sargsyan, S.H., Dynamic Model of Micropolar Elastic Thin Plates with Independent Fields of Displacements and Rotations, J. Sound Vibration, 2014, vol. 333, no. 18, pp. 4354–4375.
Sargsyan, S.H., Applied One-Dimensional Theories of Bars on the Basis of the Asymmetric Theory of Elasticity, Fiz. Mezomekh., 2008, vol. 11, no. 5, pp. 41–54.
Timoshenko, S.P., Vibration Problems in Engineering, New York: D. Van Nostrand Company, 1937.
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Original Russian Text © A.H. Sargsyan, S.H. Sargsyan, 2015, published in Fizicheskaya Mezomekhanika, 2015, Vol. 18, No. 3, pp. 25-31.
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Sargsyan, A.H., Sargsyan, S.H. Mathematical model of the dynamics of micropolar elastic thin beams. Free and forced vibrations. Phys Mesomech 19, 459–465 (2016). https://doi.org/10.1134/S1029959916040123
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DOI: https://doi.org/10.1134/S1029959916040123