# zbMATH — the first resource for mathematics

The general solution for dynamical problem of rectangular micro-polar beam vibrating at high frequency. (English) Zbl 1427.74083
Summary: In this paper, we have made an attempt to find a general solution of a problem of high frequency vibration in a micro-polar rectangular beam. We model the micro-polar beam problem in such a way that it can be reduced to Timoshenko beam problem in classical case. To solve the problem we adopted a methodology based on Hamiltonian principle with Legendre’s transformation similar to symplectic approach. It was first applied in elasticity problem in the early 1990s by Professor W. Zhong. After achieving the Hamiltonian formulation for micro-polar beam problem we no longer follow the derivation procedure of symplectic approach but make our own way to solve it in order to reduce the complexities of the problem.
##### MSC:
 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 74H45 Vibrations in dynamical problems in solid mechanics
Full Text:
##### References:
 [1] Timoshenko, S. P.; Goodier, J. N., Theory of Elasticity (1970), McGraw-Hill: McGraw-Hill New York · Zbl 0266.73008 [2] Yao, W.; Zhong, W.; Lim, C. W., Symplectic Elasticity (2009), World Scientific Publishing Co.: World Scientific Publishing Co. Singapore · Zbl 1170.74002 [3] Whittaker, E. T., A Treatise on the Analytical Dynamics (1960), Cambridge University Press · Zbl 0091.16406 [4] Arnold, V. I., Mathematical Methods of Classical Mechanics (1978), Springer-Verlag: Springer-Verlag New York · Zbl 0386.70001 [5] Feng, K.; Qin, M., Hamilton methodology for Hamiltonian dynamical systems, Prog. Nat. Sci., 1, 102-112 (1991) [6] Wang, C. M.; Lam, K. Y.; He, X. Q., Exact solutions for Timoshenko beams on elastic foundation using Green’s functions, Mech. Struct. Mach., 26, 101-113 (1998) [7] Lü, C. F.; Lim, C. W.; Yao, W. A., A new analytic symplectic elasticity approach for beams resting on Pasternak Elastic Foundations, J. Mech. Mater. Struct., 4, 1741-1754 (2009) [8] Lim, C. W.; Cui, S.; Yao, W. A., On new symplectic elasticity approach for exact bending solutions of rectangular thin plates with opposite sides simply supported, Int. J. Solid Struct., 44, 5396-5411 (2007) · Zbl 1311.74072 [9] Hu, H., Variational Principle of Elasticity and Its Applications (1981), Beijing [10] Toupin, R. A., Theories of elasticity with couple-stress, Arch. Ration. Mech. Anal., 11, 385-414 (1962) · Zbl 0112.16805 [11] Toupin, R. A., Theories of elasticity with couple-stress, Arch. Ration. Mech. Anal., 17, 85-112 (1964) · Zbl 0131.22001 [12] Mindlin, R. D.; Tiersten, H. F., Effects of couple stress in linear elasticity, Arch. Ration. Mech. Anal., 11, 415-448 (1962) · Zbl 0112.38906 [13] Koiter, W. T., Couple-stresses in the theory of elasticity Pt. I-II, Proc. Koninkl. Netherland Akad. Wetensh B, 67, 17-44 (1964) · Zbl 0124.17405 [14] Pal’mov, V. A., Fundamental equations of the theory of asymmetric elasticity, J. Appl. Mech. Math., 28, 1341-1345 (1964) · Zbl 0151.36403 [15] Eringen, A. C., Linear theory of micro-polar elasticity, J. Math. Mech., 15, 909-923 (1966) · Zbl 0145.21302 [16] Ieşan, D., On the linear theory of micropolar elasticity, Int. J. Eng. Sci., 7, 1213-1220 (1969) · Zbl 0181.53806 [17] Dyszlewicz, J., Micropolar Theory of Elasticity (2004), Springer: Springer Berlin · Zbl 0247.73002 [18] Eringen, A. C., Microcontinuum Field Theory. I. Foundations and Solids (1999), Springer: Springer New York · Zbl 0953.74002 [19] Nowacki, W., Theory of Asymmetric Elasticity (1986), Pergamon Press: Pergamon Press Oxford · Zbl 0604.73020 [20] Zhong, W., On the reciprocal theorem and adjoint symplectic orthogonal relation, ACTA Mech. Sin., 24, 432-437 (1992) [21] Nobili, A., On the generalization of the Timoshenko beam model based on the micropolar linear theory: static case, Math. Probl. Eng., Article 914357 pp. (2015), 8 pages · Zbl 1394.74011 [22] Ramezani, S.; Naghabadi, R.; Sohrabpour, S., Analysis of micropolar elastic beams, Eur. J. Mech. A/Solids, 28, 2, 202-208 (2009) · Zbl 1156.74343 [23] Asghari, M.; Kahrobaiyan, M. H.; Rahaeifard, M.; Ahmadian, M. T., Investigation of the size effects in Timoshenko beams based on the couple stress theory, Arch. Appl. Mech., 81, 7, 863-874 (2011) · Zbl 1271.74257 [24] Lim, C. W.; Xu, X. S., Symplectic elasticity: theory and applications, Appl. Mech. Rev., 63, 05802-050810 (2010) [25] Li, R.; Wang, P.; Tian, Y.; Wang, B.; Li, G., A unified analytic solution approach to static bending and free vibration problems of rectangular thin plate, Sci. Rep., 5, 17054 (2015)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.