Abstract
Plane and anti-plane dynamic problems for an elastic rectangle loaded along its sides are considered. Low-frequency perturbations to rigid body translations are calculated. The derivation involves the solutions of non-homogeneous boundary value problems for harmonic and bi-harmonic equations. The explicit solution for the harmonic problem for transverse anti-plane translation is expressed through Fourier series. The bi-harmonic problem corresponding to the longitudinal in-plane translation is studied in greater detail for an elongated rectangle, which also may be treated using the elementary theory for plate extension. The derived perturbations incorporate the variations of displacements and stresses over the interior of the rectangle, including the case of self-equilibrated loading. The latter is obviously outside the range of validity of the classical rigid body framework.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Rao, A.: Dynamics of Particles and Rigid Bodies: A Systematic Approach. Cambridge University Press, Cambridge (2006)
Milton, G.W., Willis, J.R.: On modifications of Newton’s second law and linear continuum elastodynamics. Proc. R. Soc. A Math. Phys. Eng. Sci. 463(2079), 855–880 (2007)
Huang, H.H., Sun, C.T., Huang, G.L.: On the negative effective mass density in acoustic metamaterials. Int. J. Eng. Sci. 47(4), 610–617 (2009)
Lee, S.H., Park, C.M., Seo, Y.M., Wang, Z.G., Kim, C.K.: Acoustic metamaterial with negative density. Phys. Lett. A 373(48), 4464–4469 (2009)
Yao, S., Zhou, X., Hu, G.: Experimental study on negative effective mass in a 1D mass-spring system. New J. Phys. 10(4), 043020 (2008)
Wang, X.: Dynamic behaviour of a metamaterial system with negative mass and modulus. Int. J. Solids Struct. 51(7–8), 1534–1541 (2014)
Mei, J., Liu, Z., Wen, W., Sheng, P.: Effective dynamic mass density of composites. Phys. Rev. B 76(13), 134205 (2007)
Shestakova, A.: Development of mathematical models for freight cars subject to dynamic loading. PhD Thesis, Keele University (2015)
Ansari, M., Esmailzadeh, E., Younesian, D.: Longitudinal dynamics of freight trains. Int. J. Heavy Veh. Syst. 16(1–2), 102–131 (2009)
Wu, Q., Spiryagin, M., Cole, C.: Longitudinal train dynamics: an overview. Veh. Syst. Dyn. 54(12), 1688–1714 (2016). ISO 690
Kaplunov, J., Shestakova, A., Aleynikov, I., Hopkins, B., Talonov, A.: Low-frequency perturbations of rigid body motions of a viscoelastic inhomogeneous bar. Mech. Time Depend. Mater. 19(2), 135–151 (2015)
Kaplunov, J., Prikazchikov, D., Sergushova, O.: Multi-parametric analysis of the lowest natural frequencies of strongly inhomogeneous elastic rods. J. Sound Vib. 366, 264–276 (2016)
Kaplunov, J., Prikazchikov, D.A., Prikazchikova, L.A., Sergushova, O.: The lowest vibration spectra of multi-component structures with contrast material properties. J. Sound Vib. 445, 132–147 (2019)
Şahin, O.: The effect of boundary conditions on the lowest vibration modes of strongly inhomogeneous beams. J. Mech. Mater. Struct. 14(4), 569–585 (2019)
Şahin, O., Erbaş, B., Kaplunov, J., Savšek, T.: The lowest vibration modes of an elastic beam composed of alternating stiff and soft components. Arch. Appl. Mech. 90(2), 339–352 (2020)
Kurfess, T.R. (ed.): Robotics and Automation Handbook. CRC Press, Boca Raton (2018)
Davim, J.P. (ed.): Introduction to Mechanical Engineering. Springer, Berlin (2018)
Aleksandrov, E.V., Sokolinskii, V.B.: Applied Theory and Calculation of Impact Systems. Nauka, Moscow (1969)
Meleshko, V.V.: Selected topics in the history of the two-dimensional biharmonic problem. Appl. Mech. Rev. 56(1), 33–85 (2003)
Meleshko, V.V., Gomilko, A.M.: Infinite systems for a biharmonic problem in a rectangle: further discussion. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460(2043), 807–819 (2004)
Kovalenko, M.D., Menshova, I.V., Kerzhaev, A.P.: On the exact solutions of the biharmonic problem of the theory of elasticity in a half-strip. Z. Angew. Math. Phys. 69(5), 121 (2018)
Kaplunov, J.D., Kossovitch, L.Y., Nolde, E.V.: Dynamics of Thin Walled Elastic Bodies. Academic Press, Cambridge (1998)
Gregory, R.D., Wan, F.Y.: Decaying states of plane strain in a semi-infinite strip and boundary conditions for plate theory. J. Elast. 14(1), 27–64 (1984)
Babenkova, E., Kaplunov, J.: Low-frequency decay conditions for a semi-infinite elastic strip. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460(2048), 2153–2169 (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Kaplunov, J., Şahin, O. Perturbed rigid body motions of an elastic rectangle. Z. Angew. Math. Phys. 71, 160 (2020). https://doi.org/10.1007/s00033-020-01390-w
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s00033-020-01390-w