A vibrational statement, method and algorithm to assess the damping capacity of structurally inhomogeneous viscoelastic mechanical systems, consisting of a package of rectangular plates and shells with point relations and concentrated masses at different rheological properties of deformable elements, are proposed in the paper. To describe rheological properties of the material, the linear hereditary Boltzmann-Volter theory was used. To assess the damping capacity of the system, the problem in question in each case was reduced to solving the proper problems of algebraic equations with complex parameters solved by the Muller method. The accuracy of the methods was demonstrated by comparing the calculated results with known published data and a numerical experiment. Complex natural frequency of the system was used to assess the damping capacity of inhomogeneous viscoelastic systems. Various eigenvalue problems have been solved for structurally inhomogeneous mechanical systems consisting of a package of plate and shell systems with concentrated masses and shock absorbers. A number of new mechanical effects have been discovered, related to the manifestation of the damping capacity of mechanical systems under consideration.
References
[1]
Leissa, A.W. (1969) Vibration of Plate. US Government Printing Office, Washington DC.
[2]
Leissa, A.W. (1973) The Free Vibration of Rectangular Plates. Journal of Sound and Vibration, 31, 257-293. https://doi.org/10.1016/S0022-460X(73)80371-2
[3]
Boay, C.G. (1993) Free Vibration of Rectangular Isotropic Plates with and without a Concentrated Mass. Computers and Structures, 48, 529-533. https://doi.org/10.1016/0045-7949(93)90331-7
[4]
Avalos, D.R., Larrondo, H. and Laura, A.A. (1993) Vibrations of a Simply Supported Plate Carrying an Elastically Mounted Concentrated Mass. Ocean Engineering, 20, 195-205. https://doi.org/10.1016/0029-8018(93)90035-G
[5]
Xiang, Y., Liew, K.M. and Kitipornchai, S. (1997) Vibration Analysis of Rectangular Mindlin Plates Resting on Elastic Edge Supports. Journal of Sound and Vibration, 204, 1-16. https://doi.org/10.1006/jsvi.1996.0922
[6]
Laura, P.A.A. and Grossi, R.O. (1978) Transverse Vibration of a Rectangular Plate Elastically Restrained against Rotation along Three Edges and Free on the Fourth Edge. Journal of Sound and Vibration, 59, 355-368. https://doi.org/10.1016/S0022-460X(78)80004-2
[7]
Laura, P.A.A. and Grossi, R.O. (1981) Transverse Vibrations of Rectangular Plates with Edges Elastically Restrained against Translation and Rotation. Journal of Sound and Vibration, 75, 101-107. https://doi.org/10.1016/0022-460X(81)90237-6
[8]
Wu, J.S. and Luo, S.S. (1997) Use of the Analytical-and-Numerical-Combined Method in the Free Vibration Analysis of a Rectangular Plate with Any Number of Point Masses and Translational Springs. Journal of Sound and Vibration, 200, 179-194. https://doi.org/10.1006/jsvi.1996.0697
[9]
Nicholson, J.W. and Bergman, L.A. (1986) Vibration of Damped Plate-Oscillator Systems. Journal of Engineering Mechanic, 112, 14-30. https://doi.org/10.1061/(ASCE)0733-9399(1986)112:1(14)
[10]
Gorman, D.J. (1997) Accurate Free Vibration Analysis of Shear-Deformable Plates with Torsion Elastic Edge Support. Journal of Sound and Vibration, 203, 209-218. https://doi.org/10.1006/jsvi.1996.0876
[11]
Bert, C.W., Wang, X. and Striz, A.G. (1993) Differential Quadrature for Static and Free Vibration Analysis of Anisotropic Plates. International Journal of Solids and Structures, 30, 1737-1744. https://doi.org/10.1016/0020-7683(93)90230-5
[12]
Bert, C.W., Wang, X. and Striz, A.G. (1994) Static and Free Vibration Analysis of Beams and Plates by Differential Quadrature Method. Acta Mechanica, 102, 11-24. https://doi.org/10.1007/BF01178514
[13]
Amabill M., Garziera R., Garra S. (2005) The Effect of Rotary Inertia of Added Masses on Vibrations of Empty and Fluid-Filled Circular Cylindrical Shells. Journal of Fluids and Structures, 21, 449-458. https://doi.org/10.1016/j.jfluidstructs.2005.07.018
[14]
Trotsenko, Y.V. (2006) Frequencies and Modes of Vibration of a Cylindrical Shell with Attached Rigid Body. Journal of Sound and Vibration, 292, 535-551. https://doi.org/10.1016/j.jsv.2005.08.015
[15]
Skudrzyk, E. (1968) Simple and Complex Vibratory Systems. University Park, State College, PA, London.
[16]
Mallon, N.J. (2008) Dynamic Stability of a Thin Cylindrical Shell with Top Mass Subjected to Harmonic Base-Acceleration. International Journal of Solids and Structures, 45, 1587-1613. https://doi.org/10.1016/j.ijsolstr.2007.10.011
[17]
Amabill, M. and Garziera, R. (2002) Vibrations of Circular Cylindrical Shells with No Uniform Constraints, Elastic Bed and Added Mass. Part III: Steady Viscous Effects on Shells Conveying Fluid. Journal of Fluids and Structures, 16, 795-809. https://doi.org/10.1006/jfls.2002.0446
[18]
Khalili, S.М.R., Tafazoli, S. and Malekzadeh Fard, K.F. (2011) Vibrations of Laminated Composite Shells with Uniformly Distributed Attached Mass Usid Higher Order Shell Theory Including Stiffness Effect. Journal of Sound and Vibration, 330, 535-551. https://doi.org/10.1016/j.jsv.2011.07.004
[19]
Andreev, L.V., Dyshko, A.L. and Pavlenko, I.D. (1988) Dynamics of Plates and Shells with Concentrated Masses. Moscow. (In Russian)
[20]
Sivak, V.F. and Sivak, V.V. (2002) Experimental Investigation into the Vibrations of Shells of Revolution with Added. International Applied Mechanics, 38, 623-627. https://doi.org/10.1023/A:1019770206949
[21]
Amabill, M. (2008) Nonlinear Vibrations and Stability of Shells and Plates. Cambridge University Press, Now York. https://doi.org/10.1017/CBO9780511619694
[22]
Clough, R.W. (1975) Dynamics of Structures. McGraw-Hill, New York.
[23]
Dodge, F.T. (2000) The New Dynamic Behavior of Liquids in Moving Containers. F. T. Dodge. Southwest Research Institute, San Antonio, TX.
[24]
Prabakar, R.S., Prabakar, R.S., Sujatha, C. and Narayanan, S. (2013) Response of a Quarter Car Model with Optimal Magnetorheological Damper Parameters. Journal of Sound and Vibration, 332, 2191-2206. https://doi.org/10.1016/j.jsv.2012.08.021
[25]
Choi, S.-B., Hong, S.-R., Sung, K.-G. and Sohn, J.-W. (2008) Optimal Control of Structural Vibrations Using a Mixed-Mode Magnetorheological Fluid Mount. International Journal of Mechanical Sciences, 50, 559-568. https://doi.org/10.1016/j.ijmecsci.2007.08.001
[26]
Ovidiu, P.B. (2013) Viscoelastic Model for the Rigid Body Vibrations of a Viaduct Depending on the Support Devices’ Rheological Model. Romanian Journal of Transport Infrastructure, 2, 37-43. https://doi.org/10.1515/rjti-2015-0014
[27]
Kalman, P. and Shalom, H. (1989) Firmness Indexes of Viscoelastic Bodies by Vibration Testing. Journal of Rheology, 33, 639. https://doi.org/10.1122/1.550031
[28]
Yu, A. and Rossikhin, M.V. (2010) Reflections on Two Parallel Ways in the Progress of Fractional Calculus in Mechanics of Solids. Applied Mechanics Reviews, 63, 010701-1–010701-12. https://doi.org/10.1115/1.4000246
[29]
Rossikhin, Y.A. and Shitikova, M.V. (2001) A New Method for Solving Dynamic Problems of Fractional Derivative Viscoelasticity. International Journal of Engineering Science, 39, 149-176. https://doi.org/10.1016/S0020-7225(00)00025-2
[30]
Rossikhin, Y.A. (2004) Analysis of the Viscoelastic Rod Dynamics via Models Involving Fractional Derivatives or Operators of Two Different Orders. Shock and Vibration Digest, 36, 3-26.
[31]
Safarov, I.I., Akhmedov, M.S. and Boltaev, Z.I. (2014) Waveguide Propagation in Extended Plates of Variable Thickness. Open Access Library Journal, 1, 2-9.
[32]
Safarov, I.I., Boltaev, Z.I. and Axmedov, M.S. (2015) Setting the Linear Oscillations of Structural Heterogeneity Viscoelastic Lamellar Systems with Point Relations. Applied Mathematics, 6, 228-234. https://doi.org/10.4236/am.2015.62022
[33]
Safarov, I.I., Teshaev, M.N. and Mazhidov, M. (2014) Damping of Oscillations of Mechanical Systems. LAP LAMBERT Academic Publishing, Latvia.
[34]
Safarov, I.I. (1992) Oscillations and Waves in Dissipatively Inhomogeneous Media and Structures. Fan, Tashkent, 252 p.
[35]
Safarov, I.I., Teshayev, M.K. and Axmedov, M.S. (2014) Natural Oscillations of Viscoelastic Lamellar Mechanical Systems with Point Communications. Applied Mathematics, 5, 3018-3025. https://doi.org/10.4236/am.2014.519289
[36]
Safarov, I.I., Teshaev, M.K. and Akhmedov, M.S. (2018) Free Oscillations of a Toroidal Viscoelastic Shell with a Flowing Liquid. American Journal of Mechanics and Applications, 6, 37-49. https://doi.org/10.11648/j.ajma.20180602.11
[37]
Safarov, I.I. and Teshaev, M.K. (2018) Vibration Protection of Mechanical Systems Consisting of Solid and Deformable Bodies. European Journal of Engineering Research and Science, 3, 18-28. https://doi.org/10.24018/ejers.2018.3.9.860
[38]
Safarov, I.I., Teshayev, M.K., Boltayev, Z.I. and Akhmedov, M.S. (2017) Damping Properties of Vibrations of Three-Layer Viscoelastic Plate. International Journal of Theoretical and Applied Mathematics, 3, 191-198. https://doi.org/10.11648/j.ijtam.20170306.13
[39]
Safarov, I.I., Teshayev, M.K., Nuriddinov, B.Z. and Boltaev, Z.I. (2017) Of Own and Forced Vibrations of Dissipative Inhomogeneous Mechanical Systems. Applied Mathematics, 8, 1001-1015. https://doi.org/10.4236/am.2017.87078
[40]
Mirsaidov, M.M. (2019) An Account of the Foundation in Assessment of Earth Structure Dynamics. Proceedings of 22-Interional Scientific Conference on Construction the Formation of Living Environment, 97, 1-11. https://doi.org/10.1051/e3sconf/20199704015
[41]
Mirsaidov, M.M. and Sultanov, T.Z. (2013) Use of Linear Heredity Theory of Viscoelasticity for Dynamic Analysis of Earthen Structures. Soil Mechanics and Foundation Engineering, 49, 250-256. https://doi.org/10.1007/s11204-013-9198-8
[42]
Mirsaidov, M.M. and Mekhmonov, Y. (1987) Nonaxisymmetric Vibrations of Axisymmetric Structures with Associated Masses and Hollows (Protrusions). Strength of Materials, 19, 424-430. https://doi.org/10.1007/BF01524147
[43]
Koltunov, M.A., Mirsaidov, M.M. and Troyanovskii, I.E. (1978) Transient Vibrations of Axissymmetric Viscoelastic Shells. Polymer Mechanics, 14, 233-238. https://doi.org/10.1007/BF00857468
[44]
Mirsaidov, M.M. and Troyanovskii, I.E. (1975) Forced Axisymmetric Oscillations of a Viscoelastic Cylindrical Shell. Polymer Mechanics, 11, 953-955. https://doi.org/10.1007/BF00857626
[45]
Mirsaidov, M.M., Abdikarimov, R.A. and Khodzhaev, D.A. (2019) Dynamics of a Viscoelastic Plate Carrying Concentrated Mass with Account of Physical Nonlinearity of Material: Part 1. Mathematical Model. Solution Metho and Computational Algorithm. PNRPU Mechanics Bulletin, 2, 143-153.
