Forced oscillations of a three-layer plate in an unsteady temperature field

E. I. Starovoitov ; D. V. Leonenko

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 24 (2024) no. 1, pp. 123-137.

Résumé

The effect of a constant intensity heat flux on forced oscillations of a circular three-layer plate with an asymmetric thickness is investigated. The plate is thermally insulated along the contour and the lower plane. An approximate solution of the thermal conductivity problem was used, obtained by averaging the thermophysical parameters of the materials of the layers over the thickness of the package. According to the Neumann hypothesis, free plate oscillations caused by an instantaneous drop in heat flow are summed up with forced oscillations from the power load. The deformation of the plate package corresponds to the polyline hypothesis. In relatively thin outer bearing layers, Kirchhoff's hypotheses are valid. In sufficiently thick incompressible filler, the deformed normal retains straightness and length, but rotates by an additional angle. The formulation of the corresponding initial boundary value problem includes the equations of motion obtained using the d'Alembert principle and the variational Lagrange method. The initial conditions are assumed to be homogeneous, the contour of the plate is pivotally supported. The analytical solution of an inhomogeneous system of partial differential equations is obtained using the method of expansion into a series according to a system of proper orthonormal functions. As a result, analytical expressions are written out for three desired functions – plate deflection, shear and radial displacement in the filler. An example of oscillations under the action of an instantaneously applied uniformly distributed load is considered. A numerical parametric analysis of the natural oscillation frequencies and the resulting solution depending on the intensity of the heat flux for a plate with layers of titanium alloy, fluoroplast-4, and duralumin is given.

Bibliographie

[1] Bolotin V. V., Novichkov Iu. N., Mechanics of Multilayer Structures, Mashinostroenie Publishers, M., 1980, 375 pp. (in Russian)

[2] Aghalovyan L., Prikazchikov D., Asymptotic Theory of Anisotropic Plates and Shells, World Scientific Publishing Co., Singapore, 2015, 360 pp. | DOI

[3] Carrera E., Fazzolari F. A., Cinefra M., Thermal Stress Analysis of Composite Beams, Plates and Shells: Computational Modelling and Applications, Academic Press, 2016, 440 pp.

[4] Mikhasev G. I., Altenbach H., “Free vibrations of elastic laminated beams, plates and cylindrical shells”, Thin-walled Laminated Structures, Advanced Structured Materials, 106, Springer, Cham, 2019, 157–198 | DOI | MR

[5] Bakulin V. N., Boitsova D. A., Nedbai A. Ya., “Parametric resonance of a three layered cylindrical composite rib-stiffened shell”, Mechanics of Composite Materials, 57:3 (2021), 623–634 | DOI

[6] Gorshkov A. G., Starovoitov E. I., Yarovaya A. V., “Harmonic vibrations of a viscoelastoplastic sandwich cylindrical shell”, International Applied Mechanics, 37:7 (2001), 1196–1203 | DOI | Zbl

[7] Belostochnyi G. N., Myltcina O. A., “Dynamic stability of heated geometrically irregular cylindrical shell in supersonic gas flow”, Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, 22:4 (2018), 750–761 | DOI | MR | Zbl

[8] Zemlyanukhin A. I., Bochkarev A. V., Ratushny A. V., Chernenko A. V., “Generalized model of nonlinear elastic foundation and longitudinal waves in cylindrical shells”, Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 22:2 (2022), 196–204 | DOI | MR | Zbl

[9] Blinkov Yu. A., Mesyanzhin A. V., Mogilevich L. I., “Wave occurrences mathematical modeling in two geometrically nonlinear elastic coaxial cylindrical shells, containing viscous incompressible liquid”, Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 16:2 (2016), 184–197 (in Russian) | DOI | MR

[10] Tarlakovskii D. V., Fedotenkov G. V., “Nonstationary 3D motion of an elastic spherical shell”, Mechanics of Solids, 50:2 (2015), 208–217 | DOI

[11] Tarlakovskii D. V., Fedotenkov G. V., “Two-dimensional nonstationary contact of elastic cylindrical or spherical shells”, Journal of Machinery Manufacture and Reliability, 43:2 (2014), 145–152 | DOI

[12] Belostochny G. N., Myltcina O. A., “The geometrical irregular plates under the influence of the quick changed on the time coordinate forces and temperature effects”, Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 15:4 (2015), 442–451 (in Russian) | DOI | DOI | Zbl

[13] Ivañez I., Moure M. M., Garcia-Castillo S. K., Sanchez-Saez S., “The oblique impact response of composite sandwich plates”, Composite Structures, 133 (2015), 1127–1136 | DOI

[14] Suvorov Ye. M., Tarlakovskii D. V., Fedotenkov G. V., “The plane problem of the impact of a rigid body on a half-space modelled by a Cosserat medium”, Journal of Applied Mathematics and Mechanics, 76:5 (2012), 511–518 | DOI | MR | Zbl

[15] Paimushin V. N., Gazizullin R. K., “Static and monoharmonic acoustic impact on a laminated plate”, Mechanics of Composite Materials, 53:6 (2017), 283–304 | DOI

[16] Smirnov A. L., Vasiliev G. P., “Free vibration frequencies of a circular thin plate with nonlinearly perturbed parameters”, Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 21:2 (2021), 227–237 (in Russian) | DOI | MR

[17] Paimushin V. N., Firsov V. A., Shishkin V. M., “Modeling the dynamic response of a carbon-fiber-reinforced plate at resonance vibrations considering the internal friction in the material and the external aerodynamic damping”, Mechanics of Composite Materials, 53:4 (2017), 609–630 | DOI

[18] Starovoitov E. I., Leonenko D. V., Yarovaya A. V., “Vibrations of circular sandwich plates under resonance loads”, International Applied Mechanics, 39:12 (2003), 1458–1463 | DOI | Zbl

[19] Starovoitov E. I., Leonenko D. V., Tarlakovsky D. V., “Resonance vibrations of circular composite plates on an elastic foundation”, Mechanics of Composite Materials, 51:5 (2015), 561–570 | DOI

[20] Kondratov D. V., Mogilevich L. I., Popov V. S., Popova A. A., “Hydroelastic oscillations of a circular plate, resting on Winkler foundation”, Journal of Physics: Conference Series, 944 (2018), 012057 | DOI

[21] Mogilevich L. I., Popov V. S., Popova A. A., Christoforova A. V., “Mathematical modeling of hydroelastic oscillations of the stamp and the plate, resting on Pasternak foundation”, Journal of Physics: Conference Series, 944 (2018), 012081 | DOI

[22] Bykova T. V., Grushenkova E. D., Popov V. S., Popova A. A., “Hydroelastic response of a sandwich plate possessing a compressible core and interacting with a rigid die via a viscous fluid layer”, Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 20:3 (2020), 351–366 (in Russian) | DOI | MR

[23] Ageev R. V., Mogilevich L. I., Popov V. S., “Vibrations of the walls of a slot channel with a viscous fluid formed by three-layer and solid disks”, Journal of Machinery Manufacture and Reliability, 43:1 (2014), 1–8 | DOI | Zbl

[24] Fedotenkov G. V., Tarlakovsky D. V., Vahterova Y. A., “Identification of non-stationary load upon Timoshenko beam”, Lobachevskii Journal of Mathematics, 40:4 (2019), 439–447 | DOI | MR | Zbl

[25] Rabboh S., Bondok N., Mahmoud T., El Kholy H., “The effect of functionally graded materials into the sandwich beam dynamic performance”, Materials Sciences and Applications, 4:11 (2013), 751–760 | DOI

[26] Zenkour A. M., Alghamdi N. A., “Bending analysis of functionally graded sandwich plates under the effect of mechanical and thermal loads”, Mechanics of Advanced Materials and Structures, 17:6 (2010), 419–432 | DOI

[27] Starovoitov E. I., Leonenko D. V., “Impact of thermal and ionizing radiation on a circular sandwich plate on an elastic foundation”, International Applied Mechanics, 47:5 (2011), 580–589 | DOI | MR

[28] Grover N., Singh B. N., Maiti D. K., “An inverse trigonometric shear deformation theory for supersonic flutter characteristics of multilayered composite plates”, Aerospace Science and Technology, 52 (2016), 41–51 | DOI

[29] Yarovaya A. V., “Thermoelastic bending of a sandwich plate on a deformable foundation”, International Applied Mechanics, 42:2 (2006), 206–213 | DOI | Zbl

[30] Starovoitov E. I., Leonenko D. V., “Bending of an elastic circular three-layer plate in a neutron flux by a local load”, Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 22:3 (2022), 360–375 (in Russian) | DOI | MR | Zbl

[31] Paimushin V. N., “Theory of moderately large deflections of sandwich shells having a transversely soft core and reinforced along their contour”, Mechanics of Composite Materials, 53:1 (2017), 3–26 | DOI

[32] Wang Zh., Lu G., Zhu F., Zhao L., “Load-carrying capacity of circular sandwich plates at large deflection”, Journal of Engineering Mechanics, 143:9 (2017), 04017057 | DOI

[33] S̆kec L., Jelenić G., “Analysis of a geometrically exact multi-layer beam with a rigid interlayer connection”, Acta Mechanica, 225:2 (2014), 523–541 | DOI | MR | Zbl

[34] Starovoitov E. I., Leonenko D. V., Yarovaya A. V., “Elastoplastic bending of a sandwich bar on an elastic foundation”, International Applied Mechanics, 43:4 (2007), 451–459 | DOI | MR | Zbl

[35] Pradhan M., Dash P. R., Pradhan P. K., “Static and dynamic stability analysis of an asymmetric sandwich beam resting on a variable Pasternak foundation subjected to thermal gradient”, Meccanica, 51:3 (2016), 725–739 | DOI | MR | Zbl

[36] Zadeh H. V., Tahani M., “Analytical bending analysis of a circular sandwich plate under distributed load”, International Journal of Recent Advances in Mechanical Engineering, 6:1 (2017) | DOI | MR

[37] Yang L., Harrysson O., West H., Cormier D. A., “Comparison of bending properties for cellular core sandwich panels”, Materials Sciences and Applications, 4:8 (2013), 471–477 | DOI

[38] Kudin A., Al-Omari M. A. V., Al-Athamneh B. G. M., Al-Athamneh H. K. M., “Bending and buckling of circular sandwich plates with the nonlinear elastic core material”, International Journal of Mechanical Engineering and Information Technology, 3:08 (2015), 1487–1493 | DOI

[39] Starovoitov E. I., Leonenko D. V., “Repeated alternating loading of an elastoplastic three-layer plate in a temperature field”, Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 21:1 (2021), 60–75 (in Russian) | DOI | MR

[40] Starovoitov É. I., “Description of the thermomechanical properties of some structural materials”, Strength of Materials, 20:4 (1988), 426–431 | DOI