In this paper, we consider the problem of forced oscillations of a rectangular
plate made of dissipative non-homogeneous composite materials. Based on the refined model
of Academician Yu.N. Rabotnov, integral and integro-differential dependencies between
stresses and strains with weakly singular hereditary cores and using the equations of motion of
the plate, an integro-differential equation (IDE) was obtained, which is a mathematical model of
this problem. Using the method of eliminating weakly singular features of integral and integrodifferential
equations (IDE), proposed by Professor F.B. Badalov, a methodology and algorithm
for numerically solving of a system of weakly singular IDE describing discrete models of a
problem have been developed. Using the method of fundamental systems of solutions for
solving the problem of an articulated rectangular plate, an exact solution of the IDE is obtained
that satisfies the corresponding initial conditions.
In this paper, we consider the problem of forced oscillations of a rectangular
plate made of dissipative non-homogeneous composite materials. Based on the refined model
of Academician Yu.N. Rabotnov, integral and integro-differential dependencies between
stresses and strains with weakly singular hereditary cores and using the equations of motion of
the plate, an integro-differential equation (IDE) was obtained, which is a mathematical model of
this problem. Using the method of eliminating weakly singular features of integral and integrodifferential
equations (IDE), proposed by Professor F.B. Badalov, a methodology and algorithm
for numerically solving of a system of weakly singular IDE describing discrete models of a
problem have been developed. Using the method of fundamental systems of solutions for
solving the problem of an articulated rectangular plate, an exact solution of the IDE is obtained
that satisfies the corresponding initial conditions.
| № | Author name | position | Name of organisation |
|---|---|---|---|
| 1 | Abdukarimov A.. | teacher | TSTU |
| 2 | Khaldybaeva I.. | teacher | TSTU |
| 3 | Kuralov B.. | teacher | TSTU |
| № | Name of reference |
|---|---|
| 1 | Y.N. Rabotnov. “Elements of hereditary mechanics of solid bodies”, 1977. 384 |
| 2 | A.K. Malmeistr., V.P. Tamuzh., G.A. Teters. “Resistance of composite and polymeric materials”, 1980. 572 |
| 3 | F.B. Badalov., Sh.F. Ganikhanov. Vibrations of hereditarily deformable structural elements of aircraft. “Tashkent architecture and civil engineering institute”, 2002. 230 |
| 4 | A. Abdukarimov., F.B. Badalov., A.M. Suyarov., T. Kholmatov. “Forced vibrations of rectangular plates from dissipative inhomogeneous composite materials”, 2003. 16 |
| 5 | Y.V. Suvorova., E.N. Kvasha., T.A. Chushenko. Calculation of the stress-strain state of large tires taking into account the viscous properties of the mechanics of composite materials. “Mechanics of composite materials”, 1991. 677 |
| 6 | F.B. Badalov. “Methods for solving integral and integro-differential equations of the hereditary theory of viscoelasticity”, 1987. 296 |
| 7 | F.B. Badalov., N.Y. Khuzhaerov N. “On a method for the exact solution of integrodifferential equations of dynamic problems in the linear theory of viscoelasticity”, 1988. 71 |
| 8 | A. Abdukarimov., F.B. Badalov. “Sine and cosine functions of fractional order and their application to solving dynamic problems of hereditarily deformable systems”, 2004. 156 |
| 9 | A. Abdukarimov., F.B. Badalov., A.M. Suyarov., T. Kholmatov. To the solution of IDE of nonlinear dynamic problems of structural elements of aircraft from anisotropic hereditarily deformable materials. “Methods of mathematical modeling of engineering problems”, 2001. 69 |
| 10 | A. Abdukarimov. New proof of the addition theorem for integro-differential equations of dynamic problems of hereditarily deformable systems. “Problems of mechanics”, 2004. 19 |
| 11 | A. Abdukarimov. Application of the method of fundamental systems of solutions to the numerical solution of nonlinear integro-differential equations of dynamic problems of hereditarily deformable systems. “Problems of mechanics”, 2006. 12 |
| 12 | Y.V. Suvorova., V.V. Bulatkin. Creep anisotropy of composite materials. “Mechanics of composite materials”, 1985. 927 |
| 13 | A. Abdukarimov., N.Y. Khuzhayarov. Algorithm of the addition theorem for solving integro-differential equations of dynamic problems of viscoelasticity. “Mathematical modeling and numerical methods for solving applied mathematics problems”, 1992. 29 |
| 14 | A. Abdukarimov., F.B. Badalov. Non-conservative dynamic problems of structural elements from dissipative inhomogeneous composite materials. “Composite materials”, 2002. 13 |
| 15 | A. Abdukarimov., F.B. Badalov. Non-conservative dynamic problems of structural elements from dissipatively inhomogeneous composite materials. “Composite materials”, 2002. 13 |
| 16 | A. Abdukarimov., F.B. Badalov., S. Babazhanova. Numerical-analytical solution of dynamic problems of thin-walled and bar structures made of composite materials. “Computational experiment, mathematical modeling and their application in applied mathematics and mechanics”, 1994. 9 |