 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
|
| In English (Mech. Solids): | | 4760 |
|
| << Previous article | Volume 49, Issue 2 / 2014 | Next article >> |
| A.D. Chernyshov, "Torsion of an Elastic Rod Whose Cross-Section Is a Parallelogram, Trapezoid, or Triangle, or Has an Arbitrary Shape by the Method of Transformation to a Rectangular Domain," Mech. Solids. 49 (2), 225-236 (2014) |
| Year |
2014 |
Volume |
49 |
Number |
2 |
Pages |
225-236 |
| DOI |
10.3103/S0025654414020125 |
| Title |
Torsion of an Elastic Rod Whose Cross-Section Is a Parallelogram, Trapezoid, or Triangle, or Has an Arbitrary Shape by the Method of Transformation to a Rectangular Domain |
| Author(s) |
A.D. Chernyshov (Voronezh State Technological Academy, pr-t Revolyutsii 19, Voronezh, 394000 Russia, chernyshovad@mail.ru) |
| Abstract |
A coordinate transformation is used to take the domain of the rod cross-section to a rectangular domain for which the spectra of eigenfunctions and eigenvalues are known. The torsion function is represented as a generalized Fourier series to reduce the problem to solving a closed linear system of algebraic equations for the expansion coefficients. It is shown that these Fourier series converge absolutely, because the expansion coefficients decrease by a cubic law depending on the term number. We prove that the approximate solution in the form of a finite sum of the Fourier series converges to the exact solution. This theorem is generalized to the case of a rod cross-section of arbitrary shape. |
| Keywords |
elastic rod, cross-section, parallelogram, trapezoid, triangle, fast Fourier series |
| References |
| 1. | A. I. Lurie,
The Theory of Elasticity
(Nauka, Moscow, 1970)
[in Russian]. |
| 2. | S. P. Timoshenko and J. N. Goodyear,
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| 3. | N. I. Muskhelishvili,
Several Fundamental Problems of Mathematical Theory of Elasticity
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[in Russian]. |
| 4. | N. Kh. Arutyunyan and B. L. Abramyan,
Torsion of Elastic Bodies
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[in Russian]. |
| 5. | V. A. Lomakin,
"Torsion of Rods with Elastic Properties Depending on the Stress State Form,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 4, 30-39 (2002)
[Mech. Solids (Engl. Transl.)
37 (4), 23-30 (2002)]. |
| 6. | A. V. Konovalov,
"Torsion of Cylindrical Rods and Pipes with Large Plastic Strains,"
with Large Plastic Strains,"
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No. 3, 102-111 (2001)
[Mech. Solids (Engl. Transl.)
36 (3), 86-94 (2001)]. |
| 7. | L. M. Zubov,
"The Non-Linear Saint-Venant Problem of the Torsion, Stretching
and Bending of a Naturally Twisted Rod,"
Prikl. Mat. Mekh.
70 (2), 332-343 (2006)
[J. Appl. Math. Mech. (Engl. Transl.)
70 (2), 300-310 (2006)]. |
| 8. | G. I. Bykovtsev and D. D. Ivlev,
Theory of Plasticity
(Dal'nauka, Vladivostok, 1998)
[in Russian]. |
| 9. | V. A. Il'in,
Spectral Theory of Differential Operators.
Self-Adjoint Differential Operators
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[in Russian]. |
| 10. | E. M. Kartashov,
Analytical Methods in Heat Conductivity of Solids
(Vysshaya Shkola, Moscow, 2001)
[in Russian]. |
| 11. | A. D. Chernyshov,
"Solving Nonlinear Boundary Value Problems by the Spectral Decomposition Method,"
Dokl. Ross. Akad. Nauk
411 (6), 775-778 (2006)
[Dokl. Phys. (Engl. Transl.)
51 (12), 697-700 (2006)]. |
| 12. | G. P. Tolstov,
Fourier Series
(Dover, New York, 1976; Nauka, Moscow, 1980). |
| 13. | A. N. Kolmogorov and S. V. Fomin,
Elements of the Theory of Functions and Functional Analysis
(Nauka, Moscow, 1981; Dover, New York, 1999). |
|
| Received |
04 April 2011 |
| Link to Fulltext |
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