 | | 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 |
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| N. V. Zharkova and L. V. Nikitin, "Applied problems in the dynamics of elastic rods," Mech. Solids. 41 (6), 64-79 (2006) |
| Year |
2006 |
Volume |
41 |
Number |
6 |
Pages |
64-79 |
| Title |
Applied problems in the dynamics of elastic rods |
| Author(s) |
N. V. Zharkova (Moscow)
L. V. Nikitin (Moscow) |
| Abstract |
Rod is one of the most widespread elements of various structures. It is the
simplest model of many engineering devices such as shafts, valves, piles,
columns, poles, pivots, bolts, strings, cables, bars, etc. The relative
simplicity of the mathematical description of mechanical behavior of rods
draws the attention of researchers from various fields of science and
engineering.
The present paper, which is partly of survey character, deals with the
solution of applied problems of rod dynamics, which have various
applications in physics and engineering.
The main problem considered here is the problem on the dynamics of an
elastic rod one of whose ends is subjected to a mass impact and on whose
other end, depending on the character of applications, either the
displacement (a rigid restraint), or the stress-displacement relation (an
elastic restraint), or the critical stress (a rigid-plastic restraint) is
specified. In some problems, we take into account the force of gravity and
dry friction interaction of the lateral surface with the ambient medium.
Boussinesq and Saint-Venant were the first to consider this kind of
problems. They analyzed an impact of a rigid body against an end of an
elastic rod whose other end is rigidly fixed. Their results are presented
in numerous monographs, e.g., by Love and Timoshenko [1, 2, 3].
In the present paper, we give the solution of this problem for arbitrary
time and refine the rebound time of the mass from the rod. In addition, in
this problem we take into account the force of gravity acting on the rod
and the rigid body and the dry friction drag on the lateral surface.
As a development of Boussinesq's and Saint-Venant's studies, the problem on
a die-forging hammer was considered. The hammer is modeled by a rod; one of
its ends is subjected to impact by a piston accelerated to some velocity,
and the billet to be forged is placed at the other end.
In the simplest case, this problem was stated and approximately solved in
the paper [4] so as to evaluate the stresses originating in the shaft. The
exact solution of this problem obtained in [5] was extended to the cases of
elastic and rigid-plastic billets. Judging by the numerous subsequent
publications (e.g., [6]) dealing with this problem, the paper [5] passed
unnoticed. The problem on the impact of an engine valve against the seat
(an elastic restraint), posed and considered in [7], is a special case of
this problem. |
| References |
| 1. | A. E. Love, A Treatise on the Mathematical Theory of Elasticity
[Russian translation], Gostekhizdat, Moscow, Leningrad, 1935. |
| 2. | S. P. Timoshenko and J. N. Goodier, Elasticity Theory [Russian
translation], Nauka, Moscow, 1975. |
| 3. | S. P. Timoshenko, D. H. Young, and W. Weaver, Vibration Problems in
Engineering [Russian translation], Mashinostroenie, Moscow, 1985. |
| 4. | E. P. Unksov, Experimental Determination of Stresses in Rods of
Die-Forging Hammers [in Russian], Mashgiz, Moscow, 1945. |
| 5. | L. V. Nikitin, "To the design of rods of die-forging hammers,"
Inzh. Zh., No. 3, pp. 185-188, 1961. |
| 6. | V. P. Maiboroda, A. S. Kravchuk, and N. N. Kholin, Dynamic
Deformation of Structural Materials [in Russian], Mashinostroenie, Moscow,
1986. |
| 7. | E. A. Werner and F. D. Fischer, "The stress state in a moving rod
suddenly elastically fixed at its trailing end," Acta Mech., Vol. 111,
No. 3/4, pp. 171-179, 1995. |
| 8. | L. V. Nikitin, Statics and Dynamics of Solids with External Dry
Friction [in Russian], Moskovskii Litsei, Moscow, 1998. |
| 9. | M. A. Lavrentev and B. V. Shabat, Methods of the Theory of
Functions of One Complex Variable [in Russian], Fizmatgiz, Moscow, 1958. |
| 10. | V. A. Ditkin and A. P. Prudnikov, Operational Calculus [in
Russian], Vysshaya Shkola, Moscow, 1975. |
| 11. | L. V. Nikitin, "Dynamics of Elastic Rods with External Dry
Friction," Usp. Mekh., Vol. 11, No. 4, pp. 53-106, 1988. |
| 12. | B. Hu and P. Eberhard, "Symbolic Computation of Longitudinal
Impact Waves," Computer Meth. Appl. Mech. and Eng., Vol. 190, No. 37-38,
pp. 4805-4815, 2001. |
| 13. | N. M. Gersevanov, "Theory of longitudinal elastic impact with
application to determination of pile resistance," in Collected Works. Volume 1 [in
Russian], pp. 70-94, 1948. |
| 14. | L. V. Nikitin, "Impact by a rigid body against an elastic rod
with external dry friction," Inzh. Zh. MTT, No. 2, pp. 166-170, 1967. |
| 15. | L. V. Nikitin and A. N. Tyurekhodgaev, "Wave propagation and
vibration of elastic rods with interfacial frictional slip," Wave Motion,
Vol. 12, No. 6, pp. 513-526, 1990. |
| 16. | L. V. Nikitin and A. N. Tyurekhodzhaev, "Behavior of an elastic
rod penetrating into soil under load," in Problems of Rock Mechanics [in
Russian], pp. 304-311, Nauka, Alma-Ata, 1966. |
| 17. | N. A. Veklich and B. M. Malyshev, "Wave propagation in elastic
rods sitting in a medium with dry friction," in Problems of Solid Mechanics
[in Russian], pp. 64-99, Izd-vo MGU, Moscow, 1985. |
| 18. | D. Kolymbas, "Longitudinal impacts on piles," Dynamics and
Earthquake Engineering, Vol. 10, No. 5, pp. 264-270, 1991. |
| 19. | F. Rausche, G. Goble, and G. E. Links, "Dynamic determination of
pile capacity," Geomechanical Engineering, pp. 367-383, 1985. |
| 20. | J. Grabe and U. Schuler, "Auswertung von dynamischen
Pfahlprüfungen nach verschiedenen Verfahren," Geotechnik, No. 1,
pp. 37-41, 1991. |
|
| Received |
14 September 2006 |
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