Abstract

The unbonded contact problem for bending of two-leaf spring is considered; the leaves are curved along the circular arc in their natural states. The lengths of the leaves are different; each leaf has one end clamped and the other free. The angle formed by the long leaf is less than the right one. The cross-sections of leaves are the rectangles of the same width but of the different thickness. These thicknesses are considered to vanish while the elastic lines of the leaves are analyzed geometrically. The real thicknesses affect only the bending stiffness of the leaves. The given loading is applied transversely to the leaves. There is no friction between the leaves. The bending is described by Bernoulli - Euler model. The problem is reduced to finding the density of the leaves interacting forces. This density is the sum of the piecewise-continuous part and the concentrated forces. The rigorous problem statement is formulated, the uniqueness of solution is established and the complete analytical solution of the problem is provided. This construction also proves the existence of the solution. The substantiation of the solution includes proving of the non-negativity of the contact forces and contact distances and the proving of the existence of the root of transcendental equation that gives the length of the contact segment. The proving of the non-negativity of the contact distances uses a new approach that is based on the fact that these distances can be regarded as the solutions of some variational problems. It is shown that three patterns of the leaves contact are possible: the contact along the whole short leaf; the contact at the point on the end of the short leaf; the contact along the part of the short leaf and at the point. The pattern kind depends on the given loading. The obtained results generalize the known sufficient condition for the pointwise contact of the leaves.

M A Osipenko

Perm National Research Polytechnic University, Perm, Russian Federation

Email: oma@theormech.pstu.ac.ru
29, Komsomolsky av., 614990, Perm, Russian Federation Ph.D. in Physics and Mathematics, Associate Professor, Department of Theoretical Mechanics, Perm National Research Polytechnic University

References

1. Работнов Ю.Н. Механика деформируемого твердого тела. - М.: Наука, 1988. - 711 с.
2. Osipenko M.A., Nyashin Yu.I., Rudakov R.N. A contact problem in the theory of leaf spring bending // International Journal of Solids and Structures. - 2003. - No. 40 - P. 3129-3136.
3. Osipenko M.A., Nyashin Y.I., Rudakov R.N. On the theory of bending of foot prosthesis containing the curved plates // Russian Journal of Biomechanics. - 1999. - Vol. 3 - No. 3 - P. 73-77.
4. Osipenko M.A., Nyashin Y.I., Rudakov R.N. The sufficient condition for the pointwise contact in the two-leaf curved elastic element of the foot prosthesis under bending // Russian Journal of Biomechanics. - 2000. - Vol. 4 - No. 3 - P. 33-41.
5. Расчеты на прочность в машиностроении. T. 1 / C.Д. Пономарев [и др.] - М.: Машгиз, 1956. - 884 c.
6. Пархиловский И.Г. Автомобильные листовые рессоры. - М.: Машиностроение, 1978. - 232 с.
7. Таланцев Н.Ф. Критерии оценки рессор // Автомобильная промышленность. - 1988. - № 10 - С. 20-21.
8. Пестренин В.М., Пестренина И.В., Таланцев Н.Ф. Численный анализ напряженно-деформированного состояния листовых рессор // Вычисл. мех. спл. сред. - 2009. - Т. 2, № 2 - С. 74-84.
9. Осипенко М.А., Няшин Ю.И. Новый итерационный метод расчета многолистовой рессоры // Вычисл. мех. спл. сред. - 2012. - Т. 5, № 4 - С. 371-376.
10. Осипенко М.А. Аналитический расчет статического изгиба двухлистовой рессоры с параболическим профилем короткого листа // Вестник ИжГТУ. - 2012. - № 3 (55) - С. 146-150.
11. Феодосьев В.И. Избранные задачи и вопросы по сопротивлению материалов. - М.: Наука, 1973. - 400 с.
12. Григолюк Э.И., Толкачев В.М. Контактные задачи теории пластин и оболочек. - М.: Машиностроение, 1980. - 415 с.
13. Джонсон K. Механика контактного взаимодействия. - М.: Мир, 1989. - 510 с.
14. Эльсгольц Л.Э. Дифференциальные уравнения и вариационное исчисление. - М.: Эдиториал УРСС, 2000. - 320 с.
15. Кравчук А.С. Вариационные и квазивариационные неравенства в механике; Мос. гос. акад. приборостр. и информатики. - М.,1997. - 340 с. References
16. Rabotnov Yu.N. Mekhanika deformiruemogo tverdogo tela [Mechanics of deformable solids]. Moscow: Nauka, 1988, 711 p.
17. Osipenko M.A., Nyashin Yu.I., Rudakov R.N. A contact problem in the theory of leaf spring bending. International Journal of Solids and Structures, 2003, no. 40, pp. 3129-3136.
18. Osipenko M.A., Nyashin Y.I., Rudakov R.N. On the theory of bending of foot prosthesis containing the curved plates. Russian Journal of Biomechanics, 1999, vol. 3, no. 3, pp. 73-77.
19. Osipenko M.A., Nyashin Y.I., Rudakov R.N. The sufficient condition for the pointwise contact in the two-leaf curved elastic element of the foot prosthesis under bending. Russian Journal of Biomechanics, 2000, vol. 4, no. 3, pp. 33-41.
20. Ponomaryov S.D. [et al.] Raschety na prochnost v mashinostroenii. T. 1 [Stress calculation in mechanical engineering, vol. 1]. Мoscow: Mashgiz, 1956. 884 p.
21. Parhilovsky I.G. Avtomobilnye listovye ressory [Automotive leaf springs]. Moscow: Mashinostroenie, 1978. 232 p.
22. Talantsev N.F. Kriterii otsenki ressor [The criterions for evaluation of leaf springs]. Avtomobilnaya promyshlennost, 1988, no. 10, pp. 20-21.
23. Pestrenin V.M., Pestrenina I.V., Talantsev N.F. Chislennyj analiz napryazhenno-deformirovannogo sostoyaniya listovykh ressor [The numerical analysis of stress-strain state of leaf springs]. Vychislitelnaya mekhanika sploshnykh sred, 2009, vol. 2, no. 2, pp. 74-84.
24. Osipenko M.A., Nyashin Y.I. Novyi iteratsionnyj metod rascheta mnogolistovoj ressory [The new iterational method for calculation of multi-leaf spring]. Vychislitelnaya mekhanika sploshnykh sred, 2012, vol. 5, no. 4, pp. 371-376.
25. Osipenko M.A. Analiticheskij raschet staticheskogo izgiba dvukhlistovoj ressory s parabolicheskim profilem korotkogo lista [Аnalytical calculation of the static bending of the combination two-leaf spring with the parabolic profile of the short leaf]. Vestnik Izhevskogo gosudarstvennogo tekhnicheskogo universiteta, 2012, no. 3 (55), pp. 146-150.
26. Feodosyev V.I. Izbrannye zadachi i voprosy po soprotivleniyu materialov [Selected problems and questions on the strength of materials]. Moscow: Nauka, 1973. 400 p.
27. Grigoluk E.I., Tolkachuov V.M. Kontaktnye zadachi teorii plastin i obolochek [The contact problems for plates and shells]. Moscow: Mashinostroenie, 1980. 415 p.
28. Johnson K.L. Contact mechanics. Cambridge University Press; Cambridge, New York, New Rochelle, Melbourne, Sydney, 1985. 510 p.
29. Elsholtz L.E. Differentsialnye uravneniya i variatsionnoe ischislenie [Differential equations and calculus of variations]. Мoscow: Editorial URSS, 2000. 320 p.
30. Kravtchuk A.S. Variatsionnye i kvazivariatsionnye neravenstva v mekhanike [Variational and quasi-variational inequalities in mechanics]. Moscovskaya gosudarstvennaya akademiya priborostroeniya i informatiki, 1997, 340 p.

Abstract - 40