INVESTIGATION OF ELASTIC PRESLIDING DISPLACEMENT PHENOMENA IN CONNECTION WITH THE PROBLEM OF PRECISION IN MACHINE AND INSTRUMENT ENGINEERING

V. I. Maksak, G. A. Doshchinskii

(Presented by the scientific seminar of the Department of Strength of Materials)

Ensuring maximum precision is one of the most important problems of modern machine and instrument engineering. In this connection, requirements for the rigidity of parts and structural assemblies, in particular for contact rigidity, are continually increasing. D. N. Reshetov noted in his work [1] that contact deformations substantially reduce the operating accuracy of instruments, machine tools, and other precision machines. The magnitude of contact rigidity is also great in the manufacture of high-precision machine and instrument parts. Machining tolerances for parts often do not exceed 1 μm, and in precision instrument engineering even 0.1 μm. It is significant that, for the solution of certain new technical problems, it is necessary to achieve machining accuracy equal to 0.01 μm. (From the report by I. V. Kragel’skii at the First All-Union Seminar on Contact Rigidity [2]). The magnitude of contact deformations in a number of cases reaches tens of microns.

Usually contact rigidity is considered both for the displacement of mating surfaces along the normal and for their relative displacement in the tangential direction. At present this important problem has not been studied sufficiently fully; moreover, the question of tangential rigidity, which is considered in this work, has been studied the least.

A number of works by Soviet researchers have been devoted to the study of normal contact rigidity: I. V. Kragel’skii, A. P. Sokolovskii, K. V. Votinov, D. N. Reshetov and Z. M. Levina, N. B. Demkin, E. V. Ryzhov, G. E. Chikhladze, I. R. Konyakhin [3], B. P. Mitrofanov [4], and others, as well as works by foreign authors: Opitz, Helkin, Neuweiler, Bilfeldt, and others [2].

When a shearing load acts on a contact, the onset of sliding is preceded by a certain shear due to deformation of the contact layers.

A small tangential displacement of contacting surfaces, caused by a shearing force equal in magnitude to an incomplete friction force, is called presliding displacement [5].

Questions of tangential rigidity and presliding displacement have been investigated in the works of A. V. Verkhovskii [6], Rankin [7], Tomlinson [8], S. E. Khaikin, A. E. Salomanovich and A. P. Lisovskii [9], V. S. Shchedrov [10], I. V. Kragel’skii and N. M. Mikhin [11], I. R. Konyakhin [3], and B. P. Mitrofanov [4].

Of these works, only one concludes with a computational formula that determines the magnitude of the limiting displacement (for \(P=fN\), where \(P\) is the shearing force, \(f\) is the coefficient of friction, and \(N\) is the compressive force) under plastic penetration of rigid asperities of one surface into another.

In joints of structural elements, machine parts, and instruments, repeated action of external forces often occurs, under which the contact may be regarded as elastic. In the general case, the magnitude of the elastic presliding displacement is affected both by external loads and by mechanical, geometrical, and frictional characteristics.

On the basis of an experimental study of the tangential stiffness of contact [12, 13], it has been established that an increase in the compressive force, on the one hand, increases the stiffness of the joint and, on the other hand, raises the value of the limiting displacement. By limiting displacement here is meant the displacement corresponding to the shearing force \(P=fN\), where \(f\) is the coefficient of static friction and \(N\) is the compressive force. A larger coefficient of friction also corresponds to greater joint stiffness and a larger limiting displacement, the relationship of the latter being linear. For sublimiting shearing forces \((P<fN)\), with \(P=\mathrm{const}\), this relationship is determined by a family of hyperbolas. For materials with a large shear modulus \(G\), the displacement under identical shearing forces is smaller.

The geometry of the rough surface has a considerable influence on the stiffness of the contact; when produced mechanically, this surface often has a directionality of machining marks. In this case, contact stiffness is affected not only by the height of the irregularities, with whose increase the elastic displacement increases, but also by the radii of curvature of the asperities in the longitudinal and transverse directions. Thus, for identical shearing forces, the elastic displacements along the machining marks are larger. As the roughness increases, this difference increases. The displacement vector does not coincide in direction with the vector of the shearing force. When the vectors diverge, the acute angle between the direction of the machining marks and the displacement vector decreases [14]. The magnitude of the divergence of the vectors can be determined from a formula obtained on the basis of the principle of minimum work:

\[ \tan \gamma=\frac{\chi_y f_x^2}{\chi_x f_y^2}\tan \alpha, \tag{1} \]

where \(\gamma\) is the angle between the directions of the maximum coefficient of friction \(f_x\) (across the machining marks) and of the displacement;

\(\alpha\) is the angle between the directions of the shearing force and of the coefficient \(f_x\);

\(\chi\) is a coefficient that accounts for the shape of the asperity and is determined from a diagram [15].

On the basis of R. Mindlin’s solution [16] and A. I. Lur’e’s solution [15] for the shear of an elliptical punch, and by modeling the surface as a set of ellipsoids distributed in height according to the bearing-surface curve, the work obtains computational formulas determining the magnitude of the presliding displacement of an elastic discrete contact.

Figure 1 shows, in general form, the graph of the dependence of presliding displacement on the shearing force \(P\).

When the contact is loaded by a force from zero (curve OA), the magnitude of the displacement is determined by the formula

\[ \Delta= \frac{2f h_{\max}\chi}{(1-\mu)n_a n_b} \left[ \frac{\pi n_b^{3/2}(1-\mu)(r_{\mathrm{pr}}+r_{\mathrm{pop}})^{1/2}N} {1.88\,bK_2 h_{\max}^{1/2} G A_c} \right]^{\frac{2}{2\nu+1}} \left[ 1-\left(1-\frac{P}{fN}\right)^{2/3} \right], \tag{2} \]

where \(h_{\max}\) is the maximum height of the irregularities,

\(r_{\mathrm{pr}}\) and \(r_{\mathrm{pop}}\) are the radii of the asperities in the longitudinal and transverse directions, respectively,

\(n_a\) and \(n_b\) are coefficients depending on the principal curvatures of the contacting asperities at the point of contact and on the angle between the planes of the principal curvatures [17],

\(\mu\) is Poisson’s ratio,

\(b\) and \(\nu\) are parameters of the curve of the bearing surface,

\(K_2\) is a coefficient depending on \(\nu\) [18],

\(A_c\) is the contour area.

Curve AB reflects the phenomenon of elastic recovery when the shear force is removed; BO is the second recovery, manifested when the compressive force is removed. BC determines the preliminary displacement when the contact is loaded by a shear force \(P\) of the opposite sign.

The formula determining the displacement corresponding to curve ABC (Fig. 1) has the form

\[ \Delta' = \frac{2 f h_{\max}\chi}{(1-\mu)n_a n_b} \left[ \frac{\pi n_b^{3/2}(1-\mu)(r_{\mathrm{pr}}+r_{\mathrm{pop}})^{1/2}N} {1.88\,bK_2 h_{\max}^{1/2}GA_c} \right]^{\frac{2}{2\nu+1}} \times \left[ 2\left(1-\frac{P'-P}{2fN}\right)^{2/3} - \left(1-\frac{P}{fN}\right)^{2/3} -1 \right]. \tag{3} \]

Curve CDA corresponds to a change in the shear force from \(-P'\) to \(+P'\). The displacement corresponding to this curve is determined by a formula analogous to formula (3), and may be written in the following form

\[ \Delta''=-\Delta'(-P). \tag{4} \]

The area of the loop ABCDA determines the magnitude of the energy dissipated in the contact during one loading cycle. The vibration resistance of structures depends to a considerable extent on the energy dissipated in the contact. In addition, hysteresis phenomena in joints of elements lead to a significant limitation of the accuracy of instruments [11].

Fig. 1.

The dissipation of energy, increasing with an increase in the shear force, decreases, for equal shear forces, with an increase in the coefficient of friction and the compressive force, and increases for limiting shear forces. For limiting displacements, the dependence of energy dissipation on the coefficient of friction is represented by a parabola, while for nonlimiting shear forces it is determined by a family of hyperbolas. The dependence of energy dissipation on the compressive force is analogous. The cleanliness of the machining of the surfaces and their mechanical properties affect the dissipation of energy in the same way as they affect the magnitude of the displacement.

The magnitude of energy dissipation may be determined from the formula: for

\[ P'=fN \]

\[ \omega = \frac{1.6 h_{\max} f^2 N\chi}{(1-\mu)n_a n_b} \left[ \frac{\pi n_b^{3/2}(1-\mu)(r_{\mathrm{pr}}+r_{\mathrm{pop}})^{1/2}N} {1.88\,bK_2 h_{\max}^{1/2}GA_c} \right]^{\frac{2}{2\nu+1}}; \]

for \(P'\ll fN\)

\[ \omega = \frac{0.296 h_{\max} P^3\chi}{(1-\mu)n_a n_b fN^2} \left[ \frac{\pi n_b^{3/2}(1-\mu)(r_{\mathrm{pr}}+r_{\mathrm{pop}})^{1/2}N} {1.88\,bK_2 h_{\max}^{1/2}GA_c} \right]^{\frac{2}{2\nu+1}}. \tag{5} \]

The calculated formulas obtained agree satisfactorily with the experiment, which was carried out on specimens and on the setup described in [14], and make it possible to select the optimum contact parameters.

The correct selection of the material, the assignment of the surface finish class and of the direction of the machining marks, and allowance for the acting loads and the magnitudes of the elastic displacements will make it possible to reveal a substantial reserve for increasing accuracy in machine and instrument engineering.

REFERENCES

1. D. N. Reshetov. Contact stiffness of metal-cutting machine tools. Proceedings of the First All-Union Seminar on Contact Stiffness. Tbilisi, 1966.

2. I. V. Kragel’skii. On the contact stiffness of joints. Proceedings of the First All-Union Seminar on Contact Stiffness. Tbilisi, 1966.

3. I. R. Konyakhin. Theory of preliminary displacements as applied to problems of contact between parts. Tomsk, 1965.

4. B. P. Mitrofanov. On the deformation of discrete contact (Candidate’s dissertation), Tomsk, 1964.

5. Machine Parts (Calculation and Design). Handbook, vol. 1, Mashinostroenie, Moscow, 1968.

6. A. V. Verkhovskii. The phenomenon of preliminary displacement during the starting friction of unlubricated surfaces. Journal of Applied Physics, No. 3, issue 3–4, 1926.

7. J. S. Rankin. The Elastic range of Friction. Phil. Magaz., v. (8) 2; 1926, p. 806.

8. G. A. Tomlinson. Molecular Theorie of Friction. Phil. Magaz., No. 7, 1929.

9. S. E. Khaikin, A. E. Salomonovich, L. P. Lisovskii. On dry-friction forces. Friction and Wear in Machines, vol. 1, 1939.

10. V. S. Shchedrov. Preliminary displacement in an elastic-viscous contact. Friction and Wear in Machines, collection V, 1950.

11. I. V. Kragel’skii, N. N. Mikhin. On the nature of contact preliminary displacement of solid bodies. Reports of the Academy of Sciences of the USSR, vol. 153, No. 1, 1963.

12. V. I. Maksak. Tangential stiffness of plane joints. Materials for the scientific and technical conference of March 13–14, 1969, Kemerovo, 1969.

13. I. R. Konyakhin, B. P. Mitrofanov, V. I. Maksak. Elastic preliminary displacement under a proportional change of the compressive and shearing forces. Proceedings of Higher Educational Institutions, “Machine Building”, No. 6, 1968.

14. B. P. Mitrofanov, V. I. Maksak. Anisotropy of elastic preliminary displacement. Bulletin of the Academy of Sciences of the Belorussian SSR, Series of Physical and Technical Sciences, No. 1, 1968.

15. A. I. Lur’e. Spatial Problems of the Theory of Elasticity. Gostekhizdat, 1955.

16. R. D. Mindlin. Compliance of Elastic Bodies in Contact. J. Appl. Mech., No. 16, 1949.

17. S. D. Ponomarev, V. L. Biderman, K. K. Likharev, V. M. Makushin, N. N. Malinin, V. I. Feodos’ev. Strength Calculations in Machine Building, vol. II, Mashgiz, 1958.

18. N. B. Demkin. Actual Contact Area of Solid Surfaces. Publishing House of the Academy of Sciences of the USSR, 1962.

19. L. A. Andreeva. Elastic Elements of Instruments. Mashgiz, 1962.