ENERGY DISSIPATION IN A CONTACT HAVING DIRECTIONAL MACHINING MARKS

V. I. Maksak

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

Recently, an increasing number of works have appeared devoted to questions of energy dissipation in a contact [1–7]. This is connected with the significant possibilities of structural damping. In a number of cases, energy dissipation in joints reaches 90% [7]. Until now, in studying energy dissipation it has been assumed that the mechanical properties of the surface of specimens in the plane of contact are isotropic. However, the directionality of the traces of mechanical machining of a surface often exists, and in contact processes it manifests itself not only as mechanical anisotropy, but also as a peculiarity of the deformation of roughnesses having an elongated form.

In this work, dissipation of energy in a contact is considered under conditions of an elastic preliminary displacement. The experiment was carried out under static and dynamic conditions on steel, copper, and lead specimens having different roughnesses of the contacting surfaces. The setup and specimens for investigating contact under static conditions are described in [8]. The magnitude of the energy dissipated in the contact during one loading cycle was determined from the area of the loop bounded by the displacement curves during loading of the contact by a shearing force and the subsequent unloading. Figure 1 shows plots of the preliminary displacements during shear across the machining marks (lines 1, 3, 5) and along them (lines 2, 4, 6) for a steel specimen with hardness HRC 40 units and roughness of the fifth class in contact with a steel specimen of hardness HB 220, polished to the twelfth class of finish (lines 1, 2), with lead (lines 3, 4), and with copper (lines 5, 6) specimens. The ascending line is the loading of the contact by the shearing force. The descending line is the elastic recovery corresponding to unloading. When the compressive forces are removed, it disappears and a residual displacement remains. From Fig. 1 it is seen that, for the same shearing force not exceeding the limiting value, both the magnitude of the preliminary displacement and the magnitude of the dissipated energy along the machining marks are greater than across them. This is determined not only by the difference in the coefficients of static friction in the directions considered, but also by the difference in the deformability of the roughnesses elongated along the machining marks.

For a theoretical study of energy dissipation in a contact, a rough surface is modeled by a set of ellipsoidal segments arranged in height according to the bearing-surface curve. When the contact is loaded according to the scheme (Fig. 1), the magnitude of the dissipa-

the energy dissipated in one cycle will be determined by the formula

\[ \omega=\int_{0}^{P'}(\Delta'-\Delta)\,dP, \tag{1} \]

where \(\Delta\) is the displacement corresponding to curve \(OA\) (Fig. 1), \(\Delta'\) is the elastic recovery (line \(AB\)), and \(P'\) is the shearing force.

Fig. 1. Plots of preliminary displacements under shear transverse to the machining marks (lines 1, 3, 5) and along them (lines 2, 4, 6) for a steel specimen in contact with steel (lines 1, 2), lead (lines 3, 4), and copper (lines 5, 6) specimens.

The formula for determining the displacement \(\Delta\), obtained in [9], has the form

\[ \Delta= \frac{2fh_{\max}\varepsilon \chi}{(1-\mu)n_a n_b} \left[ 1-\left(1-\frac{P}{fN}\right)^{2/3} \right], \tag{2} \]

where \(f\) is the coefficient of static friction,

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

\(\mu\) is Poisson’s ratio,

\(n_a, n_b\) are coefficients depending on the principal curvatures of the contacting asperities at the contact location and on the angle between the planes of the principal curvatures [10],

\(N\) is the compressive force,

\(\chi\) is a coefficient accounting for the influence of the shape of the asperity on its deformation and determined from the diagram [11, 12],

\(\varepsilon\) is the magnitude of the relative approach of the contacting bodies,

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

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

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

\(K_2\) is a coefficient depending on \(\nu\) [13],

\(G\) is the shear modulus,

\(A_c\) is the contour area.

Fig. 2. Oscillograms of damped vibrations of a beam, clamped as a cantilever between two copper plates in a press with a force of 50 kg: a—the roughness of the plates is directed perpendicular to the longitudinal axis of the beam; b—the roughness is directed along the axis.

In an analogous way one can obtain the magnitude of the displacement \(\Delta'\):

\[ \Delta'=\frac{2fh_{\max}\varepsilon x}{(1-\mu)n_a n_b} \left[ 2\left(1-\frac{P'-P}{2fN}\right)^{2/3} -\left(1-\frac{P'}{fN}\right)^{2/3}-1 \right], \tag{4} \]

where \(P\) is the current value of the shearing force. After substituting (2) and (4) into (1), one obtains

\[ \omega=\frac{4.8h_{\max}f^2N\varepsilon x}{(1-\mu)n_a n_b} \left\{ 1-\left(1-\frac{P'}{2fN}\right)^{5/3} -\frac{5P'}{12fN} \left[ 2+\left(1-\frac{P'}{fN}\right)^{2/3} \right] +\frac{1}{4} \left[ 1-\left(1-\frac{P'}{fN}\right)^{2/3} \right] \right\}. \tag{5} \]

For \(P'=fN\),

\[ \omega=\frac{0.48h_{\max}f^2N\varepsilon x}{(1-\mu)n_a n_b}, \]

for \(P'\ll fN\),

\[ \omega=\frac{0.11h_{\max}\varepsilon xP^3}{(1-\mu)n_a n_b fN^2}. \tag{6} \]

Under symmetric cyclic loading, formula (1) takes the form

\[ \omega=\int_{-P'}^{P'}(\Delta'-\Delta'')\,dP, \tag{7} \]

where \(\Delta''=-\Delta'(-P)\); then

\[ \omega=\frac{9.6h_{\max}f^2N\varepsilon x}{(1-\mu)n_a n_b} \left\{ 1-\left(1-\frac{P'}{fN}\right)^{5/3} -\frac{5P'}{6fN} \left[ 1+\left(1-\frac{P'}{fN}\right)^{2/3} \right] \right\}. \tag{8} \]

For \(P'=fN\),

\[ \omega=\frac{1.6h_{\max}f^2N\varepsilon x}{(1-\mu)n_a n_b}, \]

for \(P'\ll fN\),

\[ \omega=\frac{0.296h_{\max}xP^3}{(1-\mu)n_a n_b fN^2}. \tag{9} \]

Analysis of formulas (6) and (9) shows that, with increasing roughness, dissipation increases. The difference in energy dissipation when displacement occurs along and across the machining marks is determined by the values of \(f\) and \(x\) in the corresponding directions. Along the machining marks, the coefficient of static friction is usually smaller, while the value of \(x\) is always larger; therefore, at equal shearing forces \(P'<fN\), the energy dissipation along the machining marks is greater than across them. The results of the theoretical analysis agree well with experiment.

Under dynamic conditions, the theoretical dependences were checked from the results of processing oscillograms of free damped vibrations of a cantilever beam clamped in a press between two plates having a definite direction of the machining marks.

Figure 2 presents two such oscillograms, from which it is evident that when the machining marks are directed along the axis of the beam, i.e., along the line of action of the shearing forces, the damping of vibrations proceeds faster.

In conclusion it should be noted that this property of the contact may be widely used in practice as a reserve for increasing its damping properties.

REFERENCES

1. S. B. Ainbinder. On energy dissipation at a frictional contact during vibrations. Izv. AN SSSR, OTN, 1962, No. 4.

1. I. R. Konyakhin, B. P. Mitrofanov. On the dissipation of energy in elastic discrete contact. Inzhenernyi zhurnal MTT, No. 1, 1967.

2. I. L. Meerson, Yu. Kh. Gazaryan. Investigation of the damping properties of fir-tree-type lock joints of turbine blades. Thermal Strength of Materials and Structural Elements, “Naukova Dumka,” Kiev, issue 4, 1967.

3. V. V. Matveev, A. P. Yakovlev, A. D. Balyuk, L. N. Rzhavin. Investigation of the damping properties of the lock joint of compressor blades at elevated temperature. Thermal Strength of Materials and Structural Elements, “Naukova Dumka,” Kiev, issue 4, 1967.

4. K. A. Prokofiev, M. M. Epanchinkov. Damping properties of a blade with an individual fir-tree-type root. Energy Dissipation in Vibrations of Elastic Systems. “Naukova Dumka,” Kiev, 1966.

5. V. I. Gogilashvili, M. V. Khvingiya. Energy dissipation and stiffness of a shaft–bushing joint. Proceedings of the Georgian Polytechnic Institute, No. 7/112, Tbilisi, 1966.

6. R. L. Zaidelman. Increasing the reliability of turbine blades by damping vibrations. “Energiya,” Moscow, 1967.

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

8. V. I. Maksak. Quantitative assessment of elastic preliminary displacement with directional surface roughness. Izv. TPI, vol. 225, Tomsk, 1970.

9. S. D. Ponomarev, V. L. Biderman, K. K. Likharev, V. M. Makushin, N. N. Malinin, V. I. Feodosiev. Strength Calculations in Mechanical Engineering. Mashgiz, Moscow, vol. 2, 1958.

10. R. D. Mindlin. Compliance of Elastic Bodies in Contact. J. Appl. Mech. 16, 1949.

11. A. I. Lurye. Three-Dimensional Problems of the Theory of Elasticity. Gostekhizdat, Moscow, 1955.

12. N. B. Demkin. Actual Area of Contact of Solid Surfaces. Moscow, Academy of Sciences of the USSR, 1962.

13. Order 4746.