ON ENERGY DISSIPATION IN AN ELASTIC DISCRETE CONTACT

V. I. Maksak, B. P. Mitrofanov

(Presented by the Department of Strength of Materials)

Irreversible processes in contacts, which are among the most important causes of damping of vibrations in mechanical systems, are investigated in solving problems of fatigue strength and durability of the surface layers of elements. Among such processes, the dominant role belongs to phenomena associated with friction.

In recent years there has been a marked increase in attention to the problem of structural damping [1, 2, 3, 4], whose important role is most clearly manifested in mechanical systems for which it is impossible completely to eliminate resonant modes (structures of rockets, aircraft, ships, turbine blades, etc.). It is interesting to note the assumptions that are usually used in theoretical studies of structural damping. Thus, for example, in studying the distribution of friction forces between a strip and a base [3] it is assumed:

1. The friction forces are equal to their limiting value everywhere where there is slip of the elements of the strip over the base, i.e., in those regions where deformation of the strip occurs;

2. Friction forces are absent in those regions where the strip is not deformed.

Thus, the tangential forces developing on the contact surface are either equal to the limiting value of the friction force or are equal to zero.

Analyzing the premises of theoretical studies of structural damping, it should be noted that, along with those indicated above, an ideal geometry of the contacting surfaces is assumed—the smoothness of the bodies.

Below, energy dissipation is investigated for the contact of elastic bodies with a rough surface. Such a model of contact, which has been called discrete contact, better reflects the properties of real joints. In addition, we shall assume that the displacements are caused by a shearing force smaller than the friction force. Under such conditions, as is known, small displacements are observed, determined by the deformation of the rough layer and called preliminary displacements.

An explanation of energy dissipation under conditions of elastic preliminary displacement within the framework of a phenomenological theory is given in works [5, 6, 7].

To obtain the laws of energy dissipation in an elastic discrete contact, let us represent the rough surface of the bodies in contact as a set of spherical asperities [8] and use P. Mindlin’s solution for the shear of compressed spheres [9]. Then, for loading and unloading by a shearing force \(P\), the amount of energy dissipated per asperity, under the condition

\[ \frac{P}{fN} \ll 1, \]

will be

\[ \omega=\frac{K P^3}{r^{1/3} f N^{4/3}}, \tag{1} \]

where

\(K\)—coefficient taking into account the mechanical properties of the material;
\(r\)—radius of the asperity;
\(f\)—coefficient of friction;
\(N\)—compressive force.

To verify the formula obtained, a series of experiments was carried out on metallic contacts. From the graphs obtained in the experiments for the dependence of the displacement \(\Delta\) on the shearing force \(P\), for various compressive forces \(N\) and coefficients of friction \(f\), the amount of dissipated energy was calculated, which confirmed dependence (1).

Fig. 1. Graphs of the experimental dependence of the energy dissipation \(\omega\) for contact of steel specimens with surface finish \(\nabla 8\) on:
a) the shearing force \(P\), 1) \(f=0.29\); 2) \(f=0.4\); 3) \(f=0.51\); 4) \(f=0.56\); \(N=45\) kg;
b) the coefficient of friction, 1) \(P=fN\); 2) \(P=22\) kg; 3) \(P=20\) kg; 4) \(P=16\) kg; 5) \(P=12\) kg; 6) \(P=10\) kg; \(N=45\) kg

In Fig. 1a the magnitude of energy dissipation due to the shearing force is shown. In the case \(P=fN\), the dependence of energy dissipation on the coefficient of friction (curve 1, Fig. 1b) is a parabola. Indeed, for \(P=fN\), we have

\[ \omega=\frac{K f^2 N^2}{r^{1/3} N^{1/3}}. \tag{2} \]

The energy dissipation as a function of the shearing force and of the compressive force in this case is shown, respectively, in Figs. 2a and 2b. However, for \(P=\mathrm{const}<fN\), the dependence “magnitude of energy dissipation—coefficient of friction” (function 1) is represented by hyperbolas 2, 3, 4, 5, 6 (Fig. 1b).

Results confirming the validity of formula (1) were obtained in processing oscillograms of the damping of vibrations of a cantilevered

beam clamped between two rough plates. It is known that, if the influence of damping on the form of the rod’s vibration is neglected, then the damping properties determine the rate of decay of the vibrations. The energy dissipation as a result of preliminary

Fig. 2. Graphs of the experimental dependence of the energy dissipation \(\omega\) for the contact of steel specimens with surface finish \(\nabla 8\), with \(P=fN\), on: a) shear force, b) compressive force

displacements in the clamping may be represented as the sum of the dissipations of each mode of the natural vibrations. This makes it possible to analyze free vibrations for a system with one degree of freedom.

Fig. 3 shows the dependence of the logarithmic decrement of damping of the beam on the compressive force \(N\) for various amplitudes of the shearing forces.

It follows from the graph that

\[ \delta = \frac{A}{N^{4/3}}, \tag{3} \]

where \(\delta\) is the logarithmic decrement of damping; \(A\) is a certain constant quantity.

If the relation between the logarithmic decrement of damping and the energy dissipation is taken as proportional, then dependence (3) agrees well with function (1).

Using the energy method [3], the function determining the magnitude of the energy dissipated during one vibration cycle can be represented in the form

\[ \omega(y)=my^3, \]

Fig. 3. Logarithmic decrement of damping of the vibrations of a cantilever beam as a function of clamping force for various amplitudes of the shearing forces. 1) \(P=9\) kg; 2) \(P=12\) kg; 3) \(P=18\) kg; 4) \(P=24\) kg.

where \(m\) is a constant;
\(y\) is a coordinate:

and may be regarded as a first approximation to dependence (1).

A comparison of the theoretically calculated rate of decay of the oscillations with that determined from the oscillograms showed satisfactory agreement between the theoretical results and the experiments.

Thus, a function of the form (1), expressing the dependence of the amount of energy dissipated in one cycle, satisfactorily corresponds to the regularities of irreversible processes in joints of solid bodies.

REFERENCES

1. N. N. Babaev. On the internal resistances arising during vibrations of a machine body. Proceedings of the Scientific-Technical Conference on Damping of Vibrations. Publishing House of the Academy of Sciences of the Ukrainian SSR, 1960.

2. D. N. Reshetov, Z. M. Levin. Damping of vibrations in machine-tool parts. In: Study of vibrations of metal-cutting machine tools during metal cutting. Mashgiz, 1958.

3. Ya. G. Panovko. Internal friction during vibrations of elastic systems. Fizmatgiz, 1960.

4. N. G. Kalinin, Yu. A. Lebedev, V. I. Lebedeva, Ya. G. Panovko, G. I. Strakhov. Structural damping in fixed joints. Publishing House of the Academy of Sciences of the Latvian SSR, Riga, 1960.

5. I. R. Konyakhin, B. P. Mitrofanov. Procese disipative in cazul deplasăzif prealabile in limitele elasticitÿi. Studii și cercetari de Mecanica Aplicata, N 1, t. 15, 1964.

6. I. R. Konyakhin, B. P. Mitrofanov. Determination of losses due to mechanical hysteresis. Physics of Metals and Metallography, Nauka Publishing House, vol. 17, no. 6, 1964.

7. I. R. Konyakhin, B. P. Mitrofanov. On energy dissipation in an elastic discrete contact. Mechanics of Solids, Nauka Publishing House, No. 1, 1967.

8. N. B. Demkin. The actual contact area of solid surfaces. Publishing House of the Academy of Sciences of the USSR, 1962.

9. R. D. Mindlin. Compliance of elastic bodies in Contact. Jorn. Appl. Mech., vol. 16, N 3, 1949.