On the Structure and Coefficient of Friction of a Belt Drive

D. G. Stanko

When reviewing the literature data on belt drives, the first thing that catches the eye is the differing interpretation of the concept of the coefficient of friction \(\mu\) in a belt drive. In the main, three points of view on \(\mu\) may be noted:

1. The coefficient of friction \(\mu\) depends on the nature of the rubbing surfaces of the belt and pulleys, on the condition of their surfaces, and on the relative velocity of elastic slip.

2. The coefficient of friction \(\mu\) depends not only on the factors listed above, but also on the forces acting on the belt, as well as on the magnitude of the contact surface of the belt with the pulleys.

3. The third point of view is that the coefficient of friction \(\mu\) is considered in implicit form through the coefficient of transmission of the belt tension (traction coefficient)—\(\varphi\),

\[ \varphi = \frac{P}{S_{\alpha}} = \frac{S_{n} - S_{c}}{S_{n} + S_{c}}, \]

where \(S_{n}\) and \(S_{c}\) are the forces in the tight and slack spans of the belt drive, and \(P\) is the circumferential force.

The statements of a number of authors on the question of the coefficient of friction \(\mu\) are not without interest.

The well-known creator of the hydrodynamic theory of friction, N. P. Petrov [1], in polemicizing with N. M. Demyanov, writes on p. 2 of his work as follows:

“In particular, it was difficult for me to assimilate his view of the properties of the coefficient of friction, which asserts that the coefficients of friction on two pulleys are necessarily different and that even on one and the same pulley, but under different forces acting on the belt, the coefficient of friction must take on values that in some way depend on these forces.”

Further N. P. Petrov writes that:

“Delving into the circumstances accompanying the motion of pulleys connected by a flexible and extensible belt, it seemed possible to me to reconcile all known phenomena with that concept of the coefficient of friction which has long been accepted by everyone. It is therefore clear that the magnitude of the coefficient of friction depends on the nature of the rubbing bodies and the condition of their surfaces, and, if these circumstances remain unchanged, its magnitude is considered constant.”

Schulze-Pillot [2] indicates that modern theory assumes two factors of friction: pressure friction and surface friction between the pulleys and the belt. Moreover, despite the fact that experiments do not confirm the dependence of the friction force on the contact surface, he nevertheless considers it necessary to take into account the influence of the contact surface when formulating equations. True, in subsequent investigations he rejects the influence of the contact surface. He further notes that \(\mu\), in any case,

case is dependent on the relative velocity between the pulley and the belt. He then presents “data” on the coefficient of friction \(\mu\) as a function of the tension and, finally, gives average design coefficients \(\mu\).

V. A. Dobrovol’skii [3], in his works, points out the indeterminacy of the coefficient of friction \(\mu\).

Of course, the obscurity of the nature of the coefficient of friction forced the search for more reliable ways that would make it possible to judge the operation of a belt drive and to calculate it. Later a concept was introduced called the “coefficient of transmission of belt tension \(\varphi\),” and it was noted that with the aid of \(\varphi\) one can find the coefficient of friction \(\mu\), since \(S_{\mathrm{n}} = S_{\mathrm{c}} e^{\mu \alpha}\). The appearance of this quantity \(\varphi\) is quite natural, since in the experiments being carried out one can measure the sum of the tensions of the branches and the circumferential force.

V. N. Belyaev [4] developed somewhat the concept of the coefficient of transmission of belt tension—\(\varphi\), calling the latter the coefficient of traction. He introduced into the calculation the so-called “slip curves” in coordinates of relative elastic slip and \(\varphi\). At present, the method of calculating belt drives from slip curves is given in the encyclopedic reference book Mechanical Engineering, vol. 2 [13].

Such, approximately, are the views on the nature and concepts of the coefficient of friction. To judge the magnitude of \(\mu\) and the arrangement of experiments, we shall give a summary of results from various sources.

Table 1

Values of the coefficients of friction

Before turning to a critique of the views and an analysis of the coefficients of friction given in Table 1, let us examine the question: what is the coefficient of friction in a belt drive?

A belt drive, by its structure, is a simple mechanism. This deceptive simplicity has led to a not entirely clear conception of the physics of the phenomenon occurring in a belt drive.

In his time Coulomb established the dependence between the coefficient of friction \(f\), the friction force \(F\), and the normal pressure \(N\), expressed by the equation

\[ f=\frac{F}{N} \]

Let us now trace how this concept is transformed in a belt drive. Suppose that a belt under load \(N\) (Fig. 1) moves at a constant velocity \(v\). The applied force \(P\) overcomes the friction force \(F\). In this case we shall have the Coulomb coefficient of friction of the first kind in motion.

Fig. 1

If the belt is made to move over a stationary cylinder of radius \(R\) (Fig. 2), changing the direction of \(S_{\mathrm{n}}\) and \(S_{\mathrm{c}}\) in such a way that they always remain perpendicular to the corresponding radii in the new position, then again we have a connection with the Coulomb coefficient of friction of the first kind in motion. The dependence between the principal parameters in this case can be expressed by Euler’s law

\[ S_{\mathrm{n}}=S_{\mathrm{c}} e^{f\alpha}. \]

Fig. 2

By fixing the ends of the band and rotating the cylinder with constant angular velocity (Fig. 3), we obtain a brake with a flexible connection with the tension ratio

\[ S_{\mathrm{n}}=S_{\mathrm{c}} e^{f\alpha}, \]

where the Coulomb coefficient of friction \(f\)—in motion—still enters.

If the directions of the tensions \(S_{\mathrm{n}}\) and \(S_{\mathrm{c}}\) are left unchanged (Fig. 4), then in the dependence

\[ S_{\mathrm{n}}=S_{\mathrm{c}} e^{f'\alpha} \]

we have \(f'\) only as an analogue of the Coulomb coefficient of friction \(f\), since during the motion of a practical flexible connection over a stationary drum elastic slip will appear; the latter is absent in the case shown in Figs. 2 and 3.

Let us compare the operation of the brake (Fig. 3) with the operation of a flexible band on a stationary drum. With respect to one another, each of these transmissions has reversed motion, i.e., either the drum slips along the band, or the band slips along the drum.

It would seem that the reversed motion should give identity of the work of the transmissions, but this is not so. The work of the friction forces of a band sliding over a drum is “harmful,” whereas in the brake band it is “useful.”

All other conditions being equal, the specific pressure per unit area of the moving belt may be allowed to be greater than in a brake belt, since in the latter the heating factor assumes decisive importance.

As a result of the motion of the belt over the drum, centrifugal forces arise; in a brake belt they are absent. When the belt moves over the drum, elastic slip appears as a result of the nonuniform distribution of tensions in the flexible belt over the angle of wrap. With steady rotation of the brake pulley, elastic slip of the belt is absent; since the belt retains its position, there is equilibrium between the elastic forces and the friction forces.

Fig. 3

As we see even from a simple incomplete comparison, it is not one and the same thing whether the belt slips over a stationary drum, or the pulley slips over a stationary belt, and therefore it is difficult to expect that the coefficient of friction in both cases will be the same.

Although in the examples (Figs. 1, 2, 3, and 4) a flexible connection is considered, these examples by no means reflect the operation of a belt drive. A certain analogy to the operation of a belt drive is given by the following representation (Fig. 5). Suppose that we have a platform on wheels; let the motion of the platform be effected by applying a force \(P\) to the belt. Obviously, motion of the platform is possible only when the normal pressure \(N\) creates between the belt and the platform a friction force \(F > P\).

Fig. 4

Fig. 5

Determination of the coefficient of friction in this example is possible only under limiting operating conditions, i.e., when \(F=P\); this corresponds to the instant of sliding of the belt over the platform, while in Coulomb’s conception it corresponds to static friction. The normal component \(N\) may also be greater than is necessary to create the friction force \(F\); the motion of the platform will nevertheless take place, but then we cannot write the equality that

\[ f=\frac{P}{N}, \]

since \(P\) is not equal to \(F\), but is smaller.

Let us imagine that we have some elementary sector (Fig. 6) with a certain braking moment \(M_T\). Let us try to turn this sector by means of a belt. This rotation will occur only when \(P \leq F\). The normal component \(N\) may be different, depending on the tension \(S\), but the additional force \(\Delta S\), equal to \(P\), will be constant, since \(M_T=\mathrm{const}\).

Fig. 6

Fig. 7

Let us lay off, for different tensions (Fig. 7), along the \(y\)-axis \(N\), and along the \(x\)-axis \(F=Nf\). Then, for some \(N\), the belt slides over the sector; adding \(\Delta N\), the sector begins to move. The coefficient of friction \(f'\) for the belt in this case is equal to

\[ f'=\frac{P}{N+\Delta N}, \]

\(f'\) is analogous to the Coulomb coefficient of static friction.

Having made these preliminary remarks, let us note the principal features of a belt drive. A belt drive has a closed flexible member placed over the driving and driven pulleys. After an initial tension has been created, at various stages of operation of the belt drive there occurs a redistribution of the tensions of the free branches and of the friction forces over the angles of wrap of the pulleys.

For a belt drive, as a mechanism, kinematically and structurally\(^{1}\) the presence of elastic slip is inherent. Owing to the latter, redistribution of the tensions in the branches and motion of the belt as a closed contour are possible.

An inextensible flexible link, as a closed contour, cannot transmit work from one pulley to another, since it lacks the fac—

\(^{1}\) Structurally, a belt drive may be represented as a closed kinematic chain with an infinite number of connected links. Each link moves and slides over the pulleys.

factor responsible for the redistribution of the tension of the branches—elastic slip. (This observation is valid, at least, for dry and semi-dry friction.) In every section of an inextensible flexible connection the tension will remain the constant value that was created by the initial tension \(S_0\).

An ideal elastic belt is one in which the deformations obey Hooke’s law, which fits tightly against the pulley, and which works only in tension.

A belt whose deformations do not obey Hooke’s law introduces nothing new into the essence of the operation. What is important here is that the new law be stable. In practice, however, the belt will not operate normally over all portions; where the relative elongation becomes vanishingly small, i.e. the belt is, as it were, transformed into an inextensible flexible member, it will be impossible to transmit additional work from one pulley to the other by means of frictional forces, since the belt will slip.

This proposition is confirmed to a sufficient degree by an appropriate treatment of Belyaev’s experimental “slip curves” [4].

In the operation of a belt drive there are four stages. The first (Fig. 8—I) is when, with the belt stationary, an initial tension \(S_0\) is created in the free branches; the force contour\(^{1)}\) coincides with the belt contour; when it is tensioned, elastic slip occurs symmetrically over the halves of the angle of wrap; the distribution of tension over the angles of wrap will be, according to Euler’s law,

\[ S_0 = S_k e^{\frac{\mu \alpha}{2}} . \]

The second (Fig. 8—II) is the initial tension of idle running \(S'_0\); the belt is moving; the force contour is constant in character; the transmitted circumferential force is theoretically equal to zero, and elastic slip is absent in this case (in practice, owing to losses due to friction in the bearings and losses due to the stiffness of the belt, some small circumferential force will be transmitted from the driving pulley to the driven pulley).

With the redistribution of tension we have \(2S'_0 \ne 2S_0\).

The third is the working stage (Fig. 8—III). A characteristic feature of the third stage is the presence of a “working angle,” \(\alpha_0\), and an “angle of reserve,” \(\alpha'\), on the driving and driven pulleys. On the “working angles” we have continuous elastic slip.

The author was compelled to abandon the corresponding terms “arc of slip” and “arc of rest,” introduced by Grashof [9], because these terms reflect mainly only the kinematic aspect of the operation of a belt drive.

The “angle of reserve,” in the physical sense, is connected with the potential reserve of frictional forces possessed by the belt drive.

As observations show, the “angle of reserve” on the driving pulley is located on the side of the driving branch.

The sum of the “working angle” and the “angle of reserve” is equal to the full angle of wrap of the pulley by the belt.

For a transmission ratio \(i\) not equal to unity, the “working angles” on the driving and driven pulleys are equal to one another, while the angles of reserve are the complements to the full angles of wrap.

\(^{1)}\) By force contour is meant the diagram of forces laid off normal to the belt contour. The tension in each element of a stationary belt can be found from the force contour by direct measurement of the force under the belt element. In this sense the force contour coincides with the belt contour.

Fig. 8

Panels: I, II, III, IV

Visible inscriptions in the figure include:
\(S_0\), \(S'_0\), \(S_{\mathrm{н}}\), \(S'_{\mathrm{н}}\), \(S_{\mathrm{с}}\), \(S'_{\mathrm{с}}\), \(S_x\), \(S'_x\), \(2S_0\), \(2S'_0\), \(S_{\mathrm{н}}+S_{\mathrm{с}}\), \(S'_{\mathrm{н}}+S'_{\mathrm{с}}\), \(2S_0 \ne 2S'_0\), \(S_{\mathrm{н}}+S_{\mathrm{с}} \ne 2S_0\), \(S'_{\mathrm{н}}+S'_{\mathrm{с}} \ne 2S'_0\), \(\alpha\), \(\alpha_{\mathrm{с}}\), \(\alpha_{\mathrm{н}}\), \(l+\Delta l_0\).

Grasshof [9] was the first to express the idea of the existence of “sliding arcs” and “rest arcs.” In all probability, independently of Grasshof, Petrov [1; 8] developed this concept somewhat. The presence of “rest arcs” and “sliding arcs” was established by the experiments of Fieber [6], Zhukovsky [10], and Farberov [11].

Observations in a transmission with a rubber belt, carried out by the author [12], likewise confirm the presence of “rest arcs” and “sliding arcs.”

For the third stage, Euler’s law is written as follows:

\[ S_{\mathrm{n}} = S_{\mathrm{c}} e^{\mu \alpha_{o}}. \]

The redistribution of tensions and the change in circumferential force will be associated with the change in the angle \(\alpha_o\).

The fourth stage (Fig. 8—IV) corresponds to the “limiting equilibrium conditions.” These conditions are practically close to belt slippage.

In the fourth stage we have the maximum possible transmitted circumferential force and elastic sliding over the full angle of wrap of the (smaller) pulley. For the fourth stage Euler’s law is valid in the present-day interpretation, i.e.,

\[ S'_{\mathrm{n}} = S'_{\mathrm{c}} e^{\mu \alpha}. \]

Here \(\alpha\) is the full angle of wrap of the (smaller) pulley.

The operating range of a belt drive lies within the limits of the second and fourth stages. In this case the transmitted circumferential force varies from zero to a maximum owing to the change of the working angle from \(\alpha_o = \alpha\) to \(\alpha_o = 0\).

The physical meaning of the motion of the belt over the pulleys is that the belt, in the main, “runs” over the pulleys.

The actual coefficient of friction in a belt drive is, by analogy, as it were a Coulomb coefficient of friction of the first kind—“rest.” But friction in a belt drive, by its nature, has an essential difference, since in transmitting circumferential force it is associated with continuous elastic sliding of the belt over the pulley.

If we were to consider the operation of a belt drive in parts, then (98–99)% would have to be attributed to “rest” friction, and (2–1)% to sliding friction; but we cannot do this, because these two factors as a whole precisely condition the operation of the belt drive and give that new element which is physically inherent in friction in a belt drive. Determination of the actual coefficient of friction from external factors, such as the torque on the driving and driven pulley and the sum of the tensions of the free strands that cause reactions at the shaft supports, is possible only under the “limiting conditions” of operation of the belt drive.

Let us note a peculiar contradiction in the friction of a belt drive. In order to transmit a greater circumferential force, rough surfaces of the pulleys and belt would be required. But for the normal kinematics of the transmission, for the occurrence of elastic sliding, on the contrary, smooth surfaces of the pulleys and belt are needed.

It would seem that there should be no need to explain friction in a belt drive or to analyze various examples, in view of the generally known character of a number of the factors adduced. But if we consider them in the light of what is taken as the coefficient of friction in a belt drive, then at the very least rather strange notions are revealed.

For example, Bach [7] (see Table 1 of the values of \(\mu\)) took as the coefficient of friction in a belt drive a quantity obtained when the belt moves over a stationary drum; Frederich [6] considered as the coefficient of friction that which was obtained in the operation of a brake; Skutch [3] approached the question still more simply: he identified the friction of a belt over a steel ruler with the friction in a belt drive and gave his coefficients as a function of speed.

Can it even be said that Bach, Frederich, and Skutch were concerned with determining the coefficient of friction in a belt drive? The answer to this question can only be negative. They studied the friction of a belt on a stationary drum, friction in a brake, the friction of a belt on a steel ruler, and for these cases gave coefficients of friction. But the coefficients of friction obtained have nothing in common with a belt drive, since the nature of the phenomenon is different in those cases and in belt drives.

Kamerer [5], in processing his experimental data, within the limits of the second and fourth stages of operation of a belt drive, gave the correct value of the coefficient of friction $\mu$ only for the moment at which slipping begins. All the other intermediate values of $\mu$ are incorrect, since in his apparatus he could not measure the value of the “reserve angle.”

As already indicated, in experiments with a belt drive the sum of the tensions in the branches and the circumferential force can be measured. If the “reserve angle” is now ignored, then the value of the coefficients of friction will be the same as the preliminary tensions. This ignoring led a number of authors (Schulze-Pillot [2], and others) to express the idea that the coefficient of friction in a belt drive depends on the tension of the branches, on the contact surface, while citing supposedly “experimental” data. But this assertion loses its meaning if the presence of the “reserve angle” is taken into account. The presence of the “reserve angle” is confirmed by the nature of the phenomenon occurring in a belt drive. In all handbooks connected, to one degree or another, with the calculation of belt drives, Euler’s law is used. In calculating a belt drive according to Euler’s law, the quantity $\mu$ is necessary.

The variations of the “experimental” $\mu$ are so large that recently Euler’s law in its pure form has been abandoned and a “traction coefficient” $\varphi$ has been introduced into the calculation.

The use of the “traction coefficient” still does not make it possible to judge the operation of a belt drive as a whole. In essence, the use of the “traction coefficient” leads to the rejection of the interaction among all the parameters of a belt drive. In theoretical terms, the “traction coefficient” is nothing more and nothing less than an intermediate statement of experimental data, and only the subsequent processing of these data can produce one effect or another.

Belyaev [4] connected the “traction coefficient” with the relative elastic slip and the efficiency of the belt drive. In this respect he expanded the experimental observations.

The physics of the phenomenon occurring in belt operation was not analyzed by Belyaev. In particular, the reasons were not examined as to why elastic slip increases with an increase in the “traction coefficient” $\varphi$, and why slipping, at different initial tensions, begins at different $\varphi$. The mechanical characteristic of the belts, the condition of the surface of the pulleys and of the belt, etc., were entirely disregarded. All this happened not because these causes cannot be studied, but because the structure of the belt drive as a mechanism was not taken into account, the interaction among all the parameters of the belt drive was not considered, and the stages of operation of the drive were not taken into account.

All the diversity of the operation of the drive proved to be hidden in the “traction coefficient”; thereby the theoretical premises were rejected, and in the final analysis the consideration of the question was reduced to bare empiricism. Although Belyaev’s work contains extensive experimental data, the theoretical treatment of the observations leaves much to be desired.

The correctness of the coefficient of friction can be judged only by the extent to which the condition of equilibrium between the normal …

by pressure and by the force of friction, to what extent the condition of equilibrium between the circumferential force and the friction force has been correctly identified.

The considerable fluctuations in the derived values of the coefficient of friction are explained mainly by incorrect assumptions in determining \(\mu\). If we consider the value of the actual coefficient of friction, then, according to Kammerer’s data [5], for leather belts \(\mu = 0.6—0.8\); according to Belyaev’s data [4], at a relative elastic slip of \(2\%\) (which is close to the onset of belt slipping), we have

\[ \begin{aligned} &\text{for rubberized belt No. 11} && \mu = 0.54—0.60,\\ &\text{for rubberized belt No. 8} && \mu = 0.62—0.86,\\ &\text{for cotton belt No. 5} && \mu = 0.54—0.68. \end{aligned} \]

Since Belyaev did not set himself the goal in his experiments of determining the coefficient of friction \(\mu\), the values of \(\mu\) obtained by an indirect method may be considered only approximate. Nevertheless, both Kammerer’s and Belyaev’s data show that the fluctuations of \(\mu\) are not so very large. Of course, these values require clarification with respect to the influence of the condition of the pulley and belt surfaces, relative elastic slip, the influence of the properties of the belt as a flexible member, and so forth. As far as the author is aware, Eytelwein’s law is nowhere rejected in flexible-link transmission, since it reflects the operation of the transmission with sufficient accuracy.

Therefore, all the prerequisites exist for creating a theory of flexible-link transmission on the basis of Eytelwein’s law. But for this, taking into account the structure of the transmission, it is necessary to use Eytelwein’s law correctly and to determine the coefficient of friction correctly.

The foregoing arguments pursued the following aims:

1. To identify the errors that are made in determining the coefficient of friction in belt transmission.
2. To identify what the actual coefficient of friction in flexible-link transmission represents.
3. To note the structural features of belt transmission.
4. To provide the basic prerequisites for creating a theory of belt transmission on the basis of Eytelwein’s law.

REFERENCES

1. N. P. Petrov. The Influence of Friction in the Transmission of Work by an Elastic Belt. Petersburg, 1893.
2. Schulz-Pilot. A New Theory of Driving Belts. Leningrad, Gizlegprom, 1932.
3. V. A. Dobrovolsky. Belt Transmission. ONTI—NKTP, Kharkov, Kiev, 1934.
4. V. N. Belyaev. On the draft of new standards for the working capacity of driving machine belts of Union production, Mashinostroitel No. 1, 1936.
5. Kammerer, Versuche mit Riemen und Seiltrieben, Z.VDI, Bd. 51. S. 1085—91, 1907.
6. F. Reicher. Machine Parts, vol. 2, Gosmashizdat, 1933.
7. B. Ya. Shteinman, vol. 1, Gosmashizdat, 1933.
8. N. P. Petrov. The Influence of Friction in the Transmission of Work by an Elastic Belt. Confirmation of the theory presented by me in 1893. Petersburg, 1894.
9. Grashof. Teoretische Maschinenlehre, Bd. II, 1883.
10. A. D. Nevsky. The greatest power of belt transmission. Vestnik inzhenerov i tekhnikov, No. 8, 1935.
11. Farberov. Investigation of the drive of a belt conveyor. Pod’emno-transportnoe delo, No. 5, 1940.
12. D. G. Stanko. Theory and Experiment in Belt Transmission. 1940.
13. Mechanical Engineering. Encyclopedic Handbook, vol. 2, Moscow, GNTI, 1948.