On the Theory of Belt Transmission

D. G. Stanko

The basic premises of the proposed theory are the propositions set forth in the work by D. G. Stanko, “On the Structure and Coefficient of Friction of Belt Transmission.”

Among these propositions the following may be noted:

1. Belt transmission is considered as a closed contour with a distribution of tensions according to Euler’s law.

2. Structurally, belt transmission has on the pulleys a “reserve angle” and a “working angle.”

3. The coefficient of friction \(\mu\) is a quantity determined under limiting equilibrium conditions close to slipping of the closed belt contour over the pulley.

4. Analysis of the interaction of forces in the closed belt contour is carried out for four stages of operation of the transmission.

In the first stage, with the belt contour stationary, a preliminary tension is created.

The second stage corresponds to idling of the belt transmission; in this case the theoretically transmitted circumferential force is equal to zero and elastic slip is absent.

In the third, working stage, the transmitted circumferential force varies within the limits from zero to some maximum value, and elastic slip occurs only on the working angles. In this stage normal operation of the transmission takes place.

The fourth stage corresponds to the limiting operating conditions, when the belt begins to slip over the pulley, elastic slip occurs over the full angle of wrap of the (small) pulley, and the circumferential force reaches its maximum value.

Having made these preliminary remarks, we proceed to the exposition of the theory.

Suppose that we have a belt of length \(l\) (Fig. 1) and with a constant load \(S\) acting along its length. Then, for a belt with deformation obeying Hooke’s law, the increase in length \(l\) will be equal to

\[ \Delta l_0' = lSa, \tag{1} \]

where \(a\) is the increase of unit length of the belt referred to unit force.

Let us determine the increase in the length of the belt when the load is distributed according to Euler’s law.

The increase of any element on the angle of wrap, under the action of the force \(S'\) (Fig. 2), may be expressed as

\[ d(\Delta l_0'') = aS'R\,d\alpha . \]

Substituting the value of \(S'\), equal to

\[ S' = S_{\mu}e^{-\mu\alpha}, \]

we have

\[ d(\Delta l_0'') = aS_{\mu}e^{-\mu\alpha}R\,d\alpha . \]

Hence the total length of the increment under the action of the load distributed according to Euler’s law will be found after evaluating the integral

\[ \Delta l''_0=\int_0^\beta aS_hRe^{-\mu\alpha}\,d\alpha. \]

Evaluating this integral, we obtain

\[ \Delta l''_0=\frac{RaS_h}{\mu}\left(1-e^{-\mu\beta}\right) \tag{2} \]

Equations (1) and (2) are the initial ones for the case when the deformations of the belt obey Hooke’s law, i.e.

\[ a=\mathrm{const}. \]

If \(a\ne\mathrm{const}\), a new function \(a=f(S)\) enters equations (1) and (2).

Fig. 1

Fig. 2

If the function \(a=f(S)\) is not given in analytical form, or is given in cumbersome form, then the value of \(a\) in equation (1) may be taken from the stress diagram for the corresponding force \(S\).

Equation (2) is simpler to solve graphically. For this purpose, the length of the belt loaded according to Euler’s law is divided into equal segments proportional to the angles (Fig. 3), and for each segment the magnitude of the deformations is found. Then the total length of deformation will be equal to

\[ \Delta l''_0=\Delta l_\beta S_h\left(a_1e^{-\mu\beta_1}+a_2e^{-\mu\beta_2}+\ldots+a_ne^{-\mu\beta_n}\right), \tag{3} \]

or

\[ \Delta l''_0=\Delta l_\beta(\varepsilon_1+\varepsilon_2+\ldots+\varepsilon_n), \tag{4} \]

where \(\varepsilon_1,\varepsilon_2\ldots\varepsilon_n\) are the relative elongations under the corresponding forces \(S_1,S_2,\ldots,S_n\) (Fig. 4), varying according to Euler’s law.

Theoretically, a characteristic feature of the friction of a flexible link is that the distribution of tension over the angle of wrap follows Euler’s law and does not depend on what elastic properties the flexible link possesses.

In the operation of a belt transmission, the elastic properties of the belt, independently of the law defining them, are an important factor that makes it possible to realize the kinematics of the transmission.

When there is a more complex functional dependence of the coefficient of friction \(\mu\), the problem of determining the deformations can be solved in a manner analogous to the case \(a \ne \mathrm{const}\).

Let us introduce the new concept of the “free contour of the belt,” denoting the full length of the belt when it has no deformations of any kind.

Fig. 3

Fig. 4

For a transmission ratio \(i = 1\), the length of the free contour is equal to (Fig. 5)

\[ L = 2(l_0 + \pi R), \tag{5} \]

where \(l_0\) is the distance between the centers of the pulleys of the “free contour of the belt.”

In the first stage of operation of the drive, under the action of the initial tension \(2S_0\), the center \(O\) will move to a new position, \(O_2\). This displacement \(\Delta l_0\) is associated with deformation in the free branches and with elastic slip over the angles of wrap. Using equations (1) and (2), we find that

\[ \Delta l_0 = l_0 a S_0 + \frac{2aRS_0}{\mu} \left(1 - e^{-\mu \frac{\alpha}{2}}\right). \tag{6} \]

In the second term on the right-hand side of equation (6), the angle \(\frac{\alpha}{2}\) is used, because the deformation occurs symmetrically over half the angle of wrap according to Euler’s law, with

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

The contour length \(L_0\), after the initial tension \(2S_0\) has been applied, will be

\[ L_0 = 2l(1 + aS_0) + 4\left[ \frac{aRS_0}{\mu} \left(1 - e^{-\mu \frac{\alpha}{2}}\right) \right] + 2\alpha R . \tag{7} \]

As a more general case, we shall first analyze the third stage of operation of the belt drive. The expression for the displacement of the center (Fig. 6) in the third working stage may be represented as

\[ \Delta l_p = \frac{\Delta l_u + \Delta l_c}{2} + \frac{\Delta l_{u\alpha'} + \Delta l_{c\alpha'}}{2} + \frac{\Delta l_{ou} + \Delta l_{oc}}{2} \tag{7'} \]

where \(\Delta l_n\) and \(\Delta l_{n\alpha'}\) are the deformations on the working angle and on the reserve angle of the driving pulley;

\(\Delta l_c\) and \(\Delta l_{c\alpha'}\) are the deformations on the working angle and on the reserve angle of the driven pulley;

\(\Delta l_{on}\) and \(\Delta l_{oc}\) are the deformations of the incoming and outgoing branches.

Fig. 5

Fig. 6

Using equations (1) and (2), we obtain

\[ \left. \begin{aligned} \Delta l_n &= \Delta l_c = \frac{RaS_n}{\mu}\left(1-e^{-\mu\alpha_0}\right),\\ \Delta l_{n\alpha'} &= Ra\alpha' S_n,\\ \Delta l_{c\alpha'} &= Ra\alpha' S_c,\\ \Delta l_{on} &= l_0 a S_n,\\ \Delta l_{oc} &= l_0 a S_c. \end{aligned} \right\} \tag{7''} \]

Let us express \(S_n\) and \(S_c\) in terms of \(S_0\). By Euler’s law we have

\[ S_n = S_c e^{\mu\alpha_0}, \tag{8} \]

\[ S_n - S_c = P, \tag{9} \]

where \(P\) is the circumferential force transmitted from the driving pulley to the driven pulley. Since one of the centers is movable, the condition is preserved

\[ S_n + S_c = 2S_0 . \tag{10} \]

Solving the system of equations (8), (9), and (10), we find

\[ S_{\mathrm{н}}=2S_0\left(\frac{1}{1+e^{-\mu\alpha_0}}\right) \tag{11} \]

\[ S_{\mathrm{с}}=2S_0\left(\frac{1}{1+e^{\mu\alpha_0}}\right) \tag{12} \]

\[ P=2S_0\left(\frac{e^{\mu\alpha_0}-1}{e^{\mu\alpha_0}+1}\right). \tag{13} \]

Replacing \(S_{\mathrm{н}}\) and \(S_{\mathrm{с}}\) by \(S_0\) in equations (7′), and substituting the values of the deformations into equations (7′), after reduction and transformation we obtain

\[ \Delta l_p = a l_0 S_0+a\alpha' R S_0+ \frac{2RS_0a}{\mu} \left( \frac{e^{\mu\alpha_0}-1}{e^{\mu\alpha_0}+1} \right) \tag{14} \]

The length of the belt contour in the third working stage may be expressed as the sum of the quantities from the free contour and the deformations of the individual sections, i.e.,

\[ L_p=L+\Delta l_{\mathrm{н}}+\Delta l_{\mathrm{с}}+\Delta l_{\mathrm{н}\alpha'}+\Delta l_{\mathrm{с}\alpha'}+\Delta l_{0\mathrm{н}}+\Delta l_{0\mathrm{с}}, \]

or, after substitution and transformation, we have

\[ L_p = 2l_0(1+aS_0) + \frac{4aRS_0}{\mu} \left( \frac{e^{\mu\alpha_0}-1}{e^{\mu\alpha_0}+1} \right) + 2a\alpha'RS_0+2\alpha R \tag{15} \]

Having solved the general case, it is not difficult to find the contour length and the displacement of the center in the remaining stages of operation of the transmission.

In the second stage, corresponding to idling (Fig. 7), we have the distribution of forces along the contour \(S_0=\mathrm{const}\), the transmitted circumferential force \(P=0\), the working angle \(\alpha_0=0\), and the reserve angle \(\alpha'=a\).

Substituting the values of \(\alpha_0\) and \(\alpha'\) into equations (14) and (15), we find

\[ \Delta l_x=aS_0(l+\alpha R) \tag{16} \]

\[ L_x=2l_0(1+aS_0)+2\alpha R(aS_0+1), \tag{17} \]

where \(\Delta l_x\) is the displacement of the center during idling,

\(L_x\) is the belt contour length during idling.

Operation in the fourth stage takes place under limiting equilibrium conditions, close to slipping of the belt over the pulley.

In the fourth stage (Fig. 8), the forces along the contour are distributed according to Euler’s law; the circumferential force, equal to

\[ S_{\mathrm{н}}'-S_{\mathrm{с}}'=P_{\max}, \]

reaches its maximum value (in the left-hand sides of equations (11), (12), and (13), for the fourth stage one must respectively put \(S_{\mathrm{н}}'\), \(S_{\mathrm{с}}'\), and \(P_{\max}\)) at the reserve angle \(\alpha'=0\) and working angle \(\alpha_0=\alpha\).

Substituting the values of \(\alpha_0\) and \(\alpha'\) into equations (14) and (15), we obtain

\[ \Delta l_{\mathrm{п}} = a l_0 S_0+ \frac{2aRS_0}{\mu} \left( \frac{e^{\mu\alpha}-1}{e^{\mu\alpha}+1} \right), \tag{18} \]

\[ L_{\mathrm{п}} = 2l_0(1+aS_0) + \frac{4aRS_0}{\mu} \left( \frac{e^{\mu\alpha}-1}{e^{\mu\alpha}+1} \right), \tag{19} \]

where \(\Delta l_{\mathrm{п}}\) is the displacement of the center in the fourth stage,

\(L_{\mathrm{п}}\) is the belt contour length in the fourth stage.

The analysis carried out makes it possible to find the position of the moving center and the dimensions of the contour at all stages of operation of a belt transmission, when the transmission ratio is \(i = 1\). With a transmission ratio not equal to unity, all the conclusions become somewhat more complicated, in view of the fact that the displacement of the pulley center is connected with the projection of the belt deformation. The reserve angles on the driving and driven pulleys are different in magnitude.

Fig. 7

Fig. 8

Let us make the following assumptions: we shall consider that the wrap angles \(\alpha_1\) and \(\alpha\) do not change in magnitude during the displacement of the center \(O\), and that the angle \(\gamma\) is also constant for the different positions of the branches (Fig. 9). These assumptions will not cause appreciable errors, since the change in the angles is insignificant.

Taking as the transmission ratio \(i\) the ratio of the larger pulley diameter to the smaller, we express the length of the free contour \(L_i\) at a transmission ratio \(i \ne 1\); we shall have

\[ L_i = R(i\alpha_1 + \alpha) + 2l\cos\gamma . \]

Applying to the pulley center a certain geometric component

\[ 2\overline{S}_i = \overline{S}_0 + \overline{S}_0, \]

we obtain the displacement of the center \(O\) when the initial tension in the free branches is equal to \(S_0\) (first stage). The displacement of the center in the first stage will be equal to

\[ \Delta l_{0i} = (\Delta l_0 + \Delta l_{x_1} + \Delta l_{\alpha}) \frac{1}{\cos\gamma}, \tag{20'} \]

Moreover,

\[ \left. \begin{aligned} \Delta l_0 &= S_0 a l_0 \cos \gamma,\\ \Delta l_{\alpha_1} &= \frac{i a R S_0}{\mu}\left(1-e^{-\mu \frac{\alpha_1}{2}}\right),\\ \Delta l_{\alpha} &= \frac{a R S_0}{\mu}\left(1-e^{-\mu \frac{\alpha}{2}}\right),\\ S_0 &= \frac{S_i}{\cos \gamma}, \end{aligned} \right\} \tag{20''} \]

where \(\Delta l_0\) is the deformation of the free branch,
\(\Delta l_{\alpha_1}\) and \(\Delta l_{\alpha}\) are the deformations on the halves of the wrap angles of the driving and driven pulleys.

Fig. 9

Fig. 10

Substituting the values of the deformations into equation \((20')\) and replacing \(S_0\) by \(S_i\), we find

\[ \Delta l_{0i} = S_i a l + \frac{i a R S_i}{\cos^2 \gamma \, \mu} \left(1-e^{-\mu \frac{\alpha_1}{2}}\right) + \frac{a R S_i}{\cos^2 \gamma \, \mu} \left(1-e^{-\mu \frac{\alpha}{2}}\right). \tag{20} \]

The length of the belt contour in the first stage is the sum of \(L_i\) and the deformations of the individual sections, i.e.

\[ L_{oi}=L_i+2\Delta l_0+2\Delta l_{\alpha_4}+2\Delta l_{\alpha}, \]

or, after substituting the known quantities and transforming, we obtain

\[ \begin{aligned} L_{oi}={}&\cos\gamma\,2l_0\left(1+\frac{aS_i}{\cos\gamma}\right) +R(i\alpha_1+\alpha) +\frac{2aRS_i}{\mu\cos\gamma}\left(1-e^{-\frac{\mu\alpha_1}{2}}\right) \\ &+\frac{2aRS_i}{\mu\cos\gamma}\left(1-e^{-\frac{\mu\alpha}{2}}\right). \end{aligned} \tag{21} \]

As before, just as for the transmission ratio \(i=1\), we first solve the general case—the third working stage.

Let us denote the magnitudes of the deformations:

\[ \Delta l_n,\Delta l_c \]

—on the free branches,

\[ \Delta l'_{\alpha_0},\Delta l''_{\alpha_0} \]

—on the working angles,

\[ \Delta l_{\alpha'},\Delta l_{\alpha''} \]

—on the reserve angles.

With the adopted notation, the displacement of the center \(O\) in the third working stage (Fig. 10) is equal to

\[ \Delta l_{ip}=\frac{1}{\cos\gamma}\left( \frac{\Delta l_n+\Delta l_c}{2} +\frac{\Delta l'_{\alpha_0}+\Delta l''_{\alpha_0}}{2} +\frac{\Delta l_{\alpha'}+\Delta l_{\alpha''}}{2} \right), \]

where

\[ \left. \begin{aligned} \Delta l_n &= S_n a l_0\cos\gamma,\\ \Delta l_c &= S_c a l_0\cos\gamma,\\ \Delta l'_{\alpha} &= \frac{iaRS_n}{\mu}\left(1-e^{-\mu\alpha_0}\right),\\ \Delta l''_{\alpha_0} &= \frac{aRS_n}{\mu}\left(1-e^{-\mu\alpha_0}\right),\\ \Delta l_{\alpha'} &= aRa'S_c,\\ \Delta l_{\alpha''} &= Ria''S_n. \end{aligned} \right\} \tag{21'} \]

Let us express \(S_n\), \(S_c\), and \(P\), for the transmission ratio \(i\ne1\), in terms of \(S_i\). Then we have

\[ S_n=\frac{2S_i}{\cos\gamma}\left(\frac{1}{1+e^{-\mu\alpha_0}}\right), \tag{22} \]

\[ S_c=\frac{2S_i}{\cos\gamma}\left(\frac{1}{1+e^{\mu\alpha_0}}\right), \tag{23} \]

\[ P=\frac{2S_i}{\cos\gamma}\left(\frac{e^{\mu\alpha_0}-1}{e^{\mu\alpha_0}+1}\right). \tag{24} \]

Substituting the values of the deformations from equation (21) into equation (21′), replacing \(S_c\) by \(S_n\) and expressing \(S_n\) through \(S_i\), after transformation and reduction we obtain

\[ \Delta l_{ip} = al_0S_i +(i+1)\frac{RaS_i}{\cos^2\gamma\,\mu} \left(\frac{e^{\mu\alpha_0}-1}{e^{\mu\alpha_0}+2}\right) + \frac{RaS_i\left(ia''+a'e^{-\mu\alpha_0}\right)} {\cos^2\gamma\left(1+e^{-\mu\alpha_0}\right)}. \tag{25} \]

The length of the belt contour in the third stage will be found as the sum of \(L_i\) and the deformations of the individual sections. It will be equal to

\[ L_{ip}=L_i+\Delta l_n+\Delta l_c+\Delta l'_{\alpha_0}+\Delta l''_{\alpha_0}+\Delta l_{\alpha'}+\Delta l_{\alpha''}. \]

Or, after substituting the known quantities and transforming, we find

\[ L_{ip}=\cos\gamma\,2l_0\left(1+\frac{aS_i}{\cos\gamma}\right) +R(i\alpha_1+\alpha) +(i+1)\frac{2RaS_i}{\mu\cos\gamma} \left(\frac{e^{\mu\alpha_0}-1}{e^{\mu\alpha_0}+1}\right) + \frac{2S_iRa\left(i\alpha''+\alpha'e^{-\mu\alpha_0}\right)} {\cos\gamma\left(1+e^{-\mu\alpha_0}\right)}. \tag{26} \]

In the second stage—during idle running (Fig. 11), the forces along the contour are constant, i.e. \(S_0=\mathrm{const}\), the working angle \(\alpha_0=0\), the reserve angles \(\alpha''=\alpha_1\), \(\alpha'=\alpha\), and \(P=0\).

Fig. 11

Fig. 12

Replacing the values of \(\alpha_0\), \(\alpha'\), and \(\alpha''\) in equations (25) and (26), we obtain the displacement of the center—\(l_{ix}\)—and the length of the contour \(L_{ix}\) in the second stage as equal to

\[ l_{ix}=al_0S_i+\frac{RaS_i(i\alpha_1+\alpha)}{2\cos^2\gamma} \tag{27} \]

\[ L_{ix}=\cos\gamma\,2l_0\left(1+\frac{S_i a}{\cos\gamma}\right) +R(i\alpha_1+\alpha) +\frac{RS_i a}{\cos\gamma}(i\alpha_1+\alpha). \tag{28} \]

In the fourth stage—under the limiting equilibrium condition (Fig. 12), the working angle \(\alpha_0=\alpha\), the reserve angle \(\alpha'=0\) and \(\alpha''=\alpha_1-\alpha\), and the circumferential force

\[ P_{\max}=S'_{н}-S'_{с} \]

reaches its maximum value. Using equations (25) and (26), after substitution and transformation, we find

\[ \Delta l_{in}=al_0S_i+(i+1)\frac{aRS_i}{\cos^2\gamma\,\mu} \left(\frac{e^{\mu\alpha}-1}{e^{\mu\alpha}+1}\right) +\frac{RaS_i(\alpha_1-\alpha)i}{\cos^2\gamma(1+e^{-\mu\alpha})} \tag{29} \]

\[ L_{in}=\cos\gamma\,2l_0\left(1+\frac{S_i a}{\cos\gamma}\right) +R(i\alpha_1+\alpha) +(i+1)\frac{2RaS_i}{\mu\cos\gamma} \left(\frac{e^{\mu\alpha}-1}{e^{\mu\alpha}+1}\right) + \]

\[ +\frac{2S_iRa(\alpha_1-\alpha)i}{\cos\gamma(1+e^{-\mu\alpha})}, \tag{30} \]

where \(\Delta l_{in}\) is the displacement of the center,
\(L_{in}\) is the length of the contour—in the fourth stage for \(i\ne 1\).

In carrying out the analysis of the operation of the belt transmission, for \(i\ne 1\), the larger pulley was taken as the driving pulley.

If the smaller pulley is taken as the driving pulley, then the magnitude of the center displacement and the contour length, in the first and second stages, can be determined from equations (20), (21), (25), and (26).

For the third and fourth stages, retaining in the drawings of Figs. 10 and 12 the same designations of the angles \(\alpha,\alpha_1,\alpha_0,\alpha'\), and \(\alpha''\), the approaching and receding branches will interchange places, and consequently equations (25), (26), (29), and (30) will change their form somewhat. This change will affect only the last terms, namely,

\[ \Delta l'_{ip}=al_0'S_i+(i+1)\frac{RaS_i}{\mu\cos^2\gamma} \left(\frac{e^{\mu\alpha_0}-1}{e^{\mu\alpha_0}+1}\right) +\frac{RaS_i(ia''e^{-\mu\alpha_0}+\alpha')}{\cos^2\gamma(1+e^{-\mu\alpha_0})} \tag{31} \]

\[ L'_{ip}=\cos\gamma\,2l_0\left(1+\frac{aS_i}{\cos\gamma}\right) +R(i\alpha_1+\alpha) +(i+1)\frac{2RaS_i}{\mu\cos\gamma} \left(\frac{e^{\mu\alpha_0}-1}{e^{\mu\alpha_0}+1}\right) + \]

\[ +\frac{2S_iRa(ia''e^{-\mu\alpha_0}+\alpha')}{\cos\gamma(1+e^{-\mu\alpha_0})} \tag{32} \]

\[ \Delta l'_{in}=al_0S_i+(i+1)\frac{aRS_i}{\cos^2\gamma\,\mu} \left(\frac{e^{\mu\alpha}-1}{e^{\mu\alpha}+1}\right) +\frac{RaS_i(\alpha_1-\alpha)i e^{-\mu\alpha}}{\cos^2\gamma(1+e^{-\mu\alpha})} \tag{33} \]

\[ L'_{in}=\cos\gamma\,2l_0\left(1+\frac{aS_i}{\cos\gamma}\right) +R(i\alpha_1+\alpha) +(i+1)\frac{2RaS_i}{\mu\cos\gamma} \left(\frac{e^{\mu\alpha}-1}{e^{\mu\alpha}+1}\right) + \]

\[ +\frac{2S_iRa(\alpha_1-\alpha)i e^{-\mu\alpha}}{\cos\gamma(1+e^{-\mu\alpha})}. \tag{33'} \]

Let us note the special features of the fourth stage for \(i\ne 1\): if the driven pulley is the larger one, then the smaller pulley slips relative to the belt; with the larger pulley as the driving pulley, the belt slips relative to the small pulley.

The author solved the equations determining the position of the movable center for various stages of transmission operation for \(i=1\) and \(\mu=1 \div 0.1\), and it turned out that the greatest displacement occurs at idle running (the second stage) and increases with an increase in the coefficient of friction \(\mu\).

Under limiting operating conditions (the fourth stage), \(\Delta l_u\), relative to the position of the idling center, decreases the more, the larger \(\mu\) is. But this displacement does not reach the position of the moving center of the first stage.

Of considerably greater practical interest is the solution of the problem with a fixed center, after the application of the preliminary tension \(2S_0\) or \(2S_i\).

In these cases the equality is preserved

\[ l_0\Delta=\Delta l_x=\Delta l_p=\Delta l_n=\mathrm{const} \tag{34} \]

\[ \Delta l_{i0}=\Delta l_{ix}=\Delta l_{ip}=\Delta l_{in}=\mathrm{const}. \tag{35} \]

To solve the problem for \(i=1\), we transform several of equations (6) and (14).

Denote by

\[ y=\frac{l_0}{R};\qquad y_1=\frac{2}{\mu}\left(1-e^{-\mu\frac{\alpha}{2}}\right);\qquad y_2=\frac{2}{\mu}\left(\frac{e^{\mu\alpha_0}-1}{e^{\mu\alpha_0}+1}\right) \]

Taking \(R\) as unity, we have

\[ \Delta l_0=S_0a\,(y+y_1) \tag{35'} \]

\[ \Delta l_p=S_0a\,(y+y_2+\alpha') \tag{36} \]

Applying the condition \(\Delta l_0=\Delta l_p\), we may equate the right-hand sides of equations (35′) and (36), i.e.,

\[ S_0a(y+y_1)=S_0''a(y+y_2+\alpha') \tag{37} \]

But now in the right-hand side of equation (36) there will stand not \(S_0\), but a certain variable quantity \(S_0''\), corresponding to the operation of the transmission with stationary pulley centers.

Let us introduce a new notation: \(2S_0'\) is the preliminary tension of the idle run.

For idle running we have

\[ \Delta l_0=\Delta l_x;\qquad S_0=S_n=S_c;\qquad P=0;\qquad \alpha_0=0;\qquad \alpha'=\alpha;\qquad y_2=0. \]

In this case equation (37) is written as

\[ S_0a(y+y_1)=S_0'a(y+\alpha). \tag{38} \]

The fourth stage of operation of the transmission occurs under the following conditions:

\[ \Delta l_0=\Delta l_n;\qquad S_n'-S_c'=P_{\max};\qquad \alpha'=0;\qquad \alpha_0=\alpha. \]

Equation (37) then takes the form

\[ S_0a(y+y_1)=S_0''a(y+y_2). \tag{39} \]

In order to judge how the preliminary tension of the branches changes in all stages of operation of the belt transmission, let us express the relation between \(S_0\), \(S_0'\), and \(S_0''\).

According to equations (38), (37), and (39), we obtain

\[ \frac{S_0'}{S_0}=\frac{y+y_1}{y+\alpha} \tag{40} \]

\[ \frac{S_0''}{S_0'}=\frac{y+\alpha}{y+y_2+\alpha'} \tag{41} \]

\[ \frac{S_0''}{S_0'}=\frac{y+\alpha}{y+y_2} \tag{42} \]

Taking various values of \(y\) for an adopted \(\mu\), we obtain the dependence of \(S'_0\) in fractions of \(S_0\). This dependence, obtained from equation (40), is presented in Figs. 13 and 14. It is evident from the graph that, for small values of \(y\), the magnitude \(S'_0\) is considerably less than \(S_0\); for example, at \(y=2\)

Fig. 13

and \(\mu=1.0\), \(S'_0=0.69S_0\). As \(y\) increases, the difference between \(S_0\) and \(S'_0\) decreases. As \(\mu\) increases, the discrepancy between \(S'_0\) and \(S_0\) grows. In general it may be noted that for \(i=1\) one always has \(S_0>S'_0\).

Fig. 14

The dependence between \(S''_0\) in the fourth stage (equation 42) and the preliminary tension of the slack run \(S'_0\) is shown in Figs. 15 and 16. Under limiting operating conditions of the transmission, \(S''_0\), in relation to \(S'_0\), increases both for every \(y\) and for \(\mu\); in other words, the ratio

\[ \frac{S''_0}{S'_0} \]

is greater than unity.

For small \(\mu\), the value

\[ \frac{S''_0}{S'_0} \]

is close to unity. As \(\mu\) increases, the difference between \(S''_0\) and \(S'_0\) grows, especially for small \(y\); for example, at \(y=2\) and \(\mu=1.0\),

\[ \frac{S''_0}{S'_0}=1.34; \]

with an increase of \(y\) to 11, for \(\mu=1.0\) we have

\[ \frac{S''_0}{S'_0}=1.1, \]

i.e., the discrepancy is only about 10%; correspondingly, for smaller \(\mu\) this difference decreases still further.

\[ \frac{S_2'}{S_2}=\frac{y+\alpha}{y+y_2}, \qquad y_2=\frac{\rho}{m}\,\frac{e^{m\alpha'} }{e^{m\alpha'}-1} \]

\[ \alpha=\alpha_0=\frac{3}{4}, \qquad \alpha'=0 \]

Curve labels: \(m=1\), \(m=0.15\), \(m=0.5\), \(m=0.25\), \(m=0.1\). Horizontal-axis label: \(y_1\).

Fig. 15

\[ \frac{S_2'}{S_2}=\frac{y+\alpha}{y+y_2}, \qquad y_2=\frac{\rho}{m}\,\frac{e^{m\alpha'} }{e^{m\alpha'}-1} \]

\[ \alpha=\alpha_0=\frac{3}{4}, \qquad \alpha'=0, \qquad y=\frac{\ell}{R} \]

Curve labels: \(y=2\), \(y=5\), \(y=10\), \(y=15\), \(y=20\), \(y=30\). Horizontal-axis label: \(\eta\).

Fig. 16

The operation of the belt transmission takes place within the limits from idle running to the limiting equilibrium condition; in this case the working angle \(\alpha_0\) varies from \(0\) to \(180^\circ\), and correspondingly the reserve angle \(\alpha'\) varies from \(180^\circ\) to \(0^\circ\).

Thus, we have a picture of the operation of a belt transmission in all its stages for a transmission ratio equal to unity.

We shall now determine what features are present in a belt transmission with a transmission ratio not equal to unity.

Denote by

\[ y_i=\frac{l_0}{R} \]

\[ y_{i1}=\left(1-e^{-\mu \frac{\alpha}{2}}\right)\frac{1}{\mu} \]

\[ y_{i2}=\left(1-e^{-\mu \frac{\alpha_1}{2}}\right)\frac{1}{\mu} \]

\[ y_{i3}=\left(\frac{e^{\mu\alpha_0}-1}{e^{\mu\alpha_0}+1}\right)\frac{1}{\mu} \]

\[ y_{i4}=\frac{i\alpha''+\alpha' e^{-\mu\alpha_0}}{1+e^{-\mu\alpha_0}} . \]

Taking \(R=1\), after substitution and transformation of equations (20) and (25), we have

\[ \Delta l_{io}=S_i a\left[y_i+\frac{1}{\cos^2\gamma}(y_{i1}+iy_{i2})\right] \tag{43} \]

\[ \Delta l_{ip}=S'_i a\left\{y_i+\frac{1}{\cos^2\gamma}\left[y_{i3}(i+1)+y_{i4}\right]\right\} \tag{44} \]

Equating the right-hand sides of equations (43) and (44), we obtain

\[ S_i a\left[y_i+\frac{1}{\cos^2\gamma}(y_{i1}+iy_{i2})\right] = S'_i a\left\{y_i+\frac{1}{\cos^2\gamma}y_{i3}(i+1)+y_{i4}\right\}, \tag{45} \]

where \(S'_i\) is the variable half of the reaction at the shaft supports, otherwise the projection of the geometric component of half the tension of the branches in the working stage.

Analogously, as for \(i=1\), using equation (45), we find the relation between the basic parameters of the second and fourth stages.

In the second stage—under the preliminary tension of idle running \(2S_i\)—the condition is preserved:

\[ \Delta l_{io}=\Delta l_{ix};\quad P=0;\quad \alpha_0=0;\quad \alpha''=\alpha;\quad \alpha'=\alpha_1;\quad \alpha_1=0;\quad \alpha=0, \]

which gives

\[ y_{i3}=0;\quad y_{i1}=0;\quad y_{i2}=0;\quad y_{i4}=\frac{i\alpha_1+\alpha}{2}. \]

After substitution into equation (45), we obtain

\[ S_i a\left[y_i+\frac{1}{\cos^2\gamma}(y_{i1}+iy_{i2})\right] = S_i a\left[y_i+\frac{1}{\cos^2\gamma}\left(\frac{i\alpha_1+\alpha}{2}\right)\right]. \tag{46} \]

In the fourth stage, under the limiting operating conditions, we have

\[ \Delta l_{io}=\Delta l_{in};\quad P=P_{\max}=S_n-S'_c;\quad \alpha_0=\alpha;\quad \alpha'=0;\quad \alpha''=\alpha_1-\alpha, \]

then

\[ y_{i1}=\frac{i(\alpha_1-\alpha)}{1+e^{-\mu\alpha}};\qquad y_{i3}=\frac{e^{\mu\alpha}-1}{e^{\mu\alpha}+1}. \]

Substituting these values into equation (45), we find

\[ S_i a\left[y_i+\frac{1}{\cos^2\gamma}\,y_{i1}+i y_{i2}\right] = S_i''a\left\{y_i+\frac{1}{\cos^2\gamma}\left[y_{i3}(1+i)+\frac{i(\alpha_1-\alpha)}{1+e^{-\mu\alpha}}\right]\right\}. \tag{47} \]

For plotting the graphs (with the driving pulley being the larger one), we express the ratios between \(S_i\), \(S_i'\), and \(S_i''\), proceeding from equations (45), (46), and (47), and obtain

\[ \frac{S_i'}{S_i} = \frac{ y_i+\dfrac{1}{\cos^2\gamma}(y_{i1}+i y_{i2}) }{ y_i+\dfrac{1}{\cos^2\gamma}\left(\dfrac{i\alpha_1+\alpha}{2}\right) } \tag{48} \]

\[ \frac{S_i''}{S_i} = \frac{ y_i+\dfrac{1}{\cos^2\gamma}\left(\dfrac{i\alpha_1+\alpha}{2}\right) }{ y_i+\dfrac{1}{\cos^2\gamma}(y_{i1}+i y_{i1}) } \tag{49} \]

\[ \frac{S_i''}{S_i'} = \frac{ y_i+\dfrac{1}{\cos^2\gamma}\left(\dfrac{i\alpha_1+\alpha}{2}\right) }{ y_i+\dfrac{1}{\cos^2\gamma}\left[y_{i3}(1+i)+\dfrac{i(\alpha_1-\alpha)}{1+e^{-\mu\alpha}}\right] }. \tag{50} \]

If the driving pulley is the smaller one, then only equation (48) remains unchanged. Equations (49) and (50) change their form somewhat (see equations (31) and (33)) and are written as follows:

\[ \frac{S_i''}{S_i'} = \frac{ y_i+\dfrac{1}{\cos^2\gamma}\left(\dfrac{i\alpha_1+\alpha}{2}\right) }{ y_i+\dfrac{1}{\cos^2\gamma}\,[y_{i3}(i+1)+y_{i1}'] }, \tag{49'} \]

where

\[ y_{i1}'=\frac{i\alpha' e^{-\mu\alpha_0}+\alpha'}{1+e^{-\mu\alpha_0}} \]

\[ \frac{S_i''}{S_i'} = \frac{ y_i+\dfrac{1}{\cos^2\gamma}\left(\dfrac{i\alpha_1+\alpha}{2}\right) }{ y_i+\dfrac{1}{\cos^2\gamma}\left[ y_{i3}(i+1)+ \frac{i(\alpha_1-\alpha)e^{-\mu\alpha}}{1+e^{-\mu\alpha}} \right] } \tag{50'} \]

Before plotting the graphs, let us analyze the geometric ratio of the belt transmission for different transmission ratios and different distances between the centers of the pulleys.

Using the relation (Fig. 12)

\[ \cos \frac{\alpha}{2}=\frac{i-1}{y_i} \tag{51} \]

\[ \sin \gamma=\frac{i-1}{y_i} \tag{52} \]

\[ \alpha_1=360^\circ-\alpha, \]

we find the values of \(\alpha\), \(\alpha_1\), \(\gamma\), \(\cos\gamma\), and \(\alpha_1-\alpha\), and summarize the data obtained in Table 1.

Table 1

	\(i\)	2	2	2	2	5	5	5	5
	\(y_i\)	3	6	12	24	6	9	18	36
\(\alpha\)		\(141^\circ\)	\(161^\circ\)	\(170.5^\circ\)	\(175^\circ\)	\(96.5^\circ\)	\(126.5^\circ\)	\(155.5^\circ\)	\(167^\circ\)
\(\alpha_1\)		\(219^\circ\)	\(199^\circ\)	\(187.5^\circ\)	\(185^\circ\)	\(263.5^\circ\)	\(233.5^\circ\)	\(204.5^\circ\)	\(193^\circ\)
\(\alpha_1-\alpha\)		\(78^\circ\)	\(38^\circ\)	\(19^\circ\)	\(10^\circ\)	\(167.0^\circ\)	\(107.0^\circ\)	\(49.0^\circ\)	\(26^\circ\)
\(\gamma\)		\(19^\circ\)	\(9.5^\circ\)	\(4.7^\circ\)	\(2.5^\circ\)	\(41.5^\circ\)	\(26.5^\circ\)	\(12^\circ\)	\(6.5^\circ\)
\(\cos\gamma\)		0.945	0.986	0.999	0.999	0.749	0.894	0.978	0.993

The minimum value of \(y_i\) was taken on the basis of the theoretically permissible distance between the centers of the pulleys, i.e., when they touch one another by their rims. For any \(i\), \(y_{i\min}\) is equal to \(i+1\).

It is quite natural that, as \(y_i\) increases, the value of \(\cos\gamma\) approaches unity.

If the minimum permissible angle is taken as \(\alpha=150^\circ\) (which is recommended in practice), then, without any great error, \(\cos\gamma\) could be neglected by taking it equal to unity.

The influence of the angle \(\gamma\) is especially pronounced at small distances between the pulleys and at a large transmission ratio, which must be taken into account when evaluating the operation of a belt transmission.

Having made these preliminary remarks, let us construct graphs of the dependence between

\[ \frac{S_i'}{S_i},\ y_i \text{ and } \mu, \]

according to equation (48), for the transmission ratios \(i=2\) and \(i=5\).

The picture obtained is approximately the same (Figs. 17, 18, 19, and 20) as for the analogous relation at \(i=1\), but the curves run somewhat more steeply; i.e., as the transmission ratio increases, the difference between the preliminary tension \(S_i\) and the preliminary no-load tension \(S_i'\) increases; as \(y_i\) increases it decreases; and, finally, as \(\mu\) increases it also increases.

For example, this difference at \(i=2,\ y_i=3\) and \(i=5,\ y_i=6\), for \(\mu=1.0\), may reach, respectively, \(S_i'=0.66S_i\) and \(S_i'=0.54S_i\).

The manner in which the transmission ratio \(i\) affects the variation of

\[ \frac{S_i'}{S_i} \]

is shown by the graph in Fig. 21.

On the graph, the ratios

\[ \frac{S_i'}{S_i} \]

are plotted for equal distances

between the centers of the pulleys \((y_i, y)\) for different coefficients of friction \(\mu\). In all cases, for \(y_i\) and \(\mu\), as \(i\) increases the ratio

\[ \frac{S_i'}{S_i} \]

decreases.

Fig. 17

An increase in the coefficient of friction \(\mu\) leads to a decrease in the ratio

\[ \frac{S_i'}{S_i}, \]

and the decrease is greater the smaller the distance between the centers of the pulleys.

A somewhat different character of the relationship is obtained between the preliminary tension of the idle run \(S_i'\) and the projection of the geometric component of half the tensions of the branches in the fourth stage of operation of the belt transmission, \(S_i''\) (\(S_i''\) is half the reaction at the shaft supports).

Fig. 18

If, for a transmission ratio \(i\) equal to unity, this ratio was greater than unity, then for a transmission ratio not equal to unity, the rel—

…the ratio \(\dfrac{S_i'}{S_i'}\) may be, depending on \(y_i\) and \(\mu\), greater or less than unity.

Fig. 19

Graph annotations: \(\dfrac{S_i'}{S_i}\), \(i=5\); \(\mu=0.25\), \(\mu=0.50\), \(\mu=0.75\), \(\mu=1.00\); \(y_i=k\).
Inset formulas: \(\dfrac{S_i'}{S_i}=\) [[unclear: formula in inset]]; \(y_i=\{1-e^{-\mu\alpha}\}^{1/m}\); \(y_i=\{1-e^{-\mu\alpha}\}^{1/m}\) [[unclear: second exponent/argument]].

The graphs shown in Figs. 22 and 23, constructed for the gear ratio \(i=5\) at various \(y_i\) and \(\mu\), on the basis of equation (50), make it possible to judge the character of the variation of \(\dfrac{S_i''}{S_i'}\).

Fig. 20

Graph annotations: \(\dfrac{S_i'}{S_i}\), \(i=5\); \(y_i=36\), \(y_i=18\), \(y_i=12\), \(y_i=6\); \(\mu\).

In the graph (Fig. 22) the ratio \(\dfrac{S_i''}{S_i'}\) turned out to be equal to unity for the following \(y_i\) and \(\mu\):

\[ \begin{aligned} \mu &= 1, & y_i &= 14,\\ \mu &= 0.75, & y_i &= 16,\\ \mu &= 0.5, & y_i &= 20,\\ \mu &= 0.25, & y_i &= 34. \end{aligned} \]

Taking the ratio

\[ \frac{S_i''}{S_i'} = 1 \]

as the boundary, we have that the corresponding \(\dfrac{S_i''}{S_i'}\) smaller than those given yield a ratio \(\dfrac{S_i''}{S_i'}\) less than unity.

All values of \(y_i\) greater than those given yield ratios \(\dfrac{S_i''}{S_i'}\) greater than unity; this is quite natural, since with increasing \(y_i\) the belt transmission by the force contour approaches, as it were, the transmission ratio \(i = 1\), while the difference between \(\alpha\) and \(\alpha_1\) decreases and \(\cos \gamma\) is close to unity.

Fig. 21

Generally speaking, the dependence between \(S_i'\) and \(S_i''\) for a transmission ratio \(i\) not equal to unity is considerably more complicated than for \(i = 1\), since

Fig. 22

the influence of the change in \(i\), \(\alpha\), \(\alpha_1\), and \(\gamma\) makes itself felt, in contrast to the ratio \(\dfrac{S_0''}{S_0'}\) for \(i = 1\), where these quantities are constant.

In conclusion, let us note that the tensions \(S_0'\), \(S_0''\), \(S_i'\), and \(S_i''\) at all stages of operation of the belt transmission are always less than the preliminary tensions-

... values \(S_0\) and \(S_i\). (Having the ratios \(\frac{S_0}{S_0'}\), \(\frac{S_i'}{S_i}\), \(\frac{S_0''}{S_0'}\), and \(\frac{S_i''}{S_i'}\), this observation can easily be verified.)

The theoretical investigations carried out make it possible to establish the dependence among all the parameters of belt transmission.

The questions of the influence of the speed of motion of the belt, the rigidity, and the weight of the belt on the operation of the transmission have remained untouched.

Fig. 23

In essence, the questions left untouched will not affect the character of the investigation.

Taking into account a larger number of factors in the operation of a belt transmission will, to a certain degree, refine the equations obtained.

The available material is quite sufficient for entirely definite and concrete conclusions.

We proceed to the conclusions.

1. A theoretical explanation has been given of the disputed question of why the sum of the tensions of the branches in a belt transmission is a variable quantity.

It has been established that, regardless of whether or not the deformation of the belt follows Hooke’s law, the sum of the tensions of the branches is a variable quantity. The change in the tension of the branches is connected with the different stages of operation of the belt transmission and with the different character of the tension along the contour of the belt.

1. In evaluating experimental data and calculating a belt transmission it is necessary to distinguish two initial tensions: the first is the initial tension, the second is the initial tension of idle running.

The first tension, according to the accepted terminology, is created in the first stage of operation of the belt transmission, when the belt is stationary; the second is obtained in the second stage, during idle running.

1. The initial tension of idle running is less than the initial tension both when the transmission ratio is equal to unity and when it is not equal to unity. The greatest difference is obtained at the minimum distance between the centers of the pulleys, when the rims touch each other. As this distance increases, the difference between the initial tensions decreases. The difference increases with an increase in the actual coefficient of friction of the belt transmission.

For a comparative evaluation we give certain ratios of the initial tensions (2).

Table 2

\(j\)	\(\dfrac{l}{R}=y(y_i)\)	\(\dfrac{S_0'}{S_0}\ (S_i'/S_i)\), \(\mu=1.0\)	\(\dfrac{S_0'}{S_0}\ (S_i'/S_i)\), \(\mu=0.25\)
1	2	0.69	0.89
2	3	0.66	0.87
5	6	0.54	0.82

As can be seen from the table given, the difference at minimal distances between pulley centers can reach a large value.

1. The sum of the tensions of the two strands \(2S_0''\), under the limiting equilibrium conditions (the fourth stage of operation of the belt transmission), for the transmission ratio \(i=1\), is greater than the sum of the initial tension of idle running \(2S_0'\).

The ratio \(\dfrac{S_0''}{S_0'}\), as \(y=\dfrac{l}{R}\) increases, approaches unity. With an increase in the coefficient of friction \(\mu\), on the contrary, the difference between \(\dfrac{S_0''}{S_0'}\) and unity is greater. For example, at \(y=2\) and \(\mu=1.0\),

\[ \frac{S_0''}{S_0'}=1.34. \]

A somewhat different character of the relation between \(S_i''\) and \(S_i'\) is obtained when the transmission ratio is not equal to unity. The quantity \(\dfrac{S_i''}{S_i'}\) may be both greater and less than unity, in contrast to the transmission ratio \(i=1\), where this ratio is greater than unity.

1. In the second, third, and fourth stages of operation of the belt transmission, the geometric sum of the strand tensions is always less than the initial tension of the first stage.

2. It has been theoretically proved why it is disadvantageous to take small distances between pulley centers.

For transmitting one and the same maximum circumferential force, with a smaller distance between pulley centers it is necessary to apply a greater initial tension.

1. Only a complete and comprehensive study of the mechanical strength of belts can place the calculation and theoretical study of belt transmission on a firm basis.

The existing opinion that the mechanical strength of a belt cannot be connected with the operation of a belt transmission loses its meaning if Euler’s law is correctly used in belt transmission, and if the entire diversity of transmission operation is considered in mutual interrelation.

1. The equations obtained, in the presence of the mechanical characteristic of the belt and of the actual coefficient of friction of the belt transmission, will make it possible to calculate the belt transmission more accurately.