Paradoxical as it may seem at first glance, a convex curvature of a pulley causes a flat belt to keep track and not jump off!

To understand this paradoxical behavior, let’s take a closer look at the belt on a crowned pulley. For this purpose, the figure below shows a flat belt attached to a cylindrical pulley (driving pulley) and to a crowned pulley (driven pulley). The curvature is greatly exaggerated for the sake of clarity. In practice, the curvature on pulleys is hardly visible and is often only a fraction of a millimeter.

Due to the curvature, the belt is tensioned differently on the two sides. The side of the belt closer to the center is tensioned more than the other side. This difference in tension means that the belt no longer wraps symmetrically around the pulley, but curves. This effect can be demonstrated relatively easily with a rubber band or even an elastic cloth. If you pull the rubber band apart on one side, the result is not a straight line, but the band curves outwards and forms an arc!

This effect of the bulge on the more tensioned side is also seen on the belt when it wraps around the crowned pulley. The belt is, so to speak, superelevated. This effect is all the greater the greater the differences in the tensions of the two sides of the belt and thus the more the pulley is curved.

Because the pulley and the belt itself are curved, the side of the belt with the bulge hits the pulley first.

This point of the belt, which runs onto the pulley first, is also under high tension and thus sticks very strongly to the pulley. This point, however, is tempted to run along a circular path around the pulley (line drawn in white with contact points marked in red). In this way, the belt thus pulls itself up towards the center of the pulley.

As long as the belt does not lie symmetrically in the center of the pulley, the more tensioned side will continue to bulge and thus always pull the belt further upwards. This effect only disappears when the belt lies symmetrically in the center of the pulley and both sides of the belt are equally tensioned. In this case, there is no bulge and the belt does not pull in any direction. This leads to the self-centering effect of the crowned pulleys and to the fact that the belt does not jump off the pulley.

Note that such a self-centering effect is only present with flat belts as long as the belt width is significantly greater than the belt height. With round belts or belts with a square cross-section, this self-centering effect no longer works with curved pulleys.

Advantages of belt drives

Compared to a gear drive, a belt drive can be used to bridge greater distances between two shafts in a simpler way. Chain drives also offer this advantage and are therefore used for bicycles where a relatively large distance has to be covered between pedal and rear wheel.

Frictionally operating belts such as flat belts or V-belts also offer a natural overload function. In contrast to gear drives, overload simply causes the belt to slip through (sliding slip). This protects the transmission from major damage. In the worst case, only the belt needs to be replaced and not the entire gears and shafts as in the case of a damaged gear drive.

Another advantage of belt drives is the elasticity of the belts compared to rigid gears. This offers good damping characteristics (shock absorption), especially in the case of sudden torque changes. This is why belt drives are used, for example, in grinding plants or stone crushers. The starting and stopping behaviour is also damped accordingly and is not as jerky as with rigid gear drives. Note, however, that a high elasticity of the belt also results in increased elastic slip. Belts can therefore not be made too elastic, but neither can they be designed too inelastic, as otherwise the positive shock absorption properties would be missing.

An additional advantage of belt drives over gear drives is their insensitivity to angular misalignment as long as the axes continue to run in a parallel plane to each other. In many cases, such a misalignment is even deliberate. This makes it easy to redirect the direction of rotation. If the axis of the output shaft is turned by 180° and the belt is crossed, the original direction of rotation can easily be reversed. In contrast to an open belt drive, this is also referred to as a crossed belt drive.

Belt drives do not have to be lubricated in comparison to gear drives. This reduces maintenance costs accordingly. Belt drives also have lower noise emissions than gear drives, since no metallic teeth engage but only relatively soft, elastic belts drive the pulleys. This enables the transmission of high rotational speeds.

In addition, pulleys are usually not complete solid wheels, as is often the case with gears. Pulleys usually have recesses to reduce weight and manufacturing costs. As a result, belt drives are generally lighter than comparable gear drives.

Disadvantages of belt drives

However, the above-mentioned advantages of belt drives are also countered by disadvantages. Depending on the ambient conditions, belts are subject to more or less severe ageing phenomena, i.e. they lose their elastic properties over time and must be replaced. For this reason, belts can only be used within a certain temperature range. In addition, over time the belts become plastically stretched, so that they have to be re-tensioned at regular intervals.

Another disadvantage of some belt types such as flat belts or V-belts is the associated slip, which reduces the efficiency of the transmission accordingly. Slippage can only be prevented with timing belts due to the positive force transmission.

In some cases, the increased space requirement of a belt drive compared to a gear drive can also have a disadvantage. This is due to the fact that the belt pulleys cannot be placed directly against each other, while the toothed wheels of gear drives can even mesh with each other and thus be set up in a more space-saving manner. In addition, the wrap angle decreases with decreasing centre distance, so that wrapping can become unacceptably small. Although this can be compensated by idler pulley, it not only increases the design effort but may also increase the required space again.

Why mus belt tension be assured?

The importance of the belt tension for power transmission has already been explained in the article How does a belt drive work?. Despite the pre-tension (initial tension), however, the belt tension will change during operation due to plastic deformation or temperature. For this reason, belt drives must often be kept on tension by so-called tensioning systems.

It must also be borne in mind that a belt will have to be serviced over time and therefore have to be removed from the pulley and remounted. This is hardly possible when the belt is under tension, so that the tension must be removed when the belt is changed and must be tensioned again after installing by means of tensioning devices (belt tensioners).

Tensioning systems are used to generate and maintain the belt tension and thus ensure reliable power transmission!

Tensioner, idler and guiding pulleys

The belt tension can be maintained during operation, for example, by means of tensioner pulleys. Tensioning pulleys also serve to cushion heavy load changes. In addition, the wrap angle can be increased by tensioner pulleys. Tensioner pulleys are also often used for long belt lengths to prevent excessive belt vibration.

If such pulleys are merely used to deflect the belt, then these are generally referred to as idler pulleys. Idler pulleys are used, for example, in multiple drives in which one driving pulley drives several other pulleys. Idler pulleys can also be used for long belt lengths to reduce belt vibrations. Such pulleys can also take over the function of a guide at the same time, so that the belt does not jump off the pulley. Such pulleys are called guiding pulleys or guide rollers. Guiding pulleys often have protrusions (called flanged pulleys) on the left and right, between which the belt is held in track.

Tension pulleys do not yet have a tensioning effect; certain devices are required to achieve a tensioning effect (also known as tensioning systems). Tensioning devices are available in a wide variety of designs. The tensioning systems shown in the figures above were each designed with a simple spring mechanism. The springs offer the advantage that the belt tension can adapt dynamically to the operating state, e.g. strong load changes.

Eccentric tensioner pulley

Another way of generating a belt tension is by an eccentric mounting of the tensioner pulley. The desired belt tension can then be set by rotating the pulley to a certain position. In addition, torsion springs can be installed in the pulley, which then allow dynamic adjustment of the belt tension during load changes.

Hydraulic damping tensioners

Another way of tensioning a belt is by means of a lever arm to one end of which the tensioner pulley is attached and held in place by a spring.

However, if frequent load changes occur during operation, this tensioning system can be excited to strong vibrations. Hydraulic damping systems such as those found in door closers, for example, can therefore be used to absorb shocks. A piston is then placed in oil, which provides the necessary damping effect due to its viscosity.

Motor slide base

Besides the use of tension pulleys, the belt tension can also be applied by adjusting the driving pulley itself. When so-called motor slide bases are used, the entire motor is mounted on a movable slide which runs in a fixed guide.

The position of the carriage on the guide and thus the belt tension can be adjusted. However, if the belt tension decreases or if there are strong load changes, the motor slide base does not automatically adapt to the changed conditions but must be readjusted manually.

Pivoting motor base

A dynamic adaptation of the belt tension to the existing load conditions (or to plastic expansion processes) can be achieved by using a self-tightening motor base. The motor is screwed onto a pivoting motor base, whereby the centre of gravity of the entire system is designed in such a way that the motor tends to tilt backwards. At an inclination of about 15° to 20°, the motor base with its weight ensures a permanent and almost constant belt tension.

Introduction

In belt drives, power is transmitted between at least two pulleys by a belt. One pulley drives the belt (driving pulley) and the other pulley is driven by the belt (driven pulley). In belt drives, the speed is often reduced, so that in these cases the smaller of the two pulleys is the driving pulley.

Power transmission

With belt drives, the power transmission is generally friction-locked, i.e. power is transmitted through frictional forces between pulley and belt and vice versa. Flat belts or V-belts are very often used. An exception to friction-locking power transmission are toothed belts (timing belts), in which the power transmission is carried out positively by teeth attached to the belt, which engage in the pulley.

In belt drives, the power transmission is generally friction-locked by frictional forces between the belt and the pulley (exception: toothed belt drives)!

In the case of friction-locking power transmission, the belt must be pressed against the pulley with a certain contact force. This is the only way to ensure that the frictional force is large enough to prevent the belt from slipping over the driven pulley or the driving pulley from rotating faster than the belt.

Ideally, the maximum force to be transmitted corresponds to the maximum static friction force acting between the belt and pulley. If the force to be transmitted is greater than the static friction force, the belt slips over the pulley and only the lower sliding friction force is effective (called sliding slip). As a result of the relative motion between belt and pulley, the belt wears very heavily and becomes unusable in a very short time. Such a sliding slip of the belt or pulley must therefore be absolutely avoided.

The maximum transmittable force with belt drives corresponds to the static friction force between belt and pulley!

Two parameters have a special influence on the static friction force and thus on the maximum transmittable force. Firstly, the belt tension, which ensures that the belt is pressed against the pulley with a certain force and can thus generate the necessary static friction force. On the other hand, the belt must wrap the pulley sufficiently tightly so that the necessary adhesive contact can be made.

To be able to transmit high forces, the belt must be wrapped around the pulley as much as possible and the belt tension must be as high as possible!

Tight and slack side of a belt

When rotating around the pulleys, the belt is exposed to different loads. The section of the belt in which the belt is strongly pulled towards the driving pulley and is thus exposed to a large tensile load is referred to as the tight side. On the opposite section, the belt moves away from the driving pulley and is slightly relieved by its “pushing” effect. This belt section is called slack side.

On the tight side, the belt moves towards the driving pulley; on the slack side, the belt moves towards the driven pulley!

Note that on the slack side, the belt is also exposed to a tensile load! The acting tensile loads are smaller than on the tight side but still present and they must even be present. Because if no belt forces would act on the slack side, this would mean nothing else than that the belt would have no tension. However, a belt tension is absolutely necessary so that the belt can press against the pulleys and thus generate the necessary static friction for power transmission. Special tensioning systems ensure that the belt tension is maintained during operation (see section Belt tensioning systems).

The high belt forces on the tight side cause the elastic belt to be tensioned relatively strongly. The belt is stretched and will then sag slightly in the less stressed belt section on the slack side. Only if the belt drive is not under load the belt forces on both sides are equal and there will be no sagging. Note that depending on the direction of rotation of the belt drive, the tight side and slack side as well as the associated deflection are reversed (see animation above).

Wrapping

In the following, a single-stage belt drive with two pulleys that are wrapped around by a common belt is considered. The magnitude of this wrap is described by the wrap angle \(\varphi\).

The wrap angle is defined as the angle between the run-up and run-off of the belt on the pulley.

The larger the wrap angle, the more adhesive surface the belt has and the greater the frictional force or the force that can be transmitted. However, it must be noted that the pulleys of a belt drive are wrapped to different degrees by the belt if the pulleys have different diameters!

The maximum transmittable force is usually limited by the smaller of the two pulleys (usually the driving pulley), as this pulley has a smaller wrap angle compared to the larger driven pulley. In addition, the greater curvature causes greater bending stresses in the belt, which also limit the transmittable belt force.

So called idler pulleys can be used to increase the wrap angle. These are usually placed near the actual pulleys to achieve the greatest possible wrapping effect. If these pulleys are used to tension the belt at the same time, they are also called tensioner pulleys.

To increase the wrap angle and thus the transmittable force, idler pulleys can be used!

The figure below shows a belt drive of a dryer drum as an example of the use of an idler pulley (the drum itself serves as the output pulley). In order to increase the wrap angle on the drive shaft and thus ensure sufficient power transmission, a idler pulley was used.

Furthermore, it should be noted that the deflection of the slack side under load results in different wrap angles compared to the load-free state. The arrangement of the tight side and the slack side also influences the wrap angle. If the slack side is above the tight side, the wrap angle increases due to the deflection, while in the opposite case the wrap angle is reduced.

To ensure that wrapping is not reduced under load, the slack side should be arranged over the tight side!

However, the change of the wrap angle under load plays a subordinate role in practice and can often be neglected (especially with high coefficients of friction). See also the article Power transmission of a belt drive.

For the construction of belt drives it is necessary to determine both the wrap angle and the belt length for given pulley diameters and pulley spacings. The basis for this is the mounted condition of the belt on the pulleys.

The belt is then composed of the two arc-shaped sections b_s and b_l on the small and large pulley and of the two straight belt sections l. The sum of these belt sections forms the geometric belt length L_i in the mounted state, whereby this length specification then refers to the inner surface of the belt:

\begin{align}
\label{riemenabschnitte}
&L_i = 2 \cdot l + b_l + b_s \\[5px]
\end{align}

First, the wrap angle is to be determined on the basis of the pulley diameter and the distance between the pulleys. The angle of inclination \(\alpha\) is used for the derivation, which results as the angle between the center line of the pulleys and the straight line of the belt. As the figure above shows, the inclination and the wrap angle φ of the small pulley are related as follows (unless explicitly stated otherwise, the entire angle specifications and functions refer to the radians!):

\begin{align}
\label{a_p}
&\boxed{\varphi = \pi – 2 \cdot \alpha} \\[5px]
\end{align}

Now move the straight belt section l along the angular auxiliary lines to the center of the small pulley. This results in a right-angled triangle with the center distance e as hypotenuse and the span length l as ankathete, as well as the opposite cathete resulting from the pulley diameters. The angle of inclination \(\alpha\) can then obviously be determined from the sinus function by the pulley diameters and the center distance:

\begin{align}
&\sin(\alpha) =\frac{d_l-d_s}{2e} \\[5px]
\label{trumneigung}
&\boxed{\alpha =\arcsin\left(\frac{d_l-d_s}{2e}\right)} \\[5px]
\end{align}

In combination with equation (\ref{a_p}) the wrap angle φ can finally be calculated as follows by the pulley diameters d_l and d_s and the center distance e:

\begin{align}
\label{phi}
&\boxed{\varphi = \pi – 2 \cdot \arcsin\left( \frac{d_l-d_s}{2e}\right)} ~~~\text{ radian measure!} \\[5px]
\end{align}

For the derivation of the belt length, the resulting triangle in the figure above can also be used. Thus, the span length l can be determined as a function of the center distance e by the cosine function of the inclination angle \(\alpha\):

\begin{align}
\label{trumlaenge}
&\underline{l = e \cdot \cos(\alpha) } \\[5px]
\end{align}

To determine the belt section around the large pulley b_l, first determine the length of the corresponding semicircle (π/2⋅d_l). However, the two arc sections (d_l/2⋅\(\alpha\)) must also be added, which result from the inclination with the angle \(\alpha\) (Note: arc angle equals arc length divided by arc radius or arc length equals arc radius times arc angle):

\begin{align}
\label{b_g}
&\underline{b_l =\tfrac{\pi}{2}d_l+2 \cdot \tfrac{d_l}{2}\cdot \alpha} \\[5px]
\end{align}

The arc length of the belt around the small pulley b_s is determined in the analog way. Starting from the semicircle length π/2⋅d_s, however, the two arc lengths d_s/2⋅\(\alpha\) must now be subtracted due to the inclination:

\begin{align}
\label{b_k}
&\underline{b_s =\tfrac{\pi}{2}d_s-2 \cdot \tfrac{d_s}{2}\cdot \alpha} \\[5px]
\end{align}

With the equations (\ref{riemenabschnitte}), (\ref{trumlaenge}), (\ref{b_g}) and (\ref{b_k}), the length of the inner side of the belt L_i can be determined as follows on the basis of the inclination angle \(\alpha\):

\begin{align}
&L_i = 2 \cdot \underbrace{e \cdot \cos(\alpha)}_{l} + \underbrace{\tfrac{\pi}{2}d_l+2 \cdot \tfrac{d_l}{2}\cdot \alpha}_{b_l} + \underbrace{\tfrac{\pi}{2}d_s-2 \cdot \tfrac{d_s}{2}\cdot \alpha}_{b_s} \\[5px]
\label{riemenlaenge}
&\underline{L_i = 2 \cdot e \cdot \cos(\alpha) + \left(d_l + d_s \right) \tfrac{\pi}{2}+ \left(d_l-d_s \right) \cdot \alpha} \\[5px]
\end{align}

Since the inclination angle is usually relatively small, the following small angle approximations can be used for the radian measure:

\begin{align}
\label{kleinwinkel}
&\underline{\cos(\alpha) \approx 1 – \frac{\alpha^2}{2}}~~~\text{and} ~~~\sin(\alpha) = \underline{\frac{d_l-d_s}{2e} \approx \alpha} \\[5px]
\end{align}

This approximation allows the belt length to be determined with sufficient accuracy in practice as follows:

\begin{align}
&L_i  \approx 2 e \cdot \underbrace{\left(1 – \frac{\alpha^2}{2}\right)}_{\approx \cos(\alpha)} + \left(d_l + d_s \right) \tfrac{\pi}{2}+ \left(d_l-d_s \right) \cdot \underbrace{\frac{d_l-d_s}{2e}}_{\alpha} ~~~~~\text{with} ~~~\alpha \approx \frac{d_l-d_s}{2e} ~~~\text{follows:}\\[5px]
&L_i  \approx 2 e \cdot \left(1 – \frac{(d_l-d_s)^2}{8e^2}\right) + \left(d_l + d_s \right) \tfrac{\pi}{2}+ \left(d_l-d_s \right) \cdot \frac{d_l-d_s}{2e} \\[5px]
&L_i  \approx 2 e – \frac{(d_l-d_s)^2}{4e} + \left(d_l + d_s \right) \tfrac{\pi}{2}+ \frac{(d_l-d_s)^2}{2e} \\[5px]
&\boxed{L_i  \approx 2 e  + \tfrac{\pi}{2} \left(d_l + d_s \right)  + \tfrac{1}{4e} (d_l-d_s)^2} \\[5px]
\end{align}

This formula offers not only the advantage that it is clearly arranged and does not require an inclination angle, but the center distance e can now also be determined for a given belt length L_i. After mathematical transformations and solving the quadratic equation follows:

\begin{align}
& \boxed{e \approx \tfrac{1}{4} L_i – \tfrac{\pi}{8} \cdot (d_l+d_s) + \sqrt{ \left[ \tfrac{1}{4} L_i – \tfrac{\pi}{8} \cdot (d_l+d_s) \right]^2 – \tfrac{1}{8}(d_l-d_s)^2 }}
\end{align}

For flat belts, the standard center distance should be set between the following limits:

\begin{align}
& \boxed{0,7 \cdot (d_l+d_s) \le e \le 2 \cdot (d_l+d_s)}
\end{align}

Introduction

In the article Mechanical power it was shown that an object that is moved by a force F with the speed v converts the power P=F⋅v. Applied to the belt drive, this means that if the belt is moved by the circumferential force F_c at the speed v, the belt transmits the following power:

\begin{align}
\label{leistung}
\boxed{P=F_c \cdot v} \\[5px]
\end{align}

This power is transmitted from the drive pulley to the belt and then to the output pulley. Note that a belt drive (and a transmission in general) does not change the power! The belt merely serves as a “transmitter” oft the power between the input and the output pulley, so to speak. However, due to its limited strength, the belt cannot transmit any high power and it can even achieve an optimum belt speed at which the maximum power is reached, as will be explained in more detail in the following sections.

Influence of circumferential force and belt speed on the transmitted power

First of all, according to the equation (\ref{leistung}), a high power to be transmitted also means a high speed. However, high speeds lead to an increase in centrifugal forces. Since the allowable belt stress in the belt is limited, the increase in centrifugal forces is at the expense of the maximum permissible tight side force. This is also directly illustrated by the equation derived in the article Maximum belt stress, in which the sum of tight side stress σ_t, bending stress σ_b and speed-dependent centrifugal stress σ_cf(v) must not exceed the maximum permissible overall stress σ_per:

\begin{align}
&\sigma_t + \sigma_b + \sigma_{cf}(v) \le \sigma_{per} \\[5px]
\label{zulaessige}
&\boxed{\sigma_{t,per} \le \sigma_{per} -\sigma_b – \sigma_{cf}(v)} ~~~\text{and}~~~\boxed{\sigma_{cf}(v) = \rho \cdot v^2} ~~~\text{and}~~~ \boxed{\sigma_b =E_b \cdot \frac{s}{d + s}}\\[5px]
\end{align}

The permissible tensile stress σ_t,per decreases with increasing centrifugal stress σ_cf(v). This is then directly connected with a decrease of the maximum transferable circumferential force F_c,max. Because at a given tight side force (which in this case corresponds to the maximum permissible force F_t,per= σ_t,per⋅A), only a certain circumferential force F_c,max can be transmitted. The connection between the circumferential force and the tight side force is established by the so-called “gain” k and only dependent on the wrap angle φ and the friction coefficient µ:

\begin{align}
&F_{c,max} = F_{t,per} \cdot k \\[5px]
\label{nutzkraft}
&\boxed{F_{c,max} = \sigma_{t,per} \cdot A \cdot k} ~~~\text{and}~~~ \boxed{k= \left(1- \frac{1}{e^{\mu \cdot \varphi}} \right) } ~~~\text{“yield”} \\[5px]
\end{align}

According to the equation (\ref{leistung}), low circumferential forces in turn lead to a decrease in power. In extreme cases of high speeds, the centrifugal forces are so high that no circumferential force can be transmitted at all (and thus no power), as otherwise the allowable belt stress would immediately be exceeded. Excessively high belt speeds therefore prohibit the transmission of high power!

Increasing belt speeds lead to a decrease in the transmissible circumferential force, as the centrifugal force would otherwise place an unacceptably high load on the belt! The transmitted power decreases!

Conversely, at low speeds, larger circumferential forces and thus superficially higher performances can be achieved; however, if this results in the belt speed having to be reduced to such an extent (otherwise the centrifugal forces would become too high) that the belt hardly moves any more, then there is no great power behind the large circumferential force anyway!

Decreasing belt speeds lead to an increase in the transmissible circumferential force, but the transmitted power decreases!

In fact, belt drives have an economically optimal belt speed at which the maximum power P_max can be transmitted, i.e. an optimal force/speed ratio is available. This optimum belt speed will be discussed in more detail in the next section.

Optimum belt speed

Maximum transmittable power

To determine the optimum belt speed v_opt, the equations (\ref{zulaessige}), (\ref{nutzkraft}) and (\ref{leistung}) must be combined to first express the maximum transmissible power P_max as a function of the belt speed v:

\begin{align}
&P_{max} = F_{c,max} \cdot v \\[5px]
&P_{max} = \sigma_{t,per} \cdot A \cdot k \cdot v \\[5px]
&P_{max} = \left( \sigma_{per} -\sigma_b – \sigma_{cf}(v) \right) \cdot A \cdot k \cdot v \\[5px]
&P_{max} = \left( \sigma_{per} – \sigma_b – \rho \cdot v^2 \right) \cdot A \cdot k \cdot v \\[5px]
\end{align}

If the cross-sectional area A of the flat belt is expressed by its belt thickness s and its belt width b (A=b⋅s), then the maximum transmissible power P_max at a given speed v finally results as follows:

\begin{align}
\label{abs_leistung}
&\boxed{P_{max}(v) = \left( \sigma_{per} – \sigma_b – \rho \cdot v^2 \right) \cdot b \cdot s  \cdot k \cdot v} \\[5px]
\end{align}

Note that for the gain k the smallest wrap angle of the two pulleys is decisive! In general this applies to the smaller pulley, especially as the greatest bending stresses also act there. Both the wrap angle φ (relevant for the gain) and the pulley diameter d (relevant for the bending stress) therefore refer to the smallest of the pulleys.

Specific power (“power per unit belt width”)

Often the power P_max is related to the belt width b and is then called specific power p_max (“power per unit belt width”). This makes sense because the belt thickness s for calculating the bending stress σ_b must be assumed in advance anyway. This leaves the belt width as the only unknown geometric parameter b. It therefore makes sense to first express the power independently of the belt width and to state it as specific power:

\begin{align}
&p_{max} = \frac{P_{max}}{b} \\[5px]
\label{spez_leistung}
&\boxed{p_{max} (v) =\left( \sigma_{per} – \sigma_b – \rho \cdot v^2 \right) \cdot s \cdot k \cdot v} \\[5px]
\end{align}

The specific power is the power per unit belt width!

Based on this maximum transmittable specific power p_max, the required belt width b_req to transmit the nominal power P_n is then determined in practice. For safety reasons, various operating factors C are taken into account, which include the influence of impulsive torque loads and various environmental influences which could lead to friction reduction:

\begin{align}
&\boxed{b_{req} =\frac{P_n}{p_{max} \cdot C} } \\[5px]
\end{align}

Maximum power as a function of speed

The figure below shows the maximum transmittable specific power p_max as a function of the belt speed v according to the equation (\ref{spez_leistung}) for the values given in the diagram. It is now also graphically clear that neither too low nor too high belt speeds lead to an arbitrarily high transmittable power. There is a pronounced maximum in which the greatest possible maximum power can be transmitted.

To reach this maximum, the belt speed must be set to the optimum. This is also called optimum belt speed and in the present case is about 50 m/s and corresponds to a rotational speed of the smaller pulley of 4774 min^-1. As this example shows, the optimum belt speeds are often very high and are usually not achieved in practice.

There is an optimal belt speed at which the maximum power can be transmitted!

Calculation of the optimum belt speed

To calculate the optimum belt speed, the maximum value of the function p_max(v) or P_max(v) must be found. Mathematically, this corresponds to zeroing the first derivative:

\begin{align}
&\frac{\text{d}p_{max}(v)}{\text{d}v} = 0 \\[5px]
&\left( \sigma_{per} – \sigma_b – 3 \rho \cdot v^2 \right) \cdot s \cdot k = 0 \\[5px]
&\sigma_{per} – \sigma_b – 3 \rho \cdot v^2 = 0 \\[5px]
\label{v_opt}
&\boxed{v_{opt} = \sqrt{\frac{\sigma_{per} – \sigma_b  }{3 \rho}}} ~~~\text{and}~~~ \boxed{\sigma_b = E_b \cdot \frac{s}{d + s}}\\[5px]
\end{align}

At the given transmission ratio and speed of the drive pulley, the belt speed can be adjusted by larger or smaller pulleys in order to achieve the optimum belt speed. However, in order not to change the desired transmission ratio, both pulleys (driving and driven pulley) must always be changed to the same extent. Note that changing the pulley diameter changes the bending stress and thus the optimum belt speed!

The maximum specific power p_max,opt or absolute power P_max,opt, which can be transmitted at the optimum belt speed v_opt, is then obtained by applying equation (\ref{v_opt}) to equation (\ref{spez_leistung}) or (\ref{abs_leistung}):

\begin{align}
&\boxed{p_{max, opt} = k \cdot \sqrt{\frac{4 \left(\sigma_{per} – \sigma_b \right)^3 }{27 \rho}}} \\[5px]
&\boxed{P_{max, opt} = k \cdot b \cdot \sqrt{\frac{4 \left(\sigma_{per} – \sigma_b \right)^3 }{27 \rho}}} \\[5px]
\end{align}

Calculation of bearing force

The forces acting in the belt press the belt onto the pulley and thus also act on the shaft bearings. The tight side force and the slack side force is thus balanced by the bearing force of the shaft. By the law of cosines, this bearing force F_b can be determined from the tight side force F_t and the slack side force F_s as well as from the wrap angle φ:

\begin{align}
\label{lagerkraft}
F_b=\sqrt{F_t^2 + F_s^2 – 2 \cdot F_t \cdot F_s \cdot \cos(\varphi)} \\[5px]
\end{align}

As shown in the article Calculation of the belt length, the wrap angle can be calculated using the diameter of the small and the large pulley (d_s and d_l) and the center distance e:

\begin{align}
\label{phi}
&\boxed{\varphi = \pi – 2 \cdot \arcsin\left( \frac{d_l-d_s}{2e}\right)} ~~~\text{ radian measure!} \\[5px]
\end{align}

If the tight side force and slack side force is expressed by the preload force F_p and the circumferential force F_c to be transmitted (see article Power transmission of a belt drive),

\begin{align}
\label{trumkraefte}
&F_t = F_p + \tfrac{F_c}{2} ~~~~~\text{and}~~~~~ F_s =F_p – \tfrac{F_c}{2} ~\text{,}  \\[5px]
\end{align}

then the shaft load F_b can also be expressed as follows:

\begin{align}
&F_b=\sqrt{\left(F_p + \tfrac{F_c}{2} \right)^2 + \left( F_p – \tfrac{F_c}{2} \right)^2 – 2 \cdot \left(F_p + \tfrac{F_c}{2} \right) \cdot \left( F_p – \tfrac{F_c}{2} \right) \cdot \cos(\varphi)} \\[5px]
\label{F_W}
&\boxed{F_b=\sqrt{2 F_p^2 \cdot \left[1-\cos(\varphi) \right] +  \tfrac{1}{2} F_c^2 \cdot \left[1+\cos(\varphi)\right] }  } \\[5px]
\end{align}

In case the wrap angle is 180° (φ=π), the shaft load will be maximum, as the belt forces are then parallel and thus maximum effective. With cos(π)=-1 it follows directly from the equation above that the bearing load in later load operation corresponds to twice the value of the dynamic preload F_p:

\begin{align}
\label{wellenbelastung}
&F_{b,max}=2 \cdot F_p  \\[5px]
\end{align}

The maximum bearing force corresponds to twice the value of the dynamic preload!

Influence of centrifugal forces on the bearing force

No centrifugal forces have to be taken into account for the calculation of the bearing force during operating! Although the belt must be tensioned more in advance by the amount of the expected centrifugal force, the additional centrifugal belt force does not affect the bearings in later operation, since the belt is attempted to lift off from the pulley with exactly this amount of force and thus relieves the bearing again to the same extent. The centrifugal forces acting on the belt and the additional centrifugal belt force acting in the belt form a closed polygon of forces and therefor cancel each other out. Only the dynamic preload F_p is relevant for the bearing force during operation.

The centrifugal forces existing during operation are compensated by the additional preload (centrifugal belt force) and therefore do not influence the bearing force!

Relationship between maximum circumferential force and bearing force

In the article Power transmission of a belt drive it could be shown that the maximum transmittable circumferential force F_c,max is related to the dynamic preload F_p by the following equation:

\begin{align}
\label{vorspannung}
&F_{p,min} = F_{c} \cdot \frac{e^{\mu \cdot \varphi}+1}{2 \left(e^{\mu \cdot \varphi} -1 \right) } ~~~ \text{or} ~~~\underline{F_{p} = F_{c,max} \cdot \frac{e^{\mu \cdot \varphi}+1}{2 \left(e^{\mu \cdot \varphi} -1 \right) }} \\[5px]
\end{align}

If equation (\ref{vorspannung}) is applied to equation (\ref{wellenbelastung}), the following relationship results between the existing bearing force F_b and the associated maximum transmissible circumferential force F_c,max:

\begin{align}
&F_b=2 \cdot F_{c,max} \cdot \frac{e^{\mu \cdot \varphi}+1}{2 \left(e^{\mu \cdot \varphi} -1 \right) } \\[5px]
&F_b=F_{c,max} \cdot \frac{e^{\mu \cdot \varphi}+1}{e^{\mu \cdot \varphi} -1} \\[5px]
\label{durchzugsgrad}
&F_{c,max} = F_b \cdot \frac{e^{\mu \cdot \varphi}-1}{e^{\mu \cdot \varphi} +1} \\[5px]
&\boxed{F_{c,max} = F_b \cdot \phi } ~~~~~\text{where}~~~~~\boxed{\color{red}{\phi = \frac{e^{\mu \cdot \varphi}-1}{e^{\mu \cdot \varphi} +1}}} ~~~\text{as “pull facor”} \\[5px]
\end{align}

The term marked in red in the equation above is also referred to as “pull factor” ϕ. For example, a pull factor 0.8 means that a maximum of 80 % of the bearing force acting in operation is available for the power transmission (strictly speaking only applies to parallel belt spans).

The higher the “pull factor”, the higher the maximum circumferential force compared to the bearing force, i.e. a high “efficient power transmission”!

Adjusting the preload by the bearing force

The pretension of the belt can be applied by adjusting the bearing force in the load-free state. In a load-free standstill, i.e. when no circumferential force is transmitted (F_c=0), the bearing load is determined only by the total preload force F_p,total (Note that in this case the total preload force also includes the centrifugal forces to be compensated!):

\begin{align}
&\boxed{F_{b,0}=F_{p,total} \cdot \sqrt{2 \left[1-\cos(\varphi) \right] }  } \\[5px]
\end{align}

Thus, the total preload force F_p,total can be adjusted by the bearing force F_b,0 in the load-free state (or the preload force can be measured by the bearing force):

\begin{align}
&\boxed{F_{p,total}=F_{b,0} \cdot \frac{1}{\sqrt{2 \left[1-\cos(\varphi) \right] }}  } \\[5px]
\end{align}

Elastic slip

When the belt rotates around the pulleys, it is exposed to different forces. Starting from the slack side force, the belt force increases at the driving pulley (by applying the circumferential force) to the value of the tight side force. Conversely, the belt force at the driven pulley decreases again from the value of the tight side force to the value of the slack side force by releasing the circumferential force (see article Maximum belt stress). Due to the elasticity of the belt, the different forces also cause different elastic strains.

If marking lines are applied to the belt at equal distances in the load-free state, the line distances on the tight side of the belt increase due to the increased belt force during operation and decrease accordingly on the slack side due to the reduced force. The line spacings gradually adapt to the new conditions as the belt moves around the pulleys.

If a belt section between two marking lines is now looked at more closely, this section is obviously stretched on the driven pulley during rotation. The stretching belt section is pulled over the pulley, so to speak, i.e. there is relative motion between belt and pulley and thus sliding!

Conversely, the belt section coming from the tight side (and thus maximally stretched up) contracts during rotation around the driving pulley due to the decreasing belt force. The belt shrinks on the driving pulley, so to speak, and thus also results in relative motion and thus in sliding.

The belt is streched by sliding processes on the driven pulley in the direction of the tight side and contracts again on the driving pulley in the direction of the slack side!

In the animation below, additional marking lines are attached to the pulleys for better orientation. If one compares these pulley markings with the belt markings, the relative motion between belt and pulley can be seen very clearly.

Such (elastic) stretching or shrinking processes of the belt on the pulleys, which inevitably lead to relative motion, are also called elastic slip. From the belt’s point of view, elastic slip should always be kept as low as possible, otherwise enormous belt wear will occur due to the strong relative motion. The surfaces of pulleys must therefore not be too rough, as one might misleadingly assume due to the increased static friction on rough surfaces!

The relative motion on the pulleys, which is always present due to the elasticity of the belt, is called elastic slip (partial relative motion between belt and pulley)!

In addition to elastic slip, which is due to the elasticity of the belt, the belt can also slip completely over the entire driven pulley in the event of overload. This is then referred to as sliding slip. Note that every belt has a certain elasticity and therefore always results in elastic slip, whereby sliding slip should always be avoided.

Sliding slip is the complete sliding of the belt over the entire pulley in the event of overload (complete relative motion between belt and pulley)!

Belt speeds

Due to the conservation of mass (condition of continuity), the same belt mass must be moved past an imaginary point within a certain time on the tight side as well as on the slack side. Otherwise, belt mass would miraculously accumulate (or be destroyed) between the two imaginary points, since more (or less) mass is moved in over the imaginary point than is moved out again at the other point.

However, if the belt sections are now stretched on the tight side, then these sections must move faster in order to transport the same mass over the imaginary point compared to the shrunken sections on the slack side.

The belt moves faster on the tight side than on the slack side due to the greater elongation!

The belt strain and the belt speed are therefore directly related. The belt speed increases and decreases to the same extent as the belt elongation increases and decreases. The belt speed remains constant within the respective belt span, just as the forces and the elongations do. The adaptation of the different belt speeds between the slack side and the tight side takes place by stretching or shrinking processes on the pulleys.

The belt adapts to the different speeds by elastic slip on the pulleys!

Circumferential speed of the pulleys

As can be seen from the animation below, the belt is running more and more ahead due to the increasing stretching on the driven pulley. This means that the speed of an imaginary point on the belt is always slightly higher than the peripheral speed of the pulley. This corresponds to the relative motion between belt and pulley described above.

Note that the relative motion around the pulley is constantly increasing as the belt increases its speed more and more in accordance with the increasing elongation (condition of continuity!), but the pulley has a constant circumferential speed. This means that the belt speed and the peripheral speed of the pulley are only equal when the belt runs onto the driven pulley, otherwise the belt speed will be higher or the pulley speed lower.

The circumferential speed of the driven pulley is lower than the speed of the rotating belt!

Conversely, an imaginary point on the belt runs slightly behind the driving pulley due to the shrinkage of the belt (the belt is “pulled back”, so to speak, by the decreasing force). This means that the peripheral speed of the driving pulley is greater than the speed of the belt. Only when the belt runs onto the driving pulley the belt speed is equal to the peripheral speed of the pulley. Otherwise, the belt speed gradually decreases with decreasing elongation (condition of continuity!).

The circumferential speed of the driving pulley is higher than the speed of the rotating belt!

The figure below shows schematically the distribution of the speed along the belt according to the animation above.

So if the driving pulley generally moves faster than the belt and the driven pulley is slower, then the circumferential speeds of the pulleys are obviously no longer identical (this would only be the case with a complete inelastic belt). This ultimately results in a loss of circumferential speed between the drive pulley (rotating faster than the belt) and the output pulley (rotating slower than the belt).

The peripheral speed of the driven pulley is lower than that of the driving pulley! The relative loss of speed in comparison to the driving pulley is a measure of the elastic slip!

The more the belt stretches, i.e. the greater the elastic slip, the greater the difference in belt speeds and thus also in the circumferential speeds of the pulleys. Therefore, the elastic slip S can be defined by the relative loss of speed at the circumference of the input pulley (v_i) and the output pulley (v_o):

\begin{align}
\label{def_s}
\boxed{S = \frac{\Delta v}{v_i} = \frac{v_i-v_o}{v_i} = 1-\frac{v_o}{v_i}  } \\[5px]
\end{align}

Since the circumferential speeds can also be expressed by the rotational speeds n and pulley diameters d (v=π⋅d⋅n), the elastic slip can also be determined as follows:

\begin{align}
&\boxed{S = 1-\frac{n_g \cdot d_g}{n_t \cdot d_t}} \\[5px]
\end{align}

The elastic slip for belt drives (except toothed belts) is in the order of about 1 to 2 %.

Loss of power

The reduction in peripheral speed between the driving pulley and the driven pulley is directly relatet to a loss in power, because a decrease of the circumferential speed v at a transmitting circumferential force F_c means a direct decrease in power according to the P=F_c⋅v.

At the input pulley, the power P_i=F_c⋅v_i is first transferred to the belt. The power reduced by the amount of the elastic slip (P_o=F_c⋅v_o) is then taken from the output pulley. The difference in power corresponds to the power loss ΔP due to the stretching and shrinking processes of the belt (related to heating of the belt!):

\begin{align}
&\Delta P = P_i – P_o = F_c \cdot v_i –  F_c \cdot v_o = F_c \cdot \underbrace{(v_i-v_o)}_{=v_i \cdot S} = F_c \cdot v_i \cdot S = P_i \cdot S   \\[5px]
\end{align}

Thus, the elastic slip S can also be determined by the power loss ΔP with respect to the power P_i at the input pulley:

\begin{align}
&\boxed{S = \frac{\Delta P}{P_i}} \\[5px]
\end{align}

Note that the elastic slip has no effect on the circumferential force and thus does not influence the torque conversion. After all, the circumferential force is only dependent on the difference between the tight side force and the slack side force (F_c=F_t-F_s). These belt forces apply equally to both pulleys, so that the acting circumferential forces on both pulleys are identical. The different belt forces only provide different elongations, but the acting forces are still transmitted to the same extent in the belt (after all, it makes no difference whether a mass is attached to an elastic manila rope or to a rigid steel rope – the rope forces are identical, only the manila rope stretches more).

The elastic slip influences the belt speed and thus the power but not the circumferential force or the torque!

Note that the loss in power is due to the elasticity of the belt. Such elasticity is not only unavoidable but also deliberate! After all, the great advantage of belt elasticity is the associated shock absorption property at peak loads!

Sliding and adhesive areas

Due to the elasticity of the belt, slipping will always occur between belt and pulley. In such a gliding case, there is a direct relationship between the tight side force F_t and the slack side F_s according to the belt friction equation:

\begin{align}
&F_t = F_s \cdot e^{\mu_s \cdot \varphi’} \\[5px]
\end{align}

Note that in this case it is not, as previously always assumed, the static limit case in which the belt is not yet slipping. Rather, slipping is already present from the very beginning due to the elastic slip (µ_s as coefficient of sliding friction!).

In this sliding case, the connection between the two belt forces can be directly established by the belt friction equation. However, this also means that slipping does not generally take place over the entire wrap angle but only over the sliding angle φ’, which results from the given tight side force F_t and slack side force F_s as follows:

\begin{align}
&\frac{F_t}{F_s} = e^{\mu_s \cdot \varphi’} \\[5px]
&\ln\left(\frac{F_t}{F_s}\right) =\mu_s \cdot \varphi’ \\[5px]
\label{gleitwinkel}
&\boxed{\varphi’ = \frac{1}{\mu_s}\cdot \ln\left(\frac{F_t}{F_s}\right)} \\[5px]
\end{align}

The entire wrap area φ can therefore be divided into two zones. In the co-called sliding zone φ’ a relative motion takes place between belt and pulley. With the acting sliding friction, this zone ensures the transmission of the circumferential force. In the remaining adhesion zone, the belt adheres to the pulley without a relative motion and without force transmission.

The belt always runs onto the pulley in the adhesion zone and runs off the pulley in the sliding zone. The change of the belt force around the pulleys (from slack side to tight side and vice versa) only takes place in these sliding zones. This also applies to the belt speeds, which only change within the sliding zones.

The adhesion zones serve as a “safty zones” against slippage, so to speak, since the sliding zones can still increase at the expense of the adhesion zones and can be used for power transmission. Note that the sliding angle according to equation (\ref{gleitwinkel}) depends only on the ratio of the belt forces and these apply equally to both pulleys. This inevitably means that the adhesion zone and thus the safety against slippage will first be used up on the less wrapped pulley (usually the smaller drive pulley).

The adhesion zone serves as a safety measure against slippage and will usually be used up first on the smaller of the two pulleys!

The sliding angle φ’ relevant for power transmission can also be expressed by the circumferential force F_c to be transmitted. Thus follows with F_c=F_t-F_s or F_t=F_c+F_s:

\begin{align}
&\varphi’ = \frac{1}{\mu_s}\cdot \ln\left(\frac{F_c+F_s}{F_s}\right) \\[5px]
\label{4367}
&\boxed{\varphi’ = \frac{1}{\mu_s}\cdot \ln\left(1+\frac{F_c}{F_s}\right)} \\[5px]
\end{align}

This equation states that the sliding angle can be reduced with increasing slack side force at a given circumferential force and thus the safety against sliding slip is increased. The adjustment of the slack side tension can be carried out, for example, by tensioner pulleys.

Conversely, if the slack side tension is reduced or cannot be provided to the necessary extent, the sliding angle increases with increasing circumferential force and extends over the entire wrap angle. In this case, the entire circumferential force can still be transmitted, but a further increase in the circumferential force would then mean a relative motion over the entire wrap angle. In this case the limit of elastic slip is exceeded and sliding slip starts. The transition from the elastic slip to sliding slip is therefore always smooth.

Note: In the previous articles on belt drives, the sliding angle was equated with the wrap angle for reasons of conservative calculations. The statements made must then always be interpreted at the limit to sliding slip!

Calculation of the elastic slip

Equation (\ref{4367}) also shows that if no circumferential force is transmitted (F_c=0) there is no sliding zone (ln(1)=0!) but only an adhesion zone. With the transmission of a circumferential force, however, a sliding zone is created which increases with increasing circumferential force. As a result, the elastic slip also increases. The exact relationship between elastic slip S and circumferential force F_c to be transmitted is to be derived in the following sections.

As already explained in the section Belt speeds, the belt strains ε and the belt speeds v are directly interrelated. Mathematically this can be expressed as follows:

\begin{align}
& v \text{ ~ } (1+\epsilon) \\[5px]
\end{align}

With the definition of the elastic slip S as the ratio of speed loss Δv and circumferential speed of the input pulley v_i, the following formula applies v_o: circumferential speed of the output pulley):

\begin{align}
&S = \frac{\Delta v}{v_i}  = \frac{v_i-v_o}{v_i}  = \frac{v_t-v_s}{v_t} = \frac{(1+\epsilon_t) – (1+\epsilon_s)}{1+\epsilon_t} = \frac{\epsilon_t-\epsilon_s}{1+\epsilon_t}  \\[5px]
\end{align}

Note that the peripheral speed of the driven pulley v_o corresponds to the belt speed v_s on the slack side and the peripheral speed of the driving pulley v_i corresponds to the belt speed on the tight side v_t.

The strains ε can be determined as follows using the Young’s modulus E of the belt (not to be confused with the bending modulus E_b!) and the acting belt stresses σ=F/A  (with A as cross-sectional area of the belt):

\begin{align}
\frac{F}{A} &= \boxed{\sigma = E \cdot \epsilon} ~~~\text{Hooke’s law}  \\[5px]
\epsilon &= \frac{F}{E\cdot A}  \\[5px]
\end{align}

Finally, the elastic slip is calculated on the basis of the belt forces as follows:

\begin{align}
&S = \frac{\epsilon_t-\epsilon_s}{1+\epsilon_t}  = \frac{\frac{F_t}{E \cdot A}-\frac{F_s}{E\cdot A}}{1+\frac{F_t}{E\cdot A}} = \frac{F_t-F_s}{E \cdot A+F_t} \\[5px]
\end{align}

The difference in the belt forces corresponds to the circumferential force to be transmitted (F_t-F_s=F_c). Furthermore, the value of the term E⋅A is generally much higher than the tight side force F_t. For example, the Young’s modulus of flat belts is around E = 1000 N/mm² and the permissible belt tension is around σ_per=10 N/mm². The tight side force in the order of ~σ_per⋅A will therefore be significantly lower than the value of the term E⋅A. The tight side force F_t can therefore be neglected compared to the term E⋅A, so that for the elastic slip applies in a very good approximation:

\begin{align}
\label{dehnschlupf}
&S = \frac{F_c}{A \cdot E + F_t} \approx \frac{F_c}{A \cdot E} \\[5px]
&\boxed{S \approx \frac{F_c}{A \cdot E}} \\[5px]
\end{align}

The higher the circumferential forces to be transmitted and the more elastic the belt is (e.g. low Young’s modulus!), the greater the elastic slip. With the increased elastic slip, the sliding zone also includes a larger portion of the wrap angle.

The higher the circumferential forces to be transmitted and the more elastic the belt is, the higher the elastic slip!

Note that the belt speed on the tight side is determined by the peripheral speed of the driving pulley and therefore does not change even when the circumferential force is increased (provided that the rotational speed of the driving pulley is kept constant). Only the belt speed on the slack side is reduced due to the increased elastic slip and the sliding zone increases at the expense of the adhesion zone.

Flat belts

The simplest type of belt is the flat belt. It has a rectangular cross-section and was often made of leather in the early days. Today, however, steel or high-strength synthetic materials such as polyamide or aramide are used for tension cords. These force-transmitting cords are embedded in a rubber core between a top cover and a bottom cover. The bottom layer where the belt has contact with the pulley, can be coated with special rubber to increase friction and wear resistance. The top layer on the opposite side only has a protective function.

Due to their design, flat belts can in principle run on both sides around the pulleys. In this case, both sides of the belt are specially coated. The belt can then be used for multiple belt drives and for crossed belt drives.

Flat belts allow high speeds and high torques to be transmitted (and thus high power). To prevent the flat belt from jumping off the pulley, the cross-section of the pulley has a slight convex crown. Depending on the width of the pulley, this hump usually ranges between 0.3 mm and 1.2 mm. This achieves self-centering of the belt and prevents it from running off and the belt keeps track.

Flat belts generally generate very little noise. This also has positive effects on the service life and the efficiency (approx. 98 %) and thus on the maintenance of the belt. Due to the relatively small belt thickness, the belt can be very strongly bent and thus allows the use with relatively small pulleys. A disadvantage of flat belts, however, is the relatively high bearing load caused by the high pretensioning forces.

Flat belts have high efficiency and flexibility as well as low wear and low noise levels; however, they require relatively high pretensioning forces!

V-belts

The high bearing load when using flat belts can be significantly reduced by using V-belts. The wedge-shaped cross-section leads to high frictional forces on the flanks due to the “wedge effect”. Therefore, only relatively low preload forces are necessary to generate the required frictional forces for power transmission. Accordingly, the bearing load is also significantly reduced.

Conversely, with the same pretensioning forces, much higher torques can be transmitted when using V-belts instead of flat belts. To further increase the power transmission, two or more V-belts can also be arranged parallel to each other.

V-belts can transmit significantly higher torques with the same bearing load than flat belts; however, the efficiency is lower!

The so-called groove angle \(\alpha\) is 38° or 32°, depending on the pulley diameter, whereby the belt only has contact with the pulley on the inclined flanks. The V-belt must therefore not touch the groove bottom, as the contact force must only come about by the flanks. Otherwise there would be no wedge effect! For the same radial force (bearing load), the total friction force is significantly higher for V-belts.

The figure above shows that the radial force \(F_b\) must be balanced with the two resulting normal forces \(F_n\) on the flanks. In this way, a relationship between the radial force and the normal forces can be established using the groove angle \(\alpha\):

\begin{align}
\sin\left(\tfrac{\alpha}{2} \right)&= \frac{\tfrac{F_b}{2}}{F_n} = \frac{F_f}{2 \cdot F_n}     \\[5px]
F_n &= \frac{F_b}{2 \cdot \sin\left(\tfrac{\alpha}{2} \right)}      \\[5px]
\end{align}

Since the friction force \(F_f\) is proportional to the normal force \(F_n\) according to Coulomb’s friction law, the following equation applies:

\begin{align}
F_{f} & \sim F_n  \\[5px]
F_f & \sim \frac{F_b}{2 \cdot \sin\left(\tfrac{\alpha}{2} \right)}    \\[5px]
\end{align}

Note, that this frictional force acts equally on both flanks, so that there is a total of twice the frictional force. The following therefore applies to the total frictional force:

\begin{align}
&F_{f,total}  = 2 \cdot F_f  \sim F_b \cdot \frac{1}{\sin\left(\tfrac{\alpha}{2} \right)}    \\[5px]
&\boxed {F_{t,total} \sim F_b \cdot \frac{1}{\sin\left(\tfrac{\alpha}{2} \right)}} ~~~\text{valid for V-belts}    \\[5px]
\end{align}

In comparison, the frictional force of flat belts is only proportional to the radial force \(F_b\):

\begin{align}
&\boxed {F_{f,total} \sim F_b } ~~~\text{valid for flat belts} \\[5px]
\end{align}

Consequently, V-belts with the same bearing load \(F_b\) can transmit higher forces by the factor \(\frac{1}{\sin\left(\frac{\alpha}{2} \right)}\) (due to the increased frictional force). Expressed in figures, a factor of 3.1 or 3.6 for groove angles of 38° or 32°.

Since the V-belts only rest on the flanks, they are specially designed for a certain range of pulley diameters as well as for certain wrap angles. Otherwise, for example, belts designed for larger pulleys would bend too much and the flanks would buckle and then no longer rest flat.

Due to the greater belt thickness of V-belts compared to flat belts, the energy required to bend the belt around the pulleys is higher. Therefore, V-belts have a slightly lower efficiency than flat belts (approx. 95 %).

While the transmission ratio for flat belts is determined by the outer diameter of the pulleys, in the cas of V-belts the so-called pulley pitch diameter must be taken as a basis for calculating the transmission ratio due to the special geometry. The pitch diameter \(d\) is defined by the nominal belt width \(b_w\). The nominal width corresponds to the belt width at the level of the neutral axis. Thus, according to the definition of the neutral axis, the nominal width always remains constant even when the belt is bent (i.e. when rotating around different pulley diameters).

It must be noted that V-belts must run in after initial assembly before they can be put into operation. This requires a correspondingly increased preload of approx. 30 % during initial operation.

Over the course of time, different types of V-belts have developed, depending on the application. The most important ones are described in more detail in the following sections.

Standard V-belts

“Classical” V-belts are standardized in Germany according to DIN 2215 and have a height to width ratio of 1:1.6. Tension cords made of steel, aramid, polyester or glass are embedded in an elastomer core covered by a top layer. The tension cords run at the level of the nominal width (neutral axis).

To increase friction or wear resistance and to protect the belt from harmful external influences, the V-belt can be covered with a special rubber fabric. This is then referred to as a wrapped V-belt. Such wrapped V-belts are used, for example, in drives for pumps in the chemical industry to convey aggressive media.

Wrapped V-belts offer additional protection against harmful environmental influences!

If, on the other hand, such a rubber casing is missing, the edges of the belt are “raw”, so to speak, and one speaks of an raw edge V-belt. Due to this missing of the relatively stiff sheathing, raw edge V-belts thus have better flexibility. In addition, the force transmission from the pulley to the tension cords does not take place through the sheathing but is transmitted directly throught the core. This results in increased power transmission. To improve the transverse rigidity, elastomer fibres are incorporated transversely to the running direction of the belt.

The advantage of raw edge V-belts compared to classical V-belts is the lower wear due to the lack of wear-prone sheathing and the associated low-noise operation. In addition, the raw edges can be ground in comparison to wrapped V-belts so that belts with narrower tolerances can be produced.

Raw edge V-belts show less wear and higher efficiency than wrapped V-belts!

Narrow V-belts

Compared to standard V-belts, narrow V-belts have a more favourable height to width ratio of 1:1.2. The greater height (at the same width as a classic V-belt) ensures greater power transmission. Conversely, the belt width can be much smaller with the same power transmission. The associated lower belt mass of the narrow V-belt reduces the centrifugal forces occurring during operation, so that higher belt speeds can be achieved.

However, the increased belt thickness has an adverse effect on the flexibility. To compensate for this and to be able to use narrow V-belts even with relatively small pulley diameters, they are therefore cogged (so-called narrow cogged V-belts). This increases flexibility even with strong curvatures. This is why narrow V-belts are usually found in the raw edge version.

The increased power transmission combined with the high flexibility of the cogged narrow V-belts results in a relatively space-saving design of such belt drives. In addition, the lower flexural stiffness reduces the deformation energy required when the belt runs around the pulleys, which increases the efficiency compared to the classic V-belt. For this reason, classic V-belts have to give way more and more to (cogged) narrow V-belts.

Narrow V-belts offer higher transmittable powers than classic V-belts!

Wide V-belts (variable speed belts)

So-called wide V-belts with a height-width ratio of over 1:2 are used for heavy power transmissions and for applications where large speed changes occur. Therefore they are also referred to as variable speed belts.

Variable speed belts are usually cogged to reduce flexural stiffness. Such belt types are used in continuously variable transmissions in which the pulley diameter is changed by an axial shift to adjust the transmission ratio.

Double V-belts (hex-belts)

Double V-belts are basically two V-belts, that are put on top of each other. Both sides of the belt can therefore be used to transmit power. Due to their cross-sectional shape, double V-belts are also referred to as hex-belts. Hex-belts are able to drive two pulleys with opposite sense of rotation. The double V-belt can also be used when the direction of rotation is to be reversed.

Kraftbands

If several individual V-belts are connected to each other by a cover plate, this is referred to as a kraftband (“kraft” = German word for “power” and “band” = German word for ribbon). Such a combination of several V-belts ensures, among other things, that individual V-belts do not jump off the pulley under impact loads. Kraftbands usually consist of cogged narrow V-belts in the raw edge version.

Poly V-belts (serpentine belts)

The poly V-belt (also called serpentine belt or V-ribbed belt) is a mixture of a flat belt and a V-belt, whereby the tension cords run over the entire nominal width (neutral axis) in contrast to the kraftband. Such a multible ribbed belt thus combines the advantages of both belt types to a special degree, i.e. high flexibility combined with high power transmission and relatively low bearing load. Serpentine belt are used, for example, in multiple drives in which one pulley drives several other pulley. This is the case in automobiles, for example, where the engine has to drive not only the alternator but also the pump for the servomotor, the air conditioning compressor, the fan and the water pump.

Round belts

Round belts are special belts that are used almost exclusively for motion transmission and less for power transmission. Due to their symmetrical cross section, round belts can very easily be guided in different directions with the aid of guiding pulleys. The figure below shows the motion transmission of a round belt for centripetal force measurement.

Timing belts (synchronous belts)

With friction-locking belt types such as flat belts and V-belts, slippage will occur, which reduces the efficiency and control accuracy accordingly. However, this can be prevented by toothed belts, as the teeth attached to the belt surface then transmit the force positively. Slippage cannot take place. Therefore, toothed belts are always used when precise positioning is required. For this reason toothed belts are also called timing belts or synchronous belts.

The figure below shows the timing belts used to control the printhead of a 3D printer.

Circumferential force in the belt

In belt drives, forces are transmitted from the input pulley to the belt and then from the belt back to the output pulley. The force to be transmitted from one pulley to the other is also referred to as the effective force or circumferential force F_c. The amount of force F_c applied to the circumference of the input pulley depends on the torque M_i or the power P_i and the speed n_i as well as the diameter d_i of the driving pulley:

\begin{align}
&M_i = F_c \cdot \frac{d_i}{2} ~~~~~\text{and} ~~~~~ P_i = 2 \pi M_i n_i\\[5px]
\label{umfang}
&\boxed{F_c = \frac{2 M_i}{d_i} = \frac{P_i}{\pi d_i n_i}} \\[5px]
\end{align}

This circumferential force F_c causes different forces on the two belt sections and a tight side and a slack side will form. The balance of forces on a pulley generally shows that the difference between the tight side force F_t and the slack side force F_s corresponds to the transmitting circumferential force F_c:

\begin{align}
\label{fu}
&\boxed{F_c = F_t – F_s} \\[5px]
\end{align}

The decisive factor for the transmission of power in belt drives is the circumferential force (also called effective force), which results from the difference between the tight side force and the slack side force of the belt!

At the pulley with the diameter d_o driven by the belt this effectively circumferential force F_c leads to a changed torque M_o (see also the section transmission ratio):

\begin{align}
&\boxed{M_o = F_c \cdot \frac{d_o}{2}} \\[5px]
\end{align}

The circumferential force is generated at the input pulley by the torque acting there and the effective diameter of the pulley. At the output pulley, the circumferential force causes a change in torque in accordance with the effective diameter of the output pulley.

The circumferential force is transmitted at the interface between belt and pulley by frictional forces. These maximum possible frictional forces must be sufficiently high to ensure that the circumferential force can be transmitted safely. If the maximum possible frictional force is lower than the circumferential force to be transmitted, then obviously there is not enough “adhesion” to be able to transfer the circumferential force from the pulley to the belt or from the belt to the pulley according to the equation (\ref{umfang}). The pulley or belt slips (sliding slip) and the full circumferential force is no longer transmitted.

The circumferential force is transmitted by frictional forces between pulley and belt!

Maximum transmittable circumferential force

In the article Basics of power transmission it was shown with the belt friction equation that the maximum frictional force F_f,max between belt and pulley at a given tight side force F_t or slack side force F_s can be determined with the formulae listed below. These maximum frictional forces therefore also represent the limit for the maximum transmittable circumferential forces F_c,max:

\begin{align}
\label{leertrum}
& F_{f,max} = \boxed{F_{c,max} = F_s \cdot \left(e^{\mu \cdot \varphi} -1 \right)} \ge F_c \\[5px]
\end{align}

or

\begin{align}
\label{zugtrum}
&F_{f,max} = \boxed{F_{c,max} =F_t \cdot \left(1-\frac{1}{e^{\mu \cdot \varphi}} \right)} \ge F_c \\[5px]
\end{align}

Gain

In the context of the circumferential force, the so-called gain k describes the percentage of the existing tight side force F_t that could be used to transmit the maximum circumferential force F_c,max (limiting case to slippage). According to equation (\ref{zugtrum}) the gain corresponds to the term (1-1/e^µφ):

\begin{align}
&k = \frac{F_{cmax}}{F_t} = 1-\frac{1}{e^{\mu \cdot \varphi}} \\[5px]
\label{ausbeute}
&\boxed{k =1-\frac{1}{e^{\mu \cdot \varphi}} } \\[5px]
\end{align}

A gain of e.g. k = 0.6 means that a maximum of 60% of the tight side force can be used as circumferential force for power transmission. In this case, the remaining 40 % is required for maintaining the belt’s tension on the slack side (slack side force).

The gain indicates what percentage of the tight side force can be used as the maximum circumferential force; the rest is accounted for by the slack side force to maintain the belt tension!

Note that the circumferential force is transmitted by frictional forces and their formation always requires a certain pressure of the belt on the pulley. This means that there must always be an acting force on the slack side of the belt to ensure the belt tension and with it the contact pressure. The slack side must therefore not be force-free at all (see also the article Basics of power transmission)!

The relationship between a given tight side force F_t and the maximum possible circumferential force F_c,max is thus established by using the gain k as follows:

\begin{align}
&\boxed{F_{c,max} = F_t \cdot k } \\[5px]
\end{align}

According to the equation (\ref{ausbeute}), the gain depends only on the wrap angle and the coefficient of friction. While the coefficient of friction applies equally to both pulleys, the wrap angle of the smaller pulley is usually smaller. Thus, the smaller pulley (often the driving pulley) is decisive for the gain or the entire power transmission!

The smallest wrap angle of the pulleys is always decisive for power transmission!

However, based on the equation (\ref{ausbeute}) it can also be seen that the greater the coefficient of friction, the less influence the wrap angle has on the gain. The relatively small change in the wrap angle under load (due to the sagging of the slack side of the belt) therefore plays a subordinate role for large coefficients of friction in practice.

Elastic slippage

Due to the elasticity of the belt, elongation processes occur in the belt during rotation around the pulleys. These stretching processes are due to the increase (or decrease) in the belt tension at the transition from the slack side to the tight side (or vice versa) – the belt stretches according to the force acting on the pulley.

This results in relative motions between belt and pulley (called slastic slip). It is therefore not a static friction between belt and pulley but rather a sliding friction! Therefore, when applying the above equations, the sliding coefficient of friction should also be used instead of the static coefficient of friction.

The coefficient of sliding friction is decisive for power transmission!

However, it must be noted that the sliding friction depends on the (elongation) speed. Thus, the coefficient of sliding friction is also influenced by the belt speed!

Pre-load tension (initial tension of the belt)

The equations (\ref{leertrum}) and (\ref{zugtrum}) reflect the maximum possible, transmittable circumferential forces F_c,max depending on the slack side force F_s or tight side force F_t. These equations must therefore be interpreted at the limit to slippage. In principle, however, belt drives are not driven at the limit to slippage, but well below it.

The maximum possible circumferential force according to equation (\ref{leertrum}) or equation (\ref{zugtrum}) is therefore not fully utilized to transmit forces from the belt to the pulley and vice versa. However, the equations show that an increase in the slack side force or tight side force leads to an increase in the maximum frictional force.

This can be clearly explained by an increase in the contact pressure of the belt on the pulley, which increases the maximum frictional force. By increasing the belt forces, safety against slippage can be increased or larger circumferential forces can be transmitted.

The tension in the two spans of the belt (slack side and tight side) is influenced by the pre-tensioning of the belt (initial tension). The stronger the belt is already under tension in the load-free state by the so-called pre-load, the higher the belt tension will be during operation under load. Since higher belt tensions mean higher frictional forces, higher circumferential forces can be transmitted.

Preload without taking centrifugal forces into account

In principle, it must be borne in mind that the belt forces arising during operation depend on the circumferential force to be transmitted. In the load-free idle state, only the pre-load F_p in the belt initially acts. If a circumferential force F_c is introduced by the torque of the input pulley, the belt force in the tight side increases to F_t and the slack side force decreases to the same extent to F_s.

Due to the equilibrium of forces (more precisely: equilibrium of torques), the difference between the tight side force and the slack side force corresponds to the circumferential force to be transmitted. This means that the tight side force increases by just half the circumferential force and the slack side force decreases by half the circumferential force:

\begin{align}
&\boxed{F_t = F_p + \tfrac{F_c}{2}} \\[5px]
\label{F_L}
&\boxed{F_s = F_p – \tfrac{F_c}{2}} \\[5px]
&F_t – F_s = \left(F_p+\tfrac{F_c}{2} \right) – \left(F_p – \tfrac{F_c}{2} \right) = F_p + \tfrac{F_c}{2} – F_p + \tfrac{F_c}{2} = F_c \\[5px]
\end{align}

In accordance with the equation (\ref{leertrum}), the slack side force must under no circumstances drop to zero under load, as otherwise the belt tension would be lost. There would be no contact pressure between belt and pulley and there would be no friction that would be capable of transmitting circumferential forces.

The preload F_p in equation (\ref{F_L}) must therefore be chosen in such a way that the slack side force F_s does not fall below a critical value during operation, i.e. when the circumferential force F_c is transmitted. This critical case is reached when the slack side force has decreased to such an extent that the circumferential force F_c can just be transmitted with the maximum possible circumferential force according to the equation (\ref{leertrum}) [F_c,max=F_c]. The slack side force must therefore not fall below the following value:

\begin{align}
&F_{c} = F_{c,max} = F_{s,min} \cdot \left(e^{\mu \cdot \varphi} -1 \right)   \\[5px]
& F_{s,min} = F_{c} \cdot \frac{1}{e^{\mu \cdot \varphi} -1}   \\[5px]
\end{align}

With this slack side force F_s,min, which must be present at least to be able to transmit the given circumferential force F_c, the minimum preload F_p,min can then also be determined according to equation (\ref{F_L}):

\begin{align}
\label{F_V}
& F_{s,min} = F_{p,min} – \tfrac{F_c}{2} \\[5px]
& F_{p,min} = F_{L,min} + \tfrac{F_U}{2} \\[5px]
& F_{p,min} = F_{c} \cdot \frac{1}{e^{\mu \cdot \varphi} -1} + \tfrac{F_c}{2} \\[5px]
& F_{p,min} = F_{c} \cdot \left( \frac{1}{e^{\mu \cdot \varphi} -1} + \frac{1}{2} \right) \\[5px]
& \underline{F_{p,min} = F_{c} \cdot \frac{e^{\mu \cdot \varphi}+1}{2 \left(e^{\mu \cdot \varphi} -1 \right) }} ~~~\text{without taking centrifugal forces into account} \\[5px]
\end{align}

Preload with taking centrifugal forces into account

In the previous section, centrifugal forces have not yet been taken into account, but at high belt speeds v they lead to a reduction in contact pressure and thus to a reduction in frictional force. In such a case, the required circumferential force can no longer be transmitted. To compensate for this effect, the belt must be additionally preloaded by the amount of the centrifugal effect F_cf=m’⋅v² (with m’ as specific weight – “mass per unit belt length”):

\begin{align}
& F_{p,total,min} = F_{p,min} + F_cf \\[5px]
& \boxed{F_{p,total,min} = F_{c} \cdot \frac{e^{\mu \cdot \varphi}+1}{2 \left(e^{\mu \cdot \varphi} -1 \right)} + m’ \cdot v^2} ~~~\text{taking centrifugal forces into account} \\[5px]
\end{align}

Since the centrifugal force during operation is compensated by the additional preload F_cf, only the preload F_p remains as the relevant effect for contact pressure, which is therefore also referred to as dynamic preload. In contrast, the total (static) preload F_p,total also takes into account the centrifugal force to be compensated, which only acts as an additional contact force at rest. In a separate article, the concept of the belt centrifugal force is explained in more detail and the derivation of the corresponding formula is shown.

Preload measuring

The sections above showed, that the higher the circumferential force to be transmitted and the higher the belt speed, the greater the total preload must be. However, excessive pre-tensioning forces should be avoided, as this not only leads to high bearing loads but also increases belt wear. In addition, only small circumferential forces can be transmitted with high pretensioning forces, as otherwise there is a risk that the belt will tear under the high tension.

In practice, the preload force of the belt can be measured and adjusted accordingly by vibration tests. For this the belt span is picked like the string of a guitar. The frequency at which the belt span now oscillates is called natural frequency f. In the same way that the oscillation frequency (pitch) of a guitar is determined by the tension of the string, the natural frequency of the oscillating string is also directly related to the pre-load tension.

The higher the tension of the belt, i.e. the higher the preload, the higher its natural frequency. The frequency of the oscillating belt span is finally measured with the aid of an optical measuring device. In addition to the preload F_p,total, the span length l and the specific weight m’ (“mass per meter belt length”) influence the natural frequency, but these variables are generally known in advance. The following formula can then be used to determine the preload from the measured natural frequency:

\begin{align}
&F_{p,total} = 4 \cdot f^2 \cdot m’ \cdot l^2 \\[5px]
\end{align}

Or at a desired preload, the natural frequency to be set applies:

\begin{align}
& \boxed{f = \sqrt{\frac{F_{p,total}}{4 \cdot m’ \cdot l^2}} } \\[5px]
\end{align}