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Belt friction

Fig. 4.7 shows a flat belt passing over a fixed cylindrical drum. To model belt slipping let us consider interactions between segments of the belt and drum surface which are in contact. This can be taken as a one-dimensional translational problem if the belt segment as well as the adhering drum surface is straighten up as shown in Fig. 4.7. The variables can be then related to the coordinate axis .
Figure 4.7: Belt friction.

The Eytelwein's formula gives the relation between the forces and acting at the belt-segment endpoints A and B. For the case when the belt is just about to slip if pulled to the right, i.e. if , the formula is:

(4.14)

where  [-] is the coefficient of friction, and  [rad] is the angle of the belt contact. This formula is derived under the assumption that the belt segment is massless and rigid, and the friction between the belt and the drum is governed by the `law of dry friction'.

A multipole diagram of the belt friction constructed under the above assumptions is given in Fig. 4.7. The contact areas of the belt and drum segments re represented there by poles A and C, respectively. Friction is modelled by a source of friction force acting between these two poles. The rigid belt segment is modelled by the source of zero velocity placed between the poles A and B representing the belt-segment end points. At the same time, this source is employed in the model as a sensor of the tension in the belt segment. This tension force is used to control the friction-force source .

Taking into consideration the balance relations for forces at the poles

(4.15)

and substituting them into the formula we shall obtain the constitutive relation of the friction-force source
(4.16)

The function sgn was included into the formula to make it valid also for the case when the belt is pulled to the left, i.e. for . (You are encouraged to check its correctness.)

Equation (4.16) can be used both for static friction of impending slipping as well as for the slipping kinetic friction after substituting there either the coefficient of static friction or the coefficient of kinetic friction for . Fig. 4.8 shows an example of the - characteristic of combined static and kinetic friction governed by (4.16) for the tension . Note that the characteristic is asymmetric with respect to the origin.

Figure 4.8: Belt friction characteristic.

As shown in Fig. 4.7, the belt friction model from Fig. 4.7 can be completed to respect the belt segment flexibility , and mass . The belt friction model we have derived for moving belt and fixed drum applies equally well to problems involving fixed belt and rotating drum, like band brakes, and to problems involving both belt and drum rotating, like belt drives. It can be also used to systems with the belt replaced by a rope or a string.

In case of V-shaped belts (4.16) should be replaced by

(4.17)

where is the angle of the `V' of the belt cross section.

Fig. 4.9 shows an example of a hoisting machine lifting a load by means of a cable passing over an undriven sheave and wrapped around a driven drum. The rotary motion of the drum drive is transformed into the translational motion of the cable by means of an ideal transformer with the ratio , where is the radius of the drum. The first and second part of the cable and is modelled by elements characterizing the cable flexibility and damping. The element parameters of the second part of the cable depend on the length of . The friction between the cable and the sheave is represented by the belt-friction model given above. The sheave dynamics, converted from rotary to translational motion along the cable, is represented by the elements of reduced parameters

(4.18)

where is the moment of inertia of the sheave, is the damping of its bearing, and is its
Figure 4.9: Hoisting machine.



Next: Translatory-to-rotary couplings Up: Rotary-to-rotary couplings Previous: Belt and chain systems   Contents
Herman Mann 2005-05-05