BELT AND WRAP FRICTION

Rob Wendland, driving Mike Troxel's Federal-Mogul Dragster In-N-Out Burger

A Capstan-Style Winch

2Belt and Wrap Friction When flat belts, V-belts, band brakes, and line-wrap around

capstans are used in any form of product, the frictional forces developed must be determined. All forms of wrap friction are analyzed similarly. Let's first look at flat belts

T2T1

r

A flat belt has a wrap angle around a pulley of radians. Normal forces develop at each finite point of contact with a resulting frictional force acting tangent to the pulley and opposite the direction of rotation. This reduces belt tension around the pully such that tension T1

3Belt and Wrap Friction To help determine how frictional forces affect belt tension,

consider a finite element over angle . Acting on this element is slackside tension T and tightside tension T + T

Incremental forces N and F develop as a result of the applied incremental tensions. By performing an equilibrium analysis:

x

y2

TT T

NF=N

2

F x = 0

= T cos2 N T T cos2 F y = 0

= N T T sin 2 Tsin 2

4Belt and Wrap Friction

x

y2

TT T

NF=N

2

For small , sin 2 = 2 and cos 2 = 1Substituting: F x => T = N F y => N = T

Substituting N => T = T

Since this expression is developed over a very small angle, this would have to be summed around the wrap angle

iT i =

iTi

5Belt and Wrap Friction

x

y2

TT T

NF=N

2

T 1

T 2dTT=

0

d

Since this expression is developed over a very small angle, applying such a summation would be impractical

However, if we replaced the finite quantities of T and with infinitely small values of dT and d, we could integrate over as follows:

This evaluates to:

It's important to remember T2 is always larger than T1 and is in radians

T 2= T 1e

6V-Belt Friction The friction developed by a V-belt can

be developed in a similar fashion. The relationship between tightside and slackside tension for a V-belt is:

Angle must be expressed in radians

T 2= T 1e/sin /2

7Example 18 The accessory package for the engine shown below requires a

torque of 30 ft-lbf. The pulley in contact with the belt has a diameter of 8 and a coefficient of static friction of 0.30. Determine the tension in each part of the belt if the belt is not to slip and:

the system uses a flat belt the system uses a V-belt with a 38o V

and the same coefficient of friction

4040oo

6060oo

8Example 18 First determine angle of wrap. Draw

a construction line at the base of vector TB and parallel to vector TA. Angle is the difference between angles of the two vectors and is equal to 20o. This results in a wrap angle of 200o or 1.11pi radians

Since there are two unknowns, we

TA

TB

40o

60o

must develop two mathematical relationships to solve for the unknown tensions. These are the moment about the center of the pulley and the friction equation for a flat belt. Since moment is applied clockwise, tension TB is tightside tension; we will let T2 = TB and T1 = TA

9Example 18TA

TB

40o

60o

M o= T B4 in T A4 in 30 ftlb f 12 in / ft

. . . T B = T A 90 (Moments)-and-

T B = T Ae0.31.11 (Friction)

Substituting:T A 90 = T Ae

0.31.11= 2.85T A. . . T A= 48.6 lb f and T B= 139 lb f

The application of a V-belt changes only the friction equation. The 38o Vis 0.211 radians. Modifying the friction equation changes the solution to:

T A 90 = T Ae0.31.11/sin0.106= 24.5T A

. . . T A= 3.82 lb f and T B = 98.4 lb f

Notice the efficiency increase of a V-belt over that of a flat belt. The reducedtensions help increase bearing life

10

Frictional ForcesReal Life Mechanisms

45o

90o 135o

Since area does not affect the developed friction for a belt, the formula for flat belts is applicable to round line. Only angle of wrap and coefficient of friction govern developed tension

(A) = 45o

= 45180

= 0.25 rad

T 2 = T 1e0.400.25

= 1.37T 1

T2

T1

T1

T1

T2 T2

B

A

C = 0.40

(B) = 90o

= 90180

= 0.50 rad

T 2 = T 1e0.400.50

= 1.87T 1

(C) = 135o

= 135180

= 0.75 rad

T 2 = T 1e0.400.75

= 2.56T 1

(A)

11

Example 19 You are lifting an engine out of your

friends car. In your front yard is a sturdy oak tree. You wrap a rope around a tree branch to lift the engine. The engine weighs 300 lbf, the rope has a 180o wrap, and = 0.1 How much force does it take to lift the

engine from the vehicle? How much force must you apply to

lower the engine back into place? 300 lbf F

12

Example 19 Line wrap is 180o or pi radians. When

lifting, force F is the tightside of the line (T2). To solve for T2:

To lower the engine, force F becomes the slack side of the line (T1). To solve for T1:

300 lbf FT 1 = T 2e

T 1 = 300e0.1 = 219 lb f

T 2 = T 1e

T 2 = 300e0.1= 411 lb f

13

Frictional ForcesReal Life Mechanisms

Mechanisms using wrap friction are often critical life-safety devices. Such devices are used in rappelling, rock climbing, sailing, and rigging of equipment

Some devices, such as the rack, have adjustable bars. Others, like

the brake tube, rely on the number of wraps around the tube, while the anti-gybe device relies on how many passes of line through the non-adjustable bars

From left to right are: RescueTECH rappell rack; PMI brake tube for lowering long distances; and the Wichard gyb`easy to control the boom of a sailboat during a gybe. All devices must meet stringent NFPA, OSHA, ABYC, or Coast Guard requirements depending upon their application

14

Consider the adjustable rack shown. The leftmost configuration has closely spaced bars resulting in 120o of contact on each turn. The other two are spaced to provide contact arcs as shown. Determine the tension ratio for each configuration if = 0.25

Example 20

(A) (B) (C)

(A) = 800o

= 4.44 rad

T 2 = T 1e0.254.44

= 32.8T 1

(B) = 560o

= 3.11 rad

T 2 = T 1e0.253.11

= 11.5T 1

(C) = 315o

= 1.75rad

T 2 = T 1e0.251.75

= 3.95T 1

15

Building the Great Pyramids...Friction Was a Factor

=0.04bearings

=0.5roll-rope

=0.04sledge-tracks

=0.2static friction

The Egyptions and Mayans responsible for building the pyramids on their respective continents were able to reduce friction where it wasn't wanted and maximize friction where it was useful and desired. We still do this today in equipment design

Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Building the great pyramidsFriction was a factor