Belt and Wrap Friction: Determining Frictional Forces in Mechanical Systems, Study notes of Statics

Download Belt and Wrap Friction: Determining Frictional Forces in Mechanical Systems and more Study notes Statics in PDF only on Docsity! BELT AND WRAP FRICTION Rob Wendland, driving Mike Troxel's Federal-Mogul Dragster In-N-Out Burger A Capstan-Style Winch 2 Belt 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 T2 T1 θ r  A flat belt has a wrap angle around a pulley of θ 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<T2  Furthermore, since belt tension changes continuously, so do the incremental normal and frictional forces 5 Belt and Wrap Friction x y  2  TT T N F=⋅N  2 ∫ T 1 T 2 dT T = ∫ 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 ∆θ with infinitely small values of dT and dθ, we could integrate over θ as follows:  This evaluates to:  It's important to remember T2 is always larger than T1 and θ is in radians T 2= T 1e  6 V-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 α must be expressed in radians T 2= T 1e /sin  /2 α 7 Example 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 40o 60o 10 Frictional Forces Real 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 = 45⋅ 180 = 0.25 rad T 2 = T 1⋅e 0.40⋅0.25 = 1.37⋅T 1 T2 T1 T1 T1 T2 T2 B A Cµ = 0.40 (B)  = 90o = 90⋅ 180 = 0.50 rad T 2 = T 1⋅e 0.40⋅0.50 = 1.87⋅T 1 (C)  = 135o = 135⋅ 180 = 0.75 rad T 2 = T 1⋅e 0.40⋅0.75 = 2.56⋅T 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 µ = 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 π 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 F T 1= T 2e − T 1= 300⋅e−0.1⋅ = 219 lb f T 2= T 1e  T 2= 300⋅e0.1⋅ = 411 lb f