Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

1. Engineering 36 Ch08: Wedge &Belt Friction Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. Outline - Friction • The Laws of Dry Friction • Coefficient of Static Friction • Coefficient of Kinetic (Dynamic) Friction • Angles of Friction • Angle of static friction • Angle of kinetic friction • Angle of Repose • Wedge & Belt Friction • Self-Locking & Contact-Angle

3. Basic Friction - Review • The Static Friction Force Is The force that Resists Lateral Motion. It reaches a Maximum Value Just Prior to movement. It is Directly Proportional to Normal Force: • After Motion Commences The Friction Force Drops to Its “Kinetic” Value

4. Wedge Friction • Consider the System Below • Find the Minimum Push, P, to move-in the Wedge • The Wedge is of negligible Weight • Then the FBD of the Two Blocks using Newton’s 3rd Law

7. Wedge Friction • For Equilibrium of the Heavy Block • Solve for FA,n • For Equilibrium of the Wt-Less Wedge

8. Wedge Friction • In the last 2-Eqns Sub Out FA,n • Eliminating FC,n from the 2-Eqns yields an Expression for Pmin:

9. Wedge Friction • MATLAB Plots for P when W = 100 lbs

10. MATLAB Code % Bruce Mayer, PE % ENGR36 * 22Jul12 % ENGR36_Wedge_Friction_1207.m % u = 0.2 W = 100 a = linspace(0,20); P = W*((1-u*u)*sind(a) +2*u*cosd(a))./(cosd(a)-u*sind(a)) plot(a,P, 'LineWidth',3), grid, xlabel('\alpha (°)'), ylabel('P (lbs)'), title('W = 100 lbs, µ = 0.2') disp('showing 1st plot - Hit Any Key to Continue') pause % a = 10; u = linspace(0,0.3); P = W*((1-u.*u)*sind(a) +2*u*cosd(a))./(cosd(a)-u*sind(a)); plot(100*u,P, 'LineWidth',3), grid, xlabel('µ (%)'), ylabel('P (lbs)'), title('W = 100 lbs, \alpha = 10°') disp('showing 2nd plot - Hit Any Key to Continue') pause % u = linspace(0, .50); aSL =atand (2*u./(1-u.^2)); plot(100*u,aSL, 'LineWidth',3), grid, xlabel('µ (%)'), ylabel('\alpha (°)'), title('Self-Locking Wedge Angle') disp('showing LAST plot')

11. Wedge Friction • Now What Happens upon Removing P • The Wedge can • Be PUSHED OUT • STAY in Place • SelfLocking condition • Then the FBD When P is Removed • Note that the Direction of the Friction forces are REVERSED

14. Wedge Friction • For Equilibrium of the Heavy Block • Solve forFA,n • For Equilibrium of the Wt-Less Wedge

15. Wedge Friction • To Save Writing sub K for FA,n • Eliminate FC,n • Now Divide Last Eqn by Kcosα

16. Wedge Friction • Dividing by Kcosα • Recognize sinu/cosu = tanu

17. Wedge Friction • After all That AlgebraFind The Maximumα to Maintain the Block in the Static Location • Since Large angles Produce a Large Push-Out Forces, and a ZERO Angle Produces NO Push-Out Force, the Criteria for Self-Locking

18. Wedge Push-Out • SMALL PushOut Force • Likely SelfLocking • LARGE PushOut Force • Likely NOT SelfLocking

19. Wedge Friction

20. Belt Friction • Consider The Belt Wrapped Around a Drum with Contact angle . • The Drum is NOT Free-Wheeling, and So Friction Forces Result in DIFFERENT Values for T1 and T2 • To Derive the Relationship Between T1 and T2 Examine a Differential Element of the Belt that Subtends an Angle  • The Diagram At Right Shows the Free Body Diagram

21. Write the Equilibrium Eqns for Belt Element PP’ if T2>T1 Belt Friction cont • Eliminate N from the Equations

22. Combining Terms Belt Friction cont.1 • Divide Both Sides by  • Now Recall From Trig And Calculus • So in the Above Eqn Let: /2 →0; Which Yields

23. The Belt Friction Differential Eqn Belt Friction cont.2 • Integrate the Variables-Separated Eqn within Limits • T( = 0) = T1 • T( = ) = T2 • From Calculus • Now Take EXP{of the above Eqn}

24. This is a VERY POWERFUL Relationship Belt Friction Illustrated • Condsider the Case at Right. Assume • A ship Pulls on the Taut Side With A force of 4 kip (2 TONS!) • The Wrap-Angle = Three Revolutions, or 6 • µs = 0.3 • The Tension, T1, Applied by the Worker

25. WhiteBoard Work Let’s WorkThese Nice Problems

26. Engineering 36 Appendix Bruce Mayer, PE Registered Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

27. WhiteBoard Work Let’s WorkThis NiceProblem

31. Wedge Push-Out • SMALL PushOut Force • Likely SelfLocking • LARGE PushOut Force • Likely NOT SelfLocking