ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 1 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Engineering 36 Ch08: Wedge & Belt Friction

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 2 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 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

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 3 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 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

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 4 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 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 3 rd Law

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 5 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 6 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 7 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Wedge Friction  For Equilibrium of the Heavy Block  Solve for F A,n  For Equilibrium of the Wt-Less Wedge

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 8 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Wedge Friction  In the last 2-Eqns Sub Out F A,n  Eliminating F C,n from the 2-Eqns yields an Expression for P min :

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 9 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Wedge Friction  MATLAB Plots for P when W = 100 lbs

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 10 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 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')

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 11 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 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

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 12 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 13 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 14 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Wedge Friction  For Equilibrium of the Heavy Block  Solve for F A,n  For Equilibrium of the Wt-Less Wedge

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 15 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Wedge Friction  To Save Writing sub K for F A,n  Eliminate F C,n  Now Divide Last Eqn by Kcosα

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 16 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Wedge Friction  Dividing by Kcosα  Recognize sinu/cosu = tanu

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 17 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Wedge Friction  After all That Algebra Find The Maximum α 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

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 18 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Wedge Push-Out  SMALL PushOut Force Likely SelfLocking  LARGE PushOut Force Likely NOT SelfLocking

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 19 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Wedge Friction

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 20 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 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 T 1 and T 2  To Derive the Relationship Between T 1 and T 2 Examine a Differential Element of the Belt that Subtends an Angle  The Diagram At Right Shows the Free Body Diagram

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 21 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Belt Friction cont  Write the Equilibrium Eqns for Belt Element PP’ if T 2 >T 1  Eliminate  N from the Equations

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 22 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Belt Friction cont.1  Combining Terms  Divide Both Sides by   Now Recall From Trig And Calculus  So in the Above Eqn Let:  /2 →0; Which Yields

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 23 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Belt Friction cont.2  The Belt Friction Differential Eqn  Integrate the Variables-Separated Eqn within Limits T(  = 0) = T 1 T(  =  ) = T 2  From Calculus  Now Take EXP{of the above Eqn}

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 24 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Belt Friction Illustrated  This is a VERY POWERFUL Relationship  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  µ s = 0.3  The Tension, T 1, Applied by the Worker

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 25 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics WhiteBoard Work Let’s Work These Nice Problems

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 26 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Bruce Mayer, PE Registered Electrical & Mechanical Engineer Engineering 36 Appendix

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 27 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics WhiteBoard Work Let’s Work This Nice Problem

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 28 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 29 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 30 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx 31 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Wedge Push-Out  SMALL PushOut Force Likely SelfLocking  LARGE PushOut Force Likely NOT SelfLocking