Specify and determine the limiting conditions for sliding, toppling, Describe the mechanism and characteristics of dry friction,

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1 5. Friction 4 Static, 0/ Department of Mechanical Engineering, Chulalongorn Univerit Objective Student mut be able to Coure Objective Include friction into equilibrium anale Chapter Objective For friction in machine Prove, tate limitation and appl formula for impending motion in wedge, ingle continuou thread/crew, flat belt, di/clutche, pivot/collar/thrut bearing, journal bearing Analze machine with friction 3 Objective Student mut be able to Coure Objective Include friction into equilibrium anale Chapter Objective For general dr friction Decribe the mechanim and characteritic of dr friction, coefficient of friction and angle of friction Specif and determine the limiting condition for liding, toppling, free rolling and driven wheel under dr friction b appropriate FBD Specif the limiting condition for rear/front wheel drive b appropriate FBD Analze bodie/tructure with friction for unnown load/reaction b appropriate FBD Content Dr friction Characteritic, theor, coefficient and angle of friction Application Wedge Thread, crew Belt Di and Clutche Collar, pivot, journal and thrut bearing 4

2 Friction Force of reitance acting on a bod which prevent or retard lipping of the bod relative to a urface with which it i in contact. The frictional force act tangent to the contacting urface in a direction oppoed to the relative motion or tendenc for motion of one urface againt another Two tpe of friction Fluid friction eit when the contacting urface are eparated b a film of fluid. Dr friction occur between contacting urface without lubricating fluid. Equilibrium Slipping and/or tipping effect Frictional force F increae with force P. 5 7 Theor # Friction force are tangential force generated between contacting urface. Friction force arie in part from the interaction of the roughnee or aperitie of the contacting urface. Static Friction Equilibrium Impending Motion Maimum value of frictional force F on the object in equilibrium i called the limiting tatic frictional force. 6 8

3 Impending Motion Static Friction Coefficient F N F N maimum tatic frictional force coefficient of tatic friction normal force F i the maim um friction force that can be eerted b dr, contacting urface that are tation ar relat ive t o each ot he r. 9 Impending Motion Static Friction Tpical Value Contact Material Metal / ice Wood / wood Leather / wood Leather / metal Aluminum / Aluminum.0.70 How to obtain value of? Impending Motion Static Friction Angle of Static Friction N R co φ F R in φ N F Rin φ tan φ N Rco φ φ tan 0 Frictional Force More force i required to tart liding than to eep it liding can be eplained b the neceit to brea thee bond. 3

4 Kinetic Friction Coefficient & Angle Motion F N F N inetic frictional force coefficient of inetic friction normal force φ tan 3 Characteritic Frictional force act tangentiall to the contacting urface, oppoing the relative or tendenc for motion. F i independent of the area of contact, provided that the normal preure i not ver low nor great enough for deformation of the urface. In equilibrium: Impending lipping: φ φ Slipping: φ φ Ver low velocit: φ φ 5 Motion Kinetic Friction Angle of Friction At φ φ, lip i impending. At φ φ, urfa ce are lidin relative to eac h other. g φ φ 4 Motion Sliding Relative liding (tranlation motion) between two urface Toppling Fall over (rotation) about the edge Topple, tipping, rolling, tumble, trip 6 4

5 Eample Friction # Place a heet of one material on a flat board. Bond a heet of other material to bloc A. Place the bloc onto the board and lowl increae the tilt a of the board until the board, and lowl increae the tilt a of the board until the bloc lip. Show that the coefficient of tatic friction between the two material i related to the angle a b tanα Eample Friction # Will thi crate lide or topple over? 7 9 Eample Friction # α N W A F The bloc i about to lip F 0 N W coα 0 N W co α () F 0 F + W inα 0 N W in α () ()/() tan α An Eample Friction # The crate i about to lide. F F N 8 0 5

6 Eample Friction #3 The crate i about to topple. F < F ( N ) The crate i about to lide and topple. F F N Eample Hibbeler E 8- # F + 0 (80 N)co30 F 0 F 69.3 N F + 0 (80 N)in N + N 0 N 36. N C C M + O 0 (80 N)in30 (0.4 m) (80 N)co30 (0. m) + N ( ) m 9.08 mm 3 C << 0.4 m The crate will not topple. F < N (70.8 N) The crate, i cloe to but doen't not lip. C 3 Eample Hibbeler E 8- # The uniform crate ha a ma of 0 g. If a force P 80 N i applied to the crate, determine if it remain in equilibrium. The coefficient of tatic friction 0.3. Eample Hibbeler E 8-3 # The rod with weight W i about to lip on rough urface at A and B. Find coefficient of tatic friction. 4 6

7 Eample Hibbeler E 8-3 # Impending lip F N A A F N B B 3 equilibrium equation, 3 unnown ( N, N, ) A B F 0 + NA + NB co30 NB in30 0 () F 0 + NA W + NB co30 + NB in30 0 () MA 0 + NBl W co30 ( l / ) 0 (3) Eample Hibbeler E 8-4 # Find minimum coefficient of tatic friction that eep the tem in equilibrium. 5 7 Eample Hibbeler E 8-3 #3 From (3) N W From () & () B 0.8 An Eample Hibbeler E 8-4 # Smmetr of lower pipe about vertical ai paing through O. Two tpe of contact: Pipe pipe Pipe ground N N, F F, A B A B F N A, p-p A F N C, p-g C 6 8 7

8 Eample Bedford E 9.5 # Suppoe that α 0 and the coefficient of friction between the urface of the wedge and the log are 0. and 0.0. Neglect the weight of the wedge. If the wedge i driven into the log at a contant rate b vertical force F, what are the magnitude of the normal force eerted on the log b the wedge? Will the wedge remain in place in the log when the force i removed? 9 On the verge of lipping Eample Bedford E 9.5 #3 Smmetr about the center of the wedge F 0 + Nin( α / ) + Nco( α / ) 0 tan( α / ) A, wedge will remain in place. An < 3 Driven at contant rate Eample Bedford E 9.5 # Smmetr about the center of the wedge F 0 + Nin( α / ) + Nco( α / ) F 0 N N F in( α /) + ( α /) F in(0 / ) + (0.0)(0 / ) N.75 F An Eample Bedford 9.0 # The coefficient of tatic friction between the two boe and between the lower bo and the inclined urface i. What i the larget angle α for which the lower bo will not lip

9 Eample Bedford 9.0 # FBD of upper bloc F 0 N W coα 0 N W co α FBD of lower bloc F 0 N + W coα N 0 N W co α F 0 N + N W inα 0 3W coα W inα tan α / 3 or α tan (3 ) An Eample Bedford 9.30 # Impending rotation F + 0 Nw + Nf W 0 F + 0 Nw Nf 0 M 0 + O M ( Nw + N) r 0 W W Nw, Nf + + ( + ) M Wr + An Eample Bedford 9.30 # The clinder ha weight W. The coefficient of tatic friction between the clinder and the floor and between the clinder and the wall i. What i the larget couple M that can be applied to the tationar clinder without cauing it to rotate. Eample Bedford 9.33 # The di of weight W and radiu R i held in equilibrium on the circular urface b a couple M. The coefficient of tatic friction between the di and the urface i. Show that the larget value M can have without cauing the di to lip i RW M

10 Eample Bedford 9.33 # Impending lip F N F F W in α 0 F 0 + N W co α 0 MO 0 + M + FR 0 tan α, N W co α co + tan, tan α α α M NR WRco α M W R + Eample Bedford 9.66 # F 0 O + A 0 F 0 O + A 4g 0 M 0 O A ( m)co45 ) A ( m)in45 ) (4 g N)(0.5 m)co45 ) Eample Bedford 9.66 # Each of the uniform -m bar ha a ma of 4 g. The coefficient of tatic friction between the bar and the urface at B i 0.. If the tem i in equilibrium, i what i the magnitude of the friction force eerted on the bar at B. 38 Eample Bedford 9.66 #3 F 0 A + Fco30 Nin30 0 F 0 A (4 g N) + Fin30 + Nco30 0 M B 0 A ( m)co 45 ) + A ( m)in 45 ) + (4 g N)(0.5 m)co 45 ) 0 A 0, A g N, O 4 g N, O g N, N 438 N, F.63 N An 40 0

11 Eample Bedford 9.66 #4 Eample Bedford 9.63 # The crate i about to move. F cn F 0 (450 g N) + N 0 F 0 P F 0 P 0.3(450 g N) 35 g N 4 43 Eample Bedford 9.63 # The coefficient of tatic friction between the tire of the 000-g tractor and the ground and between the 450-g crate and the ground are 0.8 and 0.3, repectivel. Starting from the ret, what torque mut the tractor engine eert on the rear wheel to caue the crate to move? The front wheel can turn freel. Eample Bedford 9.63 #3 F 0 P + F 0 F 67.5 g N M O 0 (0.8 m) F T 0 T (0.8 m)(35 g N).06 N m An 4 44,

12 Eample Bedford 9.65 # The ma of the vehicle i 900 g, it ha rear-wheel drive, and the coefficient of tatic friction between it tire and the urface i The coefficient of tatic friction between the crate and the urface i 0.4. If the vehicle attempt to pull the crate up the incline, what i the larget value of the ma of the crate for which it will lip up the incline before the vehicle tire lip? Eample Bedford 9.65 #3 The crate i about to move. F cn F 0 W co0 + N 0 F 0 P F W in0 0 W.3930 P N 770. N m W / g 785 g An Eample Bedford 9.65 # Impending motion: F t, N F 0 Pco 0 + F 0 F 0 N + N (900 g N) P in0 0 M 0 O (900 g N)( m) + Pco0 (0.8 m) + Pin0 (3.7 m) N (.5 m) 0 P g N, N g N, F g N, N g N Eample Bedford 9.65 #4 What i the maimum tenion in the cord that the tractor ma tow a crate over a ramp with 45 angle of incline before the tractor topple ,

13 Eample Bedford 9.65 #5 Impending topple: N 0 M Q 0 P(.0 m) / (900 g N)(.5 m) 0 F 0 N (900 g N) P/ 0 F t, N, tire do not lip. F 0 P 955 g N An P/ + F 0 P g N, N 575 g N, F 675 g N Wedge Wedge eert a large lateral force from the face a a reult of mall angle Friction in Machine Friction help ome machine function. Machine with friction Wedge Thread, crew Belt Di and clutche Bearing Wedge Anali # The bloc i about to raie. The lip of the load and wedge are impending. F N F N, 3 3 F 0 + N 3 + N in θ + N co θ 0 F 0 + N co θ N in θ N W

14 Wedge Anali # FBD of the wedge F 0 + Nin θ Nco θ N + P 0 F 0 + N N co θ + N in θ 0 ( ) ( ) tanθ + P W tanθ Thread Geometr θ l tan π r θ r lead angle pitch of the thread (lead) mean radiu of the thread Thread/Screw Aumption: the haft ha a ingle continuou thread. Thread Motion W W l

15 Thread Moving Upward F 0 + S Rin( θ + φ) 0 F 0 + Rco( θ + φ) W 0 S W tan( θ + φ) and M Sr M Wr tan( θ + φ ) On the verge of rotating: φ φ tan Uniform upward motion: φ φ tan Thread Moving Downward # M i removed: W i upported b friction alone, providing that φ θ Self-locing condition Thread Moving Downward # F 0 + S Rin( θ φ) 0 F 0 + Rco( θ φ) W 0 S W tan( θ φ) and M Sr M Wr tan( θ φ) On the verge of rotating: φ φ tan Uniform downward motion: φ φ tan Thread Moving Downward #3 F 0 + S + Rin( φ θ) 0 F 0 + Rco( φ θ) W 0 S W tan( φ θ) and M S r Ver rough urface θ < φ : Revered motion. φ θ tan( ) M Wr

16 Eample Wedge/Thread # The ma of bloc A i 60 g. Neglect the weight of the 5 wedge. The coefficient of inetic friction between the contacting urface of the bloc A A, the wedge, the table, and the wall i 0.4. The pitch of the threaded haft i 5 mm, the mean radiu i 5 mm, the coefficient of inetic friction between the thread and the mating groove i 0.. What couple mut be eerted on the threaded haft to raie the bloc A at a contant rate? Eample Wedge/Thread #3 l 5 mm r 5 mm 0. for thread/groove φ arctan( ).30 θ arctan(/ πr ) Draw FBD of the thread Find M that will puh the thread upward: M Fr tan( θ + φ ) M ( N)(0.05 m)tan( ) M.5609 N m.56 N m An 6 63 Eample Wedge/Thread # Draw FBD of the ma A and the wedge Then, follow the ame procedure for wedge anali: ( ) ( ) tanα + F W tanα Subtitute for all nown value: F N WL 60 g N W w 0N α for all non-thread contact 6 Flat Belt Involve lippage of fleible cable, belt and rope over heave and drum. 64 6

17 Flat Belt Anali # Conider line element F 0 dn + T co dθ ( T + dt )co dθ 0 dn dt d co θ F n 0 dn T in dθ ( T + dt )in dθ 0 dn T in dθ + dt in dθ Eample Flat Belt # Friction reit the lowering of m : T mg T P mg/6 β.5( π) A force P mg/6 i required to lower the clinder at a contant β T Te mg mg e low peed with the cord maing.5 turn around the fied haft. Calculate the coefficient of friction between the cord and the haft..5 π An Flat Belt Anali # (motion impending) (ongoing movement) d d θ θ d θ 0, co : d θ dn dt co dt dθ dθ 0, in : d θ dn T in Td θ T Te β T θ θ β dt T dt β d d ln T T T 0 T Eample Flat Belt # Calculate the force P on the handle of the differential band brae that will prevent the flwheel from turning on it haft to which the torque M 50 N m i applied. The coefficient of friction between the band and the flwheel i t

18 Eample Flat Belt # Friction reit the attempt of M to rotate the flwheel in the clocwie direction: T Te T Te β (7 π /6) M center 0 (50 N m) + (0.5 m)( T T) 0 T T 000 N () Solve () and () T 300 N, T 300 N () 69 Di & Clutche Ue to connect and diconnect two coaial rotating haft The haft can upport a couple due to friction force between the di. Determine the couple b uing coefficient of tatic friction in the flat-ended thrut equation. Draw force on the plane onl 7 Eample Flat Belt #3 M O 0 + (0.5 m) T + (0.075 m)( T in30 ) + (0.45 m) P 0 P 408 N An Di & Clutche Anali # da π r dr F 0 0 P w da R P p πr dr pπr o M 0 0 M wr da R M pr π r dr pπ R o 3 3 M 3 PR

19 Di & Clutche Worn Anali # Ditribution of force on worn plate are not uniformed. For eample, linear ditribution da π r dr At r R, w 0 At r 0, w p w p( R r)/ R Bearing Pivot Bearing M 3 PR (motion impending) (ongoing movement) Di & Clutche Worn Anali # Conider the left plate F 0 P w da 0 R π p P p( R r) π r dr / R R o 3 M 0 M wr da 0 R M pr ( R r) πr dr / R o M PR Bearing Collar Bearing R R 3 R R M 3 3 i o i o (motion impending) (ongoing movement)

20 Bearing Thrut Bearing # R i R o dr da πr d πr co θ R o R i P R π ( o R Ro Ri ) A da dr π R co θ co θ dr R θ 77 Bearing Journal Bearing # P P φ 79 Bearing Thrut Bearing # F 0 P paco θ 0 P P p A co θ π R R ( o i ) dr R o R i R θ P M 0 M R ( pda ) 0 R o P π R M R dr R π co θ ( Ro Ri ) M 3 3 P R o Ri 3co θ Ro Ri Bearing Journal Bearing # R R R f M z 0 + M P( Rin φ ) 0 M P R M PR in tan φ φ φ in M PR PR i i

21 Eample Hibbeler 8- # The 00-mm-diameter pulle fit looel on a 0-mmdiameter haft for which the coefficient of tatic friction i 0.4. Determine the minimum tenion T in the belt needed to (a) raie the 00-g bloc and (b) lower the bloc. Aume that no lipping occur between een the belt and pulle and neglect the weight of the pulle. Hibbeler Eample 8- #3 CCW rotation Lower the bloc r r in φ (5 mm)in(tan 0.4).8570 mm f M 0 + P (98 N)(50 mm.86 mm) T (50 mm +.86 mm) 0 T T N 9 N An 8 83 Eample Hibbeler 8- # Raie the bloc rf r in φ (5 mm)in(tan 0.4).8570 mm M P 0 + (98 N)(50 mm mm T (50 mm.86 mm) 0 T N T.06 N An Eample Borei 0.9 # CW rotation 8 84

22 Eample Borei 0.9 # A weight W i lifted at a contant rate b appling a couple Rd to the compound djaccrew. Derive a formula for R in term of the angle θ, the weight W, the length d of the lever rod, the pitch p of the crew thread, the mean radiu r of the crew thread, and the coefficient i of inetic friction of the crew. The friction of the vertical wall guide i negligible. For θ 5, W 000 lb, d 0 in, r 0.75 in, p in, and 0.5, calculate R. Eample Borei 0.9 #4 F 0 Qco θ Tco θ 0 Q T F 0 T inθ P 0 P T in θ () () + () P W tan θ B mmetr, FBD of the left nut i mirror image of the right nut FBD Eample Borei 0.9 #3 F 0 T coθ W 0 W T co θ () Eample Borei 0.9 #5 [ M Wr tan( φ + α ) ] Rd Pr tan( φ + α) p R ( W tan θ ) r tan(tan + tan ) / d π r Subtitute number in the equation R 4.48 lb An 86 88

23 Eample Hibbeler 8-0 (modif) # The ale of the pulle fit looel in a 50-mm-diameter pinhole. If 0.30 between the pinhole and the pulle ale and 0.0 between the pulle and the cord, determine the minimum tenion T required to turn the pulle counterclocwie at a contant velocit if the bloc weigh 60 N. Neglect the weight of the pulle. 89 Cae I Journal Bearing Eample Hibbeler 8-0 (modif) #3 If the cord doe not lip over the pulle, the minimum i: f β T Te f, 60 N (54.9 N) e f, f, ( π /) b, < 0., the aumption the the cord doe not lip i valid. T 54. N An 9 Cae I Journal Bearing Eample Hibbeler 8-0 (modif) # The haft i about to rotate. φ tan b tan r r in φ mm θ β - tan ( r/ r) 45 in ( r / r ).97 F Rin( θ + β ) 0 F 0 T Rco( θ + β) 0 R N, T 54.9 N Cae II Fleible Belt Eample Hibbeler 8-0 (modif) #4 The belt i about to lide. f β T Te 60 N Te 0.( π / ) T N , f, f,

24 Cae II Fleible Belt Eample Hibbeler 8-0 (modif) #5 If the haft doe not rotate, F 0 (60 N) + R in( θ + β ) 0 F 0 T Rco( θ + β) 0 T N, θ 45, β β in ( r / r ) r.77 mm f f r r in φ φ φ tan.77 b, b, b, > 0.3, the aumpt ion that the haft doe not rotate i invalid. 93 Eample Borei 0-86 # Given T N The belt i about to lip: β T Te 0.3 π T ( N) e 5.37 N T 5.3 N An M 0 O T + ( T T) r 0 T ( T T)(0. m) N m A tenion in the belt remain unchanged between the two pu lle, TB T 33 N m An 95 Eample Borei 0-86 # A torque T i applied to pulle A, which drive pulle B. The two pulle have equal radii. The tenion in the lac ide of the belt i N. The coefficient of friction between the belt and the pulle i What i the maimum poible torque that can be applied to pulle A? What torque i tranmitted to pulle B? Eample Borei 0-98 # In ome application of collar bearing, multiple collar are ued. Two collar are ued a hown to upport the 0,000 lb load ( 0.40 and 0.0) Determine the torque T required to initiate rotation of haft S, Aume that the load i divided equall between the collar and that the preure on each collar i uniform. Wh are multiple collar ued? f

25 Eample Borei 0-98 # The load i divided equall between the collar, thu, the load on each collar i 5 ip. For each collar that i about to rotate: 3 3 R R T P 3 R R 3 3 ( ft) (0.5 ft) T 0.4(5000 lb) lb ft ( ft) (0.5 ft) Total required torque T M 3. ip ft An Le load on each collar, le wear and tear. An 97 Concept # Review Friction i the tangent reitance force which prevent or retard lipping of the bod relative to a urface with which it i in contact. Maimum friction occur that the verge of impending motion which can be lipping or toppling or both together. Some machine function b friction. To ue the formula, the phical meaning underling the friction in machine mut be undertood. 98 5