College Of Engineering Electrical Engineering Department

College Of Engineering Electrical Engineering Department Engineering Mechanics-Static Friction Lecture-1 By Dr. Salah M. Swadi 2018-2019

8.6 Frictional Forces on Collar Bearings, Pivot Bearings and Disks • Pivot and collar bearings are used to support axial load on a rotating shaft • Laws of dry friction is applied to determine the moment M needed to turn the shaft when it supports an axial force P

8.6 Frictional Forces on Collar Bearings, Pivot Bearings and Disks Frictional Analysis • The collar bearing on the shaft is subjected to an axial force P and has a total contact area π(R22 – R12) • Normal pressure p is considered to be uniformly distributed over this area – a reasonable assumption provided the bearing is new and evenly distributed • Since ∑Fz = 0, p measured as a force per unit area p = P/π(R22 – R12)

Example 8.9 The uniform bar has a total mass m. if it is assumed that the normal pressure acting at the contracting surface varies linearly along the length of the bar, determine the couple moment M required to rotate the bar. Assume that the bar’s width a is negligible in comparison to its length l. the coefficient of static friction is equal to μs.

Solution FBD of the bar is as shown. Bar has a total weight of W = mg. Intensity wo of the distributed lead at the center (x = 0) is determined from vertical force equilibrium.

Solution Since w = 0 at x = l/2, for distributed load function, For magnitude of normal force acting on a segment of area having length dx,

Solution For magnitude of the frictional force acting on the same element of area, For moment created by this force about z axis, Summation of moments by integration,

8.7 Frictional Forces on Journal Bearings • When a shaft or axle is subjected to lateral loads, a journal bearing is used for support • Well-lubricated journal bearings are subjected to the laws of fluid mechanisms • When the bearing is not lubricated, analysis of the frictional resistance can be based on the laws of dry friction • If the lateral load is P, the bearing reactive force R acting at A is equal and opposite to P

8.7 Frictional Forces on Journal Bearings • Moment needed to maintain constant rotation of the shaft can be found by the summation of moments about the z axis of the shaft, • If the bearing is partially lubricated, μk is small, μk = tanΦk ≈ sinΦk ≈ Φk • Frictional resistance M ≈ Rrμk

Example 8.10 The 100mm diameter pulley fits loosely on a 10mm diameter shaft for which the coefficient of static friction is μs = 0.4. Determine the minimum tension T in the belt needed to (a) raise the 100kg block and (b) lower the block. Assume that no slipping occurs between the belt and the pulley and neglect the weight of the pulley.

Example 8.10 Part (a) FBD of the pulley is shown. As tension T is increased, the pulley will roll around the shaft to point before motion P2 impends. Friction circle’s radius, rf = r sinΦs. Using the simplification,

Example 8.10 Part (a) For radius of friction circle, Therefore,

Example 8.10 Part (b) When the block is lowered, the resultant force R acting on the shaft passes through the point P3. Summing moments about this point,

8.8 Rolling Resistance • A rigid cylinder of weight W rolls at constant velocity along a rigid surface, the normal force is at tangent point of contact • Hard material cylinder will compresses the soft surface underneath it

8.8 Rolling Resistance • We consider the resultant of the entire normal pressure acting on the cylinder N = Nd + Nr • To keep the cylinder in equilibrium, rolling at constant rate, N must beconcurrent with the driving force P and the weight W • Summation of moment about A, Wa = P (r cosθ) Wa ≈ Pr P ≈ (Wa)/r

Example 8.11 A 10kg steel wheel has a radius of 100mm and rest on an inclined plans made of wood. If θ is increased so that the wheel begins to roll down the incline with constant velocity when θ = 1.2°, determine the coefficient of rolling resistance.

Solution FBD of the wheel is as shown. Wheel has impending motion. Normal reaction N acts at point A defined by dimension a. Summing moments about point A, Solving

QUIZ • A friction force always acts _____ to the contact surface. • Normal B) At 45° C) Parallel D) At the angle of static friction 2. If a block is stationary, then the friction force acting on it is ________ . A) s N B) = s N C)  s N D) = k N

P(A) P(B) 100 lb P(C) B A QUIZ 3. A 100 lb box with a wide base is pulled by a force P and s = 0.4. Which force orientation requires the least force to begin sliding? A) P(A) B) P(B) C) P(C) D) Not determined 4. A ladder is positioned as shown. Please indicate the direction of the friction force on the ladder at B. A)  B)  C) D)

QUIZ 5. A wedge allows a ______ force P to lift a _________ weight W. A) (large, large) B) (small, small) C) (small, large) D) (large, small) 6. Considering friction forces and the indicated motion of the belt, how are belt tensions T1 and T2 related? A) T1 > T2 B) T1 = T2 C) T1 < T2 D) T1 = T2 e

QUIZ 7. When determining the force P needed to lift the block of weight W, it is easier to draw a FBD of ______ first. A) The wedge B) The block C) The horizontal ground D) The vertical wall 8. In the analysis of frictional forces on a flat belt, T2 = T1 e  . In this equation,  equals ______ . A) Angle of contact in deg B) Angle of contact in rad C) Coefficient of static friction D) Coefficient of kinetic friction

College Of Engineering Electrical Engineering Department