M ACHINE ELEMENTS MODULE 2 Bodies in Pure Rolling Contact Module 2 LEARNING OUTCOMES 1. Compute the speed and diameter of rolling cylinders. 2. Compute the speed and cone angle of rolling cones. M ACHINE ELEMENTS PURE ROLLING CONTACT Consists of such a relative motion of two lines or surfaces that the consecutive points or elements of one come successively into contact with those of the other in their order. There is no slipping between two surfaces which have pure rolling contact; that is, all points in contact have the same linear speed. M ACHINE ELEMENTS CYLINDERS IN PURE ROLLING CONTACT β External Contact β Rotation is in opposite direction π΄πππ’πππ πππππ ππ π΄ (ππ΄ ) = 2ππ N π΄πππ’πππ πππππ ππ π΅ (ππ΅ ) = 2ππ 1 π1 πππππ π ππ‘ππ π = π π 1 = π1 π πΆπππ‘ππ π·ππ π‘ππππ πΆ πΆ = π + π 1 2πΆ = π· + π·1 M ACHINE ELEMENTS CYLINDERS IN PURE ROLLING CONTACT β Internal Contact β Rotation is in same direction π΄πππ’πππ πππππ ππ π΄ (ππ΄ ) = 2ππ N π΄πππ’πππ πππππ ππ π΅ (ππ΅ ) = 2ππ 1 π1 πππππ π ππ‘ππ π = π π 1 = π1 π πΆπππ‘ππ π·ππ π‘ππππ πΆ πΆ = π − π 1 2πΆ = π· − π·1 M ACHINE ELEMENTS SAMPLE PROBLEM 1 A cylinder 24 in. in diameter on shaft V drives by pure rolling contact another cylinder on shaft T. Has an angular speed of 600 radians per minute. Shaft T turns 148.25 rpm in the opposite direction. a. Calculate the diameter of cylinder of shaft T and the distance between the axes of the two shafts. b. Calculate the diameter of cylinder of shaft T and the distance between the axes of the two shafts if they turn in the same direction. M ACHINE ELEMENTS SAMPLE PROBLEM M ACHINE ELEMENTS CONES IN PURE ROLLING CONTACT β External Contact β Rotation is in opposite direction π΄πππ’πππ πππππ ππ π΄ (ππ΄ ) = 2ππ N π΄πππ’πππ πππππ ππ π΅ (ππ΅ ) = 2ππ π1 πππππ π ππ‘ππ π = π π 1 = π1 π πβπππ‘ π΄ππππ π = πΌ + π½ M ACHINE ELEMENTS CONES IN PURE ROLLING CONTACT π π πππ½ π πππ½ π πππ½ = = = π1 π πππΌ π ππ(π − π½) π πππ πππ π½ − πππ π π πππ½ π πππ½ ( ) πππ π½ π π‘πππ½ = = = π1 π πππ − πππ π ( π πππ½ ) π πππ − πππ π π‘πππ½ πππ π½ ∴ tan π½ = sin π (π1 Τπ) + cos π ∴ tan πΌ = sin π (πΤπ1 ) + cos π M ACHINE ELEMENTS CONES IN PURE ROLLING CONTACT β Internal Contact β Rotation is in the same direction π΄πππ’πππ πππππ ππ π΄ (ππ΄ ) = 2ππ N π΄πππ’πππ πππππ ππ π΅ (ππ΅ ) = 2ππ π1 πππππ π ππ‘ππ π = π π 1 = π1 π πβπππ‘ π΄ππππ π = πΌ − π½ tan π½ = sin π (π1 Τπ) − cos π tan πΌ = sin π πππ π − (πΤπ1 ) M ACHINE ELEMENTS SAMPLE PROBLEM 2 A turns 100 rpm and B 150 rpm. They are connected by rolling cones. Calculate the cone angle of each cone. If the base of Cone on A is 3 inches from the vertex, calculate the diameters of both cones. M ACHINE ELEMENTS
