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84 INERTIA FORCES
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Inertia forces in most cam-follower systems are the most important of all the forces analyzed, especially at high speeds Inertia forces are caused by the necessity of moving the follower masses linearly or rotationally The inertia force on a linearly moving follower is Fa = where A = acceleration, in/sec2 w = equivalent follower weight, lb The inertia force, passing through the center of gravity of the body, has a direction opposite to that of the acceleration By D Alembert s principle, we may make a free-body diagram of all forces and analyze the dynamic condition as a static problem, Fig 81a For rotating bodies, the analysis is similar If the body has an unbalanced torque, it will have an angular acceleration which will be resisted by a torque reaction The direction of this torque will be opposite to the direction of acceleration (Fig 81b) The torque is: w A lb g (81)
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220 Applied force
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Acceleration, A Center of gravity
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Acceleration, a Applied torque Ta = Ia, Inertia torque
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w Fa = g A = Inertia force (a) Translating body (b) Rotating body
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FIGURE 81
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Free-body diagrams of accelerated bodies
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Ta = Ia, lb-in where I is the moment of inertia of the body about the center of rotation, and I = m k 2 lb-in-sec2 k = radius of gyration, in a = angular acceleration, radians/sec2
(82)
85 VIBRATORY FORCES
Vibrations are generally caused by forces whose magnitudes, directions, and/or point of application vary with time These forces produce variations in elastic deformations These vibrations in turn produce stresses and forces that are superimposed on the inertia and other forces in the follower systems The magnitudes of the vibratory stresses and forces are in uenced by the acceleration characteristics, as well as the rigidity and the damping of the follower mechanism This topic is separately treated in Chaps 12 and 13
86 FRICTIONAL FORCES
Friction opposes the relative movement of contacting bodies in all machinery It is a surface phenomenon In cam-follower systems, we have both sliding and rolling friction to consider The accurate way to include frictional resistance in a design is to measure it on the actual machine or prototype Handbooks list these resistances for some combinations of materials, but the conditions under which the values are obtained seldom t the speci c conditions This is especially true in cam and follower action Also, differences in friction may be obtained under apparently similar conditions Here are the three broad categories of frictional action in cam-follower mechanisms: Pure sliding that occurs between the cam and at-faced and spherical-faced followers Rolling and some sliding in rolling-element followers and cams Linear ball-bearing guiding mechanisms that support translating followers
CAM MECHANISM FORCES
The friction force between two bodies is de ned as the force at their surface that resists their relative movement The coef cient of friction m= F Fn (83)
where F = the friction force, lb Fn = normal reaction of one body on another, lb To initiate the relative motion, the static coef cient of friction is ms = Fs Fn
where Fs is the friction force between bodies not sliding on each other To maintain the motion the kinetic coef cient of friction is mk = Fk Fn
where Fk is the friction force between bodies sliding on each other 861 Flat-Faced and Spherical-Faced Followers These kinds of followers are selected from materials compatible with the cam surface materials The following factors affect the coef cient of friction between the cam and follower surfaces: (a) cleanliness of the surfaces; (b) wear of the parts; and (c) surrounding environmental contaminants such as particles of ink, paper, dust, and textiles which may result in lubrication starvation, (d) kind of lubrication system, (e) cavitation of lubricant, (f ) elastohydrodynamic lubricant lm on surfaces, and (g) windage due to moving parts Also, multiple startups and shutdowns cause metal-to-metal contact, resulting in wear debris and affecting friction Sliding followers have a coef cient of friction mk = 012 to 020 The starting coef cient of friction ms is about 40 percent more than the kinetic mk 862 Rolling Element Cam Followers Rolling-element cam followers are made up of either ball, roller, or needle bearings encased in a steel ring and supported by a steel stem (See Fig 82) They are most popular because of their small size and low friction In addition to the factors affecting the previ-
FIGURE 82 Commercial roller follower (needle bearing) (Courtesy of McGill Manufacturing Co, South Bend, Indiana)