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Published online by Cambridge University Press: 28 July 2016
This analytical treatment gives general considerations concerning “ true rolling ” of a rolling body in a circular path : firstly, for constant speed of its centre of gravity, and, secondly, for acceleration of its centre of gravity. From these are derived the relations of the dimensions, as function of the conditions, and the corresponding constructions for both taper roller bearings and ball bearings, which are essential to produce “ true rolling” at constant and variable shaft speeds.
*1 This case must be differentiated from the case where the sphere itself is fixed to a shaft rotating with angular velocity ωy for, in this case, a particle at istance (R + x) from the Y axis has velocity (R + x)ωy and not Rωy
*2 μCf causes a pressure between the sphere and the side wall of the cage in front of the sphere— not shown in Fig. 2—producing a reaction from this wall and thus couple μCf.r.
As the cage is fixed to the shaft rotating with constant angular velocity ωy, Vc remains constant —there is no dynamic reaction.
*3 μsT causes a pressure between the sphere and the side wall of the cage behind the sphere, producing a reaction from this wall and thus couple μsT.r.
As the cage is fixed to the shaft rotating with constant angular velocity ωy, Vc remains constant —there is no dynamic reaction.
*4 The general considerations concerning the Gyroscopic Inertia Couples which arise with these two angular velocities about two axes at right angles will be given in a following paper, together with their applications to Ball Bearings, Ball and Conical Cup Type Governors, etc.
*5 “ True rolling ” denotes rolling on a generating line of the roller in the contact area between fixed track and roller.
*6 The component of velocity Vc in the direction of the geometric axis of the roller will cause oscillating dynamic pressures against the flange.
*7 There is actually only a constant mean angular velocity of a shaft, but no constant angular velocity of a shaft due to fly wheel effects, etc.
*8 In the case of orthodox taper roller bearings, where the centre of gravity lies between the central and the external end cross sections of the conical rolling surface, the sense of rotation of the couples in the plane normal to A2C due to N1μ1 and the other forces parallel to Vc will be opposite to the sense of rotation of the component along A2C of the angular acceleration dωc/dt of the roller centre of gravity about the shaft axis, which produces skewing.
*9 This will emerge clearly from the later exact determination of the required position of the centre of gravity.
*10 As P1≼N1μs, where μs is the coefficient of sliding friction, and P2, is smaller still, this force P2, will, as will be proved, be exerted with the aid of the frictional force between fixed race track and roller, if the relations of the dimensions and construction for true rolling as developed in the following are adhered to, and under the condition that, as will be derived later,
where Cf is the centrifugal force.
*11 The angular acceleration referred to is not the “ general ” angular acceleration of the roller, which consists of two components:
One, referred to above, arises during changes of the shaft speed and for a bearing constructed according to the relations of the dimensions serves only to alter the magnitude of the general angular velocity of the rolling body and will coincide with the general angular velocity vector as required for true rolling.
The second component, however, only serves to alter the direction of the general angular velocity of the rolling body, but not its magnitude, and is therefore normal to the plane containing the general angular velocity vector and the bearing (Y) axis. This means that in this case this second component of angular acceleration produces the rotation of the general angular velocity vector about the bearing axis with the same angular velocity ωc+ (dωcdt) as that of the rotation of the centre of gravity of the rolling body about the bearing axis.
This second component of angular acceleration, which only changes the direction of the general angular velocity vector must therefore be normal to the general angular velocity vector, and will always exist because the general angular velocity vector always changes its direction (rotates).
This second component of angular acceleration causes the gyroscopic inertia couples which, as shown exactly in a later paper, are nothing but the couples of the centrifugal forces on the particles of the rolling body. In that paper gyroscopic effects in application to ball bearings and ball and conical cup type governors will be treated.
*12 In general, in the steady state, only the mean angular velocity for a revolution is constant, but the actual angular velocity of the shaft varies cyclically—it oscillates about the mean value.
*13 This will emerge clearly later from the determination of the values of μ3, for various conditions.
*14 The following development will illuminate this:
The sliding friction force on a body acts in the direction of the resultant relative velocity at the point of contact between this body and the surface upon which it slides. Thus the components of the sliding friction force in two directions normal to each other are proportional to the components of the resultant relative velocity in these directions.
How large, now, must an initial force P normal to the resultant relative velocity 
be, in order to produce in the direction normal to a displacement, and thus an acceleration a and an infinitely small velocity a.dt?
The new resultant relative velocity 
will be in a different direction to and from the above consideration the friction force in the direction of force P will have the proportion to the friction force in the direction of of a.dt/V = 0. Thus the force P at right angles to the original relative velocity V will have to overcome no friction force as long as it has not produced a finite velocity in the direction normal to the original relative velocity V of sliding.
In contrast to this, if the centre of gravity of the body moves with velocity but there is no relative velocity at the point of contact between it and the plane, i.e., the body rolls truly on the plane, then an initial force P normal to the velocity of the centre of gravity will have to overcome the total sticking friction force in order to produce a small displacement, because then the resultant relative velocity at the point of contact between body and plane would be in the direction of this displacement.
*15 “ Steady state ” denotes a state in which the mean value of the shaft speed is constant—actually the angular velocity of the shaft in a steady state always varies about this mean value in a cycle between maximum and minimum values.
*16 A “ steady state ” is meant to denote a state in which the mean value of the shaft speed is constant, i.e., actually the angular velocity of the shaft varies about this mean value in a cycle. Thus, as already shown, the forces arising due to the changes of the shaft speed for a construction according to the relations of the dimensions will be useful—whereas for another construction they would produce skewing.
*17 The three cases quoted are proved in the section on “ Taper Roller Bearings.”
*18 According to Stribeck.
*19 It is already clear without the exact proof that any radial displacement due to W, which for an angle of contact α <45° will cause slackness, will for the angle of contact 60° < α <90° require a correspondingly larger displacement in the axial direction, which is therefore connected with an elastic deformation in the axial direction, etc.
*20 The bearing should have a small axial preload.
*21 The radius of the track groove will be determined according to loading considerations from the deformation formulae of Heinrich Hertz, as given in the relevant literature.
