1. Figure 18.18 shows the cross-section of a spherical roller bearing. The inner surface of the outer race has a spherical profile so that the whole inner assembly—inner race \(+\) rollers—can rotate about the bearing centre in the plane of the paper, permitting the bearing to act as a ‘pin joint’. Spherical bearings are therefore tolerant of misalignment and are used in situations where manufacturing tolerances or shaft and housing deflections make it difficult to maintain exact alignment.

The spherical inner surface demands that the rollers be barrel-shaped as shown in the figure and there is therefore always slip at some parts of the contact area. The problem is to determine the extent of this slip and hence estimate the rolling resistance of the bearing due to frictional dissipation.

(i) Choose a frame of reference in which the rollers rotate about fixed axes at a given speed \(\varOmega \) , in which case the outer race rotates in the same direction at \(\varOmega _1\) and the inner race rotates in the opposite direction at \(\varOmega _2\) . Taking \(\varOmega \) as given, determine the slip speed \(v_S\) as a function of the distance x from the centre of the roller.

(ii) Assume that the radial force N carried by one roller is uniformly distributed over the roller width c , in which case there will be a tangential frictional force fN / c per unit length into or out of the paper, depending on the sign of \(v_S\) , where f is the coefficient of friction. Determine \(\varOmega _1\) from the condition that the resultant frictional force at the contact should be zero.

(iii) Repeat steps 1, 2 for the inner race speed \(\varOmega _2\) .

(iv) The rate of energy dissipation at the roller/outer race interface is \((fN/c)(v_S)\) per unit length. Calculate the total dissipation \(E_1\) at this interface and the corresponding dissipation \(E_2\) at the inner race/roller interface.

(v) Finally, estimate the

effective coefficient of friction for the bearing, defined as

$$ f^*=\frac{(E_1+E_2)}{(\varOmega _1+\varOmega _2)RN}. $$

This is the coefficient of friction in a plane journal bearing of radius

R which would lead to the same dissipation of energy for the same relative rotational speed between the shaft and the housing.

2. Repeat the analysis of Sect.

18.2 for the case where an elastic belt is used as a speed changing mechanism between two rigid pulleys of different radii,

\(R_1, R_2\) as shown in Fig.

18.19 . In particular, determine:-

(i) the angular velocity \(\varOmega _2\) of the driven pulley, if the angular velocity \(\varOmega _1\) of the driving pulley and the input torque \(M_1\) are given.

(ii) what percentage of the input power \(M_1\varOmega _1\) is lost due to frictional slip.

Fig. 18.19 Belt drive involving pulleys of different radii

Which of the following parameters influence the energy loss:-

the initial belt tension, \(T_0\) ,

the stiffness of the belt, k ,

the coefficient of friction, f ,

the distance between pulley centres, L .

How should we choose these parameters to minimize the power loss?

3. Equation (18.53 ) for the creep ratio at incipient sliding applies only for the case where the ellipse is elongated in the rolling direction. Use the change of coordinates \(x\!\leftrightarrow \! y, a\!\leftrightarrow \!b\) and Eqs. ( 9.2 ), ( 9.3 ) to derive a corresponding expression for the case where \(b\!>\!a\) . Plot the ratio between the resulting expressions and that from Kalker’s strip theory (18.47 ) over the entire range \(0\!<\!a/b\!<\!\infty \) and show that this ratio tends to unity as \(a/b\!\rightarrow \!0\) .

4. Estimate the maximum percentage error involved in dropping the second term in (18.53 ) and hence using (18.54 ) for the maximum creep ratio.

5. A cone of angle

\(\alpha \!=\!5\deg \) rolls at constant speed on a rail as shown in Fig.

3.7 . A Hertzian analysis of the normal contact problem shows that the maximum contact pressure is

\(p_0\) and the contact area is an ellipse of semi-axes

\(a\!=\!\ell ,\, b\!=\!5\ell \) , where

b is transverse to the rolling direction, as in Fig.

18.11 . At the centre of the contact ellipse, the effective rolling radius [

r in Fig.

18.4 ] is

R .

Fig. 18.20 Brake block with an off-centre pivot

Use Kalker’s strip theory and kinematic arguments from Sect. 18.1.1 to estimate the shear traction distribution in the contact area and the energy dissipated in frictional microslip, if the coefficient of friction is f and both bodies have elastic properties \(E,\nu \) . Remember that if there is no acceleration, the tangential force \(Q\!=\!0\) .

6. The non-uniform wear of the brake block of Fig. 18.14 a can be reduced by moving the pivot point to the left as shown in Fig. 18.20 . Determine the optimal distance it should be moved, if the unworn thickness corresponds to \(b=b_0\) and the fully worn state with uniform thickness of wear for all x corresponds to \(b\!=\!b_1\) (\(0\!<\!b_1\!<\!b_0\) ). Assume that the contact pressure can be approximated in the form \(p(x)\!=\!C(t)\!+\!D(t)x\) at all times t .

7. An indenter with the periodic saw-tooth profile shown in Fig.

18.21 slides against an elastic half-plane at speed

V . Expand the initial gap function

\(g_0(x)\) as a Fourier series and use results from Sect.

18.5.3 to describe the resulting evolution of the profile due to wear, assuming that the applied pressure

\(\bar{p}\) is sufficient to ensure full contact at all times and that

\(\beta \!=\!0\) .

Fig. 18.21 Wear of a saw-tooth indenter

Show that the full contact assumption is actually unrealistic here because the required mean pressure is infinite. What do you think will really happen? Can you think of any way to estimate the time at which the last points in the profile will first make contact?

8. Use the Winkler approximation of Sect. 18.5.5 to estimate the evolution of the pressure distribution p (r , t ) for a sphere of radius R sliding against a plane at speed V and loaded by a constant normal force P . Show that the distribution becomes approximately uniform at large times, and characterize the approach to this state by an appropriate dimensionless time parameter.

9. In a particular sliding system, the distant boundaries of body 2 are cooled to maintain a bulk temperature \(T_2^\infty \!=\!0\) , but those of body 1 are insulated, so no heat can flow out of the body. The two materials have the same thermal properties. If there is a single circular contact area of radius a which moves relative to body 2 and is stationary in body 1, find the resulting steady-state bulk temperature \(T_1^\infty \) in body 1 as a function of f , V , p , a , K , k , where these symbols are defined in Sect. 18.6.1 .

Would your answer be changed if there were many contact areas with a distribution of contact radii, assuming the mean contact pressure p is the same for all?