﻿530 Mr. S. Lees on a Simple Model 



application of the force F, the origin corresponding to 

 the point (or line) of contact for the neutral condition. 

 Any equation for the surface (assuming Oy to be the 

 tangent at 0) must give z as an even function of y. To 

 simplify the algebra, we shall assume that 



* = *y\ (35) 



where a is constant and small. Thus for such displacements 

 as we have to consider, the slope of the rubbing surface for 

 any given y is given by 



tan 6 = j- = 2ay, (36) 



which is also small. 



Referring back to fig. 8, it will be noticed that the tensions 

 (or compressions) T 2 are drawn so as to pass through the 

 corresponding points of rubbing. This is done so as to 

 avoid the consideration of tilting effects of the tensions or 

 compressions T 2 on the pieces B 1? B 2 , C 1? C 2 . If such tilting 

 effects are taken into account, it is easy to see that even with 

 no slipping taking place, the F — x relationship will not be 

 exactly linear, but F will involve small terms in x 2 and 

 higher powers of x. 



Although the magnitudes of z to be considered are small, 

 it is conceivable that they will exert appreciable influence on 

 the pressure N between (say) A x and B l5 normal to the direc- 

 tion of F. We shall accordingly assume a linear relation- 

 ship between N and z, and take 



N = N„-« = N -/3y 2 , , 



where a and /3 are constants, and J3 = 2a.a. I 



For the general case of slip, the value of T 2 will be given 



hj ±T 2 = N tan (<f>+ 0), with p = tan 0. 



If we make the further assumption that //, is so small * 

 that fi tan is negligible compared with unity, we get the 

 following expressions for T 2 : — 



^ and T 2 positive, T 2 = (N -/9,z/ 2 )(/i-2a?/), . (38) 



jj£ and T 2 negative, T 2 = -~(N -/% 2 )( / u + 2ay). (39) 



* This is justifiable if the cohesive forces in metals have the large 

 values usually attributed to them as compared with the stresses here 

 dealt with. 



