1881.J 



The Action of Cutting Tools. 



135 



The same objection applies to making the fourth term a minimum by 

 putting 



+ ^=90° (B). 



Perhaps as good a value as any for is that which makes the 

 resultant force through the face of the tool parallel to the direction of 

 motion. This value is given by the equation 



cot0=w 1 (0). 



The values obtained for for this expression agree fairly with those 

 in common use, or at least are quite compatible with them, considering 

 the uncertainty of the values of y u 1 for high pressures. The elastic and 

 permanent bending is small for such values of as can be used in 

 tools for metals, the transverse strength of the shaving (which is 

 some hundred times less than that of a piece of the substance of the 

 same dimensions in its natural state) not permitting it to transmit 

 large bending forces. 



In true cutting tools, where is very small, nearly the whole force 

 acting on the blade arises from friction under the pressure on the 

 sides of the tool caused by the bending. 



It is evident, therefore, that as passes from large values to small, 

 the importance of the term in will continually increase. 



All the sources of resistance above mentioned, except the first and 

 last, which should be small, are proportional to the thickness of the 

 shaving and, of course, also to its breadth. 



Thus to remove a given volume of material will require the same 

 expenditure of work, whether it be effected by one thick cut or several 

 thinner ones. 



In practice, however, the constant friction of the machinery em- 

 ployed always make thick cuts the most economical. The construction 

 of machines, and the character of work generally, confines within 

 rather narrow limits the thickness of the cut which it is possible to 

 take, for the force required must not be large enough to sensibly bend 

 or distort the substance as a whole. 



It would be a great advantage if tools could be so held or shaped 

 that their accidental vibrations should not be sustained but extinguished 

 by the reactions which they call into play ; but to make this possible 

 the phase of the vibration of the tool must precede that of the reaction 

 which it causes by some time less than half the period of the tool, or, in 

 other words, the resistance experienced by the tool must increase with 

 its velocity. Now friction between solids being rather greater at low 

 than at high velocities, the sliding of the shaving on the tool tends to 

 keep up a vibration once started, and the same may be said of the 

 forces due to the distortion of the substance, which are at a maximum 

 just before a fresh lamina (such as e .... c, fig. 9) begins to slide, and 



VOL. xxxni. l 



