264 APPLIED MECHANICS 
232. Efficiency of Screws.—Square Thread.—Since the efficiency 
is the ratio of the effort without friction to the effort with friction, 
ff tan a 
e@ ciency = Pee aes p: 
maximum efficiency is 
n (45 — ae 1- tan’ sin $. 
n (45+£) 14tan® ~ L+sin 6 
Putting tan @=p and tan a=p/rd, where p is the pitch and d the 
this is a maximum when a= 45° -¢ and the 
2 
7 ‘ _ p(rd — pp) 
mean diameter of the screw, then efficiency = xdig ae 1)’ 
The reversed efficiency, that is, the efficiency when W becomes the 
effort and P the resistance, is =e) $) ; this is a maximum when 
r tan ( 45 — ‘) 
a= 45° +4 9? and the maximum efficiency is —--——__, which is the — 
tan (45 +5) 
same as the maximum direct efficiency. 
Triangular Thread.—Without friction P = W tan a OP : 
wad 
W(tana+n tan ¢) W(p+nprd) 
l-ntanatand wd—npp 
tana(l—ntanatan¢) p(rd—npp) 
tana+n tan d ~ dp + nprd) 
tana—ntand —7d(p—nprd)_ 
tana(l+ntanatand) p(rd+npp) 
The efficiencies of square and triangular threaded screws have been 
calculated for various 
With friction P= 
Hence, efficiency = 
The reversed efficiency is 
values of a and three ve 
different values of p, =1 
and the results have 2% ee ee ee 
been plotted in Fig. 70 ar 
403. The full curves z a | as ae 
LJ 7 = 
relate to the square go Ae Brass Ee 
threads, and the dotted Vi, Sai 
curves to the triangular i ie ral 2 ile Aare haa te = 
threads. It will be > Pes 4-7 
seen that for the same 340+—/—4 we = 
values of a and p the § | fi’ ne 
efficiency of the tri- &30)7 777% 
angular thread is not ™ | [yi 
much less than that 20777 
of the square thread ; iif 
the difference is greater 107 ‘ 
the greater the value 
of p, but where p»=0°3, . 9 5 10 15 20 25 30 35 40 45 
the greatest difference — ANGLE. GIN. DEGREES. 
is only about 4 per cent. Fria. 403. 
