304 . APPLIED MECHANICS 
265. Position of Piston for Maximum Velocity and Zero Ac- 
celeration.—Still assuming that the velocity of the crank pin is uniform, 
it is evident that when CD (Figs. 467 : 
and 470) ceases to increase, the position 0 
- for maximum velocity and zero accelera- 
tion of piston has been reached, and 
this will obviously happen when the ‘ 
angle ODA is a right angle (Fig. 475). - . 
No direct geometrical construction has - 
yet been found for drawing the figure 
so that the-angle ODA may be a right 
angle, but by analysis it may be shown 
that when ODA is a right angle, 0, the 
angle ACB, is given by the equation 
sin® 0 —n? sint 0 — n* sin? 0+ n*=0, ar p\ 22 
which is a cubic equation in sin? 6. Mr. Kee 
G. A. Burls * has solved this equation ie cal 9 xr 
for a considerable number of values of , *— x t C 1 
and placed the results in a table, of 
which the following is an abstract :— Fig. 475. 
n sr 0 n | s>r 0 n s+r 0 
fe} “ Oo “ re) “ 
1:0 |2-0000| 90 6 6 || 2-0) 0-8474| 67 48 6|| 6-0) 0-9218| 80 47-40 
1'1 |1-0530| 64 57 50 || 2-5 | 0'8550 |} 70 43-46 || 7-0| 0-9321| 82 38 3 
1:144|1:0000| 64 5 11 || 3-0| 0°8674} 73 10 31|| 8-0| 0:9389| 82 56 30 
1:2 4:0 9°0 
15 5-0 0°0 
0°9564 | 63 35 5 0:8906 | 76 43 24 0°9468 | 83 47 12 
0°8681 | 64 20 38 0°9085 | 79 6 34]|| 10°0 | 0°9524 | 84 24 59 
s is the distance of the piston from the outer end of its stroke when 
its velocity is a maximum or its acceleration zero. 
<=n+l — /1—sin? 6-— ,/n?—sin? 6. 
’ 
Students are referred to Mr. Burls’ paper for the complete discussion 
of this problem. 
For practical purposes, when m has values common in direct-acting 
engines, it is usually sufficient to assume that. the position of the piston 
for maximum velocity and zero acceleration is that for which the crank 
and connecting-rod are at right angles to one another, 
n? 2n? + 1 
n2+1 (n?+1)(n*+4n?)’ 
for values of m usual in practice, an extremely close approximation to 
the true value of 0. ; ‘ 
266. Analytical Determination of Piston Velocity and Accelera- 
tion.—Piston velocity and acceleration diagrams are most readily drawn 
by the accurate constructions which have been given in preceding Articles, 
Professor Unwin’s formula,f sin? 0= gives, 
* Proceedings Inst. C.H., vol. cxxxi. p. 338, 
+ Ibid., vol. exxv. p. 366. 
