302 APPLIED MECHANICS 
therefore OK=DE, The triangles CKL and DEH are:similar, because 
the sides of CKL are respectively perpendicular to the sides of DEH, and 
but CK = DE, therefore CL=DH. If, therefore, the point O 
‘is accessible, CL, the piston acceleration, is found by drawing CK parallel 
to OD, and KL perpendicular to AB. But for a considerable portion of 
CL_DH 
CK DE’ 
the motion of the piston the point O is 
either at an inconvenient distance or 
is quite inaccessible, and some other 
construction for finding the point K is 
desirable. 
What is generally known as Klein’s 
construction is the most convenient 
for finding KL. Klein’s construction 
is as follows. On AB as a diameter 
describe a circle. With centre B and 
radius BD describe another circle, 
cutting the former at M and N. Join MN. The line MN coincides 
with the line KL of the former construction. For, referring to K as 
, because the triangles CBK and 
found by the first construction, o 
OB AB’ 
are similar, therefore BK _ = BD. or BK-AB=BD?. 
BD AB’ 
K as found by Klein’s construction, 
BK _BK_BM_BD or BK - AB= BD? as before. 
BD BM AB AB 
For the sake of clearness, the essential lines of Klein’s construction 
are shown separately in Fig. 471. 
262. Piston Acceleration at Ends of Stroke.—When the piston is 
at either end of its stroke the crank and connecting-rod are in a straight 
Fig. 472. 
line, and Klein’s construction gives the result shown in Fig. 472 for the 
outer end of the stroke, and the result shown in Fig. 473 for the inner 
end. Referring to Fig. 472, the angular velocity of the connecting-rod 
in this position is V//, and A has an acceleration in the direction AC due _ 
to this and equal to V?/7. Also the angular velocity of the crank is V/r, 
and in the position shown A has an acceleration in the direction AC due 
a 
BC 
OB’ 
OBD are similar. Also, —~ BU BD , because the triangles CBD and OBA 
Fig. 471. 
Fig. 473. 
Referring now to 
