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Spacing the ellipse
280
G r a l z t ."
[ J. F. I.,
S P A C I N ( ; "r,E E L L I P S E . By (h.:om;E B. GRANT,Lexington, Mass. I have f r e q u e n t l y found it n e c e s s a r y in the course of m y business, which is specially confined to the m a k i n g and c u t t i n g of gear wheels, to cut the t e e t h of an elliptic gear. A l t h o u g h I have always used the g r e a t e s t care and h a v e had the a d v a n t a g e of experience and the b e s t of tools, I have seldom s u c c e e d e d in actually c u t t i n g such a gear to the c o m b i n e d satisfaction of myself and m y customer, while, from a b u s i n e s s point of View, n o t h i n g b u t a price that is almost p r o h i b i t o r y will m a k e an order for such a gear in a n y w a y desirable. T h e difficulty has been in the spacing of the curve so that the teeth could be evenly placed and give a s m o o t h action. T h e usual m e t h o d is to d r a w the curve and space it with the dividers, step b y step, and, a l t h o u g h the process is simple e n o u g h to describe, it is e x c e e d i n g l y clumsy, tedious, and inaccurate in operation. Realizing that the only i m p e d i m e n t to the general use of the elliptic g e a r for quick r e t u r n m o t i o n s alad for o t h e r variations of speed, in positions w h e r e its simplicity and its positive action makes it unrivalled b y a n y o t h e r m e c h a n i c a l m o v e m e n t , was the p r e s e n t e x p e n s e of p r o d u c i n g it and the p o v e r t y of the usual result, I u n d e r t o o k the c o n s t r u c t i o n of a special m a c h i n e for the purpose. I h a d no difficulty in d e s i g n i n g a machine that w o u l d keep the c u t t e r a c c u r a t e l y in the elliptic path, b u t it was only a f t e r a long search and m a n y e x p e r i m e n t s that I f o u n d a device for spacing the elliptic outline that w o u l d serve that p u r p o s e even approximately: T h e result of m v search was the device here described, and it can be easily applied with a degree of accuracy t h a t is far b e y o n d the r e q u i r e m e n t s of actual practice. In the figure, supp~me the ellipse to be described on t h e principle of the c o m m o n trammel, the two pins a and b
Spacing the Ellipse.
.April, 1889.]
28I
b e i n g confined to the two lines OA and OB, while the point jo draws the curve. D r a w an index circle, I, from the centre, 0, of the ellipse, and let its radius, OH, b e equal to the s u m of the radii, OA and OB, of the ellipse. Then, if the line, b a p , extended, accurately spaces the index circle at d, it will also space the ellipse with an accuracy t h a t is practically perfect for cases t h a t are in use for elliptic gears. To show the a c c u r a c y of the device take a v e r y c o m m o n e x a m p l e . Given, an ellipse with a m a j o r d i a m e t e r of ten
i \
~\ \
\ \
\
o
Z;/
inches and a m i n o r d i a m e t e r of nine inches, with seventytwo teeth. T h i s ellipse, used as a quick r e t u r n motion, w o u l d give a ratio of slow stroke to quick r e t u r n stroke of m o r e than three to one, and w h e n u s e d for variation of s p e e d , w o u l d give a variation of f a s t e s t speed to s l o w e s t speed of a b o u t six and one-half to one. It is, therefore, a practical example, elliptic gears b e i n g g e n e r a l l y m u c h more nearly circular. For this example the errors in spacing, a l t h o u g h theoretically present, are practically nothing. T h e chord of the WHOLF No. VoL. CXXVII.--(TItlRD SERIES, Vol. xcvii.)
19
282
Gra~tt.
[J. F. I.,
t o o t h space at t h e m a j o r apex A is "41433 inch, w hi l e t h a t at t h e m i n o r ap ex B is "4144I inch, g i v i n g an e r r o r of a b o u t one t w e l v e - t h o u s a n d t h of an inch. T h e g r e a t e s t e r r o r is on t h e q u a r t e r , at C and not at t h e apices, b u t t h e differenee b e t w e e n t h e g r e a t e s t and t h e l e a s t t o o t h spaces is n o t o v er one t w o - t h o u s a n d t h of an inch. If we t a k e an e x t r e m e case we shall find a g r e a t e r error. F o r d i a m e t e r s of t w e l v e and six i n c h e s a n d s e v e n t y - t w o t e e th , w h i c h w o u l d be w h o l l y useless for p r a c t i c a l purposes, t h e g r e a t e s t e r r o r is still v e r y small, n o t o v e r o n e - s e v e n t i e t h of an inch, wh ile t h e e r r or b e t w e e n t h e apices is still practically nothing. F o r t h e case s h o w n in t h e figure, an ellipse h a v i n g d i a m e t e r s of e i g h t and t e n i nches and s e v e n t y - t w o t eet h, t h e m a x i m u m e r r o r is one f o u r - h u n d r e d t h of an inch. T h i s g e a r will give a rat i o of g r e a t e s t to l e a s t s p e e d of s i x t e e n to one, wh ile as a q ui c k r e t u r n m o t i o n t he r a t i o is m o r e t h a n five to one. W i t h t h e m a c h i n e now b e i n g m a d e I c o n f i d e n t l y e x p e c t to be able to c u t elliptic g e a r s q u i t e as a c c u r a t e l y as I can n o w cu t cir cu lar gears, a n d at an e x p e n s e t h a t is b u t l i t t l e g re ater . W h e n t h e elliptic g e a r can be well and c h e a p l y made, I see no r e as on w h y its use s h o u l d not be i n c r e a s e d ten-fold, for it is one of t h e n e a t e s t a n d m o s t useful of m e c h a n i c a l devices.
G r a l z t ."
[ J. F. I.,
S P A C I N ( ; "r,E E L L I P S E . By (h.:om;E B. GRANT,Lexington, Mass. I have f r e q u e n t l y found it n e c e s s a r y in the course of m y business, which is specially confined to the m a k i n g and c u t t i n g of gear wheels, to cut the t e e t h of an elliptic gear. A l t h o u g h I have always used the g r e a t e s t care and h a v e had the a d v a n t a g e of experience and the b e s t of tools, I have seldom s u c c e e d e d in actually c u t t i n g such a gear to the c o m b i n e d satisfaction of myself and m y customer, while, from a b u s i n e s s point of View, n o t h i n g b u t a price that is almost p r o h i b i t o r y will m a k e an order for such a gear in a n y w a y desirable. T h e difficulty has been in the spacing of the curve so that the teeth could be evenly placed and give a s m o o t h action. T h e usual m e t h o d is to d r a w the curve and space it with the dividers, step b y step, and, a l t h o u g h the process is simple e n o u g h to describe, it is e x c e e d i n g l y clumsy, tedious, and inaccurate in operation. Realizing that the only i m p e d i m e n t to the general use of the elliptic g e a r for quick r e t u r n m o t i o n s alad for o t h e r variations of speed, in positions w h e r e its simplicity and its positive action makes it unrivalled b y a n y o t h e r m e c h a n i c a l m o v e m e n t , was the p r e s e n t e x p e n s e of p r o d u c i n g it and the p o v e r t y of the usual result, I u n d e r t o o k the c o n s t r u c t i o n of a special m a c h i n e for the purpose. I h a d no difficulty in d e s i g n i n g a machine that w o u l d keep the c u t t e r a c c u r a t e l y in the elliptic path, b u t it was only a f t e r a long search and m a n y e x p e r i m e n t s that I f o u n d a device for spacing the elliptic outline that w o u l d serve that p u r p o s e even approximately: T h e result of m v search was the device here described, and it can be easily applied with a degree of accuracy t h a t is far b e y o n d the r e q u i r e m e n t s of actual practice. In the figure, supp~me the ellipse to be described on t h e principle of the c o m m o n trammel, the two pins a and b
Spacing the Ellipse.
.April, 1889.]
28I
b e i n g confined to the two lines OA and OB, while the point jo draws the curve. D r a w an index circle, I, from the centre, 0, of the ellipse, and let its radius, OH, b e equal to the s u m of the radii, OA and OB, of the ellipse. Then, if the line, b a p , extended, accurately spaces the index circle at d, it will also space the ellipse with an accuracy t h a t is practically perfect for cases t h a t are in use for elliptic gears. To show the a c c u r a c y of the device take a v e r y c o m m o n e x a m p l e . Given, an ellipse with a m a j o r d i a m e t e r of ten
i \
~\ \
\ \
\
o
Z;/
inches and a m i n o r d i a m e t e r of nine inches, with seventytwo teeth. T h i s ellipse, used as a quick r e t u r n motion, w o u l d give a ratio of slow stroke to quick r e t u r n stroke of m o r e than three to one, and w h e n u s e d for variation of s p e e d , w o u l d give a variation of f a s t e s t speed to s l o w e s t speed of a b o u t six and one-half to one. It is, therefore, a practical example, elliptic gears b e i n g g e n e r a l l y m u c h more nearly circular. For this example the errors in spacing, a l t h o u g h theoretically present, are practically nothing. T h e chord of the WHOLF No. VoL. CXXVII.--(TItlRD SERIES, Vol. xcvii.)
19
282
Gra~tt.
[J. F. I.,
t o o t h space at t h e m a j o r apex A is "41433 inch, w hi l e t h a t at t h e m i n o r ap ex B is "4144I inch, g i v i n g an e r r o r of a b o u t one t w e l v e - t h o u s a n d t h of an inch. T h e g r e a t e s t e r r o r is on t h e q u a r t e r , at C and not at t h e apices, b u t t h e differenee b e t w e e n t h e g r e a t e s t and t h e l e a s t t o o t h spaces is n o t o v er one t w o - t h o u s a n d t h of an inch. If we t a k e an e x t r e m e case we shall find a g r e a t e r error. F o r d i a m e t e r s of t w e l v e and six i n c h e s a n d s e v e n t y - t w o t e e th , w h i c h w o u l d be w h o l l y useless for p r a c t i c a l purposes, t h e g r e a t e s t e r r o r is still v e r y small, n o t o v e r o n e - s e v e n t i e t h of an inch, wh ile t h e e r r or b e t w e e n t h e apices is still practically nothing. F o r t h e case s h o w n in t h e figure, an ellipse h a v i n g d i a m e t e r s of e i g h t and t e n i nches and s e v e n t y - t w o t eet h, t h e m a x i m u m e r r o r is one f o u r - h u n d r e d t h of an inch. T h i s g e a r will give a rat i o of g r e a t e s t to l e a s t s p e e d of s i x t e e n to one, wh ile as a q ui c k r e t u r n m o t i o n t he r a t i o is m o r e t h a n five to one. W i t h t h e m a c h i n e now b e i n g m a d e I c o n f i d e n t l y e x p e c t to be able to c u t elliptic g e a r s q u i t e as a c c u r a t e l y as I can n o w cu t cir cu lar gears, a n d at an e x p e n s e t h a t is b u t l i t t l e g re ater . W h e n t h e elliptic g e a r can be well and c h e a p l y made, I see no r e as on w h y its use s h o u l d not be i n c r e a s e d ten-fold, for it is one of t h e n e a t e s t a n d m o s t useful of m e c h a n i c a l devices.