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A Study on the Generating of Tooth Profiles of

Curvic Couplings

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°ê®æ¨½´Ë¤½¥q(Gleason Works Co.)©Òµo®iªº Gleason No.120 «¬±M¥Î¿i§É¾÷¡C©Ò
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I

Abstract

Curvic couplings have been widely applied to various industrial applications.
Presently the Gleason No.120 special grinding machine developed by Gleason
Works Co. is one of the commonly used machine tool for the curvic coupling
manufacturing. On the other hand, in response to the issue of improving the
domestic design and manufacturing ability, an alternative manufacturing method of
curvic coupling is introduced in this study.
In this study the geometrical characteristics of the novel gear profile are
investigated. Firstly, applying the spatial transformation matrix theorem to the
relationship between the cutting tool path and the cutting tool position, the cutting
tool profile equation of curvic coupling is successfully derived. Secondly, the
mathematical model of the envelope surface of the generating tool, or generally
being called the gear surface, is constructed based on the trajectory equation of the
generating tool motion and the tool-workpiece meshing equation. Finally, the solid
model is established based on the obtained mathematical model, and the
comparison works with the conventional curvic coupling are also carried out. The
analysis of gear surface is graphically depicted with respect to the various
machining parameters. It is believed that this thesis provides a useful tool for the
following studies of curvic couplings for the different demand of application fields.

II

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Abstract.............................................................................................................II
¥Ø¿ý ................................................................................................................ III
¹Ï¥Ø¿ý .............................................................................................................. V
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1-1 ¬ã¨s-I´º¤Î¥Øªº ..............................................................................................................1
1-2 ¤åÄm¦^ÅU ..........................................................................................................................2
1-3 ½×¤å²ÕÂ´»P³¹¸` ..............................................................................................................4
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2-1 Áp¶b¾¹ºØÃþ ......................................................................................................................5
2-2 ¦±¾¦Áp¶b¾¹Â²¤¶ ..............................................................................................................6
2-3 ®æ¨½´Ë¦±¾¦Áp¶b¾¹¥[¤u¤è¦¡ ..........................................................................................7
2-4 ¬ã¨s¬yµ{ ..........................................................................................................................9
2-5 ¥»¤åµo®iªº¦±¾¦Áp¶b¾¦§Î»s³y¤è¦¡ ............................................................................10
2-5-1 ³Ð¦¨¤M¨ã¤§¼Æ¾Ç¼Ò¦¡ ...................................................................................13
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3-1 ¥]µ¸²z½× ........................................................................................................................25
3-2 ï¿¦X¤èµ{¦¡....................................................................................................................27
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3-2-2 ¬Û¹ï³t«× .......................................................................................................32
3-2-3 ï¿¦X¤èµ{¦¡...................................................................................................36
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3-4 ¦±¾¦Áp¶b¾¹¤§¼Æ¾Ç¼Ò¦¡ ................................................................................................42
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4-1 ¾¦-±«Øºc ........................................................................................................................44
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4-2-1 ¾¦§Î¤§¤ñ¸û ...................................................................................................48
4-2-2 ¾¦§Î¥[¤u¶q ...................................................................................................54
4-2-3 ¾¦§ÎÃä½t¤ÀªR ...............................................................................................64
4-2-4 À³¥Î ...............................................................................................................71
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°Ñ¦Ò¤åÄm ......................................................................................................... 74

III

ªþ¿ý A Â½u-y¸ñ¤èµ{¦¡ ................................................................................. 77
ªþ¿ý B ¦±¾¦Áp¶b¾¹¤§¾¦-±¤èµ{¦¡ .................................................................. 87

IV

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2.1 ¦±¾¦Áp¶b¾¹ (SUDA International Gear Works Co. ) .................................................6
¹Ï 2.2 ¦±¾¦Áp¶b¾¹¤§¥Y¾¦½L¤Î¥W¾¦½L .................................................................................7
¹Ï 2.3 Gleason Works Co. No.120 .........................................................................................7
¹Ï 2.4 ®æ¨½´Ë¦±¾¦Áp¶b¾¹¤§¿i½ü¥~§Î½ü¹ø .........................................................................8
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¹Ï 3.5 ¤M¨ã»P¤u¥ó¶b¥æ¨¤¬° 135 «×®É¤§¤M¨ã-y¸ñ¹Ï .......................................................41
¹Ï 3.6 ¤M¨ã»P¤u¥ó¶b¥æ¨¤¬°160 «×®É¤§¤M¨ã-y¸ñ¹Ï .......................................................42
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¹Ï 4.11 ¦±¾¦Áp¶b¾¹¥W¾¦¡A¦b¾¦°ª mm ®É¤§¾¦§Î¥[¤u¶q .................................................55
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¹Ï 4.13 ¦±¾¦Áp¶b¾¹¥W¾¦¡A¦b¾¦°ª mm ®É¤§¾¦§Î¥[¤u¶q .................................................56
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¹Ï 4.23 ¦±¾¦Áp¶b¾¹¥Y¾¦¡A¦b¾¦°ª2.5mm ®É¤§¾¦§Î¥[¤u¶q ..............................................61
¹Ï 4.24 ¦±¾¦Áp¶b¾¹¥Y¾¦¡A¦b¾¦°ª2mm ®É¤§¾¦§Î¥[¤u¶q .................................................62
¹Ï 4.25 ¦±¾¦Áp¶b¾¹¥Y¾¦¡A¦b¾¦°ª1.5mm ®É¤§¾¦§Î¥[¤u¶q ..............................................62
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¹Ï 4.37 ¦±¾¦Áp¶b¾¹¥Y¾¦¡A¦b¾¦¥~½t³¡¤À¤§¾¦§Î¥[¤u¶q...................................................70
¹Ï 4.38 ¦±¾¦Áp¶b¾¹¥Y¾¦¡A¦b¾¦¤º½t³¡¤À¤§¾¦§Î¥[¤u¶q ...................................................70
¹Ï A.1 ¤ºÂ½u¦±½u¤Î¤º¦¸Â½u®y¼Ð¹Ï ................................................................................79
¹Ï A.2 ¥~Â½u¦±½u¤Î¥~¦¸Â½u®y¼Ð¹Ï ................................................................................80
¹Ï A.3 ¤ºÂ½u-y¸ñ¹Ï¡A( R : r : d 0 : d 1 = 6 : 2 : 4 : 1 )..............................................................83
¹Ï A.4 ¥~Â½u-y¸ñ¹Ï¡A( R : r : d 0 : d 1 = 6 : 2 : 4 : 1 )..............................................................84
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¹Ï A.7 ¥~Â½u-y¸ñ¹Ï¤§µÇÅ¦½u ( R : r : d 0 : d1 = 10 : 5 : 13 : 3 ) ..............................................85

VI

¹Ï A.8 ¥~Â½u-y¸ñ¹Ï ( R : r : d 0 : d1 = 7 : 4 : 6 : 3 ) .................................................................86
¹Ï A.9 ¤ºÂ½u-y¸ñ¹Ï ( R : r : d 0 : d1 = 7 : 4 : 6 : 3 ) .................................................................86

VII

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ªí 4.1 ¦±¾¦Áp¶b¾¹¤§³]-p°Ñ¼Æªí .........................................................................................46

VIII

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ºë«×¶V°ª¤§¯S©Ê¡A¥B¬°¤F¹F¨ì§Ö³t¿i¦Xªº®ÄªG¡A¥»¤å¥[¤u¤è¦¡¨ä¤M¨ã¤Î¤u¥ó
¾÷ºc¼Ò¦¡¬°¥æ¤e¶b«¬¦¡¤§Ãö«Y¡ «Y§Q¥Î¤M¨ã¤Î¾¦-F¤§¤¤¤ß¶b°¾Â-Ó¨¤«×¨Ó¥[
A
¤u¡C
¥»¤å¥[¤u¤è¦¡¤À¬°¨â¹Dµ{§Ç¡G²Ä¤@¹Dµ{§Ç¬°¥Ñ¿i½ü¥[¤u¥X¦±¾¦Áp¶b¾¹¤§
³Ð¦¨¤M¨ã¡C¥Ñ©ó¿i½ü¥[¤u¤M¨ã®É¡A©¼¦¹¤¤¤ß¶b¦³°¾-Ó§¨¨¤¡A¤@¦¸¶È¯à¥[¤u¤@
-Ó¾¦-±¡A©Ò¥Hµ¥¤M¨ã¾¦-F±ÛÂà¤@°é«á¶È§¹¦¨¤M¨ã¤@Ãä¾¦-±¤§¥[¤u¡A±ý¥[¤u¤M

10

¨ã¥t¤@Ãä¾¦-±¡A¨ä¿i½ü¤¤¤ß¶b¥²¶·¦A°¾¥t¤@Ãä¨¤«×¨Ó¥[¤u¥t¤@Ãä¤M¨ã¾¦-±¡A

¦p¦¹¤M¨ã¤~ºâ§¹¦¨¥[¤u¡A¦p¹Ï (2.7)¡C¨ä³Ð¦¨¤M¨ã¥[¤u¤è¦¡Ãþ¦ü©ó®æ¨½´Ë¦±¾¦
Áp¶b¾¹¤§¥[¤u¡ ¦ý¨ä®t§O¦b©ó®æ¨½ ´Ë¥[¤u¤§¿i½ü¤Î¾¦-F¤§¤¤¤ß¶b©¼¦¹¬°¥-¦æ
A
ªº¡A¤@¦¸´N¯à¥[¤u¤u¥ó¥ª¥k¨â-Ó¦±-±¡C¦Ó¥»¤å¥[¤u¤è¦¡ªº²Ä¤G¹Dµ{§Ç¬°¥Ñ¤M
¨ã¥[¤u¥X¤u¥ó¡A¤]´N¬O¦±¾¦Áp¶b¾¹¡A¦p¹Ï
(2.8)¡C

¤M¨ã¾¦-F

¤M¨ã¾¦-F

¿i½ü ¿i½ü

(a)¿i½ü»P¤M¨ã¤¤¤ß¶b°¾¨¤¥[¤u¤@¦¸¶È¯à¥[¤u¤M¨ã¤@Ãä¦±-±
(¿i½ü¥ª¶É )

¤M¨ã¾¦-F

¿i½ü
(b)¿i½ü¦A°¾¥t¤@¨¤«×¥[¤u¤M¨ã¥t¤@Ãä¦±-±
(¿i½ü¥k¶É )
¹Ï 2.7 ¥»¤å¥[¤u¦±¾¦Áp¶b¾¹µ{§Ç¤@¤§¥Ü·N¹Ï

¤M¨ã

¤u¥ó

¹Ï 2.8 ¥»¤å¥[¤u¦±¾¦Áp¶b¾¹µ{§Ç¤G¤§¥Ü·N¹Ï

11

¥Ñ©ó¥»¤åµo®iªº¦±¾¦Áp¶b¾¹¥[¤u¤è¦¡¤ñ®æ¨½´Ëªº¥[¤u¦h¤@¹DÂà´«µ{

§Ç¡A¥ý¥Ñ¿i½ü¥[¤u¥X¤M¨ã¡A¦A¥Ñ¤M¨ã¥[¤u¥X¦±¾¦Áp ¶b¾¹¡C©Ò¥H¡A¥»¤å¥[¤u¤è

¦¡»P®æ¨½´Ë¥[¤u¤è¦¡©Ò¨Ï¥Î¿i½üªº¬ã¿i-±¥¿¦n¬O¬Û¤Ïªº¡C¦Ó¥»¤å¥[¤u¤è¦¡±ý

¥[¤u¥X¦¨«~¥Y¾¦¥²¶·¨Ï¥Î¿i½ü¥~½t¬°¥[¤u-±¡¦Ó-n±o¨ì¦¨«~¥W¾¦¥²¶·¨Ï¥Î¿i
A

½ü¤º½t¬°¥[¤u-±¡A»P®æ¨½´Ë¥[¤u¤è¦¡¬Û¤Ï¡A¦b¹Ï (2.9)¦³»¡©ú¨ä¹Lµ{¡C

¤M¨ã ¤M¨ã ¤M¨ã

(a)µ{§Ç¤@¡G§Q¥Î¿i½üº½t ¥[¤u¥X¤M¨ã¬° Y¾¦
¤ ¥

¤M¨ã ¦¨«~ ¦¨«~ ¦¨«~

¦¨«~

(b)µ{§Ç¤G¡G§Q¥Î
¥Y¾¦¤M¨ã¥[¤u¥X¦¨«~¬° W¾¦ º¦±¾¦Áp¶b¾¹
¥ ª
¹Ï 2.9 ¥»¤å¥[¤u¦±¾¦Áp¶b¾¹¥W¾¦¤§µ{§Ç»¡©ú

¥»¤åªº¥[¤u¾÷ºc¼Ò¦¡«Øºc§¹¦¨«á¡A¼¶¼g AutoLISP µ{¦¡¼ÒÀÀ¤M¨ã¥[¤u¾¦
-F¤§°Ê§@¡A¥HAutoCad ÀËÅç¾¦§Î¤§´X¦ó§Îª¬§PÂ_¥»¤åªº¥[¤u¼Ò¦¡¤§¥i¦æ©Ê¡A
¨Ã¸g¹L-×¥¿«á¡A¥H±o¨ì²z·Q¤§¾¦§Î¡A¹Ï
(2.10)¬°«Ø¥ß¥»¤åªº¥[¤u¼Ò¦¡¤§ ¬yµ{
¹Ï¡C

12

«Ø¥ß ¿i½ü»P¤M¨ã
¤§¾÷ºc¼Ò¦¡

«Ø¥ß ¤M¨ã»P¤u¥ó
¤§¾÷ºc¼Ò¦¡

No

¼¶¼g AutoLISP
µ{¦¡¼ÒÀÀ¤M¨ã¤Á
«d¤u¥ó¤§°Ê§@

¥HAutoCadÃ¸
»s¾¦§Î¥H±´°Q¾÷ºc
¼Ò¦¡¤§¥i¦æ©Ê

Yes

«Ø¥ß¤M¨ã¤§
¼Æ¾Ç¼Ò¦¡

¹Ï 2.10 « Ø ¥ ß¥ » ¤ å ª º¥ [ ¤ u ¼ Ò ¦ ¡ ¤ § y µ { ¹ Ï
¬

2-5-1 ³Ð¦¨¤M¨ã¤§¼Æ¾Ç¼Ò¦¡

¥»¤å¥[¤uªº¤M¨ã¤§´X¦ó§Îª¬¼vÅTµÛ¦±¾¦Áp¶b¾¹ªº¾¦§Î¡ ¦Ó¤M¨ã¤§´X¦ó§Î
A
ª¬¥Ñ¿i½üªº½ü¹ø¨Ó¨M© ¡ ¤]´N¬O»¡¿i½ü¥~§Î½ü¹ø·|¼vÅT¦±¾¦Áp¶b¾¹ªº¾¦§Î§Î
wA
ª¬¡A¬°¤F±À¾É¤M¨ã¼Æ¾Ç¼Ò¦¡¡A»Ý¥ý±À¾É¿i½ü¤§¼Æ¾Ç¼Ò¦¡¡C§Q¥Î¾¦§Î³Ð¦¨-ì²z¡A
±À¾É¥X¤M¨ã¥~§Î½ü¹ø¤èµ{¦¡¡C-º¥ý¨D±o¿i½ü¤§¼Æ¾Ç¼Ò¦¡¡A¦A±N¿i
½ü¥[¤u¸ô®|
»P¤M¨ã¤§¦ì¸mÃö«Y²Õ¦¨®y¼ÐÂà´«¯x°}¡A³Ì«á§Q¥Î»ô¦¸®y¼ÐÂà´«¡A§Y¥i±o¨ì¤M
¨ã®y¼Ð¨t²Î¤U¤M¨ã¦±-±¤èµ{¦¡¡A¹Ï2.11)¬°±À¾É¤M¨ã¼Æ¾Ç¼Ò¦¡¤§¬yµ{¹Ï¡C
(
13

«Ø¥ß¤M¨ã¤§
¼Æ¾Ç¼Ò¦¡

No

¥HMapleÀËÅç
¤M¨ã¼Æ¾Ç¼Ò¦¡¤§¥¿
½T©Ê

Yes

¼¶¼g¤M¨ã¾¦-±¤§
ÂI¸ê®Æ²£¥Íµ{¦¡

§Q¥ÎPro/E«Ø¥ß
¤M¨ã¤§¹êÅé¼Ò«¬

±´°Q¤M¨ã¤§³]-p

¹Ï 2.11 ±À¾É¤M¨ã¼Æ¾Ç¼Ò¦¡¤§¬yµ{¹Ï

¬°¤Fº¸«á¼Æ¾Ç±À¾É¤§¤è«K¡A°w¹ï¿i½ü¡B¤M¨ã¤Î¦±¾¦Áp¶b¾¹®y¼Ð¨t²Î¤§°Ñ
¼Æ¤Î©w¸q°µ»¡©ú¡A¦p¤Uªí
(2.1~2.2)©Ò¥Ü¡G

ªí 2.1 ¥H¿i½ü³Ð¦¨¥X¤M¨ã¤§®y¼Ð¨t²Î©w¸q
®y¼Ð¨t²Î ©w¸q
St (X t , Yt , Z t ) ¿i½ü¥~¹ø»P¤M¨ã¾¦½L¬Û¤Á¤§¹B°Êª¬ºAªº°Ê®y¼Ð¨t²Î
S w (X w , Yw , Z w ) ¿i½ü¤¤¤ß¤§©T©w®y¼Ð¨t²Î
S m ( X m , Ym , Z m ) »²§U®y¼Ð¨t²Î
SG ( X G , YG , Z G ) ¤M¨ã¾¦¹ø®y¼Ð¨t²Î

14

ªí 2.2 ¥»¤å¥[¤u¤¤¤M¨ã¤Î¾¦-Fªº®y¼Ð¨t²Î¤§°Ñ¼Æý-z
°Ñ¼Æ ©w¸q
£ ¿i½ü¥~¹ø¤§À£¤O¨¤
t ªu¿i½ü¥~¹ø¤è¦V¤§°Ñ¼Æ
h ¿i½ü¥[¤u¶i¤M²`«×
rcv ¿i½ü¤º¥b®|
rcc ¿i½ü¥~¥b®|
θ ý-z¿i½ü±ÛÂà¹B°Ê¤§°Ñ¼Æ
η ý-z¿i½ü¬[³]¶b»P¿i½ü©M¤M¨ã³s¤ß½u¤§§¨¨¤
£_ ¿i½ü¹ï¤M¨ã¤§°¾Â¨¤«×
Xd ¿i½ü»P¤M¨ã¤§¤¤¤ß¶Z

w
Zt

rcv
Ot
Xt
h £

¬ã¿i-±u(t)

(a)¬ã¿i¤M¨ã¦¨¥Y¾¦¤§¿i½ü¥b-å-±

w
Z
t

r cc

Ot
Xt
£ h

¬ã¿i-±

(b)¬ã¿i¤M¨ã¦¨¥W¾¦¤§¿i½ü¥b-å-±
¹Ï 2.12 ¥»¤å¥[¤uªº¦±¾¦Áp¶b¾¹¤§¿i½ü¥~§Î½ü¹ø

15

¥»¤å¥[¤uªº¿i½ü¥~§Î¦p¹Ï (2.12)©Ò¥Ü¡A¨ä¿i½ü¬ã¿i-±ª½½u¤èµ{¦¡¬°¡G
(a)¥Y¾¦¤M¨ã¤§¿i½ü¬ã¿i-±¤èµ{¦¡

Xt = − tan aZ (2.1)

§Q¥Î(2.1)¦¡¡A¥OZ = t ϒΑ∞ι±ο♦←©↵ι⁄Μ♦©ƒ♦∞Ψƒ⁄♣↵ι∞∼≠°∇…∅×ψ↑ζ
ƒϒ Rtcv ¬°

 − (t ) tan α 
 0 
Rtcv =   (2.2)
 t 
 
 1 

(b)¥W¾¦¤M¨ã¤§¿i½ü¬ã¿i-±¤èµ{¦¡
¦P²z¡A¥i±o¨ì¬ã¿i¤M¨ã¦¨¥W¾¦¤§¿i½ü¥~¹ø°Ñ¼Æ´y-z¦¡ Rtcc ¬°

 (t ) tan α 
 0 
Rtcc =   (2.3)
 t 
 
 1 

¤M¨ã

¡÷¬ã¿i¥Y¾¦¤M¨ã¥ª ¬ã¿i¥Y¾¦¤M¨ã¥k¡ö
¦±-±¤§¿i½ü¬[³] ¦±-±¤§¿i½ü¬[³]

¹Ï 2.13 ¬ã¿i¥Y¾¦¤M¨ã¥ª¥k¦±-±¤§¿i½ü»P¤M¨ã¬[³]¥Ü·N¹Ï

16

¹Ï 2.14 ¿ i ½ ü ¥ [ ¤ u ¥ Y ¾ ¦ ¤ M ¨ ãk ° ¼ ¦ ± - ± § ® y ¼ Ð ¨ t ² Î
(¥ )¤

¹Ï 2.15 ¿ i ½ ü ¥ [ ¤ u ¥ Y ¾ ¦ ¤ M ¨ ãª ° ¼ ¦ ± - ± § ® y ¼ Ð ¨ t ² Î
(¥ )¤

17

¬°¤F±o¨ì¥Y¾¦¤M¨ã¤§¥ª¥k¦±-±¡A¥²¶·¥H¿i½ü¤º½t¨Ó¬ã¿i¤M¨ã¡A¥ª¥k¦±-±

ªº©w¸q¥Ñ¤M¨ã¾¦-F¤¤¤ß¬Ý¬ã¿i¾¦¬°¥ª¾¦©Î¥k¾¦¨Ó¬É©¡¦Ó¥Y¾¦¤M¨ã¥ª¥k¦±-±
wC

ªº²£¥Í¨Ì¨ä¿i½ü»P¤M¨ã¤§¾÷ºcÂ³]¤£¦P¨Ó¥[¤u¦Ó¦¨¡¥Y¾¦¤M¨ã¥k¦±-±¥[¤u¤§
A

¿i½üÂ³]¬°Â¶¨ä¶b¥ª¶É δ ¨¤¡A¤M¨ã¥ª¦±-±¥[¤u¤§¿i½üÂ³]¬°Â¶¨ä¶b¥k¶É δ

¨¤¡A¦p¹Ï (2.13)©Ò¥Ü¡C
¥Ñ¿i½ü®y¼Ð¨t²Î©M¤M¨ã®y¼Ð¨t²Î¶¡ªº¬Û¹ïÃö«Y¡A¦p¹Ï2.14)©Ò¥Ü¬°¿i½ü
(
¥[¤u¥Y¾¦¤M¨ã¥k°¼¦±-±)¤§®y¼Ð¨t²Î¡A§Q¥Î»ô¦¸®y¼Ð¯x°}Âà´«¤è¦¡¡A¥Ñ¿i½ü
(
»P¤M¨ã¾¦½L¬Û¤Á¤§°Ê®y¼Ð¨tSt ¡AÂà´«¦Ü¿i½ü¤¤¤ß¤§©T©w®y¼Ð¨t²Î S w ¡A¦A¥Ñ¿i

½ü¤¤¤ß¤§©T©w®y¼Ð¨t²Î S w ¡A¸g¥Ñ»²§U®y ¼Ð¨t²Î S m Âà¦Ü¤M¨ã¾¦½L¤¤¤ß¤§®y¼Ð

¨t²Î SG ¡A§Y¥i¨D±o¤M¨ã¦b¾¦-F¤¤¤ß¤§®y¼Ð¨t¤§½ü¹ø¤èµ{¦¡¦p¤U¡G

(a)¥Y¾¦¤M¨ã¦±-±¤èµ{¦¡

cv
¥Y¾¦ ¨ã¦b¾¦-F¤¤¤ß¤§®y¼Ð¨t¥k°¼¦±-±¤§½ü¹ø¤èµ{¦¡ RGr ¡G
¤M

[R ]cv
Gr = [M Gm ][M mw ][M wt ] [R cv ]
t (2.4)

¨ä¤¤ rtcv ¬°¿i½ü¤º½t¤§½ü¹ø¤èµ{¦¡¥H (2.2)¦¡±a¤J¡A¦ÓÂà´«¯x°}¬°¡G

cosθ − sin θ 0 rcv cosθ  1 0 0 0
 sin θ cosθ 0 rcv sin θ  0 cosδ sin δ 0
[M wt ] =   [M mw ] =  
 0 0 1 0  0 − sin δ cosδ 0
   
 0 0 0 1  0 0 0 1

1 0 0 − X d cosη
0 1 0 X d sin η 
[M Gm ] =  
0 0 1 h 
 
0 0 0 1 

18

cv
¦]¦¹¡A¥Y¾¦¤M¨ã¦b¨ä¾¦-F¤¤¤ß®y¼Ð¨t¥k°¼¦±-±¤§½ü¹ø¤èµ{¦¡ R Gr ¬° ¡G

 x G (t ) = (rcv ) cos θ − (t ) cos θ tan α − cosηX d
cv

 cv (2.5)
cv
RGr =  y G (t ) = (t ) sin δ + (rcv ) sin θ cos δ − (t ) cos δ sin θ tan α + sin ηX d
 z cv (t ) = (t ) cos δ + (t ) sin δ sin θ tan α − (r ) sin δ sin θ + h
 G cv

¦P²z¡A¹Ï(2.15)¬°¿i½ü¥[¤u¥Y¾¦¤M¨ã(¥ª°¼¦±-± )¤§®y¼Ð¨t²Î¡A¨ä¥Y¾¦
cv
¤M¨ã¥ª°¼¦±-±¤§½ü¹ø¤èµ{¦¡RGl ¡G

[R ]
cv
Gl = [MGm ][Mmm'][Mm'w ][M wt ] [R cv ]
t (2.6)

¨ä¤¤ Rtcv ¬°¿i½ü¤º½t¤§½ü¹ø¤èµ{¦¡¥H (2.2)¦¡±a¤J¡A¦ÓÂà´«¯x°}¬°¡G

cosθ − sinθ 0 rcv cosθ  1 0 0 0
 sin θ cosθ 0 rcv sin θ  0 cosδ − sin δ 0
[M wt ] =   [M m ' w ] =  
 0 0 1 0  0 sin δ cosδ 0
   
 0 0 0 1  0 0 0 1

 cos 2η sin 2η 0 0 1 0 0 − X d cosη
− sin 2η cos 2η 0 0 0 1 0 X d sin η 
[M mm' ] =   [M Gm ] =  
 0 0 1 0 0 0 1 h 
   
 0 0 0 1 0 0 0 1 

cv
¦]¦¹¡A¥Y¾¦¤M¨ã¦b¨ä¾¦-F¤¤¤ß®y¼Ð¨t¥ª°¼¦±-±¤§½ü¹ø¤èµ{¦¡RGl ¬° ¡G

 xG (t ) = (− cos 2η cos θ + sin 2η cos δ sin θ )(rcv − (t ) tan α ) − (t ) sin δ sin 2η − cos ηX d
cv

 cv
cv
RGl =  y G (t ) = (− sin 2η cos θ + cos 2η cos δ sin θ )(rcv − (t ) tan α ) − (t ) sin δ cos 2η + sin ηX d
 zcv (t ) = (t ) cos δ − (t ) sin δ sin θ tan α + (r ) sin δ sin θ + h
 G cv

(2.7)

(b)¥W¾¦¤M¨ã¦±-±¤èµ{¦¡
¦P²z¡A¬°¤F±o¨ì¥W¾¦¤M¨ã¤§¥ª¥k¦±-±¡A¥²¶·¥H¿i½ü¥~½t¨Ó¬ã¿i¤M
¨ã¡A¦Ó¥W¾¦¤M¨ã¥ª¥k¦±-±ªº²£¥Í¨Ì¨ä¿i½ü»P¤M¨ã¤§¾÷ºcÂ³]¤£¦P¥[¤u¦Ó

19

¦¨¡A¨ä¥ª¥k¦±-±ªº²£¥Í¤§¿i½ü»P¤M¨ãÂ³]¤è¦¡«ê»P¥Y¾¦¤M¨ã¿i½üÂ³]¬Û

¤Ï¡C

R cc
¥W¾¦¤M¨ã¦b¾¦-F¤¤¤ß¤§®y¼Ð¨t¥ª°¼¦±-±¤§½ü¹ø¤èµ{¦¡ Gl ¡G

[R ]
cc
Gl = [M Gm ][M mw ][M wt ] [R cc ]
t (2.8)

¨ä¤¤ Rtcc ¬°¿i½ü¥~½t¤§½ü¹ø¤èµ{¦¡¥H (2.3)¦¡±a¤J¡A¦ÓÂà´«¯x°}¬°¡G

 sin θ 0 cosδ 0
cosθ 0 rcv sin θ  sin δ
[M wt ] =   [M mw ] =  
 0 0 1 0  0 − sin δ cosδ 0
   
 0 0 0 1  0 0 0 1

1 0 0 − X d cosη
0 1 0 X d sin η 
[M Gm ] =  
0 0 1 h 
 
0 0 0 1 

R cv
¦]¦¹¡A¥W¾¦¤M¨ã¦b¨ä¾¦-F¤¤¤ß®y¼Ð¨t¥ª°¼¦±-±¤§½ü¹ø¤èµ{¦¡ Gl ¬° ¡G

 xG (t ) = (rcc ) cos θ + (t ) cos θ tan α − cos ηX d
cc

 cc (2.9)
cc
RGl =  y G (t ) = (t ) sin δ + (rcc ) sin θ cos δ + (t ) cos δ sin θ tan α + sin ηX d
 zcc (t ) = (t ) cos δ − (t ) sin δ sin θ tan α − (r ) sin δ sin θ + h
 G cc

cc
¦P²z¡A¥W¾¦¤M¨ã¥k°¼¦±-±¤§½ü¹ø¤èµ{¦¡ ¬°¡G
rGr

[R ]
cc
Gr = [M Gm ][M mm'][M m'w ][M wt ] [R cc ]
t (2.10)

¨ä¤¤ Rtcc ¬°¿i½ü¥~½t¤§½ü¹ø¤è µ{¦¡¥H (2.3)¦¡±a¤J¡A¦ÓÂà´«¯x°}¬°¡G

 sin θ cosθ 0 rcv sin θ  0 cosδ − sin δ 0
[M wt ] =   [M m ' w ] =  
 0 0 1 0  0 sin δ cosδ 0
   
 0 0 0 1  0 0 0 1

20

 cos 2η sin 2η 0 0 1 0 0 − X d cosη
− sin 2η cos 2η 0 0 0 1 0 X d sin η 
[M mm' ] =   [M Gm ] =  
 0 0 1 0 0 0 1 h 
   
 0 0 0 1 0 0 0 1 

cv
¦]¦¹¡A¥W¾¦¤M¨ã¦b¨ä¾¦-F¤¤¤ß®y¼Ð¨t¥k°¼¦±-±¤§½ü¹ø¤èµ{¦¡RGr ¬° ¡G

 xG (t ) = (cos 2η cos θ + sin 2η cos δ sin θ )(rcc + (t ) tan α ) − (t ) sin δ sin 2η − cos ηX d
cv

 cv
cc
RGr =  y G (t ) = (− sin 2η cos θ + cos 2η cos δ sin θ )(rcc + (t ) tan α ) − (t ) sin δ cos 2η + sin ηX d
 zcv (t ) = (t ) cos δ − (t ) sin δ sin θ tan α + (r ) sin δ sin θ + h
 G cc

(2.11)

¬°¤F¤è«K«á-±ï¿¦X¤èµ{¦¡¤§-pºâ¡A±N¥Y¾¦¤Î¥W¾¦¤M¨ã¤§¥ª¥k¦±-±¤è
µ{¦¡¡A±q¤M¨ã®y¼Ð¨tSG ¡AÂà´«¨ì¥t¤@-Ó°Ñ¦Ò¤§©T©w®y¼Ð¨t S1¡C SG ®y¼Ð¨t

©M S1 ®y¼Ð¨tªº¬ÛÃö¦ì¸m¹Ï¦p¹Ï (3.3)©Ò¥Ü¡C¨ä»ô¦¸®y¼ÐÂà´«¬°

[R ]
1
= [M 1G ][RG ]

¨ä¤¤

cos λ1 − sin λ1 0 0
sin λ cos λ1 0 0
[M1G ] =  1 
 0 0 1 0
 
 0 0 0 1

¤W¦¡®y¼ÐÂà´«¦¡¡AG ¤À§O¥H (2.5)¡B(2.7)¡B(2.9)©Î(2.11)¦¡¥N¤J¡A¥i¥H¨D
R

±o¤M¨ã¦±-±¤èµ{¦¡ªí¥Ü¦b©T©w®y¼Ð¨tS1 ¤W¡C
(a) ¥Y¾¦¤M¨ã¤§¥k°¼¦±-±¤èµ{¦¡

 x1cv (t ) = cos λ1 ((rcv ) cos θ − (t ) cos θ tan α − cos ηX d )
r

 − sin λ1 ((t ) sin δ + ( rcv ) sin θ cos δ − (t ) cos δ sin θ tan α + sin ηX d )
 y lr (t) = sin λ1 (( rcv ) cos θ − (t ) cos θ tan α − cos ηX d )
cv
(2.12)
cv
Rrl = 
 + cos λ1 (((t ) sin δ + (rcv ) sin θ cos δ − (t ) cos δ sin θ tan α + sin ηX d ))
 z1cv (t ) = (t ) cos δ + (t ) sin δ sin θ tan α − (rcv ) sin δ sin θ + h
r



21

(b) ¥Y¾¦¤M¨ã¤§¥ª°¼¦±-±¤èµ{¦¡

 x1cv (t ) = cos λ1 ((− cos 2η cos θ + sin 2η cos δ sin θ )(rcv − (t ) tan α ) − (t ) sin δ sin 2η − cos ηX d )
l

 − sin λ1 ((− sin 2η cos θ + cos 2η cos δ sin θ )(rcv − (t ) tan α ) − (t ) sin δ cos 2η + sin ηX d )
 y1cv (t ) = sin λ1 ((− cos 2η cos θ + sin 2η cos δ sin θ )(rcv − (t ) tan α ) − (t ) sin δ sin 2η − cos ηX d )
R1cv =  l
+ cos λ1 ((− sin 2η cos θ + cos 2η cos δ sin θ )(rcv − (t ) tan α ) − (t ) sin δ cos 2η + sin ηX d )
l

 z1l (t ) = (t ) cos δ − (t ) sin δ sin θ tan α + (rcv ) sin δ sin θ + h
cv



(2.13)

(c) ¥W¾¦¤M¨ã¤§¥ª°¼¦±-±¤èµ{¦¡

 x1cc (t ) = cos λ1 ((rcc ) cos θ + (t ) cos θ tan α − cosηX d )
l

 − sin λ1 ((t ) sin δ + (rcc ) sin θ cos δ + (t ) cos δ sin θ tan α + sin ηX d )
 y (t ) = sin λ1 ((rcc ) cos θ + (t ) cos θ tan α − cos ηX d )
cc
(2.14)
R1cc =  1l
+ cos λ1 ((t ) sin δ + (rcc ) sin θ cos δ + (t ) cos δ sin θ tan α + sin ηX d )
l

 z1cc (t ) = (t ) cos δ − (t ) sin δ sin θ tan α − (rcc ) sin δ sin θ + h
l



(d) ¥W¾¦¤M¨ã¤§¥k°¼¦±-±¤èµ{¦¡

 x1cv (t ) = cos λ1 ((cos 2η cos θ + sin 2η cos δ sin θ )(rcc + (t ) tan α ) − (t ) sin δ sin 2η − cos ηX d )
r

 − sin λ1 ((− sin 2η cos θ + cos 2η cos δ sin θ )(rcc + (t ) tan α ) − (t ) sin δ cos 2η + sin ηX d )
 y (t ) = sin λ1 ((cos 2η cos θ + sin 2η cos δ sin θ )(rcc + (t ) tan α ) − (t ) sin δ sin 2η − cos ηX d )
cv
R1cc =  1r
cos λ1 ((− sin 2η cos θ + cos 2η cos δ sin θ )(rcc + (t ) tan α ) − (t ) sin δ cos 2η + sin ηX d )
r

 z1cv (t ) = (t ) cos δ − (t ) sin δ sin θ tan α + (rcc ) sin δ sin θ + h
r


(2.15)

2- 5- 2 ¤M¨ã¤§¹êÅé
¼Ò«¬«Øºc¤Î¤M¨ã³]-p±´°Q

¥»¤å¥[¤u¤è¦¡ªº¤M¨ã¤Î¤u¥ó¾÷ºc¼Ò¦¡¬°¥æ¤e¶b«¬¦¡¤§¹B°ÊÃö«Y¡A©Ò¥H·í
¤M¨ã¾¦³»¬°¥--±®É¡A¥Ñ©ó¤M¨ã»P¤u¥ó©¼¦¹¶É±×Ãö«Y¡A¤u¥ó¾a¤º°¼¤Î¾¦©³¨Ã¤£
¯à³Q¤M¨ã§¹¥þ¥[¤u¨ì¡A¨£¹Ï2.16)¡C©Ò¥H¤M¨ã¦b³]-p®É»Ý¦Ò¼{»P¤u¥ó¤§¶É±×
(

22

Ãö«Y¡A¦Ó±N¤M¨ã¾¦³»³¡¤À¼W¥[¶É±×-±¥H¤Î¼W¥[¤M¨ã¾¦¼e¨Ã¦V¤º©µ¦ù¡A¨Ï±o¤u

¥ó¤§¾¦©³¤Î¤º°¼¬Ò¯à³Q¤M¨ã§¹¥þ¥[¤u¨ì¡¦p¹Ï
A (2.17)A
¡ ¦Ó¤M¨ã¤§¾¦-F¦p¹Ï(2.18)
©Ò¥Ü¡A»Ý°t¦X¤M¨ã»P¤u¥ó¤§¾÷ºc¹B°ÊÃö«Y¦Ó³]-p¡C

¤M¨ã ¤M¨ã

¤u¥ó ¤u¥ó

¹Ï 2.16 ¤M¨ã¥[¤u¤u¥ó¥Ü·N¹Ï¡A·í¤M¨ã¾¦³»¬°¥--±®É¡A
¥k¹Ï¬°¤M¨ã¤§
ºI-±©ñ¤j¹Ï

¤M¨ã
¤M¨ã

¤u¥ó ¤u¥ó

¹Ï 2.17 ¤M¨ã¥[¤u¤u¥ó¥Ü·N¹Ï¡A¤M¨ã¾¦³»¼W¥[¶É±×-±¡A¥k¹Ï¬°¤M
¤§ºI-±©ñ¤j¹Ï

23

¹Ï 2.18 ¤M¨ã¾¦-F¤§¥ßÅé¹Ï

¤M¨ã¹êÅé¼Ò«¬¤§«Øºc¡A«Y§Q¥Î«e¤@¸`
2-5-1 ¸`¤¤©Ò±À¾Éªº¥Y¾¦¤Î¥W¾¦¤§
¤M¨ã¥ª¥k¾¦-±¤èµ{¦¡«Øºc¦Ó¦¨¡A¦b¹Ï(2.19)¬°¥Y¾¦¤M¨ã¤§¹êÅé¼Ò«¬¥H¤Î¥Ñ¿i
½ü¥[¤u¤M¨ã¾¦-F©Ò°£±¼ªº³¡¤À¡A¤]´N¬O¾¦-F«Ý«õ°£³¡¤À¡A¦Ó¹Ï
(2.20)¬°¥W¾¦
¤M¨ã¤§¹êÅé¼Ò«¬¥H¤Î¥Ñ¿i½ü¥[¤u¤M¨ã¾¦-F©Ò°£±¼ªº³¡¤À¡C

¹Ï 2.19 ¥Y¾¦¤M¨ã¹êÅé¼Ò«¬»P«Ý«õ°£³¡¤À

¹Ï 2.20 ¥W¾¦¤M¨ã¹êÅé¼Ò«¬»P«Ý«õ°£³¡¤À
24

²Ä¤T³¹

¦±¾¦Áp¶b¾¹¤§¾¦-±¼Æ¾Ç¼Ò¦¡

¾¦¤M¤Á¹L¾¦½ü©Ò§Î¦¨ªº¥]µ¸-±
(Envelope)Ń¬O¾¦-±¤èµ{¦¡¡C¥»³¹§Q¥Î»ô
¦¸®y¼ÐÂà´«¤§-y¸ñ¤èµ{¦¡¤Îï¿¦X¤èµ{¦¡¨Ó±À¾É¥X¦±¾¦Áp¶b¾¹¤§¾¦-±¤èµ{¦¡¡C

3-1 ¥]µ¸²z½×

¤wª¾¤@¦±-± 1 , v 2 , φ ) ¡A v1 ©Mv 2 ¬° S ªº¦±-±®y¼Ð°Ñ¼Æ¡A
S (v φ ¬° S ªº¹B°Ê°Ñ¼Æ¡C

·í φ -ÈÅÜ°Ê®É¡A¦±-± S ªº¦ì¸m¤]¸òµÛÅÜ°Ê¡A¦]¦¹¦b¤£¦Pªº φ -È¤U·|±o¨ì¤£¦P

ªº¦±-± S ¡A³o¨Ç¦±-±ªº¶°¦X¦¨¬°¦±-±±Ú {S }¡C
φ

°²¦p¦s¦b¤@-Ó¦±-± ¡A¨Ï±oΣ ¤W¨C¤@ÂI³£¹ïÀ³©ó¦±-±±Úφ }¤¤°ß¤@ªº¦±
Σ {S
-± Sφ ¡A¦Ó¥B¦±-± Sφ ©MΣ ¦b¸ÓÂI©¼¦¹¬Û¤Á¡A«hºÙ Σ ¬°¦±-±±Ú {S }ªº¥]µ¸-±¡C
φ

®Ú¾Ú¥]µ¸-±ªº©w¸q¡A¥]µ¸-± Σ ©M¦±-± φ ¦b±µÄ²ÂI
S M ©¼¦¹¬Û¤Á¡C¦]¬°¥]

µ¸-± Σ ©M¦±-± φ ¤§¶¡¬°½u©Ê±µÄ²¡A§Y¥]µ¸-±
S Σ ©M¦±-± Sφ ¦b±µÄ²½u¤Wªº¨C¤@ÂI

³ £ ¥ ² ¶ · © ¼ ¦ ¹ ¬ Û ¤ Á ¡ A ³ o ± ø ± µ Ä ² ½ u º Ù ¬ ° ¦ ± - ± ± Ú {S φ } ¤ ¤ ¦ ± - ± Sφ ª º ¯ S ¼ x ½ u

Lφ (Characteristic Line)¡A¦p¹Ï(3.1)©Ò¥Ü¡C¦]¬°¦b¨C¤@¦±-± φ ¤W§¡¦³¤@±ø¯S¼x
S

½u©M¥]µ¸-±¬Û¤Á¡A©Ò¥H¦±-±±Ú {S }¤W©Ò¦³¯S¼x½uªº¶°¦X¡AŃºc¦¨¤F¥]µ¸-±
φ Σ ¡C

ºî¦X¥H¤Wµ²ªG¡A¥i±oª¾¯S¼x½u¦³¤T-Ó¯S©Ê¡G
1. ¦b¦±-±±Ú{S φ }¤¤¡A¨C¤@¦±-± Sφ ¤W§¡¦³¤@±ø¯S¼x½u Lφ ¡AŃ¬O¦±-± Sφ ©M¥]

µ¸-± Σ ªº±µÄ²½u¡C
2. ©Ò¦³ªº¯S¼x½uºc¦¨¤F¦±-±±Úªº¥]µ¸-± Σ ¡C
3. ¥]µ¸-± Σ ©M¦±-±±ÚS φ }¤¤ªº¨C¤@-Ó¦±-± Sφ ¡ ª Sφ ¤Wªº¯S¼x½u©¼¦¹¬Û¤Á¡C
{ Au

25

Lφ

Sφ
Σ

¹Ï 3.1 ¦ ± - ± S£p M ¥ ] µ ¸ - ± £ U ¬ Û ¤ Á ª º ¯ S ¼ x ½ u
© L£p

±N¤@-Ó¦±-±¨Ì«ü©wªº¹B°Ê¤è¦¡²¾°Ê¡A¦¹¦±-±¦b¤£¦P®É¶¡¨Ì§Ç¯d¤U-y¸ñ¡A
³o¨Ç-y¸ñªºÁ`¦X§Y§Î¦¨¥]µ¸-±¡C°²³]¦³¤@-Ó¦b®y¼Ð¨t S1 ( X 1 , Y1 , Z1 ) ¤§¦±-± R0 ¡A

·Q-n±o¨ì¥H¦¹¬°³Ð¦¨¥À-±§Î¦¨¦ì©ó®y¼Ð¨t S 2 ( X 2 , Y2 , Z 2 ) ¤W¤§¥]µ¸-± R1 ¡A¥O¥Ñ

®y¼Ð¨t S1 ( X 1 , Y1 , Z1 ) Âà´«¨ì®y¼Ð¨t S 2 ( X 2 , Y2 , Z 2 ) ¤§Âà´«¯x°}¬° M 2,1¡A§Q¥Î¦@³m²z

½× (3.1)¦¡»P»ô¦¸®y¼ÐÂà´«¯x°} (3.2)¦¡§Y¥i¥H±o¨ì¦¹¥]µ¸-±¥~§Î¡C

v v
N ⋅V 12 = 0 (3.1)

[R1 ] = [M ][R ]
1, 0 0 (3.2)

v
¦b(3.1)¦¡¤¤ªºV 12 ¥H¤Î (3.2)¦¡¤¤ªº M 1, 0 ¥]§t¤F¨â®y¼Ð¨t¶¡ªº¬Û¤¬¹B°ÊÃö

«Y¡A§Y¾÷ºc©Ò»Ýªº¿é¤J»P¿é¥XÃö«Y¡A¦b³Ð¦¨ªk¤¤¡A¦¹¹B°ÊÃö«Y§Y¬°¤M¨ã»P¤u

¥ó¶¡ªº¬Û¹ï¹B°ÊÃö«Y¡A¦Ó R0 ¬°¤M¨ã¥~§Î¡C±N (3.1)§Y(3.2)¦¡Áp¥ß¡A§Y¥i±o¨ì¦b

S 2 ( X 2 , Y2 , Z 2 ) ®y¼Ð¨tªº¥]µ¸¥~§Î¡A¤]´N¬O¤M¨ã©Ò§Î¦¨ªº¥]µ¸-±¤§¤u¥ó¥~§Î R1 ¡C

26

3-2 ï¿¦X¤èµ{¦¡

ï¿¦X¤èµ{¦¡Equation of Meshing)¬O¾¦½ü-ì²z¤¤-«-nªº²z½×¤§¤@¡ ¥»¤å§Q
( C
¥Î³Ð¦¨ªk(Generation Method)¨Ó±À¾É¦±¾¦Áp¶b¾¹¤§´X¦ó¼Æ¾Ç¼Ò¦¡¡C¦Ó³Ð¦¨ªk
±À¾É¦±¾¦Áp¶b¾¹ªº¼Æ¾Ç¼Ò¦¡¹Lµ{¤¤¡A¥Ñ©ó²o¯A¨ì¤M¨ã»P¤u¥ó¶¡ªº¹B°ÊÃö«Y¡A
¦]¦¹»Ý¥ý±À¾É¨âªÌ¶¡¤§ï¿¦X¤èµ{¦¡¡C
³]ªÅ¶¡¤¤¦³¥ô·N¨â-Ó¤¬¬Ûï¿¦X¹B°Êªº¦±-±Σ 1 ♥Μ 2 ϒΑƒπ≠∉ 3.2 ©Ò¥Ü¡A©¼¦¹
Σ

¬Û¤Á©óÂI M¡AM ÂI¦P®É¤]¬O³o¨â-Ó¦@³m¹B°Ê¹ïªºÀþ®É±µÄ²ÂI¡A¨â-Óï¿¦X¦±
v v
-±¦b¨ä¦@¤ÁÂI M ÂI¨ã¦³¦@¦P¤§¦±-±ªk¦V¶q ¡A V (12) «hªí¥Ü¦±-± Σ 1 ♥Μ 2 ƒβ M
N Σ

ÂIªº¬Û¹ï³t«×¡C
¥Ñ©ó Σ 1 ♥Μ 2 ♠≡±∝⊗″≠Λ∝{←Ο≥σ⊗∫♠≡ϒΑ♦®ƒ±↑±↵ƒΞ≠Β°⊇→⊃ϒΑ←ϑ⁄≤⁄ℵℜ∞⁄≤
Σ

⊗…″∂ι⁄ϑ∞τ⁄≅ƒ±↑±⁄≡ϒΑƒ]ƒ≠⁄≤⋅♦®ƒ±↑±←ΟℜΙ±∝⊗″♥∈υ±∝⊗″ϒΑ♦®ƒ±↑±ƒβƒ≅ƒΠ♠κ

ƒς∂θ♠≡⁄ƒς♦℘∝Λ←⇔≠≠Β°⊇ϒΑ♦™ƒβ ÂIªº¬Û¹ï³t«×¥²¦b¨â¦±-±¤§¦@¦P¤Á¥--±
M
¤W¡C
¦Ó¨â¦±-±¤§¦@¦Pªk¦V¶q¥²©M¤Á¥--±¬Û¤¬««ª½¡A¦]¦¹¨âï¿¦X¹B°Ê¦±-±¨ä¬Û
v v
¹ï³t«× V (12) ©M¦@¦Pªk¦V¶q N ¡A¦b¨ä¦@¦P±µÄ²ÂI M ³B¥²¤¬¬Û««ª½¡C©Ò¥H¦b
(3.1)
¦¡ªºï¿¦X¤èµ{¦¡¥²¦¨¥ß

v v
N ⋅V 12 = 0

¦¹¤èµ{¦¡´N¬O¾¦½üï¿¦X-ì²z¤¤±´°Q¦@³m¹B°Ê¹ïï¿¦X®É¥²¶·º¡¨¬ªºï¿¦X
¤èµ{¦¡¡C

27

Σ1

v T
N
v Σ2
V 12
M

¹Ï 3.2 ï¿¦X¦±-±¤§Ãö«Y¥Ü·N¹Ï

¥»¬ã¨sµo®iªº¦±¾¦Áp¶b¾¹¥[¤u¤è¦¡«Y¥H¤M¨ã§Q¥Î³Ð¦¨ªk»s³y¥X¨Óªº¡ ©
AÒ
¥H¦±¾¦Áp¶b¾¹©M¤M¨ã¦b¥[¤u¹Lµ{¤¤¬O©¼¦¹¦@³mï¿¦Xª¡ ¦]¦¹¦b±À¾É¦±¾¦Áp¶b
ºC
¾¹¾¦-±¼Æ¾Ç¼Ò¦¡®É¡A¥²¶·¥ý¨D±o¤M¨ã»P¦±¾¦Áp¶b¾¹¤§ï¿¦X¤èµ{¦¡¡CÂÇ¥Ñï¿¦X
¤èµ{¦¡¤Î¤M¨ã»P¤u¥ó©ó³Ð¦¨¹Lµ{¤¤¤M¨ã¤§-y¸¡ «K¥i¥H±o¨ì¦±¾¦Áp¶b¾¹ªº¥~
ñA
§Î¡C
¥»¬ã¨s¤§¤M¨ã»P¤u¥ó¶¡ªº¹B°Ê¾÷ºc¬OÄÝ©ó¥æ¤e¶b¤§¦@³m¹B°ÊÃö«Y¡C¦pªG
¤M¨ã©M¤u¥óªº¨¤³t«×¤wª¾¡A¦P®É¤M¨ã¤Î¤u¥ó¨â¶b¤¤¤ß¶Z©M¥æ¤e¨¤§¡¤wª¾¡A«h
¤M¨ã»P¤u¥ó¶¡¤§¹B°Ê³t«×§Y¥i¨D¥X¡A¶i¦Ó¥i¨D±o¨âªÌ¤§ï¿¦X¤èµ{¦¡¡C
v
¥Ñï¿¦X¤èµ{¦¡(3.1)ª¾¡A»Ý¥ý¨D¥X
¤M¨ã¤§¦±-±ªk¦V¶q ¤Î¤M¨ã»P¤u¥ó¶¡¤§
N
v v
¬Û¹ï³t«× V (12) ¡A¥Ñ©ó¦b¤M¨ã»P¤u¥óï¿¦X¤§¦@¦P±µÄ²ÂI M ¤W¡A¨äªk¦V¶q »P
N
v
¬Û¹ï³t«× V (12) §e¤¬¬Û««ª½¡A¦p¦¹«K¥i¨D¥Xï¿¦X¤èµ{¦¡¡C

28

3 - 2- 1 ¤M¨ã¦±-±¤§ªk¦V¶q

v
®Ú¾Ú·L¤À´X¦ó²z½× [Spivak,1979]±oª¾¡A¤M¨ã¦±-± G ªºªk¦V¶q¥i¥Ñ¤U¦¡¨D
R

±o¡G

v v
r ∂RG ∂RG
NG = × (3.3)
∂t ∂θ

¨Ì¦¹¡A¥i¥H§Q¥Î¤W¦¡¨Ó¤À§O¨D±o¤M¨ã¥Y¾¦¤Î¥W¾¦¤§¥ª°¼»P¥k°¼¦±-±ªºªk
¦V¶q¡C
(a) ¥Y¾¦¤M¨ã¥k°¼¦±-±¤§ªk¦V¶q¡G
-º¥ý¡A±N¥Y¾¦¤M¨ã¥k°¼¦±-±¤èµ{¦¡
(2.5)¹ï¨ä¦±-±°Ñ¼Æ¤À§O°¾·L¤À±o

v cv
∂RGr r r r
= [− cos θ tan α ]i + [sin δ − sin θ cos δ tan α ] j + [sin θ sin δ tan α + cos δ ]k
∂t
(3.4)

v cv r r
∂RGr [(t ) sin θ tan α − (r ) sin θ ]i + [(r ) cos θ cos δ − (t) cos θ cos δ tan α ] j
= r
∂θ + [( t ) sin δ cos θ tan α − ( r ) sin δ cos θ ]k

±N(3.4)¥N¤J(3.3)¦¡¡A§Y¥iÀò±o¥Y¾¦¤M¨ã¥k°¼¦±-±¤§ªk¦V¶q¦p¤U¡G

v cv v v v
N Gr = [ A1 A4 − A2 A3 ] i + [A1 (cos θ tan α ) + A2 A5 ] j + [ A3 (− cos θ tan α ) − A4 A5 ] k
(3.5)

¨ä¤¤

A1 = t sin d cos ? tan a − rcv sin d cos ?

A2 = sin d sin ? tan a + cos d

29

A3 = rcv cos δ cos θ − t cos δ cosθ tan α

A4 = sin δ − cos δ sin θ tan α

A5 = t sin θ tan α − rcv sin θ

v v v
i ¡B j ©Mk ¬° SG ®y¼Ð¨t²Îªº¤T-Ó««ª½¶bªº³æ¦ì¦V¶q¡C

(b) ¥W¾¦¤M¨ã¥ª°¼¦±-±¤§ªk¦V¶q¡G
¦P²z¡A§Q¥Î¥W¾¦¤M¨ã¥ª°¼¦±-±¤èµ{¦¡
(2.9)¤À§O¹ï¨ä¦±-±°Ñ¼Æ·L¤À±o

r cc
∂RGl r r r
= [cos θ tan α ] i + [sin δ + sin θ cos δ tan α ] j + [− sin θ sin δ tan α + cos δ ] k
∂t
(3.6)

r cc r r
∂RGl [− (t ) sin θ tan α − (r ) sin θ ] i + [(r ) cos θ cos δ + (t) cos θ cos δ tan α ] j
= r
∂θ + [− ( t ) sin δ cos θ tan α − ( r ) sin δ cos θ ] k

±N(3.6)¥N¤J(3.3)¦¡¡A§Y¥iÀò±o¥W¾¦¤M¨ã¥ª°¼¦±-±¤§ªk¦V¶q¦p¤U¡G

v cc
N Gl = [A A
'
1
'
4
] [
v
] [
v
] v
− A '2 A'3 i + A1' ( − cos θ tan α ) + A'2 A5' j + A3' (cos θ tan α ) − A '4 A'5 k
(3.7)

¨ä¤¤

A1' = − t sin d cos ? tan a − rcc sin d cos ?

A'2 = − sin d sin ? tan a + cos d

A'3 = rcc cos δ cos θ + t cos δ cosθ tan α

A'4 = sin δ + cos δ sin θ tan α

30

A'5 = − t sin θ tan α − rcc sin θ

¦Ó¦P¼Ëªº¡A¥Y¾¦¤M¨ã¥ª°¼¦±-±¤§ªk¦V¶q
¤Î¥W¾¦¤M¨ã¥ª°¼¦±-±¤§ªk¦V
¶q¤]¥i¤À§O¨D±o¦p¤U¡G
(c) ¥Y¾¦¤M¨ã¥ª°¼¦±-±¤§ªk¦V¶q¡G

v cv
N Gl = [− A B
1 3
] [
v v
] v
− A '2 B1 i + A1 B4 + A2' B2 j + [B4 B1 − B3 B2 ] k (3.8)

¨ä¤¤

B1 = (sin 2η sin θ + cos 2η cos δ cos θ )( rcv − t tan α )

B2 = (− cos 2η sin θ + sin 2η cos δ cos θ )( rcv − t tan α )

B3 = − tan α (cos 2η cos δ sin θ − sin 2η cos θ ) − cos 2η sin δ

B4 = − tan α (sin 2η cos δ sin θ + cos 2η cosθ ) − sin 2η sin δ

(d) ¥W¾¦¤M¨ã¥k°¼¦±-±¤§ªk¦V¶q¡G

v cv
N Gr = [− A B
'
1
'
3
] [
v v
] [ ] v
− A2 B1' i + A1' B '4 + A2 B '2 j + B '4 B '1 − B '3 B '2 k (3.9)

¨ä¤¤

B1' = (sin 2η sin θ + cos 2η cos δ cos θ )( rcc + t tan α )

B '2 = ( − cos 2η sin θ + sin 2η cos δ cosθ )( rcc + t tan α )

B '3 = tan α (cos 2η cos δ sin θ − sin 2η cos θ ) − cos 2η sin δ

31

B '4 = tan α (sin 2η cos δ sin θ + cos 2η cos θ ) − sin 2η sin δ

v v v
¦b¦¡¤l (3.5)¡B(3.7)¡B(3.8)¥H¤Î(3.9)¤¤ªº i ¡B j ©Mk ¬Ò¬° SG ®y¼Ð¨t²Îªº

¤T-Ó««ª½¶bªº³æ¦ì¦V¶q¡C

3- 2- 2 ¬Û¹ï³t«×

v
¬°¤F¨D±o¤M¨ã¤Î¦±¾¦Áp¶b¾¹¦±-±ªº¬Û¹ï³t«×V (12) ¡A»Ý¥ýÁA¸Ñ¤M¨ã¤Î¦±¾¦
Áp¶b¾¹ï¿¦X®É¤§¾÷ºc¹Ï¤Î¨ä¬Û¹ï¹B°ÊªºÃö«Y¡ ¤M¨ã¤Î¦±¾¦Áp¶b¾¹¤§¾÷ºc¹Ï¦p
C

¹Ï (3.3)©Ò¥Ü¡C

Y3
Y4
β YC
ω1 Z Z Z 3 Z2 λ2
G¡A1
OG¡AO1 v O2¡A3 Z Z
O
YG R3 ¡AC ω 2
4 O4 Oc
¡A
?1 X 2¡A3
X Y2 X4
X1 Y1
XG XC

v
v R2
R1

M

¹Ï 3.3 ¤M¨ã¤Î¤u¥ó¤§¹B°Ê¾÷ºc¹Ï

32

¹Ï¤¤¤M¨ã«Y©T©w©ó S G ( X G , YG , Z G ) ® y ¼ Ð ¨ t ¡ A ¦ Ó ¤ u ¥ ó « h ¬ O © T © w © ó

S C ( X C , YC , Z C ) ®y¼Ð¨t¡A¤S®y¼Ð¨t S1 ¡B S 2 ©MS 3 §¡¬°°Ñ¦Ò®y¼Ð¨t¡F±Û Âà¨¤ λ1 ←°⁄Μ

♦© SG ®y¼Ð¨t )¥HZ1 ¶b¬°±ÛÂà¶b¡A¬Û¹ï©ó
( S1 ®y¼Ð¨tªº±ÛÂà¨¤«×¡F±ÛÂà¨¤ λ 2 ←°⁄υ

∞⌠( SC ®y¼Ð¨t )¥HZ 4 ¶b¬°±ÛÂà¶b¡A¬Û¹ï©ó S 4 ®y¼Ð¨tªº±ÛÂà¨¤«×¡F β ¬°¤M¨ã»P¤u

¥ó±ÛÂà¶b¤§§¨¨¤¡C¤S ϖ 1 ⁄∈ ϖ 2 ⁄ℵ♣Ο←°⁄Μ♦©♥Μ⁄υ∞⌠♠≡±⇔ℜ◊♦⁄≥τ↔⋅ϒΑƒ©♦™ℜ◊≥τ⁄〉

←°⁄Μ♦©♥Μ⁄υ∞⌠⁄♣ƒ…∅♠≡⁄∉⁄〉ϒΧ⁄Μ♦©♥Μ⁄υ∞⌠⁄♣∞⁄≅ƒ≅ƒΠ±∝⊗″ℜΙ ( X , Y , Z ) ϒΑ↑Ψ
M
v v
♠∞⇐♥⌠→ψ…∠♦τ SC ¡A«h¨ä¦ì¸m¦V¶q¤À§O¬° R1 ©M R2 ¡A¦Ó R3 ¬°¤M¨ã»P¤u¥ó¤§
SG ¤Î
v v
¶ê¤ß¶Z¡C-Y V1(1) ¬°¤M¨ã¦±-±¤W¸Ó±µÄ²ÂIªº³t«×ªí¥Ü©ó S1 ®y¼Ð¨t¡F V1( 2) ¬°¤u¥ó¦±

-±¤W¸Ó±µÄ²ÂIªº³t«×ªí¥Ü©ó S1 ®y¼Ð¨t¡A¥Ñ¹Ï (3.3)¥iª¾

v r r
V1(1) = ϖ 1 × R1 (3.10)

v r r
V1( 2) = ϖ 2 × R2
r r r
= ϖ 2 × ( R1 + R3 ) (3.11)
r r r r
= ϖ 2 × R1 + ϖ 2 × R3

¦]¦¹¡A¤M¨ã»P¤u¥ó¬Û¹ï³t«×¬°

v v v
V1(12) = V1(1) − V1( 2)
r r r r r (3.12)
= (ϖ 1 − ϖ 2 ) × R1 − ϖ 2 × R3

¥Ñ©ó¦b S1 ®y¼Ð¨t¤¤

r r
ϖ 1 =ϖ1k

r r r
ϖ2 = − ϖ 2 sin( π − β ) j − ϖ 2 cos(π − β ) k
r r
= − ϖ 2 sin β j + ϖ 2 cos β k (3.13)

33

¨ä¤¤¤§°Ñ¼Æ¦p¹Ï (3.4)©Ò¥Ü¡G

r r v r v
R3 = − ( R3 sin ν ) j − ( R3 cosν ) k

R3 = ao _1 + a o − 2 × a o × ao _1 × cos τ
2 2 2 2

3
τ = π −β −µ
2

¤M ¤u
¨ã ¥ó

¹Ï 3.4 ¤M¨ã»P¤u¥ó¤§¬Û¹ïÃö«Y

34

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