CN104392641A - Demonstration teaching aid of Euler's rotation theorem - Google Patents
Demonstration teaching aid of Euler's rotation theorem Download PDFInfo
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- CN104392641A CN104392641A CN201410673877.7A CN201410673877A CN104392641A CN 104392641 A CN104392641 A CN 104392641A CN 201410673877 A CN201410673877 A CN 201410673877A CN 104392641 A CN104392641 A CN 104392641A
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- G—PHYSICS
- G09—EDUCATION; CRYPTOGRAPHY; DISPLAY; ADVERTISING; SEALS
- G09B—EDUCATIONAL OR DEMONSTRATION APPLIANCES; APPLIANCES FOR TEACHING, OR COMMUNICATING WITH, THE BLIND, DEAF OR MUTE; MODELS; PLANETARIA; GLOBES; MAPS; DIAGRAMS
- G09B23/00—Models for scientific, medical, or mathematical purposes, e.g. full-sized devices for demonstration purposes
- G09B23/02—Models for scientific, medical, or mathematical purposes, e.g. full-sized devices for demonstration purposes for mathematics
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Abstract
A demonstration teaching aid of Euler's rotation theorem belongs to the technical field of scientific education. The demonstration teaching aid of Euler's rotation theorem consists of a first device and a second device, wherein the first device is able to set a posture of a reference rigid body; after setting, a Euler axis position and a primary Euler rotation angle are obtained by measuring rotation angles of three rotations in the first device, and substituting the rotation angles into a direction cosine matrix for calculation; the second device is used for verifying the theorem; a corresponding position with respect to the angle obtained from the previous calculation is adjusted in the second device, so as to demonstrate the position of the Euler axis; the rigid body is returned to the initial posture by adjusting a spherical hinge; and then the rigid body in the initial posture is rotated around the Euler axis; if the rotation angle is the primary Euler rotation angle, the rigid body moves from the initial posture to the posture that is set in the first device, thereby the theorem is verified to be proved. The demonstration teaching aid of Euler's rotation theorem is more 'visual and intuitive' in demonstration effect; the determination way of the Euler axis does not affect the rotation among each portion of the devices; and the data can be read out at any time; the devices can be fixed after setting any position; the operation is simple; and the devices are portable.
Description
Technical field
The invention belongs to education of science technical field, be specifically related to a kind of teaching aid demonstrating Euler's rotation theorem.
Background technology
Euler's rotation theorem is proposed by the famous mathematician of Switzerland and physicist Lai Anghade Euler.Euler's rotation theorem is described below: rigid body rotates can be realized by the once rotation walking around this some root axle around the arbitrary finite of fixed point.Also can simply be expressed as: rigid body is turned to any one attitude from an attitude and can be realized by the once rotation around certain root axle.The difficulty understanding this theorem is mainly: one is how to find rotation axis, i.e. Euler's axle; Two is how to determine the corner around Euler's axle, i.e. Euler's corner.
At present, market there is no the relevant teaching aid of Euler's rotation theorem, classroom is explained Euler's rotation theorem abstract hard to understand, curricular advancement is very fast, content of attending class dullness, and mere theoretical knowledge is difficult in student's short time accept, and needs the cost plenty of time to learn by oneself after class and understands.And understand the study of Euler's rotation theorem to many-body dynamics and have very important significance, greatly simplifying the calculating to rotating by the concept of its derivative hypercomplex number, being widely used in space industry and computer graphics.Therefore, Euler's rotation theorem demonstration teaching aid is applied to actual classroom, helping student deeply to understand becomes the task of top priority.
Summary of the invention
The object of the invention is to the difficulty for overcoming instruction Euler rotation theorem, helping student to understand this theorem, a kind of Euler's rotation theorem demonstration teaching aid is provided.
The present invention is made up of device one A and device two B, device one A is made up of base I 1, groove I 2, wing nut I 3, double threaded screw I 4, square universal joint 5, groove II 6, axle I 7, hollow rigid body I 8, nut I 9, nut II 10, double threaded screw II 11, wing nut II 12, wing nut III 25, double threaded screw III 26, double threaded screw IV 27 and wing nut IV 28, wherein affixed with base I 1 center bottom groove I 2; Axle I 7 lower end is affixed through nut II 10 and groove II 6 top mesopore I 32, and axle I 7 upper end is affixed bottom nut I 9 with hollow rigid body I 8; Square universal joint 5 four sides is threaded with the inner of double threaded screw I 4, double threaded screw II 11, double threaded screw III 26 and double threaded screw IV 27 respectively; The hole I 29 of double threaded screw I 4 outer end left side vertical plate in groove I 2 is threaded with wing nut I 3; The hole II 30 of double threaded screw II 11 outer end right side riser in groove I 2 is threaded with wing nut II 12; The hole IV 33 of double threaded screw III 26 outer end front side riser in groove II 6 is threaded with wing nut III 25; The hole III 31 of double threaded screw IV 27 outer end rear side riser in groove II 6 is threaded with wing nut IV 28.
Device two B is made up of base II 13, axle II 14, dish axle sleeve 15 with a scale, axle III 16, detachable groove 17, axle IV 18, hollow rigid body II 19, nut III 20, nut IV 21, nut V 22, axle V 23 and wing nut V 24, affixed with base II 13 center bottom its axis II 14, dish axle sleeve 15 with a scale is placed on axle II 14 top, and fixes with nut V 22; Axle III 16 inner with dish axle sleeve 15 with a scale on the left of affixed; Axle III 16 outer end is flexibly connected with the hole VI 37 of left riser 36 bottom of detachable groove 17; Axle V 23 inner with dish axle sleeve 15 with a scale on the right side of affixed; Axle V 23 outer end is threaded with wing nut V 24 through the hole V 34 of right riser 35 bottom of detachable groove 17; The mesopore 38 of axle IV 18 lower end transverse slat 39 in nut IV 21 with detachable groove 17 is affixed; Axle IV 18 upper end is threaded with the bottom outlet of ball pivot assembly D bottom bracket axle 44; Connecting rod 47 upper end of ball pivot assembly D is affixed bottom nut III 20 with hollow rigid body II 19.
Described detachable groove 17 is made up of right riser 35, left riser 36 and transverse slat 39, and wherein transverse slat 39 is provided with mesopore II 38, and right riser 35 bottom is provided with hole V 34, and left riser 36 bottom is provided with hole VI 37; Left riser 36 upper end through screw I 40 and screw II 41 affixed with transverse slat 39 left end, right riser 35 upper end through screw III 42 and screw IV 43 affixed with transverse slat 39 right-hand member.
Described ball pivot assembly D is made up of axle sleeve 44, screw V 45, ball pivot 46, connecting rod 47, screw VI 48, axle sleeve top cover 49 and screw VII 50, wherein ball pivot 46 warp beam sleeve top cover 49, screw V 45, screw VI 48 and screw VII 50 are movably connected on axle sleeve 44 center, ball pivot 46 top and connecting rod 47 lower end affixed.
The technical solution adopted for the present invention to solve the technical problems is:
General structure of the present invention is aluminum, is made up of two devices.
Device one, for setting the attitude of object of reference rigid body, after setting, by the corner that in measurement mechanism one, three times are rotated, substitutes into direction cosine matrix, obtains Euler's shaft position and Euler's corner after calculating.
Device two is for theorem proving, the angle that previous calculations obtains, after in device two, relevant position is adjusted, can demonstrate out the position of Euler's axle, adjustment ball pivot, makes rigid body get back to initial attitude, then rigid body is made to be rotated around Euler's axle by initial attitude, rotational angle is Euler's corner, then rigid body is moved to the attitude of setting in device one by initial attitude, thus checking theorem is set up.
The invention has the beneficial effects as follows: " object colour terms " demonstrating effect is more obvious; Theorem implementation is adopted an original approach, and the determination patten's design of Euler's axle is ingenious, and rotation is independent of each other, and gets final product sense data by self scale; Design optional position stationary installation, simple, convenient; Device is light, detachably, easily takes.
Accompanying drawing explanation
Fig. 1 is the structural representation of Euler's rotation theorem demonstration teaching aid
Fig. 2 is the front view of device one
Fig. 3 is the front view of device two
Fig. 4 is the side view of device one
Fig. 5 is the C-C cut-open view in Fig. 2
Fig. 6 is the side front view of device two
Fig. 7 is the structural representation of groove I
Fig. 8 is the structural representation of groove II
Fig. 9 is the front view of detachable groove
Figure 10 is the vertical view of detachable groove
Figure 11 is the front view of ball pivot assembly
Figure 12 is the vertical view of ball pivot assembly
Figure 13 is E-E cut-open view in Fig. 6
Device including: a. b. A 2 D. Ball hinge component 1. The base I 2. Grooves I 3. The butterfly nut I 4. Double screw I 5. 6. Cube universal joint grooves Ⅱ 7. Shaft 8. I hollow rigid body I 9. Nut I 10. Nut Ⅱ 11. Double screw Ⅱ 12. Butterfly nut Ⅱ Ⅳ 13. Base Ⅱ 14. Shaft Ⅱ 15. With dial shaft sleeve 16. Shaft Ⅲ 17. 18. The detachable type groove shaft IV 19. The hollow rigid II, nut Ⅲ 21. Nut Ⅳ 22. Nut Ⅴ 23. The shaft Ⅴ 24. The butterfly nut Ⅴ 25. The butterfly nut Ⅲ 26 double-headed screw Ⅲ 27 double-headed screw Ⅳ 28. The butterfly nut IV 29 Ⅰ 30 hole. The hole Ⅱ 31. Hole Ⅲ 32. Hole Ⅰ 33. Hole Ⅳ 34 hole Ⅴ 35 vertical plate 36 right. The left vertical plate with hole Ⅵ 38. The hole in the Ⅱ 39. The rung 40. Screw Ⅰ 41. The screw Ⅱ 42. The screw Ⅲ 43. The screw Ⅳ 44. 45. The shaft sleeve screw Ⅴ 46. The ball joint 47. 48. The connecting rod bolt Ⅵ 49. The shaft sleeve top 50. Screw Ⅶ
Embodiment
Time initial, two device rigid-body attitude are consistent, and are in horizontality, by air level leveling.
During demonstration:
In device one A, square universal joint 5 rotates around double threaded screw I 4, double threaded screw II 11, groove II 6 rotates around double threaded screw III 26, double threaded screw IV 27, axle I 7 rotates and (unclamps respective nut during rotation in groove II 6 central plate mesopore I32, fix by respective nut after rotation completes, rotation is any rotation), determine the attitude of hollow rigid body I 8 in space.Substitute into Direct cosine matrix and Euler's rotating shaft vector by measurements and calculations, and finally obtain Euler's axle and Euler's corner of the rotation of hollow rigid body I 8, obtain three angular values a1, a2, a3.
In device two B: dish axle sleeve 15 with a scale is overlapped and turns over a1 around axle II 14, detachable groove 17 is made to turn over a2 around axle III 16 and axle V23, adjustment ball pivot, rigid body is made to get back to initial attitude, make axle V23 anglec of rotation a3 in detachable groove 17 mesopore II38, hollow rigid body II 19 and hollow rigid body I 8 can be made to become same attitude (angle is read by the index dial at this place, unclamps clamp nut, again tighten after adjustment during adjustment angle).
In above process, device one A sets any attitude of a rigid body, by calculating Euler's axle and Euler's corner, obtain three angular values a1, a2, a3, namely in device two B, a1, a2 two angles are rotated in adjustment is the position determining Euler's axle, adjustment ball pivot, rigid body is made to get back to initial attitude, finally make rigid body turn over Euler corner a3 around Euler's axle, the rigid body in this timer two B is identical with rigid-body attitude in device one A, demonstrates the establishment of theorem.
Claims (3)
1. Euler's rotation theorem demonstration teaching aid, it is characterized in that being made up of device one (A) and device two (B), device one (A) is by base I (1), groove I (2), wing nut I (3), double threaded screw I (4), square universal joint (5), groove II (6), axle I (7), hollow rigid body I (8), nut I (9), nut II (10), double threaded screw II (11), wing nut II (12), wing nut III (25), double threaded screw III (26), double threaded screw IV (27) and wing nut IV (28) form, wherein groove I (2) bottom is affixed with base I (1) center, axle I (7) lower end is affixed through nut II (10) and groove II (6) top mesopore I (32), and axle I (7) upper end is affixed with hollow rigid body I (8) bottom through nut I (9), square universal joint (5) four sides is threaded with the inner of double threaded screw I (4), double threaded screw II (11), double threaded screw III (26) and double threaded screw IV (27) respectively, the hole I (29) of double threaded screw I (4) outer end left side vertical plate in groove I (2) is threaded with wing nut I (3), the hole II (30) of double threaded screw II (11) outer end right side riser in groove I (2) is threaded with wing nut II (12), the hole IV (33) of double threaded screw III (26) outer end front side riser in groove II (6) is threaded with wing nut III (25), the hole III (31) of double threaded screw IV (27) outer end rear side riser in groove II (6) is threaded with wing nut IV (28), device two (B) is made up of base II (13), axle II (14), dish axle sleeve with a scale (15), axle III (16), detachable groove (17), axle IV (18), hollow rigid body II (19), nut III (20), nut IV (21), nut V (22), axle V (23) and wing nut V (24), its axis II (14) bottom is affixed with base II (13) center, dish axle sleeve (15) with a scale is placed on axle II (14) top, and fixes with nut V (22), axle III (16) inner and with a scale dish axle sleeve (15) left side is affixed, axle III (16) outer end is flexibly connected with the hole VI (37) of left riser (36) bottom of detachable groove (17), axle V (23) inner and with a scale dish axle sleeve (15) right side is affixed, axle V (23) outer end is threaded with wing nut V (24) through the hole V (34) of right riser (35) bottom of detachable groove (17), the mesopore II (38) of axle IV (18) lower end transverse slat (39) in nut IV (21) with detachable groove (17) is affixed, axle IV (18) upper end is threaded with the bottom outlet of ball pivot assembly (D) bottom bracket axle (44), connecting rod (47) upper end of ball pivot assembly (D) is through nut III (20)) affixed with hollow rigid body II (19) bottom.
2. by Euler's rotation theorem demonstration teaching aid according to claim 1, it is characterized in that described detachable groove (17) is made up of right riser (35), left riser (36) and transverse slat (39), wherein transverse slat (39) is provided with mesopore II (38), right riser (35) bottom is provided with hole V (34), and left riser (36) bottom is provided with hole VI (37); Left riser (36) upper end through screw I (40) and screw II (41) affixed with transverse slat (39) left end, right riser (35) upper end through screw III (42) and screw IV (43) and transverse slat (39) right-hand member affixed.
3. by Euler's rotation theorem demonstration teaching aid according to claim 1, it is characterized in that described ball pivot assembly (D) is made up of axle sleeve (44), screw V (45), ball pivot (46), connecting rod (47), screw VI (48), axle sleeve top cover (49) and screw VII (50), wherein ball pivot (46) warp beam sleeve top cover (49), screw V (45), screw VI (48) and screw VII (50) are movably connected on axle sleeve (44) center, ball pivot (46) top and connecting rod (47) lower end affixed.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201410673877.7A CN104392641B (en) | 2014-11-19 | 2014-11-19 | Euler's rotation theorem demonstration teaching aid |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201410673877.7A CN104392641B (en) | 2014-11-19 | 2014-11-19 | Euler's rotation theorem demonstration teaching aid |
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| Publication Number | Publication Date |
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| CN104392641A true CN104392641A (en) | 2015-03-04 |
| CN104392641B CN104392641B (en) | 2016-11-09 |
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| Application Number | Title | Priority Date | Filing Date |
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| CN201410673877.7A Expired - Fee Related CN104392641B (en) | 2014-11-19 | 2014-11-19 | Euler's rotation theorem demonstration teaching aid |
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Cited By (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN112435548A (en) * | 2020-12-01 | 2021-03-02 | 吉林大学 | Finite rotation tensor demonstration teaching aid |
Citations (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN1358993A (en) * | 2001-12-24 | 2002-07-17 | 北京控制工程研究所 | variable parameter dynamics and control experiment system and method making experiment using said system |
| CN201359006Y (en) * | 2008-12-30 | 2009-12-09 | 中国科学院沈阳自动化研究所 | Hinge device |
| CN202331993U (en) * | 2011-09-09 | 2012-07-11 | 哈尔滨工业大学（威海） | Building block type planar structure mechanical property universal experiment bench |
| CN103034237A (en) * | 2012-12-27 | 2013-04-10 | 北京航空航天大学 | Spacecraft attitude maneuver control method using two single-frame control moment gyroscopes |
| CN204257067U (en) * | 2014-11-19 | 2015-04-08 | 吉林大学 | Euler's rotation theorem demonstration teaching aid |
-
2014
- 2014-11-19 CN CN201410673877.7A patent/CN104392641B/en not_active Expired - Fee Related
Patent Citations (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN1358993A (en) * | 2001-12-24 | 2002-07-17 | 北京控制工程研究所 | variable parameter dynamics and control experiment system and method making experiment using said system |
| CN201359006Y (en) * | 2008-12-30 | 2009-12-09 | 中国科学院沈阳自动化研究所 | Hinge device |
| CN202331993U (en) * | 2011-09-09 | 2012-07-11 | 哈尔滨工业大学（威海） | Building block type planar structure mechanical property universal experiment bench |
| CN103034237A (en) * | 2012-12-27 | 2013-04-10 | 北京航空航天大学 | Spacecraft attitude maneuver control method using two single-frame control moment gyroscopes |
| CN204257067U (en) * | 2014-11-19 | 2015-04-08 | 吉林大学 | Euler's rotation theorem demonstration teaching aid |
Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN112435548A (en) * | 2020-12-01 | 2021-03-02 | 吉林大学 | Finite rotation tensor demonstration teaching aid |
| CN112435548B (en) * | 2020-12-01 | 2021-08-27 | 吉林大学 | Finite rotation tensor demonstration teaching aid |
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| Publication number | Publication date |
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| CN104392641B (en) | 2016-11-09 |
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Granted publication date: 20161109 Termination date: 20211119 |