LU500269B1 - A Demonstration System For Parametric Equations And Polar Coordinate Equations Of Higher Mathematics - Google Patents

A Demonstration System For Parametric Equations And Polar Coordinate Equations Of Higher Mathematics Download PDF

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Publication number
LU500269B1
LU500269B1 LU500269A LU500269A LU500269B1 LU 500269 B1 LU500269 B1 LU 500269B1 LU 500269 A LU500269 A LU 500269A LU 500269 A LU500269 A LU 500269A LU 500269 B1 LU500269 B1 LU 500269B1
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LU
Luxembourg
Prior art keywords
gear
rod
angle ring
ring
length
Prior art date
Application number
LU500269A
Other languages
French (fr)
Inventor
Jianping Wu
Jianjun Wang
Xiaodong Wang
Xiaoling Meng
Limin Shi
Xinyue Zhang
Sisheng Zhang
Original Assignee
Univ Zhengzhou Aeronautics
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Priority to LU500269A priority Critical patent/LU500269B1/en
Application granted granted Critical
Publication of LU500269B1 publication Critical patent/LU500269B1/en

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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
    • 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
    • G09B23/04—Models for scientific, medical, or mathematical purposes, e.g. full-sized devices for demonstration purposes for mathematics for geometry, trigonometry, projection or perspective

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  • Physics & Mathematics (AREA)
  • General Physics & Mathematics (AREA)
  • Computational Mathematics (AREA)
  • Mathematical Analysis (AREA)
  • Mathematical Optimization (AREA)
  • Pure & Applied Mathematics (AREA)
  • Engineering & Computer Science (AREA)
  • Mathematical Physics (AREA)
  • Business, Economics & Management (AREA)
  • Algebra (AREA)
  • Educational Administration (AREA)
  • Educational Technology (AREA)
  • Theoretical Computer Science (AREA)
  • Geometry (AREA)
  • Transmission Devices (AREA)

Abstract

The invention relates to a demonstration system of higher mathematics parameter equation and polar coordinate equation, which effectively solves the problem that the existing teachers can not demonstrate to students visually when explaining polar coordinate and parameter equation; The technical solution is that:According to the teaching requirements, the device can switch the corresponding demonstration mode, which can be used for the demonstration of parametric equation or polar coordinate equation, and can also vividly show the drawing process of Archimedes helix to students, so that students can deepen their understanding of polar coordinates.

Description

SPECIFICATION A Demonstration System For Parametric Equations And Polar LU500269 Coordinate Equations Of Higher Mathematics Technical Field The invention relates to the technical field of advanced mathematics teaching, in particular to a demonstration system for parametric equations and polar coordinate equations of advanced mathematics.
Technical Background Polar coordinate system is an important tool in higher mathematics. It can transform with rectangular coordinate system; In higher mathematics, the polar coordinate system 1s used to calculate double integral and Archimedes helix.Because there is no special polar coordinate teaching aids for teachers to show polar coordinate equation better when explaining polar coordinate equation, students can only understand polar coordinate equation according to the polar coordinate curve drawn by teachers on the blackboard, resulting in boring teaching, which 1s not conducive to improving teachers' teaching efficiency; The traditional method can not draw the spiral of Archimedes through the existing teaching tools and can not show the drawing process of the spiral of Archimedes derived from polar coordinates, which is not conducive to students" understanding of polar coordinates.
A demonstration system for parametric equations and polar coordinate equations of higher mathematics is provided to solve the above problems.
Summary of the Invention In order to overcome the defects of the prior art, the present invention can switch the demonstration mode according to the teaching requirements, and can demonstrate the parametric equation or polar coordinate equation, and can also show the drawing process of Archimedes helix.
The utility model relates to a demonstration system for advanced mathematical parametric equations and polar coordinate equations, comprising a demonstration box, which is characterized in that a transverse sliding frame is 1
SPECIFICATION arranged between longitudinal two side walls of the demonstration box and a LU500269 longitudinal movable first angle ring is arranged on the adjusting frame; A first length rod is arranged on the first angle ring rotating in the same axis, and a second length rod is sliding arranged in the first length rod. A telescopic transmission device connected with the second length rod is arranged in the first length rod, and the telescopic transmission device is connected with a unidirectional gear which is rotated and arranged on the first angle ring. A positioning device for positioning the unidirectional gear is arranged on the first angle ring.
A rectangular coordinate bar is arranged on the demonstration box.
Beneficial effects of the invention: The device can switch the demonstration mode according to the teaching demand, and can demonstrate the parameter equation or polar coordinate equation. It can also display the drawing process of Archimedes spiral line vividly, deepen the understanding of polar coordinates, improve the enthusiasm of students in class, and also help to improve the efficiency of classroom teaching.
Descriptions of the Drawings Fig. 1 1s the overall structure diagram of the invention; Fig. 2 is a schematic diagram of the structure from another perspective when the invention is in use; Fig. 3 is a schematic diagram of the structure of the demonstration box of the invention after partial sectionalization; Fig. 4 is a schematic diagram of the invention when several arc rods are fitted together; Fig. 5 1s a schematic diagram of the matching relationship between the spine gear and the first gear of the invention.
Fig. 6 is a schematic diagram of the relationship between the first length rod and the second length rod of the invention; Fig. 7 is a schematic diagram of the matching relationship between the end face gear and the expansion gear of the invention; 2
SPECIFICATION Fig. 8 is the schematic diagram after the lifting rod and the screw rod are LU500269 separated.
The best embodiment for implementing the invention The contents mentioned in the following embodiments are referred to in the drawings.
In the first embodiment, refer to Fig. 1, the utility model comprises a demonstration box, an adjusting frame 2 is horizontally sliding between the two longitudinal walls of the demonstration box, and a first angle ring 3 is longitudinally movable on the adjusting frame 2 (the circumferential surface of the first angle ring 3 is provided with an angle scale), refer to Fig. 1, the demonstration box is equipped with a rectangular coordinate rod 6, and initially the first angle ring 3 and the direct coordinate rod are set at the same axis (the center of the rectangular coordinate rod 6 coincides with the center of the demonstration box). When the polar coordinates are explained, the first length rod 4 is rotated and its angle relative to the rectangular coordinate rod 6 is adjusted (reflected in real time on the first angle ring 3), A scale value is set on the first length rod 4, that is, the polar coordinates of the point are expressed by the distance between a point on the first length rod 4 and the center of the first angle ring 3 (which can be read out by the scale value on the first length rod 4) and the angle of the first length rod 4 relative to the rectangular coordinate rod 6 (which can be obtained from the first angle ring 3); The polar coordinates of the point are (r, t), r is the distance from the point to the center of the first angle ring 3, t is the angle between the line connecting the point and the center of the first angle ring 3 and the positive half axis of the rectangular coordinate rod 6 (extending along the X axis), The polar coordinate expressions of different points on the first length rod 4 are displayed (the polar coordinate expressions of different points are displayed by rotating the angle of the first length rod 4 relative to the rectangular coordinate rod 6 and the length of different points from the first angle ring 3); 3
SPECIFICATION
When the drawing process of Archimedes helix is shown, the first length rod LU500269 4 can be rotated at an equal speed along the set direction (only the first length rod 4 can be rotated along the set direction, the telescopic transmission device can be coordinated with the unidirectional gear to drive the telescopic transmission device; On the contrary, the unidirectional gear can not drive the telescopic transmission device to move, and the initial unidirectional gear is not positioned by the positioning device), so that when the first length rod 4 rotates around the first angle ring 3, the telescopic transmission device arranged on the first angle ring 3 cooperates with the unidirectional gear to drive the second length rod 5 to slide out of the first length rod 4 at a certain speed, The first length rod 4 can be driven by a micro motor (not shown in the figure, the micro motor is connected with an external power supply through a wire). When the micro motor drives the first length rod 4 to rotate at a certain speed, it synchronously drives the second length rod 5 to slide out at a certain speed through a one-way gear and a telescopic transmission device, which vividly demonstrates the drawing process of Archimedes spiral, The above actions can realize the drawing of Archimedes spiral;
When the second length rod 5 needs to be retracted into the first length rod 4, the positioning device on the first angle ring 3 is used to position the unidirectional gear.
At this time, the micro motor is controlled to rotate in the direction by the microcontroller to drive the second length rod 5 to retract into the first length rod 4; By rotating the first length rod 4 along the set direction, the position of the second length rod 5 can be obtained, so as to demonstrate the polar coordinate expression of the point farther from the center of the first angle ring 3;
When displaying the parameter equation, refer to Fig.2, by moving the adjusting bracket 2 laterally and adjusting the longitudinal position of the first angle ring 3 relative to the adjusting bracket 2, the center of the first angle ring 3 and the center of the rectangular coordinate bar 6 no longer coincide.
At this time, the coordinate of the center of the first angle ring 3 relative to the rectangular
4
SPECIFICATION coordinate bar 6 can be determined by rotating the first length bar 4 (Rotate the LU500269 first length rod 4 to be perpendicular to the Y-axis direction of the right-angle sitting post to obtain the Y-axis coordinate b of the center of the first angle ring 3. Rotate the first length rod 4 to be perpendicular to the X-axis of the right-angle sitting post to obtain the X-axis coordinate a at this time); Then according to the angle of the first length rod 4 relative to the positive half axis of the rectangular coordinate rod x 6, the coordinates of a point on the first angle ring 3 are obtained, that is, x = a + RCOs 6, y=b+Rsin 0, where R is the radius of the first angle ring 3, the above expression is the circle parameter equation with the center coordinates of (a, b) and the radius of R.
In the second embodiment, on the basis of embodiment 1, the lower end face of the first angle ring 3 is fixed with a U-shaped rod 29 through its center, and the first length rod 4 is rotationally arranged on the U-shaped rod 29. When adjusting the position of the second length rod 5, it only needs to drive the first length rod 4 to rotate along the set direction (the direction meets the requirement that when the first length rod 4 is rotated, the, The end gear 8 meshed with the telescopic gear 9 is connected with the unidirectional gear through the outer ring gear 11, and cannot drive the unidirectional gear to rotate at this time (that is, the end gear 8 cannot rotate); With the rotation of the first length rod 4, the expansion gear 9 arranged in the circular plate 7 is rotated along with the end face gear 8, that is, the expansion gear 9 is driven to rotate. With the rotation of the expansion gear 9, the expansion lead screw 10 drives the second length rod 5 to move within the first length rod 4.
If rod 4 of the first length is rotated against the set direction, rod 5 of the second length will not move.
In the third embodiment, on the basis of embodiment 2, when the first length rod 4 rotates in the counter clockwise direction (setting direction) as shown in Fig. 8, the outer ring gear 11 arranged in the same axis as the end face gear 8 drives the spur gear 12 to rotate in the counter clockwise direction through the first gear 16 and the second gear 17; Due to the cooperation of the pawl 14 and the spur gear 12,
SPECIFICATION the spur gear 12 can not rotate. With the rotation of the first length rod 4, the end LU500269 gear 8 can not rotate synchronously with the circular plate 7, so that the telescopic gear 9 arranged in the circular plate 7 and the end gear 8 rotate relatively, driving the telescopic gear 9 to rotate; If the first length rod 4 is driven to rotate in the clockwise direction as shown in Fig. 8, the outer ring gear 11, through the first gear 16 and the second gear 17, has the tendency to drive the spiny gear 12 to rotate in the clockwise direction as shown in Fig. 8, while the spiny gear 12 can rotate clockwise around the shaft
13.At this point, the thread frictional resistance between the telescopic lead screw and the second length rod 5 is greater than the rotational resistance of the driving spine gear 12. When the first length rod 4 rotates clockwise, the second length rod 5 will not be driven, and relative rotation will occur between the spine gear 12 and shaft 13.
In the fourth embodiment, on the basis of embodiment 3, when the spur gear 12 1s not positioned, the positioning rods 20 arranged on both sides of the axial direction of the lifting rod 19 are not inserted into the positioning holes, and the lifting rod and the screw 18 are axially sliding arranged, and a return spring is connected between them (not shown in the figure); Under the action of the return spring, the positioning rods 20 arranged on both sides of the axial direction of the lifting rod are just above the positioning hole 21; When the second length rod 5 is retracted, the screw 18 is rotated (to move the screw 18 in the direction of retracting to the shaft 13) to drive the lifting rod 19 to rotate synchronously, and the screw 18 moves in the direction of retracting to the shaft 13; The return spring is in a compressed state (the lower end face of the positioning rod 20 collides with the upper end face of the ring), so that the positioning rod 20 arranged on both sides of the axial direction of the lifting rod 19 is adjusted to the corresponding position with the positioning hole 21; Under the action of the return spring, the positioning rod 20 is inserted into the positioning hole 21 to position the spur gear 12. At this time, the spur gear 12 can not rotate 6
SPECIFICATION relative to the shaft 13. At this time, the first length rod 4 is driven to rotate in the LU500269 clockwise direction as shown in Fig. 8 to shrink the second length rod 5 into the first length rod 4, release the positioning of the spur gear 12, lift the lower rod 19, and then rotate the screw 18 in the reverse direction, Move the screw 18 away from the U-shaped rod 29 until the lower end of the positioning rod 20 is above the ring.
In the fifth embodiment, on the basis of embodiment 1, when displaying the parametric equation of the ellipse, the two groups of matched arc-shaped rods retracted into the annular groove 23 can be rotated outward. Initially, the two groups of arc-shaped rods retracted into the annular groove 23, and the two groups of arc-shaped rods (the first arc-shaped rod 25 and the second arc-shaped rod 28) are expanded (when one of the first arc-shaped rods 25 is rotated; The second arc rod is driven to rotate synchronously by the matching meshing gear 26, one of which can be driven by a motor or manually rotated by the teacher. An ellipse 1s formed, and the radius of the outer circle is the radius of the first angle ring 3.
The radius of the second angle ring 27 is the same as that of the inscribed circle of the ellipse. The parameter equation of the ellipse can be determined by the first angle ring 3 (circumscribed circle of the ellipse), the second angle ring 27 (inscribed circle of the ellipse) and the angles on the first angle ring 3 and the second angle ring 27. The above image cooperation can demonstrate the parameter equation of the circle and ellipse; A coarse scale is set on the second angle ring 27, and a fine scale is set on the first angle ring 3. Through the cooperation of the first angle ring 3 and the second angle ring 27, the angle reading is more accurate; The demo box is slided horizontally with a writing board 22, and a line pen 24 is arranged on the second length rod 5. In the process of displaying Archimedes spiral line drawing, the graphics can be drawn on the writing board. In order to facilitate the pulling of the writing board, a handle 32 is set on the writing board, and the thread between the line pen and the second length rod 5 is installed to adjust the distance between the line pen 24 and the writing board 22; 7
SPECIFICATION One of the adjusting frames 2 is equipped with a transmission rod 30 which is LU500269 threaded with the U-shaped rod 29, and the transmission rod 30 is rotated to adjust the longitudinal position of the first angle ring 3. The bottom of the demonstration box is equipped with a transverse lead screw 31 which is threaded with the adjusting frame 2 below to adjust the transverse position of the first angle ring 3.
The above description is only for the purpose of explaining the invention, it should be understood that the invention is not limited to the above embodiments, and various alternative forms conforming to the idea of the invention are within the protection scope of the invention.
8

Claims (5)

1. The utility model relates to a demonstration system for advanced LU500269 mathematical parametric equations and polar coordinate equations, comprising a demonstration box (1), which is characterized in that Demo box (1) between the longitudinal two side walls transverse sliding is provided with an adjusting frame (2) and the adjusting frame (2) on the longitudinal movable installation of the first angle ring (3), the first angle ring (3) on the same axis rotation is provided with a first length rod (4) and the first length rod (4) within the sliding installation of the second length rod (5); The first length rod (4) is provided with a telescopic transmission device connected with the second length rod (5) and the telescopic transmission device is connected with a rotating unidirectional gear arranged on the first angle ring (3), the first angle ring (3) is provided with a positioning device for unidirectional gear positioning; The demonstration box (1) is provided with a right angle sitting pole (6).
2. According to claim 1, the method is characterized in that, the first length rod (4) is integrally connected with a circular plate (7), and the circular plate (7) and the first angle ring (3) are rotationally arranged with a coaxial center. The telescopic transmission device comprises a face gear (8) which is rotationally arranged in the circular plate (7) with a coaxial center, and the face gear (8) is matched with a telescopic gear (9) which is rotationally arranged in the circular plate (7); The coaxial sleeve of the telescopic gear (9) is provided with a telescopic lead screw (10) which is matched with the thread of the second length rod (5). The end gear (8) is integrated with the shaft center, and an outer gear ring (11) is arranged outside the circular plate (7), and the outer gear ring (11) is connected with the unidirectional gear.
3. According to claim 2, the method is characterized in that, the unidirectional gear comprises a spur gear (12) which is rotationally mounted on a first angle ring (3), a shaft (13) which is rotationally mounted with the spur gear (12) is fixed on the first angle ring (3), a pawl (14) which is matched with the spur gear (12) 1s rotationally mounted on the shaft (13), and an elastic part (15) which conflicts with 1
CLAIMS the pawl (14) is fixed on the shaft (13); The spiny gear (12) is engaged with the LU500269 first gear (16) mounted on the first angle ring (3) and the first gear (16) is coaxially rotated with the second gear (17), and the second gear (17) is engaged with the outer ring gear (11).
4. According to claim 3, the method is characterized in that, the positioning device comprises a screw rod (18) which is matched with the screw thread of the shaft (13), and a lifting rod (19) is arranged in the screw rod (18) by axial sliding. The positioning rod (20) which extends vertically is arranged on both sides of the axial direction of the lifting rod (19), and the positioning hole (21) which is matched with the positioning rod (20) is arranged on both sides of the axial direction of the spur gear (12).
According to claim 1, the method is characterized in that, the first angle ring (3) is provided with an annular groove (23) in the same axis, and two groups of arc rods are respectively rotated and arranged on both sides of the axial direction of the annular groove (23). Each group of arc rods includes a first arc rod (25), a second arc rod (28), and the first arc rod (25) and the second arc rod (28) are coaxially rotated with a meshing gear (26), The two meshing gears (26) in the same group mesh with each other, and the two groups of arc rods cooperate to form an ellipse, and the first angle ring (3) 1s provided with a second angle ring (27) on the same axis.
2
LU500269A 2021-04-09 2021-04-09 A Demonstration System For Parametric Equations And Polar Coordinate Equations Of Higher Mathematics LU500269B1 (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
LU500269A LU500269B1 (en) 2021-04-09 2021-04-09 A Demonstration System For Parametric Equations And Polar Coordinate Equations Of Higher Mathematics

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
LU500269A LU500269B1 (en) 2021-04-09 2021-04-09 A Demonstration System For Parametric Equations And Polar Coordinate Equations Of Higher Mathematics

Publications (1)

Publication Number Publication Date
LU500269B1 true LU500269B1 (en) 2021-12-10

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Family Applications (1)

Application Number Title Priority Date Filing Date
LU500269A LU500269B1 (en) 2021-04-09 2021-04-09 A Demonstration System For Parametric Equations And Polar Coordinate Equations Of Higher Mathematics

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LU (1) LU500269B1 (en)

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Effective date: 20211210