Perimeter measuring device for large rotary component

Through the combination of encoder roller and cleaning friction wheel, the operation difficulty and error problems of traditional tape measure when measuring the circumference of large rotary components is solved, and high-precision and automated circumference measurement is achieved, adapting to a variety of environments, improving measurement efficiency and safety.

CN223307536UActive Publication Date: 2025-09-05TANGSHAN UDO TECH LTD
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Patent Information

Application Number
CN202422714203.9
Authority / Receiving Office
CN · China
Patent Type
Utility models(China)
Current Assignee / Owner
Filing Date
2024-11-07
Publication Date
2025-09-05
Estimated Expiration
2034-11-07

AI Technical Summary

Technical Problem

When traditional tape measure the circumference of large rotary components, it is difficult to operate, and there are errors in manual reading, making it difficult to meet the needs of high-precision and automated production.

Method used

The encoder roller measurement method is adopted, combined with the stroke switch and the cleaning friction wheel, to realize automatic measurement, use the measurement wheel and the stroke switch to record the number of turns, the cleaning friction wheel removes surface impurities, and wirelessly control and data storage through mobile phones.

Benefits of technology

It realizes high-precision measurement without shutting down the equipment, reduces the professional level requirements for technicians, improves the accuracy of measurement and operation convenience, adapts to various environments, and reduces manual errors.

✦ Generated by Eureka AI based on patent content.

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Abstract

The utility model relates to the technical field of large-scale rotary component detection, and provides a perimeter measuring device for a large-scale rotary component, which comprises a base. The measuring wheel is rotationally arranged relative to the base, is used for abutting against the wheel belt part and is provided with a built-in measuring element, the wheel belt part rotates to drive the measuring wheel to rotate, and the measuring wheel is used for rotationally measuring the perimeter of the wheel belt part; the travel switch is arranged on the base and used for measuring the number of rotation turns of the wheel belt part. By means of the technical scheme, the problems that in the prior art, a traditional measuring tape measuring mode is not easy to operate, and errors exist in manual reading are solved.
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Description

Technical Field

[0001] The utility model relates to the technical field of large-scale rotating component detection, and in particular to a circumference measuring device for large-scale rotating components. Background Art

[0002] In industrial production and other fields, measuring the circumference of large rotating parts is of great significance. It is an important inspection and measurement item related to the core equipment of the building materials industry. The measurement object is generally a large rotary kiln. The current industry inspection method is to use a tape measure to circle the rotating part. The current traditional method has the following problems:

[0003] Manual reading is a relatively primitive measurement method that is not easy to operate and cannot be adapted to large rotating parts, especially when the equipment cannot be inspected during operation. Manual readings may result in errors and require high technical skills from the operator. In addition, for some high-precision occasions, ordinary measurement methods may not be able to meet the accuracy requirements and may produce large errors. This makes it difficult to match the automated production process and cannot meet the needs of modern industry for efficient and accurate measurement. This has prompted the development and improvement of circumference measurement devices specifically for large rotating parts. Utility Model Content

[0004] The utility model provides a circumference measuring device for large rotary components, which solves the problems in the related art of the traditional measuring method with a tape measure, which is difficult to operate and has errors in manual readings.

[0005] The technical solution of the utility model is as follows:

[0006] A circumference measuring device for large rotating components, used for measuring the circumference of a wheel belt portion, comprising:

[0007] base;

[0008] A measuring wheel, the measuring wheel being rotatably arranged relative to the base, the measuring wheel being used to abut against the wheel belt portion, the measuring wheel having a built-in measuring element, and the wheel belt portion rotating to drive the measuring wheel to rotate, and the measuring wheel being used to rotate and measure the circumference of the wheel belt portion;

[0009] A travel switch is provided on the base and is used to measure the number of rotations of the wheel belt portion.

[0010] As a further technical solution, the travel switch has a measuring slot and further includes:

[0011] The shifting rod is arranged stationary relative to the wheel belt portion, and the wheel belt portion drives the shifting rod to pass through the measuring slot after rotation.

[0012] As a further technical solution, it also includes:

[0013] A cleaning friction wheel is rotatably arranged relative to the base, the cleaning friction wheel is used to abut the wheel belt portion, and the rotation direction of the cleaning friction wheel is opposite to the rotation direction of the measuring wheel.

[0014] As a further technical solution, the measuring wheel has a rotating shaft 1 and further includes:

[0015] Two support rods 1 are both arranged on the base, the two support rods 1 are arranged at intervals, and the rotating shaft 1 passes through the two support rods 1 and is located between the two support rods 1.

[0016] As a further technical solution, it also includes:

[0017] Two support rods 2, one of the support rods 2 is arranged on one of the support rods 1, and both ends of the cleaning friction wheel are rotatably connected to the two support rods 2 respectively.

[0018] As a further technical solution, the cleaning friction wheel is located above the measuring wheel, and the support rod 2 is swingably arranged relative to the support rod 1. The swing axis of the support rod 2 is coaxial with the rotating shaft 1, and the rotating shaft 1 passes through the two support rods 2. After the support rod 2 swings, it drives the cleaning friction wheel to abut or cancel the abutment against the wheel belt portion.

[0019] As a further technical solution, the second support rod has a support portion, both ends of the cleaning friction wheel have support holes, and the support portion is located in the support hole, and further includes:

[0020] a transmission wheel, the transmission wheel being sleeved on the support portion;

[0021] A transmission belt, wherein the transmission belt is sleeved on the transmission wheel and the first rotating shaft, and when the first rotating shaft rotates, the transmission belt drives the transmission wheel to rotate;

[0022] A transmission member is provided, and after the transmission wheel rotates, the cleaning friction wheel is driven to rotate through the transmission member.

[0023] As a further technical solution, the transmission wheel has a toothed ring portion on the side close to the cleaning friction wheel and at both ends of the cleaning friction wheel, and the transmission member is provided at both ends of the cleaning friction wheel, and the transmission member includes:

[0024] Two gears, both of which are rotatably arranged on the support rod 2, the two gears are coaxially arranged and spaced apart, the gear rotation axis is perpendicular to the axis of the rotating shaft, the two gears are located between the transmission wheel and the cleaning friction wheel, and the gear ring portion of the transmission wheel and the gear ring portion at one end of the cleaning friction wheel are both engaged with the gears, and after the transmission wheel rotates, the cleaning friction wheel is driven to rotate through the gears.

[0025] As a further technical solution, it also includes:

[0026] Support rod three, the support rod three is arranged on the support rod two, and the gear is rotated by the support rod three and arranged on the support rod two.

[0027] As a further technical solution, the first support rod is L-shaped, and the second support rod is linear.

[0028] The working principle and beneficial effects of the utility model are as follows:

[0029] 1. This product adopts encoder roller measurement method, which has high precision and can be measured without stopping the equipment;

[0030] 2. The measurement process is automated, requiring less expertise from technicians;

[0031] 3. The use of mobile phone wireless remote control improves the safety and convenience of operation.

[0032] 4. Measurement results are automatically generated and saved. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] The preferred implementation scheme will be described below in a clear and understandable manner with reference to the accompanying drawings to further illustrate the above-mentioned characteristics, technical features, advantages and implementation methods of the present invention.

[0034] Figure 1 This is a schematic structural diagram from a first perspective of a circumference measuring device for large rotating components in the present utility model;

[0035] Figure 2 This is a schematic structural diagram of a circumference measuring device for large rotating components in the present utility model from a second perspective;

[0036] Figure 3 For this utility model Figure 1 Enlarged view of part A in the middle;

[0037] Figure 4 For this utility model Figure 2 Enlarged view of middle part B;

[0038] Figure 5 This is a diagram showing the coordination of the measuring wheel and the cleaning friction wheel in the present utility model;

[0039] Figure 6 For this utility model Figure 5 Enlarged view of middle C part;

[0040] Figure 7 For this utility model Figure 5 Enlarged view of the middle D part;

[0041] Figure 8 This is a schematic diagram of the three structures of the support rod in this utility model.

[0042] In the figure: 1. wheel belt portion, 2. base, 3. measuring wheel, 301. rotating shaft 1, 4. travel switch, 401. measuring slot, 5. shift rod, 6. cleaning friction wheel, 601. supporting hole, 7. support rod 1, 8. support rod 2, 801. supporting portion, 9. transmission wheel, 10. transmission belt, 11. gear, 12. support rod 3. DETAILED DESCRIPTION

[0043] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the specific implementation methods of the present invention will be described below with reference to the accompanying drawings. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings and other implementation methods can be obtained based on these drawings without inventive work.

[0044] To simplify the drawings, only the parts relevant to the utility model are schematically shown in each figure; they do not represent the actual structure of the product. Furthermore, to simplify the drawings and facilitate understanding, in some figures, only one of the components with the same structure or function is schematically shown or labeled. In this document, "one" not only means "only one" but also "more than one," and "several" includes "two" and "more than two."

[0045] It should be noted that, unless otherwise specified or limited, the terms "mounted," "connected," and "connected" should be understood broadly. For example, they can refer to fixed, detachable, or integral connections; mechanical or electrical connections; direct or indirect connections through an intermediary; and internal communication between two components. Those skilled in the art will understand the specific meanings of these terms in this utility model based on the specific circumstances.

[0046] In addition, in the description of the present application, the terms "first", "second", etc. are only used to distinguish the description and cannot be understood as indicating or implying relative importance.

[0047] Reference Figures 1 to 8, which is the first embodiment of the utility model, proposes a circumference measuring device for large rotating components, which is used to measure the circumference of the wheel belt portion 1, including: a base 2; a measuring wheel 3, which is rotatably arranged relative to the base 2, and is used to abut the wheel belt portion 1. The measuring wheel 3 has a built-in measuring element, and the measuring wheel 3 is driven to rotate when the wheel belt portion 1 rotates. The measuring wheel 3 is used to rotate to measure the circumference of the wheel belt portion 1; a limit switch 4, which is arranged on the base 2 and is used to measure the number of rotations of the wheel belt portion 1.

[0048] In this embodiment, to replace the traditional manual reading method and improve measurement efficiency and accuracy, a measuring device assembled with higher-precision electronic components is used. Automatic circumference counting and wireless transmission are used to detect the circumference of a large rotating component, such as the belt portion 1 of a large rotary kiln. The specific steps are as follows:

[0049] 1. Fix the lever 5 to the outer circle or side of the large rotating component and fix the travel switch 4 at a fixed position to ensure that the lever 5 can contact the fork of the travel switch 4 every time the wheel belt part 1 rotates one circle;

[0050] 2. The measuring wheel 3 with a built-in measuring element in the form of an encoder is rotatably mounted on the base 2 via a bearing so that it contacts the surface of the wheel belt portion 1 and rolls relative to it.

[0051] 3. Turn on the measuring instrument, connect your phone to the instrument's Wi-Fi hotspot, open the APP, enter the name of the rotating component to be measured - wheel belt part 1. The built-in measuring element has been electrically connected to the measuring instrument. Click to start measurement;

[0052] 4. After starting the measurement, you can view the data fed back by the instrument on the mobile phone APP. The measurement process is automated. After measuring 3 circles, the APP prompts that the measurement is completed.

[0053] 5. After the measurement is completed, click to obtain the measurement results. The measured circumference and diameter of each circle will be displayed on the mobile phone APP, and the average circumference and diameter of the three measured circles will be calculated.

[0054] 6. Click Save Measurement Data to save the data in the instrument. Connect the computer to the instrument's Wi-Fi hotspot, open the accompanying computer software, and click Read Measurement File to read the measurement file into the computer.

[0055] The advantages of this measuring device are:

[0056] High measurement accuracy: The measuring wheel 3 has a built-in measuring element, which can accurately measure its own rotation, thereby accurately obtaining data related to the circumference of the wheel belt portion 1. The travel switch 4 can accurately measure the number of rotations of the wheel 3 belt portion 1. Combined with the measurement of the measuring wheel 3, the accuracy of the circumference measurement is further improved. The encoder roller measurement method is used, which has high accuracy and can be measured without stopping the equipment.

[0057] Strong adaptability: The measuring wheel 3 is rotatably arranged relative to the base 2, and can flexibly adapt to the rotation of the wheel belt portion 1. It can well abut the wheel belt portion 1 of large rotating components for measurement, is not overly restricted by factors such as component size, and has low requirements on the professional level of technicians.

[0058] Easy to operate: The overall device structure is reasonably designed. When measuring, you only need to contact the measuring wheel 3 with the belt part 1. The rotation of the belt part 1 can drive the measuring wheel 3 to rotate for measurement. The operation is simple and convenient, without the need for complicated operating procedures, and the measurement results are automatically generated and saved.

[0059] Improved Efficiency: This device can quickly and accurately measure the circumference of the belt portion of the wheel (3), reducing the time and labor costs of manual measurement and improving measurement efficiency. It is particularly suitable for industrial production, where frequent measurements are required or where high measurement efficiency is crucial. Furthermore, it is well-suited for automated production processes, improving overall production efficiency.

[0060] Furthermore, the travel switch 4 has a measuring slot 401 and further includes: a lever 5 , which is stationary relative to the wheel belt portion 1 , and drives the lever 5 to pass through the measuring slot 401 after the wheel belt portion 1 rotates.

[0061] In this embodiment, the specific travel switch 4 is configured with a measuring slot 401, which facilitates the smooth passage of the shift fork in the rotating wheel belt portion 1, interacting with the travel switch 4. When the wheel belt portion 1 rotates, it drives the shift rod 5 through the measuring slot 401. In this way, the travel switch 4 can accurately record the number of times the shift rod 5 passes through the measuring slot 401, thereby more accurately measuring the number of rotations of the wheel belt portion 1. Compared with other methods of measuring the number of rotations, which may be less direct or accurate, this design greatly improves the accuracy of the number of rotations measured, thereby providing a more reliable guarantee for the accuracy of circumference measurement.

[0062] Furthermore, it also includes: a cleaning friction wheel 6, which is rotatably arranged relative to the base 2, and is used to abut the wheel belt portion 1, and the rotation direction of the cleaning friction wheel 6 is opposite to the rotation direction of the measuring wheel 3.

[0063] In this embodiment, in order to improve the measurement accuracy of the belt portion 1 and avoid the accumulation of impurities, rust and other substances that affect the measurement after long-term work, initially only a cleaning brush is fixed on the base 2, but the cleaning effect is general. Therefore, a cleaning friction wheel 6 is adopted to effectively remove impurities, dust, oil and other pollutants on the surface of the belt portion 1 that may affect the contact accuracy between the measuring wheel 3 and the belt portion 1 by means of sliding friction. Keeping the surface of the belt portion 1 clean can ensure close and accurate contact between the measuring wheel 3 and the belt portion 1, thereby improving the accuracy of circumference measurement and reducing measurement errors caused by surface uncleanness.

[0064] The cleaning friction wheel 6 rotates in the opposite direction to the measuring wheel 3 . This saves driving energy and increases the relative sliding speed of the cleaning friction wheel 6 , resulting in a better cleaning effect. The cleaning friction wheel 6 is also supported by the base 2 .

[0065] Furthermore, the measuring wheel 3 has a rotating shaft 301 and also includes: two support rods 7, both of which are arranged on the base 2, the two support rods 7 are arranged at intervals, and the rotating shaft 301 passes through the two support rods 7 and is located between the two support rods 7.

[0066] In this embodiment, two support rods 7 of an L-shaped steel structure welded to the base 2 are used to jointly support the rotation of the measuring wheel 3. Specifically, the rotating shaft 301 of the measuring wheel 3 passes through the two support rods 7 and is located therein, and bearings are installed to provide stable support.

[0067] Furthermore, it also includes: two support rods 2 8, one support rod 2 8 is set on one support rod 1 7, and both ends of the cleaning friction wheel 6 are rotatably connected to the two support rods 2 8 respectively.

[0068] In this embodiment, further support for the cleaning friction wheel 6 is also achieved by two support rods 2 8 of the steel structure. Specifically, the two ends of the cleaning friction wheel 6 are respectively connected in rotation through the support parts 801 of the support rod 2 8. The support parts 801 are inserted into the support holes 601 at the ends. While completing the support, the support rod 2 8 is fixedly mounted on the support rod 1 7 by fasteners to improve the overall operating stability.

[0069] Furthermore, the cleaning friction wheel 6 is located above the measuring wheel 3, and the support rod 2 8 is set to swing relative to the support rod 1 7. The swing axis of the support rod 2 8 is coaxial with the rotating shaft 1 301, and the rotating shaft 1 301 passes through the two support rods 2 8. After the support rod 2 8 swings, it drives the cleaning friction wheel 6 to abut or cancel the abutment with the wheel belt portion 1.

[0070] In this embodiment, support rod 2 8 is set to swing relative to support rod 1 7. On the one hand, it is convenient to disassemble and replace the new cleaning friction wheel 6. On the other hand, the swing can keep it in contact with the wheel belt portion 1. After the friction layer on it becomes thinner, it can be more closely attached to the wheel belt portion 1 under the action of its own gravity. Therefore, support rod 2 8 can be installed on support rod 1 7 through bearing swinging, specifically installed on rotating shaft 1 301.

[0071] Furthermore, the support rod 2 8 has a support portion 801, and both ends of the cleaning friction wheel 6 have support holes 601, and the support portion 801 is located in the support hole 601, and also includes: a transmission wheel 9, the transmission wheel 9 is mounted on the support portion 801; a transmission belt 10, the transmission belt 10 is mounted on the transmission wheel 9 and the rotating shaft 301, and after the rotating shaft 301 rotates, the transmission wheel 9 is driven to rotate through the transmission belt 10; a transmission member, after the transmission wheel 9 rotates, the cleaning friction wheel 6 is driven to rotate through the transmission member.

[0072] In this embodiment, since the measuring wheel 3 does not require additional power to drive and rotates with the belt portion 1, the power can be used to actively drive the cleaning friction wheel 6 that is expected to reverse, saving the cost of an external motor. Therefore, a transmission wheel 9 is mounted on the support portion 801, and using the principle of a circular belt, a transmission belt 10 in the form of a belt is mounted on the transmission wheel 9 and the rotating shaft 301. The diameter ratio of the rotating shaft 301 to the transmission wheel 9 is configured to achieve the rotation of the rotating shaft 301 to drive the transmission wheel 9 to rotate, and at the same time, the power is transmitted to the reversed cleaning friction wheel 6 through the transmission member.

[0073] Furthermore, the transmission wheel 9 has a toothed ring portion on one side close to the cleaning friction wheel 6 and at both ends of the cleaning friction wheel 6. Transmission parts are provided at both ends of the cleaning friction wheel 6. The transmission parts include: two gears 11. The two gears 11 are rotatably set on the support rod 2 8. The two gears 11 are coaxially set and spaced apart. The rotation axis of the gear 11 is perpendicular to the axis of the rotating shaft 301. The two gears 11 are both located between the transmission wheel 9 and the cleaning friction wheel 6, and the toothed ring portion of the transmission wheel 9 and the toothed ring portion at one end of the cleaning friction wheel 6 are both engaged with the gear 11. After the transmission wheel 9 rotates, the cleaning friction wheel 6 is driven to rotate through the gear 11.

[0074] In this embodiment, the form of the transmission member can adopt a structure similar to that of a ship's paddle blade, so the transmission member is composed of two gears 11, and a gear ring portion is welded on the side of the transmission wheel 9 close to the cleaning friction wheel 6 and on both ends of the cleaning friction wheel 6. The rotation axes are perpendicular to each other, thereby realizing the rotation of the transmission wheel 9, and driving the cleaning friction wheel 6 to reverse through the two gears 11.

[0075] The device can be used to measure the circumference of large rotating components in a variety of working environments. Even in harsh environments such as those with high dust and oil pollution, the cleaning friction wheel 6 can ensure measurement accuracy and stability. This improves the environmental adaptability of the device and expands its application range.

[0076] Furthermore, it also includes: a third support rod 12, the third support rod 12 is set on the second support rod 8, and the gear 11 is rotated by the third support rod 12 and is set on the second support rod 8.

[0077] Furthermore, the first support rod 7 is L-shaped, and the second support rod 8 is linear.

[0078] In this embodiment, the support rod 3 12 welded to the support rod 2 8 provides stable support for the gear 11, which is installed and connected via a bearing.

[0079] It should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit the present invention. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that the technical solutions of the present invention may be modified or replaced by equivalents without departing from the spirit and scope of the technical solutions of the present invention, and all of these should be included in the scope of the claims of the present invention.

Claims

1. A circumference measuring device for a large rotating component, used for measuring the circumference of a wheel belt portion (1), characterized in that: include: Base (2); A measuring wheel (3), the measuring wheel (3) being rotatably arranged relative to the base (2), the measuring wheel (3) being used to abut against the wheel belt portion (1), the measuring wheel (3) having a built-in measuring element, the wheel belt portion (1) rotating to drive the measuring wheel (3) to rotate, and the measuring wheel (3) being used to rotate and measure the circumference of the wheel belt portion (1); A travel switch (4), the travel switch (4) being arranged on the base (2), and the travel switch (4) being used to measure the number of rotations of the wheel belt portion (1).

2. A circumference measuring device for large rotating components according to claim 1, characterized in that: The travel switch (4) has a measuring slot (401) and further comprises: A shifting rod (5) is provided stationary relative to the wheel belt portion (1), and the wheel belt portion (1) rotates to drive the shifting rod (5) to pass through the measuring slot (401).

3. The circumference measuring device for large rotating components according to claim 1, characterized in that: Also includes: A cleaning friction wheel (6) is rotatably arranged relative to the base (2), the cleaning friction wheel (6) is used to abut against the wheel belt portion (1), and the rotation direction of the cleaning friction wheel (6) is opposite to the rotation direction of the measuring wheel (3).

4. A circumference measuring device for large rotating components according to claim 3, characterized in that: The measuring wheel (3) has a rotating shaft (301) and further comprises: Two support rods (7), both of which are arranged on the base (2), the two support rods (7) are arranged at intervals, and the rotating shaft (301) passes through the two support rods (7) and is located between the two support rods (7).

5. A circumference measuring device for large rotating components according to claim 4, characterized in that: Also includes: Two support rods 2 (8), one support rod 2 (8) is arranged on one support rod 1 (7), and both ends of the cleaning friction wheel (6) are rotatably connected to the two support rods 2 (8) respectively.

6. A circumference measuring device for large rotating components according to claim 5, characterized in that: The cleaning friction wheel (6) is located above the measuring wheel (3), and the second support rod (8) is swingably arranged relative to the first support rod (7). The swing axis of the second support rod (8) is coaxial with the first rotating shaft (301), and the first rotating shaft (301) passes through the two second support rods (8). After the second support rod (8) swings, it drives the cleaning friction wheel (6) to abut or cancel the abutment with the wheel belt portion (1).

7. A circumference measuring device for large rotating components according to claim 6, characterized in that: The second support rod (8) has a support portion (801), both ends of the cleaning friction wheel (6) have support holes (601), the support portion (801) is located in the support hole (601), and further includes: A transmission wheel (9), the transmission wheel (9) being sleeved on the support portion (801); A transmission belt (10), wherein the transmission belt (10) is sleeved on the transmission wheel (9) and the rotating shaft (301), and when the rotating shaft (301) rotates, the transmission wheel (9) is driven to rotate via the transmission belt (10); A transmission member, wherein the transmission wheel (9) rotates to drive the cleaning friction wheel (6) to rotate via the transmission member.

8. The circumference measuring device for large rotating components according to claim 7, characterized in that: The transmission wheel (9) has a toothed ring portion on one side close to the cleaning friction wheel (6) and at both ends of the cleaning friction wheel (6). The transmission member is provided at both ends of the cleaning friction wheel (6). The transmission member includes: Two gears (11), both of which are rotatably arranged on the support rod 2 (8), the two gears (11) are coaxially arranged and spaced apart, the rotation axis of the gears (11) is perpendicular to the axis of the rotating shaft 1 (301), the two gears (11) are located between the transmission wheel (9) and the cleaning friction wheel (6), and the toothed ring portion of the transmission wheel (9) and the toothed ring portion at one end of the cleaning friction wheel (6) are both engaged with the gears (11), and after the transmission wheel (9) rotates, the cleaning friction wheel (6) is driven to rotate through the gears (11).

9. The circumference measuring device for large rotating components according to claim 8, characterized in that: Also includes: Support rod three (12), the support rod three (12) is arranged on the support rod two (8), and the gear (11) is rotatably arranged on the support rod two (8) through the support rod three (12).

10. The circumference measuring device for large rotating components according to claim 5, characterized in that: The support rod 1 (7) is L-shaped, and the support rod 2 (8) is linear.