Impeller shaping device and impeller shaping process based on the device
By designing an impeller shaping device that includes a clamping platform and a shaping tool, and utilizing the relative movement of the shaping shaft and the bearing seat, efficient and precise shaping of the impeller is achieved, solving the problems of low shaping efficiency and insufficient precision caused by thermal deformation after welding, and significantly improving the shaping accuracy and efficiency.
Patent Information
- Application Number
- CN202310182450.6
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2023-03-01
- Publication Date
- 2025-09-16
- Estimated Expiration
- 2043-03-01
AI Technical Summary
The existing impellers are thermally deformed after welding, resulting in low shaping efficiency and insufficient precision. In addition, the traditional shaping device has poor stability, which affects the operating status and maintenance costs of equipment such as fans and pumps.
An impeller shaping device is designed, which includes a rigid base bracket, a clamping platform, a correction tool and a shaping tool. The relative movement of the shaping shaft and the bearing in the bearing seat is used to achieve flexible rotation and high-precision shaping of the impeller. The runout and the position of excessive runout of the impeller are gradually adjusted through dial indicator detection and cyclic operation of the shaping tool.
The precision and efficiency of impeller shaping are improved, the shaping time is shortened, and the shaping precision can reach less than 1mm for the end face runout and less than 0.5mm for the radial runout, which significantly improves the shaping efficiency.
Smart Images

Figure CN116372588B_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to a technical solution for quality inspection and shaping correction in mechanical manufacturing, and in particular to a device and process for efficiently shaping an impeller after welding and assembly. Background Art
[0002] Impellers are core components in rotating fluid machinery, such as fans and pumps. In addition to designing the blade radius and shape according to performance curve requirements, the impeller also requires welding and assembling separately processed components, including the base plate, blades, and front disc. Due to the varying degrees of thermal deformation associated with welding, fan impellers can affect their operational performance after assembly, resulting in abnormally high noise levels, increased bearing damage, and even downtime.
[0003] To this end, each welded impeller requires a complete shaping process to meet the precision parameters required for assembly of equipment such as fans and pumps. Currently, impeller shaping is performed using a stepped sleeve that fits within the impeller's inner bore, positioning the impeller flat against the ground. This is supplemented by a calibration fixture, which simultaneously rotates the impeller to identify locations where runout exceeds the specified value and then performs the shaping. However, in practice, the stepped sleeve and calibration fixture exhibit poor relative stability to the impeller, resulting in inaccurate alignment data. The entire process is labor-intensive and inefficient. Some even omit the shaping process, placing a significant cost burden on after-sales maintenance for fans, pumps, and other equipment. Summary of the Invention
[0004] In order to solve the above technical problems, the purpose of the present invention is to propose an impeller shaping device and an impeller shaping process based on the device, so as to improve the shaping efficiency and the shaping accuracy of the impeller product.
[0005] The technical solution of the present invention to achieve one of the above-mentioned purposes is: an impeller shaping device, having a rigid base support and a top panel horizontally arranged thereon, characterized in that: the device is composed of a clamping platform based on the base support, an auxiliary operation correction tool and a shaping tool, wherein the clamping platform includes at least one group of bearing seats, bearings and shaping shafts positioned and matched on the top panel, and either end of the shaping shaft is provided with a mounting column corresponding to an impeller inner hole specification and a matching pressure plate and screw for locking the impeller shaft disc end face, and the mounting column is exposed in the lateral peripheral space of the top panel; the correction tool includes a dial seat and a dial indicator connected by a dial rod, and the dial surface is movably positioned toward one or more places to be corrected of the impeller; the shaping tool includes a manually operated knocking hammer, a flame heater and an automatic turning machine.
[0006] The above-mentioned impeller shaping device further comprises the clamping platform comprising two sets of matching bearing seats, bearings and shaping shafts, wherein the two shaping shafts are stacked in the height direction, and the projections of their respective central axes on the top panel are orthogonal.
[0007] The above-mentioned impeller shaping device, further, has a pair of low-position bearing seats and a pair of support pad groups on both sides of the axial connecting line fixedly connected to the surface of the top panel, a pair of high-position bearing seats fixedly connected to the top of the support pad group, the first shaping shaft at a lower position is positioned and assembled in the low-position bearing seats through bearings, and the second shaping shaft at an upper position is positioned and assembled in the high-position bearing seats through bearings, wherein the height of the support pad group satisfies that the surfaces of the two shaping shafts are separated by a certain distance.
[0008] The above-mentioned impeller shaping device further has the mounting posts at both ends of the first shaping shaft corresponding to the impeller inner hole specifications of φ38 and φ42 respectively, and the mounting posts at both ends of the second shaping shaft corresponding to the impeller inner hole specifications of φ48 and φ55 respectively, and the center concave of the outer end surface of each mounting post is provided with a mounting hole adapted for screw rotation.
[0009] The technical solution of the present invention to achieve the above-mentioned another object is: an impeller shaping process based on the above-mentioned device, characterized by comprising the steps of:
[0010] S1. Put the impeller to be shaped after welding on the mounting column of the corresponding specification of the shaping shaft, and use the pressure plate and screw to lock the end face of the impeller shaft disc, and adjust the flexibility of the impeller rotation through the bearing seat;
[0011] S2. Attach the dial indicator to the dial base via the dial indicator stem. First, adjust the position and angle of the dial indicator surface toward the impeller base plate for runout correction. Operate the impeller to perform a dial test. Mark the position where the runout amplitude exceeds the standard as reported by the dial indicator. Use a shaping tool to tap and shape the marked position on the base plate. Repeat the dial test, marking, and shaping process until the impeller base plate is shaped to meet the requirements.
[0012] S3. Adjust the position and angle of the dial gauge seat and the percentage surface to the impeller front disc runout correction, operate the impeller to perform dial gauge detection, mark the position where the runout amplitude exceeds the standard according to the dial gauge feedback, and use the shaping tool to tap and shape the marked position of the front disc. Repeat the dial gauge detection, marking, and shaping operations until the impeller front disc is shaped to meet the requirements.
[0013] S4. Then adjust the position and angle of the dial gauge seat and the radial runout correction of the percentage surface to the impeller outer circle, operate the rotating impeller to perform dial gauge detection, mark the position where the runout amplitude exceeds the standard according to the feedback of the dial gauge, and use the shaping tool to perform turning shaping on the marked position of the outer circle, and repeat the dial gauge detection, marking, and shaping operations until the impeller outer circle is shaped to meet the requirements;
[0014] S5. Finally, adjust the position of the dial seat and the position and angle of the radial runout correction of the percentage surface to the impeller inlet, operate the rotating impeller to perform dial detection, mark the position where the runout amplitude exceeds the standard according to the feedback of the dial indicator, and use the shaping tool to knock and shape the marked position of the inlet, and repeat the dial detection, marking, and shaping operations until the impeller inlet is shaped to meet the requirements.
[0015] The above-mentioned impeller shaping process further comprises the following steps: the device is provided with mounting posts corresponding to two or more impeller inner hole specifications, and the impeller mounted on each mounting post performs multi-station synchronous or asynchronous rotation, meter detection, marking, and shaping operations.
[0016] The shaping device and process of the present invention have outstanding substantial characteristics and significant progress: using this device, the rotation of the impeller during the shaping process is achieved through the relative movement of the shaping shaft and the bearing in the bearing seat, which is flexible and labor-saving and eliminates the position error caused by stability; it can achieve rapid alignment, and the impeller shaping accuracy can reach end face runout of less than 1mm and radial runout of less than 0.5mm, while the operation time of the entire shaping process is reduced by 70%, and the efficiency is significantly improved. BRIEF DESCRIPTION OF THE DRAWINGS
[0017] Figure 1 It is a schematic diagram of the assembly structure of the impeller shaping device of the present invention.
[0018] Figure 2 It is a schematic diagram of the exploded structure of the bearing seat used in the impeller shaping device of the present invention.
[0019] Figure 3 It is a schematic structural diagram of the correction tool used in the impeller shaping device of the present invention.
[0020] Figure 4 It is a schematic diagram of the operation flow of the impeller shaping process of the present invention. Implementation Method
[0021] The specific implementation methods of the present invention will be further described below in conjunction with the accompanying drawings to make the technical solutions of the present invention easier to understand and grasp, thereby making a clearer definition of the protection scope of the present invention.
[0022] The present invention provides an impeller shaping device with a simple structure, high assembly strength, stable assembly of the shaping object, and greatly improved shaping accuracy, as well as an impeller shaping process based on the device. It can not only reduce the fatigue strength of the shaping operation and maintain stable rotation of the impeller during the shaping operation, but also use a dial indicator for high-precision meter detection to improve the shaping efficiency and accuracy.
[0023] From the structural composition of the impeller shaping device and the general characteristics of the functional design of each part, we can understand that Figure 1 、 Figure 2 and Figure 3 The structural display, from the overall to the detailed details, shows that the impeller shaping device comprises a rigid base support 1 and a horizontally mounted top panel 2. This section constitutes the foundational structure of conventional industrial equipment or devices, designed to provide ease of operation for operators while also ensuring the structural strength of the work platform to offset the various stresses generated during actual operation. The structural features in this section are not the subject of this application's protection claim and are provided solely as a basic overview. As the key feature requested for protection, the device consists of a clamping platform 3 based on a base bracket 1, an auxiliary operation correction tool 4 and a shaping tool. Here, the clamping platform 3 includes at least one set of bearing seats 31, bearings 32 and shaping shafts positioned and matched on the top panel, and either end of the shaping shaft is provided with a mounting column corresponding to an impeller inner hole specification and a matching pressure plate 34 and a screw 35 for locking the end face of the impeller shaft disk. The mounting column is exposed in the lateral peripheral space of the top panel to ensure that the impeller is separated from the top panel, rotates freely and is not interfered with after assembly and fixation; the above-mentioned correction tool includes a gauge seat 41 and a dial indicator 43 connected by a gauge rod 42, and the dial indicator 43 is designed for movable positioning at more than one position to be corrected on the impeller 5. The specific implementation of its positioning and angle adjustability can be found in the detailed description below; and the above-mentioned shaping tools include but are not limited to manually operated percussion hammers, flame-focusing heaters and automatic turning machines. These shaping tools are all existing conventional instruments, and their functions only exist as supporting accessories for the shaping device. The functions of each part are briefly described as follows. The striking hammer can be an iron hammer, a wooden mallet, a rubber hammer, etc. of various specifications and pounds, and the main function of the flame-focusing heater is to perform local heating on the impeller to be shaped, so as to soften it and facilitate shaping; the automatic turning machine includes hand-held grinders or industrial lathes of various specifications, etc., and its function is to process and correct the appearance exceeding the limit caused by local deformation during the impeller production and welding process.
[0024] From the above overview, it can be seen that the impeller shaping device of the present invention is different from the traditional one that uses a stepped shaft sleeve set upright on the ground. The impeller is directly sleeved and operated to rotate to implement detection and shaping operations. The relative movement of the main shaft (i.e., the shaping shaft) and the bearing in the bearing seat maximizes the simulation of the actual assembly state of the impeller, making its operation more flexible and more stable, and can completely avoid the shaping error caused by the axial offset between the traditional stepped shaft sleeve and the impeller inner hole.
[0025] The preferred embodiment illustrated in the figure provides a more detailed understanding of the device's structural features: To improve device efficiency, the clamping platform 3 can include two sets of matching bearing seats, bearings, and shaping shafts, allowing operators to attach and shape the impeller in different directions. The two shaping shafts are stacked in the vertical direction, with their respective central axes projected orthogonally on the top panel. This ensures that the operator's position and the impeller assembly direction are staggered, and the axial directions do not interfere with each other during impeller rotation. More specifically, a pair of low-position bearing seats 31a and a pair of support pad groups 36 on both sides of the axial connecting line are fixed to the surface of the top panel 2. A pair of high-position bearing seats 31b are fixed to the top of the support pad group 36. The first shaping shaft 33a at the lower position is positioned and attached to the low-position bearing seat 31a through the bearing 32, and the second shaping shaft 33b at the upper position is positioned and attached to the high-position bearing seat 31b through the bearing 32. The height of the support pad group 36 satisfies that the surfaces of the two shaping shafts are separated by a certain distance, thereby ensuring that the surfaces of the orthogonal shaping shafts do not contact or interfere with each other when they rotate.
[0026] In the illustrated embodiment, the device satisfies the simultaneous shaping operation of at least two workstations. The impellers generally have diverse specifications, so the mounting posts at both ends of each shaping shaft are designed to have a variety of specifications. Specifically, the mounting post 331a provided at one end of the first shaping shaft 33a corresponds to the impeller inner hole specification of φ38, while the mounting post (not shown) at the other end corresponds to the inner hole specification of φ42. Similarly, the mounting posts 331b and 332b provided at both ends of the second shaping shaft 33b correspond to the impeller inner hole specifications of φ48 and φ55, respectively, and the center of the outer end surface of each mounting post is concave with a mounting hole for screwing with the adaptor screw 35, so that the impeller is connected to the mounting post through the pressure plate 34 as a whole and the required structural strength is achieved.
[0027] To facilitate assembly of the clamping platform and its shaping shaft, the bearing seat 31 utilizes a split design, primarily consisting of a seat body 311 and a top cover 312. The bearing 32 is embedded between the two joined and locked parts, providing axial positioning. The seat body 311 is attached to the top panel and support pad assembly via high-strength bolts and nuts. Once the shaping shaft and bearing are properly positioned, the top cover 312 can be secured to it, ensuring axial stability for the shaping shaft while providing slight adjustability for rotational flexibility.
[0028] The main body of the base support 1 is configured as a hollow quadrangular prism formed by bending and sealing a metal plate, and the hollow quadrangular prism is integrally encapsulated with the top and bottom panels to form a box. A rigid frame can be optionally installed within the hollow quadrangular prism to further enhance assembly strength and absorb stress to ensure the stability of impeller rotation. Access holes 11 for clamping platform assembly are provided on either side of the hollow quadrangular prism, allowing tools to be inserted from the bottom of the top panel to tighten bolts and nuts during assembly of the bearing seat.
[0029] The dial holder 41 in the above-mentioned calibration tool 4 is integrated with an electromagnetic block with a switch, which can avoid the situation where the dial holder is difficult to disassemble when a strong magnetic block is used. When the connection position of the dial holder needs to be changed, the dial holder can be removed by simply turning off the switch for demagnetization. After the position is shifted and positioned, the switch can be turned on again to restore the magnetism and connect with the iron carrier with high strength. From the perspective of the positioning of the dial holder of the calibration tool, it can be directly magnetically positioned on the side wall of the base bracket, or a movable auxiliary bracket can be provided next to the base bracket, and the dial holder can be magnetically positioned on the auxiliary bracket. The adjustable angle of the dial indicator is achieved by the orientation of the dial holder and the two or more sections of the zigzag positioning of the dial rod. Each joint of the dial rod is provided with a joint lock that can be manually tightened and loosened.
[0030] In addition to the preferred embodiment shown in the figure, the clamping platform can also include three or more groups of matching bearing seats, bearings and shaping shafts. Under the premise that the spatial scene is not restricted, the above-mentioned base bracket and its top panel are rectangular (the length meets the distribution requirements of the clamping platform) and three pairs of bearing seats are fixedly connected along the length of the surface. The three shaping shafts are arranged side by side in the plane parallel to the top panel, and their respective central axes are parallel to each other. In this implementation, the specifications of the mounting columns provided at both ends of each shaping shaft can be consistent or can be distributed in a discrete manner. In actual circumstances, each shaping shaft is a replaceable assembly setting, so the specifications of its mounting columns have the variability required to meet the shaping product.
[0031] In order to understand the functional realization of the impeller shaping device of the present invention, the following flowchart and detailed description are used to specifically demonstrate the implementation of the shaping process. Figure 4 The process mainly includes the following steps.
[0032] S1. Different from the traditional method of directly fitting on the stepped shaft sleeve, the impeller to be shaped after welding is fitted on the mounting column of the corresponding specification of the shaping shaft, and the end face of the impeller shaft disc is locked with a pressure plate and a screw. The shaping operator manually turns the impeller to rotate, and adjusts the flexibility of the impeller rotation through observation and the bearing seat, making solid technical preparations for the subsequent meter inspection of the shaping operation.
[0033] S2. Attach the dial indicator to the base via the dial rod. First, adjust the position and angle of the dial indicator face toward the impeller baseplate for runout correction. The baseplate is magnetically connected to the top panel surface near the impeller baseplate, and the dial indicator is adjusted close to the impeller baseplate with the dial head facing the surface. Operate the impeller to perform a dial test. Mark the locations where the runout exceeds the standard based on the runout amplitude reported by the dial indicator. Use a hammer to tap the marked locations on the baseplate to reshape the impeller baseplate. Repeat the dial test, marking, and reshaping process until the impeller baseplate is reshaped to meet the required standards.
[0034] S3. Adjust the position and angle of the dial gauge base and the dial indicator face toward the impeller front disc for runout correction. The dial gauge base is magnetically connected to the side wall of the base bracket near the bottom of the impeller, and the dial indicator is adjusted close to the impeller front disc with the dial gauge facing the face. Rotate the impeller for a dial gauge test in the same manner. Mark the locations where the runout exceeds the standard based on the runout amplitude reported by the dial indicator. Use a shaping tool (also a hammer) to tap the marked locations on the front disc for shaping. Repeat the dial gauge test, marking, and shaping process until the impeller front disc is shaped to meet the requirements.
[0035] S4. Then adjust the position and angle of the dial gauge seat and the percentage surface to correct the radial runout of the impeller outer circle. The position of the dial gauge remains unchanged. Only the dial gauge needs to be adjusted back from the impeller front disc to the radial surface side of the impeller outer circle. Continue to operate the impeller to rotate the dial gauge to check. According to the runout amplitude feedback from the dial gauge, mark the position that exceeds the standard. Use an automatic turning machine to perform turning and shaping on the marked position of the outer circle. Repeat the dial gauge inspection, marking, and shaping operation until the impeller outer circle is radially shaped to meet the requirements.
[0036] S5. Finally, adjust the position and angle of the dial seat and the radial runout correction of the percentage surface toward the impeller inlet. The dial seat needs to be demagnetized and transferred to the auxiliary bracket of the peripheral device next to the base bracket. Then adjust the dial indicator to the impeller inlet and face the edge of the inlet. Continue to operate the rotating impeller for meter detection. Mark the position that exceeds the standard according to the runout amplitude fed back by the dial indicator. Use a hammer to perform tapping and shaping on the marked position of the inlet again. Repeat the meter detection, marking, and shaping operations until the impeller inlet is shaped to meet the requirements. In the above steps, if the impeller needs to be softened by using a hammer, a flame heater can be used to heat and soften the impeller locally, so that the tapping and shaping can be completed with high quality within a more detailed range, avoiding additional deformation forces on parts that do not require shaping.
[0037] In view of the fact that the above-mentioned device is distributed with mounting columns corresponding to more than two impeller inner hole specifications, multiple shaping workers can stand in a distributed manner (whether in a cross-shaped or side by side position) and perform multi-station synchronous or asynchronous rotation, meter detection, marking, and shaping operations on the impellers mounted on each mounting column, thereby improving overall efficiency.
[0038] In summary, from the detailed description of the impeller shaping device of the present invention and the impeller shaping process based on the device combined with the illustrated embodiment, it can be seen that the present solution has many significant advantages. By using the device, the rotation of the impeller during the shaping process is achieved by the relative movement of the shaping shaft and the bearing in the bearing seat, which is flexible and labor-saving and eliminates the position error caused by stability; it can achieve rapid alignment, and the impeller shaping accuracy can reach less than 1mm of end face runout and less than 0.5mm of each radial runout, while the operation time of the entire shaping process is reduced by 70%, and the efficiency is significantly improved.
[0039] In addition to the above embodiments, the present invention may also have other implementation methods. Any technical solutions formed by equivalent replacement or equivalent transformation fall within the scope of protection required by the present invention.
Claims
1. An impeller shaping device having a rigid base support and a top plate disposed horizontally thereon, characterized in that: The device consists of a clamping platform based on a base bracket, a correction tool for auxiliary operations and a shaping tool. The clamping platform includes two sets of matching bearing seats, bearings and shaping shafts. A pair of low-position bearing seats and a pair of support pad groups on both sides of the axial connection line are fixedly connected to the surface of the top panel. A pair of high-position bearing seats are fixedly connected to the top of the support pad group. The first shaping shaft at the lower position is positioned and installed in the low-position bearing seat through the bearing, and the second shaping shaft at the upper position is positioned and installed in the high-position bearing seat through the bearing, wherein the height of the support pad group meets The two shaping shafts are stacked in the height direction, and the projections of the central axes of the two shaping shafts on the top panel are orthogonal. The two ends of the first shaping shaft are provided with mounting columns corresponding to the impeller inner hole specifications of φ38 and φ42 and their matching pressure plates and screws for locking the end faces of the impeller shaft disc. The two ends of the second shaping shaft are provided with mounting columns corresponding to the impeller inner hole specifications of φ48 and φ55 and their matching pressure plates and screws for locking the end faces of the impeller shaft disc. All mounting columns are exposed in the lateral peripheral space of the top panel, and the center of the outer end face is concave with a mounting hole adapted for screw rotation. The correction tool includes a dial base and a dial indicator connected by a dial rod, and the dial surface is movably positioned toward one or more locations of the impeller to be corrected; the shaping tool includes a manually operated percussion hammer, a flame-focusing heater and an automatic turning machine.
2. The impeller shaping device according to claim 1, characterized in that: The clamping platform can also have three sets of matching bearing seats, bearings and shaping shafts. The top panel is rectangular and has three pairs of bearing seats fixedly connected along the length of the surface. The three shaping shafts are arranged side by side in a plane parallel to the top panel, and their respective central axes are parallel to each other.
3. The impeller shaping device according to claim 1 or 2, characterized in that: The bearing seat is a split assembly consisting of a seat body and a top cover, and the bearing is embedded between the seat body and the top cover that are spliced and locked together for axial positioning.
4. The impeller shaping device according to claim 1, characterized in that: The main body of the base bracket is set as a hollow quadrangular prism formed by bending a metal plate and sealing the edges, and the hollow quadrangular prism is integrally packaged with the top panel and the bottom panel to form a box body, wherein any side wall of the hollow quadrangular prism is provided with an inspection hole for clamping the platform assembly.
5. The impeller shaping device according to claim 1, characterized in that: The dial base in the calibration tool is integrated with an electromagnetic block with a switch. The dial base is magnetically positioned on the side wall of the base bracket or the auxiliary bracket of the peripheral device, and the position and angle of the dial indicator are adjustable.
6. An impeller shaping process based on the device according to any one of claims 1 to 5, characterized in that Including steps: S1. Put the impeller to be shaped after welding on the mounting column of the corresponding specification of the shaping shaft, and use the pressure plate and screw to lock the end face of the impeller shaft disc, and adjust the flexibility of the impeller rotation through the bearing seat; S2. Attach the dial indicator to the dial base via the dial indicator stem. First, adjust the position and angle of the dial indicator surface toward the impeller base plate for runout correction. Operate the impeller to perform a dial test. Mark the position where the runout amplitude exceeds the standard as reported by the dial indicator. Use a shaping tool to tap and shape the marked position on the base plate. Repeat the dial test, marking, and shaping process until the impeller base plate is shaped to meet the requirements. S3. Adjust the position and angle of the dial gauge seat and the percentage surface to the impeller front disc runout correction, operate the impeller to perform dial gauge detection, mark the position where the runout amplitude exceeds the standard according to the dial gauge feedback, and use the shaping tool to tap and shape the marked position of the front disc. Repeat the dial gauge detection, marking, and shaping operations until the impeller front disc is shaped to meet the requirements. S4. Then adjust the position and angle of the dial gauge seat and the radial runout correction of the percentage surface to the impeller outer circle, operate the rotating impeller to perform dial gauge detection, mark the position where the runout amplitude exceeds the standard according to the feedback of the dial gauge, and use the shaping tool to perform turning shaping on the marked position of the outer circle, and repeat the dial gauge detection, marking, and shaping operations until the impeller outer circle is shaped to meet the requirements; S5. Finally, adjust the position of the dial seat and the position and angle of the radial runout correction of the percentage surface to the impeller inlet, operate the rotating impeller to perform dial detection, mark the position where the runout amplitude exceeds the standard according to the feedback of the dial indicator, and use the shaping tool to knock and shape the marked position of the inlet, and repeat the dial detection, marking, and shaping operations until the impeller inlet is shaped to meet the requirements.
7. The impeller shaping process according to claim 6, characterized in that: The device is distributed with mounting posts corresponding to more than two impeller inner hole specifications, and the impellers mounted on each mounting post perform multi-station synchronous or asynchronous rotation, meter detection, marking, and shaping operations.
Citation Information
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