Anti-chatter lathe fixture

CN123000536APending Publication Date: 2026-10-30CRRC CHANGZHOU AUTO PARTS CO LTD +1
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Patent Information

Application Number
CN202611048668.2
Authority / Receiving Office
CN · China
Patent Type
Applications(China)
Current Assignee / Owner
Filing Date
2026-07-15
Publication Date
2026-10-30

AI Technical Summary

Technical Problem

然而,上述装夹方式在抑制颤振方面存在明显不足:

Benefits of technology

[0019]有益效果:1、本发明的防颤振的车床工装,能显著降低车削时工件弯曲变形量,通过将工件主体容纳于安装套筒内,并设置周边多点支撑,能大幅缩短受扭与悬空跨度,使工件有效悬空跨度从194mm大幅缩减至代加工叶柄尺寸,约10-15mm。根据材料力学,工件的弯曲变形量与跨度成正比,因此理论上本发明的防颤振的车床工装可使工件弯曲变形降低1-2个数量级,有效抑制切削力主导的工件弯曲变形。

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Abstract

The present application relates to a kind of anti-chattering lathe tool, belong to construction technology field, the present application includes installation sleeve, first support, second support and fixed assembly, the bottom of installation sleeve is fixed to lathe spindle and with same coaxial line arrangement;First support is arranged in the bottom center of installation sleeve inside, second support is arranged along the circumferential direction of installation sleeve axis;Fixed assembly includes several pressing blocks, is installed in installation sleeve along the direction perpendicular to the spindle axis, and is arranged at intervals around the axis, and the block body on the axis side is enclosed into workpiece clamping end;Workpiece is installed in the inner cavity of installation sleeve, one end is connected with first support, the workpiece body in sleeve is supported by second support, workpiece to be processed end extends outside sleeve and is stopped by tailstock center of lathe;Workpiece is clamped after clamping end, and is coaxially arranged with lathe spindle under the action of first support and second support.The present application can realize efficient, high-precision "lathe instead of grinding" process.
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Description

Technical Field

[0001] This invention belongs to the field of precision grinding technology, specifically relating to a lathe tooling for preventing chatter. Background Technology

[0002] Nickel-based superalloys such as GH696 are widely used in hot-end components such as rotatable stationary blades of aero-engine compressors due to their excellent high-temperature performance. These parts have extremely high requirements for shaft fit accuracy (usually requiring h6 grade), cylindricity (≤5μm), and surface roughness (Ra≤0.6μm), but the material has poor machinability, resulting in problems such as high cutting force, high cutting temperature, and rapid tool wear.

[0003] In traditional machining, the precision machining of the stalk 2 of the rotatable stationary blade 1 is mostly carried out using grinding processes. For example... Figure 1 However, grinding processes have drawbacks such as long processing cycles (2-3 hours per piece), rapid wheel wear, high costs, and susceptibility to grinding burns, making them unsuitable for mass production. In recent years, the "turning a lathe instead of grinding" process has gradually gained attention due to its advantages of high efficiency and low cost, but it places higher demands on the rigidity and stability of the process system.

[0004] In turning, rotatable stationary blades exhibit "flexible shaft" characteristics under stress. For example, radial cutting forces easily lead to bending deformation, while tangential cutting forces easily induce torsional deformation. More critically, due to the slender shape and insufficient rigidity of the rotatable stationary blade workpiece, chatter is highly likely to occur under high-speed cutting conditions. Chatter refers to the self-excited vibration generated between the tool and the workpiece during the cutting process. It not only causes chatter marks and deterioration of the surface roughness (failing to achieve Ra≤0.6μm), but also accelerates tool wear, reduces dimensional accuracy, and may even force machining interruption. Therefore, suppressing chatter is one of the core challenges in realizing the "turning instead of grinding" process.

[0005] To position the workpiece, current turning processes typically employ a clamping method of "one clamp and one center" or "double centers." However, these clamping methods have significant shortcomings in suppressing chatter. (a) Shortcomings of the "one clamp, one top" method In the "one-clamp-one-cushion" method, the chuck directly clamps the small end 3 of the rotatable stationary blade, while the tailstock tip presses against the large end 4 on the other side. This method has the following problems: First, the thickness of the chuck and jaws results in a large suspended span of the tooling (approximately 200mm), significantly reducing the natural frequency of the process system and causing it to fall into the vibration range of cutting excitation, which easily induces chatter; Second, the chuck clamping force generates a large radial clamping force, which can easily cause clamping deformation of thin-walled or slender workpieces, resulting in out-of-tolerance roundness of the workpiece after unloading; Third, there is a repeatability error between the chuck and the tooling, affecting the consistency of machining; Fourth, the clamping force cannot be automatically adjusted according to thermal expansion during machining, which can easily cause workpiece bending or dimensional instability.

[0006] (II) Shortcomings of the "Double Top" Approach The "double-center" method relies solely on the center holes at both ends for positioning, without any auxiliary support. While this avoids chuck deformation, the system suffers from severe rigidity because the entire workpiece is suspended in the air. During high-speed cutting, the workpiece is highly susceptible to bending and torsional vibrations under cutting force, resulting in continuous chatter. Figure 2 The red dashed line indicates bending deformation of the workpiece when using a double-center method. In addition, traditional centers are mostly rigidly clamped and lack axial floating compensation capability. Not only can they not suppress vibration, but they may also exacerbate the instability of the process system due to rigid clamping.

[0007] In summary, existing clamping methods such as "one clamp and one top" and "double top" have significant shortcomings in terms of process system stiffness, natural frequency, and vibration suppression. They cannot effectively suppress chatter and are insufficient to meet the requirements of machining stability and surface quality for precision parts such as nickel-based superalloy rotatable stationary blades in the "turning instead of grinding" process. Therefore, there is an urgent need to design an anti-chatter lathe fixture that can reconstruct the stiffness chain of the process system, raise the natural frequency to avoid the vibration range, and has the ability to actively suppress chatter. Summary of the Invention

[0008] The purpose of this invention is to provide a lathe tooling with anti-chatter to solve the problems mentioned in the background art.

[0009] To achieve the above objectives, the present invention provides the following technical solution: a lathe fixture for preventing chatter, comprising a mounting sleeve, a support assembly, and a fixing assembly, wherein: The mounting sleeve is a hollow structure with an opening on one side, and its bottom is fixed to the lathe spindle and set coaxially with it; The support assembly includes a first support member and a second support member. The first support member is located at the center of the bottom inside the mounting sleeve, and the second support member is arranged circumferentially along the axis of the mounting sleeve. The fixing assembly includes several pressure blocks, which are installed on the mounting sleeve in a direction perpendicular to the spindle axis and are arranged at intervals around the axis. The blocks located on the axis side form a workpiece clamping end. The workpiece is installed in the inner cavity of the mounting sleeve, with one end connected to the first support member. The workpiece body located inside the sleeve is supported by the second support member. The end of the workpiece to be processed extends out of the sleeve and is stopped by the center of the lathe tailstock. After the workpiece is clamped by the clamping end, it is set coaxially with the lathe spindle under the action of the first and second support members.

[0010] Preferably, the mounting sleeve includes a disc base and a sleeve. The sleeve is a hollow structure with openings on both sides. The disc base is fixedly installed on one side opening of the sleeve. Several mounting holes are provided on the base. The disc base is detachably installed on the lathe spindle through the mounting holes and connecting bolts. The disc base, the sleeve and the lathe spindle are arranged coaxially.

[0011] Preferably, the first support is a small-end tip, the disc base has a connecting hole at the center of the base, the small-end tip has a threaded hole facing the body of the disc base, the small-end tip is fixedly installed at the center of the disc base by passing through the connecting hole with the threaded hole through the connecting bolt; the small-end tip abuts against the tip hole of the workpiece.

[0012] Preferably, there are two pressure blocks, which are installed opposite each other in the opening on the side of the sleeve away from the disc base. The opening side of the sleeve has bolt holes at the position of the pressure blocks, and the pressure blocks have mounting holes at the corresponding positions. The pressure blocks are fixedly installed on the sleeve by bolts passing through the mounting holes and being threaded into the bolt holes. The block on the axial side of the pressure block has a V-shaped clamping groove. The clamping grooves of the two pressure blocks form a workpiece clamping end with a shape that matches the shape of the workpiece.

[0013] Preferably, the mounting hole of the pressure block is a strip-shaped adjustment hole, and the pressure block is installed at different positions of the mounting hole by bolts to adjust the clamping shaft diameter of the workpiece clamping end.

[0014] Preferably, the sleeve is made of lightweight POM; the pressure block is made of steel; and the disc base and small end tip are made of 40Cr alloy steel, which is heat-treated and stress-relief annealed.

[0015] Preferably, the second support is a counterscrew, which is threaded to the wall of the mounting sleeve, and there are several counterscrews. The inner end of the counterscrew abuts against the outer circumferential surface of the workpiece.

[0016] Preferably, several supporting screws are arranged in pairs, with each pair symmetrically distributed along the outer periphery of the workpiece.

[0017] Preferably, the side wall of the mounting sleeve is provided with an opening that penetrates the side wall, which also serves as a clamping observation port and a discharge port for cutting fluid and chips.

[0018] Preferably, the lathe tailstock is a servo tailstock with a floating center at its front end; the servo tailstock has a constant clamping force, and the floating center can float elastically along the axial direction to compensate for the thermal expansion of the workpiece.

[0019] Beneficial effects: 1. The anti-chatter lathe fixture of the present invention can significantly reduce the bending deformation of the workpiece during turning. By accommodating the main body of the workpiece in the mounting sleeve and setting multiple points of support around it, the torsional and overhang span can be greatly shortened, reducing the effective overhang span of the workpiece from 194mm to the size of the blade shank being machined, approximately 10-15mm. According to the mechanics of materials, the bending deformation of the workpiece is proportional to the span. Therefore, theoretically, the anti-chatter lathe fixture of the present invention can reduce the bending deformation of the workpiece by 1-2 orders of magnitude, effectively suppressing the bending deformation of the workpiece dominated by the cutting force.

[0020] 2. Building upon the foregoing, the anti-chatter lathe fixture of this invention can suppress torsional deformation and coaxiality errors. The aforementioned feature of housing most of the workpiece length within the mounting sleeve allows the torque generated at the blade shank cutting point to be directly borne by the mounting sleeve without passing through a long, suspended section, significantly shortening the torque transmission path. This effectively suppresses torsional deformation of the workpiece under tangential force, preventing spiral machining marks caused by torsion. Furthermore, two opposing clamping blocks are used at the workpiece blade shank for top-to-top clamping, instead of the traditional single-sided chuck clamping. This method eliminates the eccentric bending moment caused by chuck clamping and establishes a stable force couple balance system at the blade shank, effectively counteracting the rotational torque generated by the tangential cutting force and systematically suppressing circumferential hysteresis errors, thereby ensuring coaxiality requirements throughout the entire length of the workpiece.

[0021] 3. Building upon the foregoing, the anti-chatter lathe fixture of this invention achieves stress-free positioning with a balance of rigidity and flexibility. This invention employs a scheme where the mounting sleeve is directly mounted on the lathe spindle, coupled with constant pressure clamping from the lathe servo tailstock. On one hand, the disc base of the mounting sleeve is directly fixed to the lathe spindle, eliminating repetitive positioning errors caused by traditional chuck clamping and reducing the overhang span by approximately 200mm. On the other hand, the lathe servo tailstock provides a constant and axially floating clamping force. This "rigidity-flexibility" design ensures reliable axial positioning while automatically compensating for thermal expansion of the workpiece during processing, preventing the workpiece from bending.

[0022] 4. The anti-chatter lathe fixture of this invention can effectively suppress cutting chatter at its source. This invention adopts a steel-plastic composite structure, with the sleeve made of POM engineering plastic and the pressure block made of steel. Calculations show that the total weight of the fixture can be reduced from approximately 18 kg of traditional steel fixtures to 3.6 kg, a weight reduction of 80%. The lightweight design of this invention significantly increases the natural frequency of the machining system, moving it away from the cutting excitation frequency range in turning, thereby fundamentally eliminating the conditions for chatter occurrence, effectively suppressing regenerative chatter, and ensuring the integrity of the machined surface.

[0023] 5. Building upon the foregoing, the anti-chatter lathe fixture of this invention ensures long-term high precision and stability. Key load-bearing and positioning components, such as the disc base and small-end center, are made of 40Cr alloy steel and undergo heat treatment and stress-relief annealing, effectively eliminating machining and heat treatment stresses and ensuring the fixture's dimensional stability and wear resistance under external forces. The strip-shaped adjustment hole design of the mounting holes for the support screws and pressure blocks allows for flexible adjustment based on the actual outer diameter error of the workpiece blank, avoiding forced clamping deformation of the workpiece and further ensuring consistency in batch processing.

[0024] 6. The anti-chatter lathe fixture of this invention significantly improves machining efficiency and has wide applicability. The combined effects enable a stable "turning instead of grinding" process, reducing the machining cycle of a single rotatable stationary blade shank from 2-3 hours in traditional grinding to approximately 20 minutes, increasing efficiency by 6-9 times. It also reduces the cost of tool and grinding wheel wear. The design principle of this lathe fixture is not only applicable to lathes but can also be extended to high-precision grinding machines, milling machines, and other rotary machining equipment to solve machining problems of various slender shafts and weakly rigid parts. Attached Figure Description

[0025] Figure 1 This is a schematic diagram of the structure of a rotatable stationary blade in the prior art; Figure 2 This is a schematic diagram illustrating the bending deformation that occurs during turning of a workpiece using a double-center clamping method in the prior art. Figure 3 This is a schematic diagram of the lathe tooling for clamping workpieces, as shown in the embodiment. Figure 4 The diagram shows the structure of the anti-chatter lathe fixture installed on the lathe (the lathe spindle is not shown, and only the main body of the floating center is shown on the tailstock). Figure 5 This is a top view of the lathe tooling for anti-chatter during workpiece clamping, as shown in the embodiment. Figure 6 for Figure 5 Sectional view at point II. Detailed Implementation

[0026] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings. However, the scope of protection of the present invention is not limited to the embodiments. For clarity, the workpiece in this embodiment is a GH696 high-temperature alloy rotatable stationary blade, with a total length of approximately 200 mm and a diameter transitioning from Ф22 mm to Ф12 mm. The cylindricity is required to be ≤5 μm and the roughness Ra≤0.6 μm.

[0027] Please see Figures 3-6 This embodiment provides a lathe tooling for chatter prevention, including a mounting sleeve, a support assembly, and a fixing assembly, wherein: The mounting sleeve is a hollow structure with an opening on one side, and its bottom is fixed to the lathe spindle and set coaxially with it; The support assembly includes a first support member and a second support member. The first support member is a small-end tip 7, which is located at the center of the bottom of the inner side of the mounting sleeve. The second support member is a support screw 8, which is arranged circumferentially along the axis of the mounting sleeve. The fixing assembly includes two pressure blocks 9, which are installed on the mounting sleeve in a direction perpendicular to the spindle axis and are arranged at intervals around the axis. The blocks located on the axis side form a workpiece clamping end. The rotatable stationary blade 12 is installed in the inner cavity of the mounting sleeve, with its small end connected to the small end tip. The blade inside the sleeve is supported by the abutment screw. The blade shank of the rotatable stationary blade extends out of the sleeve and is abutted by the tip of the lathe tailstock. After the rotatable stationary blade is fixed by the clamping end, it is set coaxially with the lathe spindle under the action of the small end tip and the abutment screw.

[0028] The mounting sleeve in this embodiment includes a disc base 5 and a sleeve 6. The sleeve 6 is a hollow structure with openings on both sides. A bolt hole penetrating the sleeve is provided on the open side of the sleeve located on the disc base side. A threaded hole is provided on the disc base corresponding to the bolt hole. The disc base 5 is mounted on the open side of the sleeve, and the two are fixedly connected by a connecting bolt 11 penetrating the bolt hole of the sleeve and threadedly connected to the threaded hole of the disc base. In addition, several mounting holes are provided on the base of the disc base. The disc base is detachably mounted to the lathe spindle by installing connecting bolts in the mounting holes. The disc base, sleeve, and lathe spindle are arranged coaxially.

[0029] Furthermore, in this embodiment, the disc base is provided with a connecting hole at the center of the base body, and the small end tip is provided with a threaded hole facing the body of the disc base. The small end tip is fixedly installed at the center of the disc base by passing through the connecting hole and being threadedly connected to the threaded hole by the connecting bolt 10; the small end tip abuts against the tip hole of the small end of the rotatable stationary blade.

[0030] In this embodiment, two clamping blocks are mounted top-to-top at the opening on the side of the sleeve away from the disc base. Bolt holes are provided on the sleeve at the positions of the clamping blocks, and mounting holes are provided on the clamping blocks at corresponding positions. The clamping blocks are fixed to the sleeve by bolts 13 passing through the mounting holes and threadedly connected to them. The block on the axial side of the clamping block has a V-shaped clamping groove. The clamping grooves of the two clamping blocks form a workpiece clamping end whose shape matches the shape of the rotatable stationary blade shank. Furthermore, the mounting holes of the clamping blocks in this embodiment are strip-shaped adjustment holes. The clamping blocks are mounted at different positions in the mounting holes by bolts to adjust the clamping shaft diameter of the workpiece clamping end. Simultaneously, when clamping or disassembling the workpiece, the bolts can be loosened, and the clamping blocks can be moved without removing them to create space in the sleeve opening, facilitating workpiece clamping or disassembly.

[0031] Furthermore, in this embodiment, the sleeve is made of lightweight POM; the pressure block is made of 45# steel; and the disc base and small end tip are made of 40Cr alloy steel, which has undergone quenching and tempering and stress-relieving annealing. Meanwhile, in this embodiment, eight support screws are threadedly connected to the cylinder wall of the mounting sleeve, with the inner ends of the support screws abutting against the outer circumferential surface of the rotatable stationary blade. The eight support screws are arranged in pairs, with four pairs of support screws symmetrically distributed along the outer circumferential surface of the blade.

[0032] In addition, the sleeve in this embodiment has an opening 14 that penetrates the side wall, which serves as both a clamping observation port and a discharge port for cutting fluid and chips. The lathe tailstock in this embodiment is a servo tailstock with a floating center at its front end; the servo tailstock has a constant clamping force of 1.0KN, and the floating center can elastically float 0.5-1mm along the axial direction to compensate for the thermal expansion of the workpiece.

[0033] Working principle: The machining operation process of the anti-chatter lathe fixture in this embodiment is as follows: 1. Clamping the workpiece: Insert the rotatable stationary blade into the hollow cavity of the sleeve from the small end of the blade until the tip hole at the small end of the blade contacts the tip at the small end of the disc base.

[0034] 2. Tailstock clamping: Start the lathe's servo tailstock and move it forward until the floating center on the tailstock aligns and contacts the center hole at the large end of the rotatable stationary blade shank. Set the servo tailstock to provide a constant clamping force of 1.0 kN and allow for an axial float of 0.8 mm.

[0035] 3. Large end clamping: Manually push the two clamping blocks so that they press against the outer circle of the large end of the rotatable stationary blade from both sides, and then tighten the connecting bolts to secure the clamping blocks. The elliptical mounting hole design of the clamping blocks allows for fine adjustment of the clamping position.

[0036] 4. Auxiliary Support: Tighten the peripheral support screws on the sleeve in sequence until their inner ends lightly contact the front and back outer circumferential surfaces of the blade. After contact, tighten each screw by half a turn to prevent excessive force from deforming the workpiece. The operator can visually inspect the tightening process through the opening on the side wall of the sleeve.

[0037] 5. Perform cutting: Start the lathe and perform high-speed turning according to the optimized "turning instead of grinding" cutting parameters.

[0038] 6. Disassembling the workpiece: After processing is completed, loosen the peripheral support screws, loosen the connecting bolts of the pressure block, and retract the servo tailstock in sequence to remove the processed workpiece.

[0039] Using the above tooling and process, 100 blade shanks of GH696 rotatable stationary blades were continuously machined. Test results showed that the workpiece cylindricity was consistently controlled within 4.2 μm; the surface roughness Ra was consistently below 0.55 μm; online monitoring using a laser displacement sensor showed a maximum overall deformation of 1.8 μm at the cutting point; no visible chatter was observed during machining, and the process system operated smoothly; the turning cycle (including clamping) for a single blade shank was shortened to approximately 20 minutes, representing a 6-9 times increase in efficiency compared to traditional grinding (2-3 hours).

[0040] The anti-chatter lathe fixture of this invention reconstructs the rigidity chain of the machining system through "double centers + multi-point auxiliary support". The disc base is directly connected to the lathe spindle, reducing cantilever and error; the servo tailstock constant force floating center achieves stress-free axial positioning; the POM lightweight sleeve significantly reduces weight and increases the natural frequency, suppressing chatter; the clamping block and peripheral support screws provide the main clamping force and reverse prestress respectively, controlling workpiece deformation to the micron level. The three elements work together to provide a reliable tooling solution for "turning instead of grinding".

[0041] As described above, although the invention has been shown and described with reference to specific preferred embodiments, it should not be construed as limiting the invention itself. Various changes in form and detail may be made without departing from the spirit and scope of the invention as defined in the appended claims.

Claims

1. A lathe fixture for preventing chatter, comprising a mounting sleeve, a support assembly, and a fixing assembly, characterized in that: The mounting sleeve is a hollow structure with an opening on one side, and its bottom is fixed to the lathe spindle and coaxially arranged with it; The support assembly includes a first support member and a second support member. The first support member is located at the center of the bottom inner side of the mounting sleeve, and the second support member is radially movably mounted on the mounting sleeve. The fixing component includes several pressure blocks, which are installed on the mounting sleeve in a direction perpendicular to the main shaft axis and are arranged at intervals around the axis. The blocks located on the axis side form a workpiece clamping end. The workpiece is installed in the inner cavity of the mounting sleeve, with one end connected to the first support member. The workpiece body located inside the sleeve is supported by the second support member. The end of the workpiece to be processed extends out of the sleeve and is stopped by the center of the lathe tailstock. After the workpiece is clamped by the clamping end, it is set coaxially with the lathe spindle under the action of the first and second support members.

2. The anti-chatter lathe fixture according to claim 1, characterized in that, The mounting sleeve includes a disc base and a sleeve. The sleeve is a hollow structure with openings on both sides. The disc base is fixedly installed on one side opening of the sleeve. Several mounting holes are provided on the base. The disc base is detachably installed on the lathe spindle through the mounting holes and connecting bolts. The disc base, the sleeve and the lathe spindle are arranged coaxially.

3. The anti-chatter lathe fixture according to claim 2, characterized in that, The first support member is a small-end tip. The disc base has a connecting hole at the center of the base body. The small-end tip has a threaded hole facing the body of the disc base. The small-end tip is fixedly installed at the center of the disc base by passing through the connecting hole and being threadedly connected to the threaded hole by a connecting bolt. The small-end tip abuts against the tip hole of the workpiece.

4. The anti-chatter lathe fixture according to claim 2, characterized in that, The number of pressure blocks is two. The two pressure blocks are installed in an up-to-up manner at the opening on the side of the sleeve away from the disc base. The opening side of the sleeve has bolt holes at the position of the pressure blocks. The pressure blocks have mounting holes at the corresponding positions of the bolt holes. The pressure blocks are fixedly installed on the sleeve by bolts passing through the mounting holes and being threadedly connected to the bolt holes. The block body on the axial side of the pressure block has a V-shaped clamping groove. The clamping grooves of the two pressure blocks form a workpiece clamping end with a shape that matches the shape of the workpiece.

5. A chatter-resistant lathe fixture according to claim 4, characterized in that, The mounting hole of the pressure block is a strip-shaped adjustment hole. The pressure block is installed at different positions in the mounting hole by bolts to adjust the clamping shaft diameter of the workpiece clamping end.

6. The anti-chatter lathe fixture according to claim 3, characterized in that, The sleeve is made of lightweight POM; the pressure block is made of steel; the disc base and the small end tip are made of 40Cr alloy steel, which has been tempered and stress-relieved annealed.

7. The anti-chatter lathe fixture according to claim 1, characterized in that, The second support member is a support screw, which is threaded to the wall of the mounting sleeve. There are several support screws, and the inner end of the support screw abuts against the outer circumferential surface of the workpiece.

8. The anti-chatter lathe fixture according to claim 7, characterized in that, The aforementioned supporting screws are arranged in pairs, with each pair symmetrically distributed along the outer circumference of the workpiece.

9. A lathe fixture for preventing chattering according to claim 1, characterized in that, The side wall of the mounting sleeve is provided with an opening that penetrates the side wall. This opening also serves as a clamping observation port and a discharge port for cutting fluid and chips.

10. A lathe fixture for preventing chattering according to claim 1, characterized in that, The lathe tailstock is a servo tailstock with a floating center at its front end; the servo tailstock has a constant clamping force, and the floating center can float elastically along the axial direction to compensate for the thermal expansion of the workpiece.