Shield expansion system and control method thereof

Abstract

The application discloses a shield expansion system and a control method thereof. Through proportional flow control and flow meter flow feedback, a differential fitting approximation strategy is adopted to solve the automatic control and control precision of the over-excavation cutter, and automatic continuous curve expansion and reliable expansion accuracy of the expansion system are realized. The application has the advantages of high control precision and convenient installation and use.

Description

Technical Field

[0001] This invention relates to the field of tunnel boring machine (TBM) construction technology, specifically to a TBM widening and excavation system and its control method. Background Technology

[0002] With economic development and the continuous acceleration of urbanization, tunnels for urban subways, water conservancy and hydropower, highways, railways and other projects all need to be excavated using shield tunneling. Shield tunneling machines have gradually become popular due to their many advantages such as safety and speed. Therefore, shield tunneling machines are widely used in tunnel construction for tunnel excavation.

[0003] In tunnel boring machine (TBM) construction, over-excavation is an essential process that plays a crucial role in enabling the TBM to navigate curves and turns. When the TBM reaches a tunnel bend, over-excavation cutters extend radially from both ends of the cutterhead to widen the tunnel diameter, preventing the shield from jamming during the turn and ensuring safe maneuvering. Furthermore, when the TBM deviates from the design axis during excavation, over-excavation cutters are used to correct the deviation in the tunneling path.

[0004] Currently, over-digging cutters are mainly operated manually and can only perform simple expansion digging, unable to achieve automatic continuous curve expansion digging. This results in low expansion digging accuracy and a limited range of expansion digging methods.

[0005] The information disclosed in this background section is intended only to enhance the understanding of the background technology of this disclosure and should not be construed as an admission or in any way implying that the information constitutes prior art known to those skilled in the art. Summary of the Invention

[0006] In view of at least one of the above technical problems, this disclosure provides a shield tunneling excavation system and its control method. By using proportional flow control and flow meter flow feedback, and employing a differential fitting approximation strategy, the problem of automatic control and control accuracy of the over-digging cutter is solved, thereby realizing automatic continuous curve excavation and reliable excavation accuracy of the excavation system.

[0007] According to one aspect of this disclosure, a shield tunneling widening system is provided, including a cutterhead, an oil tank, and a widening cylinder with an over-digging cutter disposed on the cutterhead and whose axis forms a certain angle with the cutterhead surface, a balancing valve correspondingly connected to the chamber of the widening cylinder, a proportional valve correspondingly connected to the oil tank for controlling the flow rate entering the balancing valve, a flow sensor disposed between the balancing valve and the proportional valve, and a controller correspondingly electrically connected to the flow sensor and the proportional valve.

[0008] In some embodiments of this disclosure, a rotary joint is also included to prevent oil circuit entanglement when the cutter head rotates.

[0009] In some embodiments of this disclosure, the oil tank is equipped with a motor and a pump source for supplying oil to the proportional valve.

[0010] In some embodiments of this disclosure, a safety valve is provided between the oil tank and the proportional valve.

[0011] In some embodiments of this disclosure, the controller is a PLC or a microcontroller.

[0012] According to another aspect of this disclosure, a method for controlling tunnel boring machine enlargement is provided, comprising the following steps:

[0013] (1) Divide the excavation trajectory into N equal parts, and calculate the change in distance ΔL of the excavation cutter extending or retracting within each part of the trajectory using the controller;

[0014] (2) Calculate the extension or retraction time ΔT of the digging cutter within each trajectory range based on the cutterhead rotation speed r;

[0015] (3) Calculate the required flow rate ΔQ of the rodless or rod-type chamber of the reaming cylinder corresponding to the change in the extension or retraction distance ΔL of the reaming cutter;

[0016] (4) Obtain the correlation coefficient K between current and flow rate from the proportional valve product manual, and calculate the proportional valve control current I corresponding to the required flow rate ΔQ of the rodless or rod chamber of the excavation cylinder;

[0017] (5) Select the differential measurement period Δt within the range of each excavation trajectory, and the corresponding time error P;

[0018] (6) Determine the corresponding control current I applied to the proportional valve through the controller within a single measurement cycle Δt for n=1, 2... n =nI, and the flow rate monitored by the flow sensor within the time interval Δt is Δq. n At that time, the corresponding extension time t of the excavation cylinder n =(ΔQ-Δq (n-1) ) / Δq n The relationship between ΔT+P-(n-1)Δt and t continues until t n <ΔT+P-(n-1)Δt;

[0019] (7) Determine t n Is it less than ΔT-P-nΔt? If it is, adjust I. n+m =I n -0.5 m I (m=1, 2...) up to t n+m >ΔT-P-(n+m)Δt, determine t n+m If it is less than ΔT+P-(n+m)Δt, then adjust I. n+m+a =I n+m +0.5 aI (a=1, 2...) up to t n+m+a <ΔT+P-(n+m+a)Δt, and so on, until ΔT-P-bΔt<t b <ΔT+P-bΔt, complete the control of the extension or retraction of the excavation cutter within the corresponding excavation trajectory range.

[0020] In some embodiments of this disclosure, in step (3), the required flow rate ΔQ of the rodless chamber of the reaming cylinder is the product of the circular cross-sectional area of ​​the rodless chamber and the change in the extension distance ΔL of the corresponding reaming cutter; the required flow rate ΔQ of the rod chamber of the reaming cylinder is the product of the difference between the circular cross-sectional area of ​​the rod chamber and the circular cross-sectional area of ​​the rod body and the change in the retraction distance ΔL of the corresponding reaming cutter.

[0021] One or more technical solutions provided in the embodiments of this application have at least one of the following technical effects or advantages:

[0022] 1. By adopting a hydraulically driven over-digging cutter head and controller, the driving and control problems of the over-digging cutter head are effectively solved, thereby realizing the automatic control of the over-digging cutter head by the controller.

[0023] 2. By using a flow sensor for feedback, the technical problem of low flow control accuracy of proportional valves is effectively solved, thereby achieving precise control.

[0024] 3. By adopting a control strategy of piecewise adjustment and approximation using infinitesimal elements, the technical problem of control accuracy was effectively solved, ensuring the control accuracy of continuous trajectory excavation. Attached Figure Description

[0025] Figure 1 This is a diagram showing the arrangement of over-digging cutters in the cutterhead in one embodiment of this application.

[0026] Figure 2 This is a schematic diagram showing the location of the over-digging cutter in one embodiment of this application.

[0027] Figure 3 This is a schematic diagram of the shield tunneling system in one embodiment of this application.

[0028] In the above figures, 1 is the cutterhead, 2 is the excavation cylinder, 3 is the motor, 4 is the pump source, 5 is the proportional valve, 6 is the flow sensor, 7 is the balance valve, 8 is the rotary joint, 9 is the safety valve, and 10 is the oil tank. Detailed Implementation

[0029] In the description of this application, it should be understood that the terms "upper," "lower," "front," "rear," "left," "right," "top," "bottom," "inner," "outer," "vertical," "horizontal," "clockwise," and "counterclockwise," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are used only for the convenience of describing this application and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on this application. Furthermore, the terms "connection" and "linkage" in this application, unless otherwise specified, include both direct and indirect connections (linkages).

[0030] The programs involved or relied upon in the following embodiments are all conventional or simple programs in this technical field. Those skilled in the art can make conventional selections or adaptive adjustments according to specific application scenarios.

[0031] Unless otherwise specified, the components, structures, mechanisms, or sensors involved in the following embodiments are all commercially available products.

[0032] To better understand the technical solution of this application, the above technical solution will be described in detail below with reference to the accompanying drawings and specific embodiments.

[0033] This example discloses a shield tunneling excavation system, including a cutterhead, an excavation cylinder, a rotary joint, a flow sensor, a proportional valve, a safety valve, a pump source, a motor, and an oil tank.

[0034] See Figure 1 The head of the excavation cylinder is equipped with an excavation cutter for tunnel widening. When the tunnel boring machine is tunneling normally, the excavation cylinder retracts, which in turn drives the excavation cutter to retract and not exceed the end face of the cutterhead, without affecting the normal tunnel diameter excavation. When the tunnel boring machine needs to widen the tunnel, the excavation cylinder extends its internal rod under the drive of hydraulic oil, which in turn drives the excavation cutter to extend beyond the end face of the cutterhead. The excavation cutter protruding from the end face of the cutterhead rotates with the cutterhead to achieve widening.

[0035] In this implementation, see Figure 2 The reaming cylinder is fixed on the cutterhead. Due to geological constraints, the central axis of the reaming cylinder forms a certain spatial angle with the cutterhead surface to facilitate reaming. In some other embodiments, the central axis of the reaming cylinder is arranged radially along the cutterhead, i.e., parallel to the cutterhead.

[0036] A balance valve allows the actuator to remain stationary under a certain load and effectively controls its smooth movement under load, preventing uncontrolled movement due to load variations. Furthermore, in the event of a hydraulic line rupture or severe leakage, the balance valve installed on the actuator can prevent uncontrolled movement of the load. Since the reaming cylinder needs to drive the reaming cutter to extend outward for reaming, it experiences stress during extension. This stress is transmitted to the reaming cylinder and compresses the oil in the rodless chamber, causing oil to flow back into the hydraulic tank. This leads to the reaming tank retracting, preventing further reaming. Therefore, a balance valve is installed after the reaming cylinder to ensure that the extended cylinder remains in position and does not retract, thus ensuring contact between the reaming cutter and the rock mass and ensuring smooth reaming. In addition, this balance valve also drives the reaming cylinder to extend and retract smoothly, preventing damage to the cylinder structure from the impact of synchronous extension and retraction.

[0037] A rotary joint is a rotating hydraulic connection device suitable for engineering machinery equipped with a rotary support structure. In the shield tunneling excavation system disclosed herein, the excavation cutterhead and the excavation cylinder need to rotate together with the shield machine cutterhead. However, due to space limitations, the oil tank supplying hydraulic oil to the excavation cylinder cannot be located behind the cutterhead and rotate with it. Therefore, a hose needs to be connected between the excavation cylinder and the oil tank to supply hydraulic fluid. Since the excavation cylinder rotates with the cutterhead, it causes the hose to rotate around its axis. This can lead to the hose becoming entangled on the cutterhead shaft after repeated rotations, potentially causing the excavation system to fail, or even hose rupture and damage to the shield machine cutterhead. Therefore, in this embodiment, see... Figure 3 A rotary joint is installed after the balance valve to solve the problem of the hydraulic hose getting tangled after the excavation cylinder rotates with the cutterhead.

[0038] A proportional valve is a hydraulic control device that can continuously and proportionally control the pressure, flow rate, or direction of hydraulic fluid remotely according to an input electrical signal. In this embodiment, to control the extension and retraction of the excavation cylinder at different times, it is necessary to adjust the amount of hydraulic fluid entering / leaving the excavation cylinder to ensure precise operation. Therefore, a proportional valve is provided to regulate the amount of hydraulic fluid entering / leaving the excavation cylinder. Furthermore, because the proportional valve is designed for control accuracy and economic reasons, it cannot provide highly precise control over the flow rate.

[0039] To address the accuracy issue of proportional valves, this embodiment includes a flow sensor installed between the rotary joint and the proportional valve. This sensor monitors the flow rate through the proportional valve, generates feedback, and adjusts the control signal of the proportional valve to achieve precise control.

[0040] In this embodiment, a PLC is used as the controller, which is electrically connected to the proportional valve and the flow sensor respectively, to control and adjust the flow of the proportional valve and achieve precise control of the flow.

[0041] In addition, the oil tank pumps oil to the excavation cylinder through a motor and pump source. A safety valve is installed between the oil delivery and recovery circuits to prevent safety issues caused by the system exceeding the set working pressure. When the system pressure exceeds the working pressure, the oil pressure between the oil delivery and recovery circuits increases, which in turn exceeds the safety valve's opening pressure. The overpressure is released through the safety valve to ensure the safe operation of the system.

[0042] See Figure 3 When the excavation cylinder needs to extend, the motor drives the pump to rotate. The pressurized oil in the tank then flows through the proportional valve, flow sensor, rotary joint, and balance valve into the rodless chamber of the excavation cylinder, using the hydraulic oil pressure to push the excavation cylinder to extend. Meanwhile, the oil in the rod chamber of the excavation cylinder returns to the tank through the balance valve, rotary joint, and proportional valve.

[0043] When the excavation cylinder needs to retract, the motor drives the pump to rotate, and the pressurized oil in the tank then flows through the proportional valve, the rotary joint, and the balance valve into the rod chamber of the excavation cylinder. The hydraulic pressure then causes the excavation cylinder to retract. Meanwhile, the oil in the rodless chamber of the excavation cylinder returns to the tank through the balance valve, the rotary joint, the flow sensor, and the proportional valve.

[0044] This embodiment also discloses a control method for a tunnel boring machine (TBM) widening and excavation system, including the following steps:

[0045] (1) Divide the excavation trajectory into N equal parts, and calculate the change in distance ΔL of the excavation cutter extending or retracting within each part of the trajectory using PLC.

[0046] To achieve precise control of the excavation trajectory, the trajectory is divided into N equal parts, resulting in arc segments with different radii. Then, using appropriate trigonometric functions, the PLC calculates the excavation trajectory line within each segment, as well as the change in the extension or retraction distance ΔL of the excavation cutter within each segment. The N-parted excavation trajectory can be viewed as a combination of N trajectories, each of which can be approximated as a straight line. A higher degree of division results in a trajectory that closely approximates the design trajectory, leading to a higher degree of fit and excavation accuracy. However, excessively high division increases the computational load on the PLC. Since the excavation trajectory has a certain allowable error, it is only necessary to ensure that the trajectory remains within this allowable range.

[0047] (2) Calculate the time ΔT for the excavation cutter to extend or retract within each trajectory range based on the cutter head rotation speed r.

[0048] Since the excavation trajectory is divided into N equal parts, the angle of each excavation trajectory is α=360° / N. By the cutterhead rotation speed r, the time required for the excavation cylinder to extend or retract for each part of the trajectory can be obtained as ΔT=360° / Nr.

[0049] (3) Calculate the required flow rate ΔQ of the rodless or rod-type chamber of the reaming cylinder corresponding to the change in the extension or retraction distance ΔL of the reaming cutter.

[0050] When the reamer extends, hydraulic oil enters the rodless chamber, generating the thrust for extension. Based on the radius R1 of the circular cross-section of the rodless chamber, the circular cross-sectional area A1 = πR1 of the rodless chamber is calculated. 2 Then, the required hydraulic oil volume for the rodless cavity is calculated using the cylindrical volume formula: ΔQ = ΔL × A1 = ΔLπR1 2 .

[0051] When the reamer retracts, hydraulic oil enters the rod chamber, generating a retraction thrust. Based on the radius R2 of the circular cross-section of the rod chamber, the circular cross-sectional area A2 = πR2 is calculated. 2 Furthermore, the presence of the rod occupies a certain volume space within the cavity, which can be determined using the cylinder volume formula and the rod radius R. 3， Calculate the required hydraulic fluid volume for the rodless chamber: ΔQ = ΔLπ(R²) 2 -R3 2 ).

[0052] (4) Obtain the correlation coefficient K between the current and flow rate of the proportional valve through the product manual corresponding to the proportional valve, and calculate the proportional valve control current I corresponding to the required flow rate ΔQ of the rodless or rod chamber of the excavation cylinder.

[0053] The hydraulic oil volume ΔQ required for the extension or retraction of the excavation cylinder has been calculated in step (3). In order to know the opening degree of the proportional valve required to deliver the hydraulic oil of volume ΔQ, i.e. the control current of the proportional valve, the correlation coefficient K between the flow rate through the proportional valve and its control current can be obtained by consulting the product technical manual of the proportional valve. Based on the relationship between the two, ΔQ=KI, the proportional valve control current I corresponding to the flow rate ΔQ required for the rodless or rod chamber of the excavation cylinder can be calculated.

[0054] (5) Select the differential measurement period Δt within the range of each excavation trajectory, and the corresponding time error P.

[0055] Given the extension or retraction distance ΔL and extension time ΔT for each excavation trajectory, to determine whether the excavation cylinder can reach the predetermined position at the predetermined time, a time error P is selected, and the infinitesimal method is used to differentiate several measurement periods Δt within the range of each excavation trajectory. The smaller the time error P, the more accurately and quickly the excavation cylinder can extend or retract to the predetermined position, resulting in higher excavation accuracy. Furthermore, for the measurement period Δt, the smaller the value, the higher the feedback frequency of the flow sensor, leading to more accurate fitting of the actual trajectory.

[0056] (6) Determine the corresponding control current I applied to the proportional valve through the controller within a single measurement cycle Δt for n=1, 2... n =nI, and the flow rate monitored by the flow sensor within the time interval Δt is Δq. n At that time, the corresponding extension time t of the excavation cylinder n =(ΔQ-Δq (n-1) ) / Δq n The relationship between ΔT+P-(n-1)Δt and t continues until t n <ΔT+P-(n-1)Δt.

[0057] Based on the proportional valve control current I corresponding to the required flow rate ΔQ of the rodless or rod chamber of the excavation cylinder calculated in step (4), firstly, apply current I1=I to the proportional valve through the PLC, and obtain the flow rate Δq1 monitored by the flow sensor within the first measurement cycle Δt. Calculate the total time t1=ΔQ / Δq required for the excavation cylinder to extend or retract under this trajectory by using the opening of the proportional valve under the control current I1. 1， The process involves determining whether the estimated extension or retraction time of the reaming cylinder is less than ΔT+P. This involves comparing the estimated extension or retraction time of the reaming cylinder with the required extension or retraction time. Specifically, it determines whether the control current I1 can drive the reaming cylinder to the required position via hydraulic oil within the operating time of the reaming cutter on its trajectory. If the estimated time is greater than the required time, it indicates that the control current I1 is too small, which in turn indicates that the flow rate through the proportional valve is too small, and the response speed does not meet the expected requirements. Therefore, the control current I is increased, setting I2=2I. Then, it is determined whether the proportional valve opening within the second measurement cycle Δt satisfies the requirement that the extension or retraction time t2=(ΔQ-Δq1) / Δq2 is less than ΔT+P-Δt. Here, ΔQ-Δq1 is used to eliminate the influence of the first measurement cycle with a known action speed on subsequent accuracy. This process is then repeated for each remaining measurement cycle until the estimated time t is reached. n =(ΔQ-Δq (n-1) ) / Δq n If the current is less than ΔT+P-(n-1)Δt, it means that the control current is large enough to make the excavation cylinder reach a certain response speed.

[0058] (7) Determine t n Is it less than ΔT-P-nΔt? If it is, adjust I. n+m =I n -0.5 m I (m=1, 2...) up to t n+m >ΔT-P-(n+m)Δt, determine t n+m Is it less than ΔT+P-(n+m)Δt? Otherwise, adjust I. n+m+a =I n+m +0.5 a I (a=1, 2...) up to t n+m+a Less than ΔT+P-(n+m+a)Δt, and so on, until ΔT-P-bΔt<t. b <ΔT+P-bΔt, complete the control of the extension or retraction of the excavation cutter within the corresponding excavation trajectory range.

[0059] The control current I has been set in step (6). n The control current I was adjusted to a sufficiently large value to achieve a fast enough response speed. However, an excessively fast response speed could cause impact damage to the reaming cylinder and other components. Therefore, further adjustment of the control current I is necessary. n This makes the movement of the hydraulic cylinder for widening the dredging process smoother. First, the response speed within the nth measurement cycle Δt is used to determine whether the estimated time is too fast, i.e., to determine if t is too fast. n Is it less than ΔT-P-nΔt? If it is, it indicates that the response speed is too fast, and the control current I in the (n+1)th measurement cycle needs to be adjusted. n+1 =I n -0.5I, by reducing the control current to a certain extent, determine the estimated time t of the reaming cylinder under the proportional valve flow rate driven by this control current. n+1 Can the excavation cylinder respond and reach the predetermined position within the shortest possible action time of ΔT-P-(n+1)Δt? If so, i.e., less than ΔT-P-(n+1)Δt, it indicates that the excavation cylinder is moving too fast and the proportional valve control current is too large. In the next measurement cycle, continue to reduce the control current, letting I... n+2 =I n -0.5 2 I. Determine the response of the excavation cylinder under the control current, and so on, until the control current I is adjusted. n+m =I n -0.5 m I (m=1, 2, ...) makes the estimated response time t n+m The reaming cylinder exhibits a smooth response, exceeding the shortest action time ΔT-P-(n+m)Δt. Furthermore, to prevent over-adjustment and subsequent sluggish response, the control current I is... n+m If the estimated response time exceeds the maximum required response time, the response in the (n+m)th measurement period needs to be reassessed, i.e., the response time t needs to be determined.n+m Whether the response is less than ΔT+P-(n+m)Δt indicates whether the response is fast enough. If it is greater, it indicates a slow response, and the control current needs to be increased slightly in the (n+m+1)th measurement cycle, so that I... n+m+1 =I n+m +0.5I, to determine if the response speed is fast enough, i.e., to determine t. n+m+1 If the current is less than ΔT + P - (n + m + 1)Δt, otherwise continue to increase the control current, let I n+m+a =I n+m +0.5 a I, and so on, until ΔT-P-bΔt<t b <ΔT+P-bΔt, ensuring the response speed is within a certain required range, thus achieving smooth and rapid control of the excavation cylinder.

[0060] Although some preferred embodiments of the invention have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments as well as all changes and modifications falling within the scope of the invention.

[0061] Obviously, those skilled in the art can make various modifications and variations to this invention without departing from its spirit and scope. Therefore, if these modifications and variations fall within the scope of the claims of this application and their equivalents, this invention also intends to include these modifications and variations.

Claims

1. A method of shield overbreak control, characterized by, The application discloses a shield expansion system, which comprises a cutter head, an oil tank, an expansion cylinder provided with an expansion cutter and arranged on the cutter head and having a central axis at an angle with a cutter head surface, a balance valve corresponding to a cavity of the expansion cylinder, a proportional valve connected with the oil tank and used for controlling flow into the balance valve, a flow sensor arranged between the balance valve and the proportional valve, a rotary joint arranged behind the balance valve and used for preventing oil line winding when the cutter head rotates, and a controller electrically connected with the flow sensor and the proportional valve. The shield expansion control method comprises the following steps: (1) dividing an expansion track of an expansion site into N parts, and calculating a distance change amount ΔL of the expansion cutter in each track part by a controller; (2) calculating an expansion or retraction time ΔT of the expansion cutter in each track part according to a cutter head rotating speed r; (3) calculating a required flow ΔQ of a rodless cavity or a rod cavity of the expansion cylinder corresponding to the expansion or retraction distance change amount ΔL of the expansion cutter; (4) obtaining a correlation coefficient K between a proportional valve current and flow through a product manual or a parameter manual corresponding to the proportional valve, and calculating a proportional valve control current I corresponding to the required flow ΔQ of the rodless cavity or the rod cavity of the expansion cylinder; (5) selecting a differential measurement period Δt in each expansion track part and a corresponding time error P. (6) judge n = 1, 2… Single differential measurement period Δt, through the controller to the proportional valve corresponding to the control current is I n =nI, and Δt time within the flow sensor monitoring flow is Δq n When, compare the corresponding extension or retraction time of the cutting and filling cylinder t n =(ΔQ-Δq (n-1) ) / Δq n And ΔT+P-(n-1)Δt size relationship, until t n <ΔT+P-(n-1)Δt; (7) judge t n whether less than ΔT-P-nΔt, if less than adjust I n+m =I n -0.5 m I (m=1, 2,...) until t n+m > ΔT-P-(n+m)Δt, judge t n+m whether less than ΔT+P-(n+m)Δt, otherwise adjust I n+m+a =I n+m +0.5 a I (a=1, 2,...) until t n+m+a < ΔT+P-(n+m+a)Δt, and so on, until ΔT-P-bΔt < t b < ΔT+P-bΔt, complete the corresponding share of the range of expansion of the expansion of the knife to extend or retract control.

2. The shield expansion control method according to claim 1, characterized by, The oil tank is provided with a motor and a pump source for supplying oil to the proportional valve.

3. The method of claim 1, wherein, A safety valve is arranged between the oil tank and the proportional valve.

4. The method of claim 1, wherein, The controller is a PLC or a single-chip microcomputer.

5. The method of claim 1, wherein, In the step (3), the required flow ΔQ of the rodless cavity of the expansion cylinder is a product of a circular cross-sectional area of the rodless cavity and the expansion or retraction distance change amount ΔL of the expansion cutter; and the required flow ΔQ of the rod cavity of the expansion cylinder is a product of a difference between a circular cross-sectional area of the rod cavity and a circular cross-sectional area of a rod body and the expansion or retraction distance change amount ΔL of the expansion cutter.

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