A theoretical model of spiral self-feeding of pipe

Abstract

This invention discloses a theoretical model for spiral self-feeding straightening of pipes, including, under the assumption that the rotation axis of the pipe coincides with the bending plane and axial roll of the pipe is ignored, establishing the total kinetic energy of the pipe based on the thin-wall assumption. T The theoretical formula is derived; neglecting the plastic dissipation of the pipe, a simplified potential energy model for the pipe is established based on the unified curvature theorem of reciprocating bending and the bidirectional effect. V The theoretical formula is used to model the motion trajectory of the pipe based on the Lagrange equation, thereby obtaining a set of motion trajectory equations. These equations include the circumferential bending angle of the pipe, the helical azimuth angle of the pipe, and the axial self-feed displacement. s The coupling relationship between them; where the Lagrange equation is, Q q For the generalized forces acting on the pipe, L For Lagrange units, q For a generalized coordinate system, , for q The first derivative.

Description

Technical Field

[0001] This invention relates to the field of pipe straightening, and in particular to a theoretical model for spiral self-feeding straightening of pipes. Background Technology

[0002] Pipes, as key structural components, are widely used in transportation, construction, shipping, mining, and scientific research. With the rapid development of the pipe extrusion industry, the market demands increasingly stringent dimensional accuracy and surface quality for pipes. Straightening is a core process for improving the straightness, roundness, and overall forming quality of pipes; the performance of its processes and equipment directly determines the final product's pass rate and performance.

[0003] Pipe straightening processes typically employ upper and lower rollers without threaded grooves, achieving bending and straightening solely through radial compression of the pipe. This lack of axial propulsion during the straightening process prevents the application of axial force. Consequently, continuous pipe feeding necessitates an independent propulsion mechanism. This not only increases equipment complexity and cost but also interferes with or even hinders the pipe's rotational movement during straightening, affecting the uniformity and stability of the straightening force. Ultimately, this can lead to decreased straightening accuracy, surface scratches, or reduced efficiency.

[0004] In related technologies, although a spiral self-feeding correction scheme has been proposed, there is still no theoretical model that can quantify the coupling mechanism of spiral self-feeding motion, which makes it difficult to achieve coordinated optimization of accuracy, efficiency and stability in process development and roll design. Summary of the Invention

[0005] One objective of this invention is to provide a theoretical model for the spiral self-feeding straightening of pipes, which can quantify the coupling mechanism of the spiral self-feeding motion and provide a theoretical basis for the spiral self-feeding straightening process and roller design of pipes.

[0006] To achieve the above objectives, the technical solution adopted by this invention is as follows: a theoretical model for spiral self-feeding straightening of pipes, comprising: assuming that the rotation axis of the pipe coincides with the bending plane, and neglecting the axial roll of the pipe, establishing the total kinetic energy of the pipe based on the thin-wall assumption. T The theoretical formula is derived; neglecting the plastic dissipation of the pipe, a simplified potential energy model for the pipe is established based on the unified curvature theorem of reciprocating bending and the bidirectional effect. V The theoretical formula; based on the Lagrange equation, the motion trajectory of the pipe is modeled to obtain a set of motion trajectory equations, which includes the circumferential bending angle of the pipe. Spiral azimuth angle of the pipe Axial self-feed displacement s The coupling relationship between them; where the Lagrange equation is: , Qq For the generalized forces acting on the pipe, L For Lagrange units, , q For a generalized coordinate system, , for q The first derivative.

[0007] As a preferred option, the total kinetic energy T Including translational kinetic energy and rotational kinetic energy Among them, neglecting the radial variation of the pipe, based on the thin-wall assumption, the translational kinetic energy is... Assuming the pipe's axis of rotation coincides with the bending plane and axial roll of the pipe is neglected, based on the thin-wall assumption, the rotational kinetic energy... ; to transfer the translational kinetic energy The theoretical formula and the rotational kinetic energy Substituting the theoretical formula into the total kinetic energy T The theoretical formula is obtained. ,in, I 1 represents the moment of inertia of the pipe section. , r The outer radius of the pipe. t For the wall thickness of the pipe, m For the quality of the pipes, R The radius of the roller-shaped groove, for s The first derivative, for The first derivative, for The first derivative.

[0008] As a preferred option, the potential energy of the pipe is considered when the plastic dissipation of the pipe is neglected. ,in, u The strain energy density of the pipe material. , E The Young's modulus of the roller. k For the local curvature of the pipe, y The distance between the neutral axes of the pipes. A Let be the cross-sectional area of ​​the pipe; based on the unified curvature theorem for reciprocating bending and the bidirectional effect, the local curvature is... k Coupled axial curvature and helical curvature ,in, , The local curvature is obtained according to the vector superposition theorem. ;Will Substitute the strain energy density of the pipe uBy treating the pipe as a uniform beam segment, a simplified potential energy of the pipe is obtained. ;in, , K Let be the radius of curvature of the roller-shaped groove.

[0009] As a preferred option, the Lagrange equation is... Components in coordinate system , for Q q exist Components in coordinate system ,in, For pipe materials in Centrifugal force in curvilinear motion under coordinates The position vector of the pipe right The partial derivative, In curvilinear motion, the centrifugal force ,in, m For the quality of the pipes, K The radius of curvature of the roller-shaped groove is... for s The first derivative, Let be the normal unit vector in the spiral coordinate system. , α The helix angle of the roller-shaped groove; Substitute the centrifugal force ,get .

[0010] As a preferred option, the Lagrange equation is... Components in coordinate system , for Q q exist Components in coordinate system ,in, For pipe materials in Centrifugal force in helical motion under coordinate system The position vector of the pipe right The partial derivative, In spiral motion, the centrifugal force ,in, m For the quality of the pipes, K The radius of curvature of the roller-shaped groove is... R The radius of the roller-shaped groove, for The first derivative, A radial unit vector in a spiral coordinate system. ,α The helix angle of the roller-shaped groove; Substitute the centrifugal force ,get .

[0011] As a preferred option, the Lagrange equation is... s Components in coordinate system , for Q q exist s Components in coordinate system ,in, For pipe materials in s The helical thrust experienced in the coordinate system. The position vector of the pipe right s The partial derivative, The spiral thrust ,in, m For the quality of the pipes, R The radius of the roller-shaped groove, for The first derivative, for The first derivative, As a tangential unit vector, in a spiral coordinate system, ;Will Substitute the centrifugal force ,get .

[0012] As a preferred embodiment, the equations of the motion trajectory are as follows:

[0013] ;

[0014] ;

[0015] ;

[0016] in, m For the quality of the pipes, R The radius of the roller-shaped groove, α The helix angle of the roller-shaped groove, E The Young's modulus of the roller. K The radius of curvature of the roller-shaped groove; I 1 represents the moment of inertia of the pipe section. , r The outer radius of the pipe. t The wall thickness of the pipe; , y The distance between the neutral axes of the pipes. A This represents the cross-sectional area of ​​the pipe. for s The first derivative, for s The second derivative, for The first derivative, for The second derivative, for The first derivative, for The second derivative of .

[0017] As a preferred option, when introducing Poisson's ratio Under the given conditions, based on the Mises criterion and Hooke's law, a theoretical formula for the initiation condition of plastic deformation of pipe is established. The theoretical formula for the initiation condition of plastic deformation is Poisson's ratio. The radius of curvature of the roller-shaped groove K The yield strength of the pipe material The relationship is obtained by solving the theoretical formula for the initiation condition of plastic deformation to obtain the critical curvature of the roll groove. The expression.

[0018] As a preferred option, assuming the pipe is in a state of pure bending, the principal stresses of the pipe are obtained based on Hooke's Law. ;in, E The Young's modulus of the roller. For the elastic strain of the pipe, , K The radius of curvature of the roller-shaped groove is... R The radius of the roller-shaped groove; when introducing Poisson's ratio Under these conditions, the deviation stress components of the pipe are obtained. Based on the equivalent stress formula and yield function of the Mises criterion, the yield strength of the pipe is obtained. ,in This represents the equivalent stress of the pipe.

[0019] As a preferred option, according to Solving for the radius of curvature K The threshold value is used to obtain the critical curvature of the roller-shaped groove. .

[0020] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0021] The trajectory of the pipe is modeled based on the Lagrange equation to obtain a set of trajectory equations, which includes the circumferential bending angle of the pipe. Spiral azimuth angle of the pipe Axial self-feed displacements The coupling relationship between them can be used to quantify the coupling mechanism of the spiral self-feeding motion, providing a theoretical basis for the spiral self-feeding straightening process and roller design of pipes. Attached Figure Description

[0022] Figure 1 This is a schematic diagram of the self-feeding straightening of pipes between spiral roller groups in related technologies.

[0023] Figure 2 This is a schematic diagram of the parameters of a roller according to some embodiments of this application.

[0024] Figure 3 These are the solution trajectories of coupled ordinary differential equations according to some embodiments of this application.

[0025] In the diagram: 1. Upper convex roller; 2. Lower concave roller. Detailed Implementation

[0026] The present invention will now be further described in conjunction with specific embodiments. It should be noted that, without conflict, the various embodiments or technical features described below can be arbitrarily combined to form new embodiments.

[0027] The terms “comprising” and “having”, and any variations thereof, in the specification and claims of this application are intended to cover non-exclusive inclusion, for example, a process, method, system, product, or device that includes a series of steps or units is not necessarily limited to those steps or units that are explicitly listed, but may include other steps or units that are not explicitly listed or that are inherent to such process, method, product, or device.

[0028] In related technologies, a scheme for spiral self-feeding straightening of pipes has been proposed. For example... Figure 1As shown, the spiral self-feeding straightening device consists of several rollers, specifically, one spiral upper convex roller 1 and two spiral lower concave rollers 2. The upper convex roller 1 has a convex structure with a central outer diameter larger than its two end outer diameters; the lower concave rollers have a concave structure with a central outer diameter smaller than their two end outer diameters. The convex and concave curvatures of the upper convex roller 1 and the lower concave roller 2 are matched. Furthermore, the upper convex roller 1 and the lower concave roller 2 rotate in opposite directions. When the pipe passes between the spiral rollers, the circumferential and axial sections of the pipe undergo multiple reciprocating bends under the action of the rollers, i.e., bidirectional reciprocating bends, thereby achieving elastic-plastic deformation and straightening. Simultaneously, the surfaces of the upper convex roller 1 and the lower concave roller 2 are provided with spiral grooves, i.e., roller-shaped grooves, which convert the rotational motion of the rollers into a spiral thrust that drives the pipe to continuously advance along its own axial direction, enabling the pipe to achieve self-feeding during the straightening process. However, no theoretical model has yet been proposed in the relevant technologies to quantify the "bidirectional reciprocating bending-self-feeding" coupling mechanism of the spiral self-feeding motion, making it difficult to achieve coordinated optimization of accuracy, efficiency and stability in process development and roll design.

[0029] Based on the above, in order to quantify the coupling mechanism of the helical self-feeding motion and provide a theoretical basis for the helical self-feeding straightening process and roller design of pipes, this application proposes a theoretical model for the helical self-feeding straightening of pipes, such as... Figure 2 As shown, this includes: assuming the pipe's rotation axis coincides with the bending plane and neglecting the pipe's axial roll, establishing the total kinetic energy of the pipe based on the thin-wall assumption. T The theoretical formula is derived; neglecting the plastic dissipation of the pipe, a simplified potential energy model for the pipe is established based on the unified curvature theorem of reciprocating bending and the bidirectional effect. V The theoretical formula; based on the Lagrange equation, the motion trajectory of the pipe is modeled to obtain a set of motion trajectory equations, which includes the circumferential bending angle of the pipe. Spiral azimuth angle of the pipe Axial self-feed displacement s The coupling relationship between them can be used to quantify the coupling mechanism of the helical self-feeding motion, providing a theoretical basis for the helical self-feeding straightening process and roller design of pipes. Specifically, the Lagrange equation is: ,in, Q q For the generalized forces acting on the pipe, L For Lagrange units, , q For a generalized coordinate system, , for q The first derivative.

[0030] In some embodiments, total kinetic energy T Including translational kinetic energy and rotational kinetic energy Among them, translational kinetic energy This represents the kinetic energy of the tube's core moving linearly along the helical path. Specifically, a cylindrical coordinate system is established along the roller axis. ,in, R Where is the radius of the roller-shaped groove, in other words, R The distance from the center trajectory line of the roller groove to the axis of the roller shaft, such as... Figure 2 As shown. Ignoring radial variations in the pipe, and based on the thin-wall assumption, the position vector of the pipe's center of gravity can be obtained. After simplification, we can obtain .

[0031] Furthermore, in the helical coordinate system, the velocity vector of the tube's material center... Among them, velocity vector axial component , for s The first derivative, i.e., the pipe material in s The axial self-feed velocity driven by the helical thrust in the coordinate system. Velocity vector. Circumferential component , It is the pipe material The tangential velocity formed by circumferential bending in the coordinate system, where, for The first derivative, i.e., the bending angular velocity. Velocity vector. helical component ,in, for The first derivative, i.e., the derivative of the pipe material in The azimuth angular velocity formed by the spiral motion in coordinate system ,like Figure 3 As shown, α The helix angle of the roller profile groove; the lead of the roller is... P That is, the distance from point A to point B when the roller-shaped groove rotates around once is P .

[0032] Furthermore, according to We can obtain the square of the velocity of the tube's center moving linearly along the spiral path in the spiral coordinate system. Furthermore, the translational kinetic energy of the pipe can be obtained according to the kinetic energy formula. The theoretical formula is: ,in, m For the quality of the pipes.

[0033] rotational kinetic energy Let ω be the kinetic energy of the pipe rotating about its local axis as a rigid beam segment. Specifically, assuming the pipe's axis of rotation coincides with the bending plane and neglecting the pipe's axial roll, let ω be the angular velocity. According to Euler's kinematic equations, Euler's angular velocity can be obtained. ,in, and This is the unit vector in the corresponding direction.

[0034] It should be understandable that, based on the assumption of a symmetrical thin-walled cylinder, the inertial tensor can be... I Simplified to the diagonal form, i.e., the moment of inertia of the section. And in the length direction Based on the above analysis, the moment of inertia of the pipe section can be obtained. , ;in, , r The outer radius of the pipe. t This refers to the wall thickness of the pipe. Further, [the following will be determined]... and I Substituting into the rotational kinetic energy formula, we can obtain the rotational kinetic energy of the pipe. The theoretical formula is: .

[0035] In summary, the total kinetic energy can be obtained. .

[0036] In some embodiments, the potential energy of the pipe is determined by neglecting plastic dissipation of the pipe. This is mainly due to the elastic bending deformation of the pipe. Specifically, the strain of the pipe... ,in, y The distance is the neutral axis distance. k This refers to the local curvature of the pipe. The stress in the pipe. ,in, E Let be the Young's modulus of each roller in the spiral roller assembly. and Substituting into the strain energy density formula, the strain energy density of the pipe can be obtained. ,in, , A This represents the cross-sectional area of ​​the pipe.

[0037] Furthermore, the pipe extends along its length l Continuous deformation, therefore the potential energy of the pipe Among them, local curvature k It is about x The function, for small deformation cases, k The second derivative is approximately: .

[0038] Furthermore, in helical self-feeding motion, the local curvaturek Coupled axial curvature and helical curvature Based on the unified curvature theorem for reciprocating bending and the two-way effect, , , K The radius of curvature of the roller-shaped groove, in other words, such as Figure 2 As shown, the cross-section of the roller-shaped groove is approximately a concave arc-shaped surface. K Let be the radius of curvature of the arc-shaped surface; therefore, according to the vector superposition theorem, the local curvature can be obtained. It should be understandable that if the pipe is considered as a uniform beam segment, then a uniform beam segment per unit length satisfies... .Will Substitute the strain energy density of the pipe u In other words, will and Substitution The potential energy of the simplified pipe can be obtained. .

[0039] In summary, Lagrange daily quantity .

[0040] In some embodiments, the Lagrange equation is Components in coordinate system , for Q q exist Components in coordinate system. Specifically, due to the Lagrange... L In The term exists only in total kinetic energy T Therefore, Furthermore, ,in, for The second derivative. Due to the Lagrange... L In The term exists only in potential energy V Therefore, .Will and Substitution , can be obtained .

[0041] Furthermore, according to the generalized force formula, we can obtain... ,in, For the core of the pipe material Centrifugal force in curvilinear motion under coordinates The position vector of the pipe right The partial derivative. Specifically, in curvilinear motion, the acceleration of the tube's center of gravity includes tangential acceleration and normal acceleration, where the normal acceleration... , , According to Newton's second law, the centrifugal force of the tube's center of gravity during curvilinear motion can be obtained. ,in, It is the normal unit vector.

[0042] Furthermore, as mentioned earlier, the position vector of the tube material's center of gravity... Position vector right partial derivatives It should be understandable that the unsimplified position vector is used. right Taking partial derivatives can improve subsequent... The accuracy of the calculation. and Substitution , can be obtained .

[0043] It should be understandable that in a helical coordinate system, due to the existence of a helical angle... α Normal unit vector Approximately After further simplification, we can obtain .Will Substitute centrifugal force In other words, will Substitution , can be obtained Q q exist Components in coordinate system .

[0044] In conclusion, The coupling equations in coordinate system are: .

[0045] In some embodiments, the Lagrange equation is Components in coordinate system , for Q q exist Components in coordinate system. Specifically, due to the Lagrange... L In The term exists only in total kinetic energy T Therefore, Furthermore, ,in, for The second derivative. Due to the Lagrange... L In The term exists only in potential energy V Therefore, .Will and Substitution , can be obtained .

[0046] Furthermore, according to the generalized force formula, we can obtain... ,in, For the core of the pipe material Centrifugal force in helical motion under coordinate system The position vector of the pipe right The partial derivative. Specifically, due to the bending of the pipe, the core of the pipe material is... A spiral motion occurs in a coordinate system, and the angular velocity of the spiral motion generates centrifugal force. It can be understood that during the pipe straightening process, the pipe is constrained by the radius of curvature of the concave arc surface of the roller groove and the radius of curvature of the curved cylindrical surface of the roller shaft. The straightening effects of these two factors are superimposed. Therefore, it is assumed that the total curvature of the roller groove is... Effective radius of curvature ;because Therefore Furthermore, based on the centrifugal acceleration formula, the centrifugal acceleration of the tube's center of gravity during spiral motion can be obtained. According to Newton's second law, the centrifugal force of the tube's center of gravity during spiral motion can be obtained. ,in, It is a radial unit vector.

[0047] Furthermore, as mentioned earlier, the position vector of the tube material's center of gravity... Position vector right partial derivatives It should be understandable that the unsimplified position vector is used. right Taking partial derivatives can improve subsequent... The accuracy of the calculation. and Substitution , can be obtained Q q exist Components in coordinate system .

[0048] It should be understandable that in a helical coordinate system, due to the existence of a helical angle... α Radial unit vector Approximately After further simplification, we can obtain .Will Substitute centrifugal force In other words, will Substitution , can be obtained .

[0049] In conclusion, The coupling equations in coordinate system are: .

[0050] In some embodiments, the Lagrange equation is s Components in coordinate system , for Q q exist s Components in coordinate system. Specifically, due to the Lagrange... L In The term exists only in total kinetic energy T Therefore, Furthermore, ,in, for s The second derivative. Due to the Lagrange... L The absence of s Therefore, .Will Substitution , can be obtained .

[0051] Furthermore, according to the generalized force formula, we can obtain... ,in, For pipe materials in s The helical thrust experienced in the coordinate system. The position vector of the pipe right s The partial derivative. Specifically, the pipe material in The circumferential bending motion under the coordinate system results in tangential acceleration. Pipes in It undergoes spiral motion in the coordinate system, thereby generating azimuth angular velocity. tangential acceleration and azimuth angular velocity The interaction between the two generates a helical thrust along the circumference of the pipe. In other words, the nonlinear coupling effect of "bending-rotation" generates a helical thrust along the circumference of the pipe. As mentioned earlier, the core of the pipe material is... Tangential velocity formed by circumferential bending in coordinate system azimuth velocity of the tube material's core Therefore, according to the centripetal force formula, the pipe material in... s Helical thrust under coordinates ,in, It is the normal unit vector.

[0052] Furthermore, as mentioned earlier, the position vector of the tube material's center of gravity... Position vector right s partial derivatives It should be understandable that the unsimplified position vector is used. right s Taking partial derivatives can improve subsequent... The accuracy of the calculation. and Substitution , can be obtained .

[0053] It should be understandable that in a helical coordinate system, due to the existence of a helical angle... α Normal unit vector Approximately After further simplification, we can obtain .Will Substitute centrifugal force In other words, will Substitution , can be obtained Q q exist s Components in coordinate system .

[0054] In conclusion, s The coupling equations in coordinate system are: .

[0055] In other words, the equations of the trajectory are:

[0056] ;

[0057] ;

[0058] .

[0059] In some embodiments, when introducing Poisson's ratio Under these conditions, based on the Mises criterion and Hooke's law, a theoretical formula for the initiation condition of plastic deformation in pipes is established. The theoretical formula for the initiation condition of plastic deformation is Poisson's ratio. The radius of curvature of the roller-shaped groove K The yield strength of the pipe material The relationship is established; the theoretical formula for the initiation condition of plastic deformation is solved to obtain the critical curvature of the roll groove. The expression.

[0060] It is understandable that, due to the coupled axial and circumferential stresses on the pipe, equivalent stress measurement is necessary. Determine if the pipe has undergone plastic deformation. According to the Mises criterion, when the equivalent stress... When the material yields to its required strength, the pipe begins to undergo plastic deformation.

[0061] Specifically, based on Hooke's Law, the stress of the pipe can be obtained. ,in, This represents the elastic strain of the pipe. During bending deformation, the elastic strain of the fibers in the pipe cross-section is... Originating from local curvature k Local curvature k The elastic strain of the pipe is linearly distributed along the neutral axis; therefore, the elastic strain of the pipe... .

[0062] It is understandable that bending deformation of the pipe will cause lateral strain. Introducing Poisson's ratio Under the condition of transverse strain Define the deviation stress tensor as ,in, The trace of the stress tensor. Let Kronecker function be the principal stress of the pipe under the assumption of pure bending. radial stress of pipe Axial stress of the pipe Furthermore, by introducing Poisson's ratio Under these conditions, the deviation stress components of the pipe can be obtained. .

[0063] Furthermore, based on the Mises criterion, the equivalent stress formula is: Equivalent Stress ,in, This is an inherent factor for the principal stress deviation in the equivalent stress formula. and Substitution , can be obtained Furthermore, and Substitution , can be obtained .

[0064] Furthermore, the yield function based on the Mises criterion can be obtained. .according to Solving for the radius of curvature K The threshold value can be used to obtain the critical curvature of the roller groove. The expression is: , where the critical curvature radius of curvature K The maximum allowed value.

[0065] It should be understandable that the circumferential bending angle of the pipe is included in the equations of motion trajectory. Spiral azimuth and axial self-feed displacement s These equations are mutually coupled; in other words, the system of equations for the motion trajectory is a coupled ODE (Ordinary Differential Equation) system. In some embodiments, transforming the system of equations for the motion trajectory yields:

[0066] ;

[0067] ;

[0068] .

[0069] From the modified form of the above system of equations for the trajectory of motion, it can be seen that... by The impact, by The influence of this on the coupling relationship is further defined as follows: The coupled ODE system of helical self-feeding motion is defined as follows: ,in, p Let be a state vector containing variables and their derivatives. f It is a nonlinear function. Transforming the above system of equations for the trajectory into a first-order form, we obtain... , .

[0070] In at least one embodiment, the initial value Pick: , , , , The parameters of the pipe are: weight Moment of inertia of cross section The parameters of the spiral roller assembly are: the radius of the roller groove. The radius of curvature of the roller-shaped groove helix angle Time range The number of equally spaced points is 500, that is .

[0071] Furthermore, integration is performed using the SciPy solver, where the relative tolerance... absolute tolerance The solution results are as follows: Figure 3 As shown. Specifically, by Figure 3 It can be seen that the blue curve corresponds to From the beginning start The oscillation amplitude shows a decreasing trend, exhibiting small oscillations and gradually stabilizing, reflecting the feedback suppression effect; the final circumferential bending angle of the pipe The red curve corresponds to... Under the effect of coupling acceleration, it shows a slow upward trend; the final helical azimuth angle of the pipe The green curve corresponds to ,exist Under the influence of [the action of] [the force], it exhibits a slight linear increase; the final axial self-feed displacement of the pipe [is as follows]. .

[0072] Furthermore, the initial local curvature of the pipe .exist hour, Although the curvature value has increased, its oscillation amplitude has decreased. A 50% decrease conforms to the law of exponential decay. ,in, The second cycle demonstrates that the coupled ODE system possesses the ability to suppress fluctuations and stabilize local curvature. k The ability.

[0073] Under the decoupling comparison benchmark, i.e. hour, It is in a state of free oscillation. The coupled ODE system has no suppression mechanism. As mentioned above, Q q exist Components in coordinate system Therefore, in the actual coupled state, hour, , making Less than 0, therefore It approaches 0. That is, when the axial self-feed speed... When it increases, It can suppress bending deformation and reduce the local curvature of the pipe. k Approaching 0. The above results verify the "feedback annihilation effect," meaning that when the pipe has initial local curvature, the helical self-feeding motion generates an inhibitory effect through dynamic coupling. , Acting in the opposite direction of deformation, resulting in local curvature k Approaching zero, thereby achieving uniformity and consistency in system performance.

[0074] Going further, , hour, To verify the effect of geometric parameters on... Sensitivity analysis was conducted to assess the impact of the event. ,visible along with α It increases with the increase of, for example, compared to ,when hour, An increase of 25%. Meanwhile, as mentioned earlier, Q q exist Components in coordinate system It should be understandable that when the bending angular velocity... When stable, The radius of curvature of the roller groove K It is directly proportional to the bending angular velocity. The square of the number is directly proportional to the sum of its squares, therefore When the radius of curvature of the roller groove K A 10% increase, that is At that time, the new This represents a 10% increase compared to the previous version.

[0075] Furthermore, For generalized force Q q exist The components in the coordinate system are used to drive the pipe in... When it undergoes spiral motion in coordinate system, As the value increases, due to the nonlinear coupling effect of "bending-rotation" mentioned above, the azimuth angular velocity will increase. Increase, and azimuth angular velocity The increase was greater than The increase. For example, the original ,when After increasing by 10%, the new This represents a 15% increase compared to the previous figure. Furthermore, as mentioned earlier, Therefore ,Will Substituting the values ​​will yield new results. Compared to the original It increased by 15%, in other words, the radius of curvature of the roller groove... K The coupling effect is indirectly amplified through nonlinear multiplication, making This represents a further increase of approximately 15%.

[0076] Under the decoupling comparison benchmark, i.e. hour, It has no propulsive effect. However, in the actual coupled state, It exhibits linear growth, with axial self-feed speed The above results verify the "geometry-self-feed coupling effect," that is, the radius of curvature of the roller groove... K With helix angle α The nonlinear product terms of the coupled ODE system are amplified together to continuously generate This enables stable and continuous self-feeding motion.

[0077] In summary, the solution results of the motion trajectory equations of this application quantitatively confirm the effectiveness of the "bidirectional reciprocating bending-self-feeding" coupling mechanism, which can provide a theoretical basis for the spiral self-feeding straightening process and roller design of pipes.

[0078] The basic principles, main features, and advantages of this invention have been described above. Those skilled in the art should understand that this invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely principles of the invention. Various changes and modifications can be made without departing from the spirit and scope of the invention, and all such changes and modifications fall within the scope of the invention as claimed. The scope of protection claimed by this invention is defined by the appended claims and their equivalents.

Claims

1. A method for quantifying the coupling mechanism of helical self-feeding motion, characterized in that, include: Assuming the pipe's axis of rotation coincides with the bending plane and neglecting axial roll, the total kinetic energy of the pipe is established based on the thin-wall assumption. T Theoretical formula; Neglecting plastic dissipation in the pipe, a simplified potential energy of the pipe is established based on the unified curvature theorem of reciprocating bending and the bidirectional effect. V Theoretical formula; The trajectory of the pipe is modeled based on the Lagrange equation to obtain a set of trajectory equations, which includes the circumferential bending angle of the pipe. Spiral azimuth angle of the pipe Axial self-feed displacement s The coupling relationship between them; Among them, the Lagrange equation is , Q q For the generalized forces acting on the pipe, L For Lagrange units, , q For a generalized coordinate system, , for q The first derivative.

2. The method for quantifying the coupling mechanism of the spiral self-feeding motion according to claim 1, characterized in that, The total kinetic energy T Including translational kinetic energy and rotational kinetic energy Among them, neglecting the radial variation of the pipe, based on the thin-wall assumption, the translational kinetic energy is... ; Assuming the pipe's rotation axis coincides with the bending plane and neglecting axial roll, based on the thin-wall assumption, the rotational kinetic energy... ; The translational kinetic energy The theoretical formula and the rotational kinetic energy Substituting the theoretical formula into the total kinetic energy T The theoretical formula is obtained. ,in, I 1 represents the moment of inertia of the pipe section. , r The outer radius of the pipe. t For the wall thickness of the pipe, m For the quality of the pipes, R The radius of the roller-shaped groove, for s The first derivative, for The first derivative, for The first derivative.

3. The method for quantifying the coupling mechanism of the spiral self-feeding motion according to claim 1, characterized in that, Neglecting plastic dissipation in the pipe, the potential energy of the pipe ,in, u The strain energy density of the pipe material. , E The Young's modulus of the roller. k For the local curvature of the pipe, y The distance between the neutral axes of the pipes. A This represents the cross-sectional area of ​​the pipe. Based on the unified curvature theorem for reciprocating bending and the bidirectional effect, the local curvature k Coupled axial curvature and helical curvature ,in, , The local curvature is obtained according to the vector superposition theorem. ; Will Substitute the strain energy density of the pipe u By treating the pipe as a uniform beam segment, a simplified potential energy of the pipe is obtained. ;in, , K Let be the radius of curvature of the roller-shaped groove.

4. The method for coupling mechanism of quantized spiral self-feeding motion according to any one of claims 1-3, characterized in that, Lagrange equations in Components in coordinate system , for Q q exist Components in coordinate system ,in, For pipe materials in Centrifugal force in curvilinear motion under coordinates The position vector of the pipe right The partial derivative, ; In curvilinear motion, the centrifugal force ,in, m For the quality of the pipes, K The radius of curvature of the roller-shaped groove is... for s The first derivative, Let be the normal unit vector in the spiral coordinate system. , α The helix angle of the roller-shaped groove; Will Substitute the centrifugal force ,get .

5. The method for coupling mechanism of quantized spiral self-feeding motion according to any one of claims 1-3, characterized in that, Lagrange equations in Components in coordinate system , for Q q exist Components in coordinate system ,in, For pipe materials in Centrifugal force in helical motion under coordinate system The position vector of the pipe right The partial derivative, ; In spiral motion, the centrifugal force ,in, m For the quality of the pipes, K The radius of curvature of the roller-shaped groove is... R The radius of the roller-shaped groove, for The first derivative, A radial unit vector in a spiral coordinate system. , α The helix angle of the roller-shaped groove; Will Substitute the centrifugal force ,get .

6. The method for coupling mechanism of quantized spiral self-feeding motion according to any one of claims 1-3, characterized in that, Lagrange equations in s Components in coordinate system , for Q q exist s Components in coordinate system ,in, For pipe materials in s The helical thrust experienced in the coordinate system. The position vector of the pipe right s The partial derivative, ; The spiral thrust ,in, m For the quality of the pipes, R The radius of the roller-shaped groove, for The first derivative, for The first derivative, As a tangential unit vector, in a spiral coordinate system, ; Will Substituting the aforementioned helical thrust ,get .

7. The method for coupling mechanism of quantized spiral self-feeding motion according to any one of claims 1-3, characterized in that, The equations of the motion trajectory are as follows: ; ; ;in, m For the quality of the pipes, R The radius of the roller-shaped groove, α The helix angle of the roller-shaped groove, E The Young's modulus of the roller. K The radius of curvature of the roller-shaped groove; I 1 represents the moment of inertia of the pipe section. , r The outer radius of the pipe. t The wall thickness of the pipe; , y The distance between the neutral axes of the pipes. A This represents the cross-sectional area of ​​the pipe. for s The first derivative, for s The second derivative, for The first derivative, for The second derivative, for The first derivative, for The second derivative of .

8. The method for coupling mechanism of quantized spiral self-feeding motion according to any one of claims 1-3, characterized in that, Introducing Poisson's ratio Under the given conditions, based on the Mises criterion and Hooke's law, a theoretical formula for the initiation condition of plastic deformation of pipe is established. The theoretical formula for the initiation condition of plastic deformation is Poisson's ratio. The radius of curvature of the roller-shaped groove K The yield strength of the pipe material The relationship is obtained by solving the theoretical formula for the initiation condition of plastic deformation to obtain the critical curvature of the roll groove. The expression.

9. The method for quantifying the coupling mechanism of the spiral self-feeding motion according to claim 8, characterized in that, Under the assumption of pure bending of the pipe, the principal stresses of the pipe are obtained based on Hooke's Law. ;in, E The Young's modulus of the roller. For the elastic strain of the pipe, , K The radius of curvature of the roller-shaped groove is... R The radius of the roller-shaped groove; Introducing Poisson's ratio Under these conditions, the deviation stress components of the pipe are obtained. Based on the equivalent stress formula and yield function of the Mises criterion, the yield strength of the pipe is obtained. ,in This represents the equivalent stress of the pipe.

10. The method for quantifying the coupling mechanism of the spiral self-feeding motion according to claim 9, characterized in that, according to Solving for the radius of curvature K The threshold value is used to obtain the critical curvature of the roller-shaped groove. .

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