CN113094820B - Rigidity calculation and check method for rigid elliptic cylinder spiral pipeline under bias force - Google Patents

Rigidity calculation and check method for rigid elliptic cylinder spiral pipeline under bias force Download PDF

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CN113094820B
CN113094820B CN202110491714.7A CN202110491714A CN113094820B CN 113094820 B CN113094820 B CN 113094820B CN 202110491714 A CN202110491714 A CN 202110491714A CN 113094820 B CN113094820 B CN 113094820B
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孙志成
宋晓东
张欢
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Beijing Institute of Technology BIT
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Abstract

The invention discloses a rigidity calculation and check method for a rigid elliptic cylinder spiral pipeline under bias force, and belongs to the technical field of aerospace. According to the invention, the metal hard filling pipeline of the on-orbit docking filling mechanism is wound into an elliptic cylinder spiral configuration, so that the additional resistance of the pipeline expansion to the driving source in the docking process is greatly reduced. The method for calculating the rigidity of the elliptical hollow pipeline and checking the stress of the elliptical hollow pipeline is formed, and can be used as a design basis for the filling pipeline of the on-orbit butt filling mechanism, so that the influence of the resistance of the filling pipeline on the butt joint process is minimum, and the pipeline is not broken.

Description

Rigidity calculation and check method for rigid elliptic cylinder spiral pipeline under biasing force
Technical Field
The invention discloses a rigidity calculation and check method for a rigid elliptic cylinder spiral pipeline under bias force, and belongs to the technical field of aerospace.
Background
With the development of aerospace technology, the in-orbit docking and filling technology has become one of the mainstream research contents, and the technology relates to two aspects of docking and filling. The process of on-track filling is butt joint and then filling, a filling pipeline at the tail part in the butt joint process can also stretch out and draw back step by step, the pipelines are basically all rigid pipelines made of titanium alloy materials in consideration of the high pressure property of filling fuel and the compatibility of fuel-pipelines, the rigid pipelines inevitably bring additional resistance to a butt joint mechanism in the stretching process, and the additional force is usually not negligibly small. And aiming at linear telescopic motion, the additional resistance can be reduced to the maximum extent by coiling the pipeline into a spiral spring shape. Meanwhile, the layout space of the pipeline may be irregular, the axial rigidity of the elliptic cylinder spiral pipeline is superior to that of a cylindrical spiral pipeline under certain conditions, and the offset of the pipeline connecting point can increase the envelope of the pipeline so as to further reduce the rigidity. However, for the oval pipeline, the existing method can only calculate the rigidity according to finite element simulation, and a set of complete theoretical method is not provided. Therefore, a general rigidity calculation and check method for the rigid elliptic cylinder spiral pipeline needs to be designed, the rigidity of the pipeline can be calculated and checked according to given geometric parameters and material parameters, and a theoretical basis is provided for the design of the filling pipeline of the on-orbit butt-joint filling mechanism.
Disclosure of Invention
The invention solves the technical problems that: aiming at a rigid filling pipeline of an on-orbit butt-joint filling mechanism, a rigidity calculation and check method of a rigid elliptic cylinder spiral pipeline subjected to a biasing force is provided, so that the rigidity of the filling pipeline is calculated and the stress is checked, and the minimum influence of the resistance of the filling pipeline on the butt-joint process is ensured, and the filling pipeline is not broken.
The technical scheme of the invention is as follows: the rigid elliptic cylinder spiral pipeline rigidity calculation and check method under the bias force comprises the following steps:
step one, calculating the rigidity of an elliptical hollow pipeline;
checking the stress of the elliptical hollow pipeline;
the invention relates to a rigidity calculation and stress check method for an elliptic cylinder spiral pipeline biased by force, which comprises the following steps:
1. according to the elliptic cylinder spiral pipeline model, namely the formula (1), the geometric relation between the center O of the ellipse and any section Q on the ellipse, namely the L is found out through the graphs (4) and (5)2And alpha2；
An ellipse equation:
Figure BDA0003052585090000011
wherein a is the major semi-axis of the elliptical envelope, b is the minor semi-axis of the elliptical envelope,
Figure BDA0003052585090000021
parameters of an elliptical envelope parameter equation;
derivation is carried out on the ellipse equation (1) to obtain the tangent slope k of any section Q on the ellipse1：
Figure BDA0003052585090000022
I.e. the slope k of the perpendicular to the tangent of the arbitrary section Q on the ellipse2：
Figure BDA0003052585090000023
Slope k of connecting line between ellipse center O and arbitrary section Q on ellipse3：
Figure BDA0003052585090000024
The perpendicular ON of the tangent line Q of any section ON the ellipse obtained by the formulas (1) to (4)2And the included angle between the center of the ellipse and the connecting line OQ of any cross section Q on the ellipse2OQ(α2)：
α2＝arctan(k2)-arctan(k3) (5)
Distance L between the center O of the ellipse and any section Q on the ellipse2：
Figure BDA0003052585090000025
2. Obtaining the shearing force, the torque and the bending moment of any section Q on the ellipse under the action of the force F at the center O of the ellipse through the geometric relationship obtained in the step 1;
shear force:
FT＝F (7)
torque:
T＝F·L2·|cosα2| (8)
bending moment:
M＝F·L2·|sinα2| (9)
3. obtaining the working of the external force F and the strain energy V of the pipeline by an energy method and a infinitesimal methodεObtaining a rigidity coefficient K expression of the pipeline by using the conservation equation;
according to the law of conservation of energy, the work W applied by the external force F is equal to the strain energy V of the pipelineεThe change of (c):
Figure BDA0003052585090000026
wherein, lambda is the deformation length of the pipeline under the action of an external force F;
under the action of the external force F, the elliptic cylindrical spiral pipeline converts the work done by the external force into torsional strain energy VTorsion barBending strain energy VElbow bendAnd the forms of, namely:
Vε＝Vtorsion bar+VElbow bend (11)
Method for solving torsional strain energy V by infinitesimal methodTorsion bar：
Figure BDA0003052585090000031
Wherein v isTorsion barIs torsional strain energy density, V is volume under stress, dVTorsion barThe volume of the infinitesimal volume is under the action of torsional stress; torsional strain energy density vTorsion bar：
Figure BDA0003052585090000032
Wherein G is the shear modulus of the material, tau is the shear stress, and gamma is the shear strain;
shear stress τ:
Figure BDA0003052585090000033
wherein T is torque, IPThe inertia moment of the section to the center of a circle is shown, and rho is the infinitesimal distance which is rho away from the center of the section of the hollow circle;
Figure BDA0003052585090000034
wherein D is the inner diameter of the pipeline, and D is the outer diameter of the pipeline;
infinitesimal volume dV of pipeline under torsional stress actionTorsion bar：
Figure BDA0003052585090000035
Wherein dA isTorsion barThe infinitesimal area under the action of torsional stress, dS is the infinitesimal length and d theta is the infinitesimal included angle;
the expression (12) is taken in from formulas (13) to (16):
Figure BDA0003052585090000041
wherein N is the effective number of turns of the pipeline;
simplifying the formula (17) to obtain:
Figure BDA0003052585090000042
wherein Elliptics K is a first type of complete elliptic integral;
similarly, the bending strain energy V is obtained by infinitesimal methodBend：
Figure BDA0003052585090000043
Wherein v isElbow bendIs bending strain energy density, dVElbow bendThe infinitesimal volume under the action of bending stress;
flexural strain energy density vElbow bend：
Figure BDA0003052585090000044
Wherein E is the elastic modulus of the material, sigma is tensile and compressive stress, and epsilon is tensile and compressive strain;
tensile and compressive stresses σ:
Figure BDA0003052585090000045
wherein M is a bending moment, IZThe inertia moment of the cross section to the neutral axis, and z are the distance between any chord and the center of the hollow circular section;
Figure BDA0003052585090000046
infinitesimal volume dV of pipeline under bending stress actionBend：
Figure BDA0003052585090000051
Wherein dA isElbow bendArea of infinitesimal under bending stress, dS length of infinitesimal, LStringIs the chord length of the corresponding infinitesimal, r is r1And r2General term of1Is the inner radius of the pipeline (i.e. r)1＝0.5d)、r2Is the outer radius of the pipeline (i.e. r)2＝0.5D)；
Since the pipeline model is a hollow circle, and the density of the hollow part needs to be subtracted from the density of the whole circle when the bending stress density is obtained for the pipeline, the following (20) to (23) are substituted into (19):
Figure BDA0003052585090000052
wherein z is1The distance z between any chord of the hollow part of the pipeline and the center of the hollow circular section2The distance between any chord of the solid part of the pipeline and the center of the section of the hollow circle is the distance;
the formula (24) is simplified to obtain
Figure BDA0003052585090000053
Substituting the formula (18) and the formula (25) into the formula (10) to obtain the rigidity K of the elliptical hollow pipeline;
Figure BDA0003052585090000054
wherein Elliptics K is a first type of complete elliptic integral; EllipticE is a second class of complete elliptic integrals;
when the pipeline is a special type of round hollow pipeline, i.e. a ═ b ═ R, formula (26) is simplified as:
Figure BDA0003052585090000055
wherein R is the average radius of the coil.
The invention relates to a rigidity calculation and stress check method for an elliptic cylinder spiral pipeline biased by force, which comprises the following steps:
1. solving the geometric relation between any stress point P in the ellipse and any section Q on the ellipse;
slope k of connecting line between any stress point P in ellipse and any cross section Q on ellipse4：
Figure BDA0003052585090000061
Wherein, (m, n) is the coordinate of any stress point P in the ellipse;
the following equations (3) and (28) can be used:
perpendicular PN of Q tangent line of arbitrary section on ellipse1The included angle N between the point P of any stress in the ellipse and the connecting line PQ of any cross section Q1PQ(α1)：
α1＝arctan(k2)-arctan(k4) (29)
Distance L' between any stress point P in the ellipse and any section Q on the ellipse:
Figure BDA0003052585090000062
2. obtaining the shearing force, the bending moment and the torque of any section Q on the ellipse under the action of the force F at any stress point P in the ellipse through the geometric relationship obtained in the step two 1;
shear force:
FT′＝F (31)
torque:
T′＝F·L′·|cosα1| (32)
bending moment:
M′＝F·L′·|sinα1| (33)
3. calculating the maximum stress borne by any section Q on the ellipse, and checking whether the pipeline material meets the requirements or not;
the stress check of the elliptic cylindrical spiral pipeline under the bias force comprises the check of two stresses of shear stress and tensile stress. Wherein, the offset force forms shear stress to the shearing force and torque formed by the pipeline, and the bending moment forms tensile and compressive stress;
and (3) stress checking is carried out by adopting a third strength theory, namely:
Figure BDA0003052585090000063
wherein, WzIs the bending resistance section coefficient, [ sigma ]]Allowable stress for the material;
substituting the formulas (32) and (33) into the formula (34) to obtain
Figure BDA0003052585090000071
In formula (35), only L 'is variable when other parameters are determined, so that the maximum value L' of L 'is determined'maxSuch that:
Figure BDA0003052585090000072
a maximum value L 'of the L'maxThe solving method comprises the following steps:
an ellipse equation:
Figure BDA0003052585090000073
l' distance expression:
Figure BDA0003052585090000074
combined vertical type (37) and type (38) (L')2Derivation is carried out, and a derivative function is made to be 0 to obtain an extreme point:
Figure BDA0003052585090000075
order [ (L')2]' -0, squared on both sides:
Figure BDA0003052585090000076
finishing formula (40) to obtain:
(a2-b2)2x4-2a2m(a2-b2)x3-a2[(a2-b2)2-a2m2-b2n2]x2+2a4m(a2-b2)x-a6m2＝0 (41)
formula (41) is a unitary quartic equation set about x, and x is calculated to be a positive real root, a negative real root and two virtual roots; the value of x is opposite to the sign of m, and then x is substituted into an elliptic equation, namely, y is obtained by the formula (1); similarly, y is inverted from n, and L 'is obtained by substituting the maximum point (x, y) into formula (38)'max(ii) a Finally L'maxAnd (7) taking back to check the pipeline stress.
Advantageous effects
1. The invention carries out spiral winding treatment on the hard metal filling pipeline, greatly reduces the additional resistance of the pipeline expansion and contraction to a driving source in the butt joint process, and derives a rigidity calculation and check method of the rigid elliptic cylinder spiral pipeline under the bias force.
2. Aiming at the 'rigidity calculation and check method of the rigid elliptic cylinder spiral pipeline under the bias force' described in the effect 1, the existing rigidity calculation of the spiral pipeline can only be realized through finite element simulation, and a set of complete theoretical method is not provided. The rigidity derivation simplification is carried out on the rigid elliptical pipeline biased by force, a complete theoretical system is formed, and a concise expression form is obtained.
3. Aiming at the 'rigidity calculation and check method of the rigid elliptic cylinder spiral pipeline under the bias force' described in the effect 1, the rigidity calculation and check formula obtained by the invention is very suitable for programming calculation, the calculation speed is far higher than that of the traditional calculation methods such as finite element analysis and the like, the research and development period is shortened, and the research and development cost is reduced.
4. Aiming at the rigidity calculation and check method of the rigid elliptic cylinder spiral pipeline under the bias force, which is described in the effect 5, the invention not only provides a theoretical basis for the rigidity calculation and check of the rigid elliptic cylinder spiral pipeline, but also has the advantages of calculation error within 2 percent and high precision through simulation verification, and can meet the requirements of engineering application.
5. Aiming at the 'rigidity calculation and check method of the rigid elliptic cylinder spiral pipeline under the bias force' in the effect 5, the rigidity calculation and check method can be applied to rigidity calculation and check under various permutation and combination conditions of ellipse, circle, hollow, solid, tension bias and non-bias under various parameters, and has high universality.
Drawings
FIG. 1 is an isometric view of an embodiment of the invention in its entirety
1-service end of on-orbit butt joint filling mechanism, 2-satellite end of on-orbit butt joint filling mechanism.
FIG. 2-a service end isometric view of an embodiment of the invention
1.1-driving mechanism, 1.2-five-degree-of-freedom tolerance adjustment self-resetting mechanism, 1.3-filling pipeline, 1.4-service end shell, 1.5-service end liquid butt joint, 1.6-service end electric connector, 1.7-service end electric connector mounting seat and 1.8-fixing seat.
FIG. 3 is an isometric view of a satellite according to an embodiment of the invention
2.1-satellite end electric connector, 2.2-satellite end liquid butt joint, 2.3-satellite end quick disconnect, 2.4-satellite end shell, 2.5-satellite end electric connector mount pad.
FIG. 4 is a schematic diagram of a cross-sectional view of a pipeline and related parameters of the present invention
FIG. 5 is a schematic view showing the geometric relationship between the stress point P and the arbitrary section Q of the pipeline
FIG. 6 is a schematic diagram showing the geometric relationship between the elliptic center O and the arbitrary interface Q of the pipeline
FIG. 7 is a schematic diagram of the solution of torsional stress density by infinitesimal method
FIG. 8 is a schematic diagram showing the expression of the elliptic arc infinitesimal elements of the present invention
FIG. 9 is a schematic diagram of the solution of bending stress density by infinitesimal method
FIG. 10 is a cloud diagram of finite element simulation displacements according to an embodiment of the present invention
FIG. 11 is a cloud diagram of the maximum equivalent stress of finite element simulation according to an embodiment of the present invention
Detailed Description
The rigidity calculation and check method of the rigid elliptic cylinder spiral pipeline subjected to the biasing force is applied to a rigid filling pipeline of an on-orbit butt-joint filling mechanism, so that the rigidity of the filling pipeline is calculated and the stress is checked, and the butt-joint process is not influenced by the resistance of the filling pipeline or is not broken by the filling pipeline.
The on-track butt joint filling mechanism is shown in figures 1-3:
the on-orbit butt joint filling mechanism service end (1) is fixed on a filling spacecraft through a mounting hole on a service end shell (1.4) and is an active butt joint end; the satellite end (2) of the in-orbit docking mechanism is fixed on a filled satellite through a mounting hole on a satellite end shell (2.4) and is a passive docking end. The service end (1) of the on-rail butt-joint filling mechanism drives the service end liquid butt joint head (1.5) and the service end electric connector (1.6) to synchronously extend out through the driving mechanism (1.1). The tail of the service end liquid butt joint (1.5) is connected with a filling pipeline (1.3), the bottom surface of the filling pipeline (1.3) is fixedly connected with the service end shell (1.4), and the end connector is connected with a filling source. When in butt joint, the driving mechanism (1.1) drives the five-degree-of-freedom tolerance adjustment self-resetting mechanism (1.2) to drive the service end liquid butt joint (1.5), the service end electric connector (1.6), the service end electric connector mounting seat (1.7) and the filling pipeline (1.3) to realize synchronous motion, so that the filling pipeline (1.3) is coiled into a spiral shape similar to a spring, and the additional force of the filling pipeline (1.3) on the service end liquid butt joint (1.5) in the stretching-out process can be reduced to the greatest extent. After five processes, the service end quick disconnect connector (1.5.3) and the satellite end quick disconnect connector (2.3) are synchronously butted in place with the service end electric connector (1.6) and the satellite end electric connector (2.1), and then filling work is started. The filling liquid flows into a service end quick disconnector (1.5.1) in a service end liquid butt joint (1.5) through a filling pipeline (1.3), and then flows into a filled satellite through a satellite end quick disconnector (2.3). The working principle of the on-rail butt joint filling mechanism is as above.
In addition, the additional resistance brought by the filling pipeline (1.3) needs to be considered in the butt joint process, so that the rigidity calculation and the stress check of the filling pipeline (1.3) need to be carried out. Now, it is known that: the additional driving force F provided by the driving mechanism (1.1)EThe movement stroke of the filling line (1.3) is 30N, 41 mm.
According to the foregoing, due to constraints such as layout space, the filling line (1.3) is designed as a rigid, biased, elliptical cylindrical spiral configuration, the relevant parameters of which are ultimately determined as follows:
1. the major semi-axis a of the elliptical section is 70 mm;
2. the short semi-axis b of the elliptic section is 35 mm;
3. the outer diameter D of the pipeline is 4 mm;
4. the diameter d in the pipeline is 2.4 mm;
5. the effective number of turns N of the pipeline is 4;
6. the helix angle beta of the pipeline is 2 degrees;
7. the pitch p of the pipeline is 14.4 mm;
8. the stress point of the pipeline deviates from the position P (-10mm, -7 mm);
9. pipeline material: titanium alloy TC 4;
10. titanium alloy TC4 allowable stress [ sigma ] 825 Mpa;
11. the elastic modulus E of the titanium alloy TC4 is 110 Gpa;
12. the shear modulus G of the titanium alloy TC4 is 41 Gpa.
And (3) solving the rigidity coefficient K of the pipeline according to a formula (26):
Figure BDA0003052585090000101
substituting the relevant parameters into the formula to obtain:
K＝0.248N/mm
because the motion stroke of the filling pipeline (1.3) is 41mm, the additional resistance F brought by the filling pipeline (1.3) is as follows:
F＝0.248×41＝10.168N
and the driving mechanism (1.1) can provide an additional driving force FE＝30N>F10.168N, the additional resistance of the filler pipe (1.3) does not affect the docking process.
Then, L 'is obtained by simultaneous solution of the formulas (37), (38) and (41)'max：
Figure BDA0003052585090000102
Calculating the longest distance L 'formed by the stress point P and an arbitrary point Q on the ellipse envelope'maxThe coordinate of point Q of (2.89, 1.96) (mm) is L 'at this time'max＝80.39mm。
L'maxChecking whether the stress on the pipeline satisfies allowable stress [ sigma ]]：
Figure BDA0003052585090000103
Substituting the relevant parameters to obtain
Maximum stress sigma to the pipelinemax＝146.99Mpa<[σ]The pipeline can not be broken when the pressure is 825 Mpa.
And (3) analyzing the rigidity and the maximum equivalent stress of the pipeline model by using finite element software Abaqus, wherein the axial displacement lambda of the pipeline is 41mm, and the obtained result is as follows:
rigidity
Figure BDA0003052585090000111
Maximum equivalent stress sigmamax126.9Mpa, the simulated displacement and the maximum equivalent stress cloud chart are shown in fig. 10 and 11.
The results were analyzed as shown in table 1.
TABLE 1 comparison of the results of the pipeline examples
Figure BDA0003052585090000112
To sum up: the axial rigidity K of the filling pipeline (1.3) is 0.248N/mm, the additional resistance F is 10.168N, and the maximum equivalent stress sigma is obtained through calculation by the rigid elliptic cylinder spiral pipeline rigidity calculation and check method under the biasing forcemax146.99 MPa; axial rigidity K of the filling pipeline obtained through finite element simulation is 0.244N/mm, additional resistance F is 10N, and maximum equivalent stress sigma is obtainedmax146.99 MPa. The error of the axial rigidity K of the two is 1.61 percent, and the maximum equivalent stress sigma ismaxThe error of (2) is 13.67%. The reason why the maximum equivalent stress error is large is that the theoretical result adopts the third intensity theory, and the finite element simulation result adopts the fourth intensity theory. And theoretical analysis is carried out by adopting a fourth strength theory, and the result is as follows: 127.29Mpa, the error with the finite element simulation result is 0.31%, and the correctness of the theoretical method is verified. At the same time, the additional resistance F of the pipeline is 10.168N<Additional driving force FE30N, maximum equivalent stress σmax＝127.29Mpa<Allowable stress [ sigma ] of TC4 material]825Mpa, the configuration is verified to be reasonable.
The above detailed description is intended to illustrate the objects, aspects and advantages of the present invention, and it should be understood that the above detailed description is only exemplary of the present invention and is not intended to limit the scope of the present invention, and any modifications, equivalents, improvements and the like made within the spirit and principle of the present invention should be included in the scope of the present invention.

Claims (2)

1. The rigidity calculation and check method of the rigid elliptic cylinder spiral pipeline under the bias force is characterized by comprising the following steps:
step one, calculating the rigidity of the elliptical hollow pipeline
1) According to an elliptic cylinder spiral pipeline model, namely an equation (1), finding out the geometric relation between the center O of an ellipse and any section Q on the ellipse, namely L2And alpha2；
An ellipse equation:
Figure FDA0003678991740000011
wherein a is the major semi-axis of the elliptical envelope, b is the minor semi-axis of the elliptical envelope,
Figure FDA0003678991740000012
parameters of an elliptical envelope parameter equation;
derivation is carried out on the ellipse equation (1) to obtain the tangent slope k of any section Q on the ellipse1：
Figure FDA0003678991740000013
I.e. the slope k of the perpendicular to the tangent line of any section Q on the ellipse2：
Figure FDA0003678991740000014
Slope k of connecting line between center O of ellipse and arbitrary section Q of ellipse3：
Figure FDA0003678991740000015
A perpendicular ON to a tangent line Q of an arbitrary section ON the ellipse obtained by the equations (1) to (4)2And the included angle between the center of the ellipse and the connecting line OQ of any section Q on the ellipseN2OQ(α2)：
α2＝arctan(k2)-arctan(k3) (5)
Distance L between the center O of the ellipse and any section Q on the ellipse2：
Figure FDA0003678991740000016
2) Obtaining the shearing force, the torque and the bending moment of any section Q on the ellipse under the action of the force F at the center O of the ellipse through the geometric relationship obtained in the step 1);
shear force:
FT＝F (7)
torque:
T＝F·L2·|cosα2| (8)
bending moment:
M＝F·L2·|sinα2| (9)
3) obtaining the working of the external force F and the strain energy V of the pipeline by an energy method and a infinitesimal methodεThe conservation equation of the system, and further obtaining a rigidity coefficient K expression of the pipeline;
according to the law of conservation of energy, the work W applied by the external force F is equal to the strain energy V of the pipelineεThe change of (c):
Figure FDA0003678991740000021
wherein, lambda is the deformation length of the pipeline under the action of an external force F;
under the action of an external force F, the elliptic cylindrical spiral pipeline converts the work done by the external force into torsional strain energy VTorsion barBending strain energy VElbow bendAnd the form of, namely:
Vε＝Vtorsion bar+VElbow bend (11)
Torque strain energy V obtained by infinitesimal methodTorsion bar：
Figure FDA0003678991740000022
Wherein v isTorsion barIs torsional strain energy density, V is volume under stress, dVTorsion barThe infinitesimal volume under the action of torsional stress;
torsional strain energy density vTorsion bar：
Figure FDA0003678991740000023
Wherein G is the shear modulus of the material, tau is the shear stress and gamma is the shear strain;
shear stress τ:
Figure FDA0003678991740000024
wherein T is torque, IPThe inertia moment of the cross section to the center of a circle is shown, and rho is the infinitesimal distance which is rho away from the center of the cross section of the hollow circle;
Figure FDA0003678991740000025
wherein D is the inner diameter of the pipeline and D is the outer diameter of the pipeline;
infinitesimal volume dV of pipeline under torsional stress actionTorsion bar：
Figure FDA0003678991740000026
Wherein dA isTorsion barThe area of a infinitesimal under the action of torsional stress, dS the length of the infinitesimal and d theta the included angle of the infinitesimal;
the formula (12) is derived by bringing the formulae (13) to (16):
Figure FDA0003678991740000031
wherein N is the effective number of turns of the pipeline;
simplifying the formula (17) to obtain:
Figure FDA0003678991740000032
wherein Elliptics K is a first type of complete elliptic integral;
similarly, the bending strain energy V is obtained by infinitesimal methodBend：
Figure FDA0003678991740000033
Wherein v isElbow bendTo bending strain energy density, dVElbow bendIs the infinitesimal volume acted by bending stress;
flexural strain energy density vElbow bend：
Figure FDA0003678991740000034
Wherein E is the elastic modulus of the material, sigma is the tensile and compressive stress, and epsilon is the tensile and compressive strain;
tensile and compressive stresses σ:
Figure FDA0003678991740000035
wherein M is a bending moment, IZThe inertia moment of the cross section to the neutral axis and z are the distance between any chord and the center of the section of the hollow circle;
Figure FDA0003678991740000041
pipe bending stressOf a infinitesimal volume dVBend：
Figure FDA0003678991740000042
Wherein dA isElbow bendArea of infinitesimal under bending stress, dS length of infinitesimal, LStringIs the chord length of the corresponding infinitesimal, r is r1And r2General term of1Is the inner radius of the pipeline, r1＝0.5d，r2Is the outer radius of the pipeline, r2＝0.5D；
Since the pipeline model is a hollow circle, and the density of the hollow part needs to be subtracted from the density of the whole circle when the bending stress density is obtained for the pipeline, the following steps (20) to (23) are substituted into (19):
Figure FDA0003678991740000043
wherein z is1Is the distance z between any chord of the hollow part of the pipeline and the center of the hollow circular section2The distance between any chord of the solid part of the pipeline and the center of the section of the hollow circle is the distance;
the formula (24) is simplified to obtain
Figure FDA0003678991740000044
Substituting the formula (18) and the formula (25) into the formula (10) to obtain the rigidity K of the elliptical hollow pipeline;
Figure FDA0003678991740000045
wherein Elliptics K is a first type of complete elliptic integral; EllipticE is a second class of complete elliptic integrals;
when the pipeline is a special type of round hollow pipeline, i.e. a ═ b ═ R, formula (26) is simplified as:
Figure FDA0003678991740000051
wherein R is the average radius of the spring ring;
the second step is specifically as follows:
1) solving the geometric relation between any stress point P in the ellipse and any section Q on the ellipse;
slope k of connecting line between any stress point P in ellipse and any section Q on ellipse4：
Figure FDA0003678991740000052
Wherein, (m, n) is the coordinate of any stress point P in the ellipse;
the following equations (3) and (28) can be used:
perpendicular PN of Q tangent line of arbitrary section on ellipse1The included angle N between the point P of any stress in the ellipse and the connecting line PQ of any cross section Q1PQ(α1)：
α1＝arctan(k2)-arctan(k4) (29)
Distance L' between any stress point P in the ellipse and any section Q on the ellipse:
Figure FDA0003678991740000053
2) obtaining the shearing force, the bending moment and the torque of any section Q on the ellipse under the action of the force F at any stress point P in the ellipse through the geometric relationship obtained in the step two 1);
shear force:
FT′＝F (31)
torque:
T′＝F·L′·|cosα1| (32)
bending moment:
M′＝F·L′·|sinα1| (33)
3) calculating the maximum stress borne by any section Q on the ellipse, and checking whether the pipeline material meets the requirements or not;
the stress check of the elliptic cylindrical spiral pipeline under the bias force comprises the check of two stresses of shear stress and tensile stress; the offset force forms shear stress on the shearing force and the torque formed by the pipeline, and the bending moment forms tensile stress and compressive stress;
and (3) stress checking is carried out by adopting a third strength theory, namely:
Figure FDA0003678991740000061
wherein, WzIs the bending resistance section coefficient, [ sigma ]]Allowable stress for the material;
substituting the formulas (32) and (33) for the formula (34) to obtain
Figure FDA0003678991740000062
In formula (35), only L 'is variable when other parameters are determined, so that the maximum value L' of L 'is determined'maxSo that:
Figure FDA0003678991740000063
maximum value L 'of L'maxThe solving method comprises the following steps:
an ellipse equation:
Figure FDA0003678991740000064
l' distance expression:
Figure FDA0003678991740000065
combined vertical type (37) and type (38) (L')2Derivation is carried out, and a derivative function is made to be 0 to obtain an extreme point:
Figure FDA0003678991740000066
order [ (L')2]' -0, squared on both sides:
Figure FDA0003678991740000067
finishing formula (40) to obtain:
(a2-b2)2x4-2a2m(a2-b2)x3-a2[(a2-b2)2-a2m2-b2n2]x2+2a4m(a2-b2)x-a6m2＝0 (41)
formula (41) is a unitary quartic equation set about x, and x is calculated to be a positive real root, a negative real root and two virtual roots; the value of x is opposite to the sign of m, and then x is substituted into an elliptic equation, namely, y is obtained by the formula (1); similarly, y is inverted to n, and L 'is obtained by substituting the maximum value point (x, y) into formula (38)'max(ii) a Finally L 'is prepared'maxAnd (7) taking back to check the pipeline stress.
2. The method of claim 1, wherein: the method as claimed in claim 1 is used to calculate and check the rigidity of the pipeline to be stretched so as to ensure that the additional resistance of the pipeline does not affect the stretching motion and the pipeline does not break during the motion.
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