CN113051718B - Static analysis method for packet topology radial loaded ring with extension hypothesis - Google Patents

Static analysis method for packet topology radial loaded ring with extension hypothesis Download PDF

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CN113051718B
CN113051718B CN202110248281.2A CN202110248281A CN113051718B CN 113051718 B CN113051718 B CN 113051718B CN 202110248281 A CN202110248281 A CN 202110248281A CN 113051718 B CN113051718 B CN 113051718B
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王世宇
王久霞
夏春花
刘晨
魏振航
王一凡
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Tianjin University
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Abstract

The invention discloses a static analysis method for a radial loaded ring of an extended hypothesis packet topology, which comprises the following steps: firstly, intercepting a micro-segment on a circular ring, and establishing a static model of the circular ring under the action of a single radial load by adopting a Dalnbell principle and a cross-section method; secondly, establishing a distribution function of each static parameter of the ring under the action of a single radial load with an extension hypothesis through the static model; and thirdly, obtaining a function of each static parameter of the grouped topological radial loaded circular ring based on the extension hypothesis by using an overlap method.

Description

Static analysis method for packet topology radial loaded ring with extension hypothesis
Technical Field
The invention relates to the field of stress distribution and elastic deformation of materials, in particular to a method for superposing stress and deformation of a grouped topological radial loaded ring based on an extension hypothesis.
Background
The annular structure is widely applied to the fields of machinery, hydraulic pressure, chemical engineering, water conservancy, traffic, aerospace and the like, such as structures of gear rings, automobile hubs, centrifugal pump impellers, valve plates of plunger pumps, stators and rotors of motors, gyroscopes, aircraft engines and the like. Their operating states can be divided into stationary, fixed axis rotation and multi-axis rotation. In engineering practice, the static and its dynamic properties are of major concern, especially in the case of high loads, high speeds and high precision requirements. When the annular structure bears unbalanced radial force and tangential force, the stress distribution state can be changed, so that the structure precision is reduced, the vibration is more severe, and the service life is even shortened. Therefore, in order to make the structure have the best use performance, the stress distribution of the structure under the action of radial force and tangential force needs to be firstly researched, so that the structure obtains the best strength, rigidity and stability.
The influence of the radial magnetic tension generated by permanent magnets on the stress distribution is investigated in the literature (Sakamoto S, hirata T, kobayashi T, et al, simulation analysis and characterization of harmonics of electromagnetic forces J IEEE T Magn,1999,35 (3): 1662-1665). The results show that the radial magnetic tension is the main reason for generating unbalanced stress distribution of the annular stator and is also the main reason for generating vibration and radiating noise of the stator.
The problem of solving the forces in a hyperstatic torus without the extensional assumption is studied in the literature (Barber J R, force and displacement functions for the circular ring. J Journal of stress Analysis,1978,13 (2): 77-81). A general method is established, so that various complex situations under the condition of no extension can be obtained through a distribution function given by a document.
It is worth mentioning that, at present, many researchers have studied the ring based on the inextensible assumption that the length of the loaded ring structure neutral circle remains the same. In practice, the structure subjected to external loads must be deformed, and in particular in the case of static studies, the inextensible assumption cannot be satisfied in practice.
Disclosure of Invention
Aiming at the problems of statics analysis and the like of the radial loaded ring, a statics model is established on a micro-segment of the ring based on the extension assumption, and the stress distribution of the packet topology radial loaded ring based on the extension assumption is calculated by adopting an superposition method, so that the obtained result is closer to the engineering practice.
The purpose of the invention is realized by the following technical scheme:
a static analysis method for a radial loaded ring of an extended hypothesis packet topology comprises the following steps:
firstly, intercepting a micro-segment on a circular ring, and establishing a static model of the circular ring under the action of a single radial load by adopting a Dalnbell principle and a cross-section method;
secondly, establishing a distribution function of each static parameter of the ring under the action of a single radial load with an extension hypothesis through the static model;
and thirdly, obtaining a function of each static parameter of the packet topology radial loaded ring based on the extension hypothesis by using an superposition method.
Further, the static model of the ring under the action of the single radial load is specifically as follows:
Figure BDA0002964960800000021
Figure BDA0002964960800000022
Figure BDA0002964960800000023
theta is the angle of a certain mass point on the ring, R is the neutral radius of the ring, F ef For radial loading, F ff For virtual loads, F sf For radially internal forces, F tf As a tangential internal force, M bm Is a bending moment.
Further, the distribution function of each static parameter of the ring under the action of a single radial load based on the assumption of extension is specifically as follows:
Figure BDA0002964960800000024
Figure BDA0002964960800000025
Figure BDA0002964960800000026
Figure BDA0002964960800000031
Figure BDA0002964960800000032
in the formula, F tfθ As tangential stress, F sfr For radial stress, v radial deformation, u tangential deformation, a (a = bh) the cross-sectional area of the ring, h radial thickness, b axial thickness, E young's modulus, I moment of inertia, μ poisson's ratio.
Further, the function of each static parameter of the packet topology radial loaded ring based on the extended assumption is specifically as follows:
Figure BDA0002964960800000033
Figure BDA0002964960800000034
Figure BDA0002964960800000035
Figure BDA0002964960800000036
Figure BDA0002964960800000037
in the formula, N 1 Number of sets of radial loads, N 2 Number of radial loads in each group, N (N = N) 1 N 2 ) Is the total number of radial loads, α i,j Is the position angle of the jth radial load in the ith group, F tfθN Tangential stress of the ring under N radial loads, F sfrN For radial stress of the ring under N radial loads, M bm Bending moment of the ring under N radial loads, v N For radial deformation of the ring under N radial loads, u N Is the tangential deformation of the ring under N radial loads.
Compared with the prior art, the technical scheme of the invention has the following beneficial effects:
1. the method adopts the Dalabel principle to establish a statics model on the micro-segment of the circular ring by a cross section method based on the extension assumption, then solves the distribution function of each statics parameter of the circular ring under the action of a single radial load based on the extension assumption by an operator method, thereby obtaining the distribution function and simplifying the solving process of a differential equation by the operator method;
2. the invention solves the function of each statics parameter of the grouped topological radial loaded ring based on the extension hypothesis by using the superposition method by utilizing the superposition principle, and provides a new thought and method for providing reference for solving each statics parameter of the loaded ring under the complex condition;
3. the invention has the characteristics of universality, convenience, accuracy, novelty and the like. According to the method, the relation between the radial load and each static parameter can be researched, for example, the influence of the radial load action on the tangential stress of the ring, the relation between the radial load and the radial deformation of the ring and the like, and the function of each static parameter under the action of various complex radial loads, such as the radial stress, the tangential deformation and the like, can also be solved, and the subsequent research on the statics and the dynamics can be facilitated by the basic parameters and the relation among the parameters.
Drawings
FIGS. 1a and 1b are schematic diagrams illustrating the distribution of forces on a full ring and a micro-segment under a single radial load according to the present invention;
FIG. 2 shows the rotation θ provided by the present invention k The distribution diagram of the force on the whole ring under the action of a single radial load based on the extension hypothesis of the angle;
FIG. 3a is a schematic diagram of the distribution of tangential and radial stresses and deformations of a ring under a single radial load based on the assumption of elongation according to the present invention;
3 b-3 f are schematic diagrams of the magnitude of the tangential and radial stresses and deformations of the ring under a single radial load based on the assumption of elongation according to the present invention;
FIG. 3g is a rotation θ provided by the present invention k A schematic diagram of the distribution of tangential and radial stresses and deformations of the ring under the action of a single radial concentrated load with an assumption of extension behind the angle;
fig. 4 is a schematic diagram of the distribution of force on the radial loaded ring of the packet topology based on the extended assumption provided by the present invention.
Detailed Description
The invention is described in further detail below with reference to the figures and the specific embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
The embodiment of the invention provides a method for stacking stress of a packet topology radial loaded ring. The method includes the steps that a statics model is built on a micro-segment of a circular ring by applying the Dalabel principle, a cross section method is adopted to obtain a distribution function of each statics parameter of the circular ring under the action of a single radial load with an extension hypothesis, then a superposition method is utilized to obtain a function of each statics parameter of a grouped topological radial loaded circular ring with the extension hypothesis, the accuracy, universality and authenticity of the statics analysis of the circular ring are obviously improved, and the method is better close to the engineering practice. The method can also be used for solving each statics parameter of a typical periodic structure such as a gear, a gear ring, a stator and a rotor of a rotating motor, an annular component in a precision instrument and the like.
The ring is subjected to the radial load of the packet topology; the superposition method of the distribution function of the statics parameter is basically characterized in that: the method adopts an overlay method to realize the parameter solution of the circular ring, and comprises the following specific steps:
(1) By utilizing a cross section method, a static model of the ring under the action of single radial concentrated force is established on the micro-section of the ring according to the force and moment balance principle:
Figure BDA0002964960800000051
Figure BDA0002964960800000052
Figure BDA0002964960800000053
where θ is the angle of a certain mass point on the ring, R is the neutral radius of the ring, and F ef For radial concentration of forces, F ff To simulate holding force, F sf For radially internal forces, F tf As a tangential internal force, M bm Is a bending moment.
Fig. 1a and 1b are distribution diagrams of forces on the ring and the micro-segment under a single radially concentrated force. As shown in FIG. 1a, the neutral radius of the ring is R, the radial width of the ring is h, and the axial height of the ring is b. The ring is subjected to a radially concentrated force F at θ =0 ef Acting in a leftward direction. A uniform virtual support is distributed on the circumference of the ring, and the virtual support generates a virtual force F ff The direction is to the right. The ring maintains balance under the action of a concentrated force and a uniformly distributed virtual force. In order to research the stress distribution of the ring under the action of a single radial concentrated force, a micro-segment of d theta is cut at the position of the ring theta (theta epsilon (0, 2 pi)), and the micro-segment is subjected to stress analysis by adopting a cross section method, as shown in figure 1b, O and O' are respectively the geometric center of the ring and the midpoint of the micro-segment, F sf 、F tf And M bm Respectively radial internal force, tangential internal force and bending moment.
(2) Since the study is of a circle in a micro-segment, d θ is a trace amount, and the concept of limit is utilized, a trigonometric function containing a trace amount can be simplified as follows:
Figure BDA0002964960800000061
Figure BDA0002964960800000062
Figure BDA0002964960800000063
Figure BDA0002964960800000064
(3) Substituting the formulas (4) to (7) into the formulas (1) to (3) can be simplified:
Figure BDA0002964960800000065
Figure BDA0002964960800000066
dM bm ＝RF sf dθ (10)
(4) Solving radial internal force
The radially inner force F is obtained from the formulae (8) and (9) sf With radial concentration force F ef The relationship of (1):
Figure BDA0002964960800000067
equation (11) is a second-order heterogeneous differential equation of the radially inner force, and a characteristic equation can be obtained from equation (11):
λ sf 2 +1＝0 (12)
in the formula, λ sf Is the eigenvalue of the eigen equation, the solution of which yields lambda sf1,2 (= ± i, i) is an imaginary unit.
To solve a general solution of a differential equation, let a particular solution of the differential equation be:
F sf * ＝θ(a 1 cosθ+b 1 sinθ) (13)
in the formula, a 1 And b 1 Are real numbers.
Substituting formula (13) into formula (11) can simplify:
Figure BDA0002964960800000068
a is obtained by undetermined coefficient method 1 And b 1 Respectively as follows:
Figure BDA0002964960800000069
b 1 ＝0 (16)
thus, the special solution of the differential equation is:
Figure BDA0002964960800000071
in the formula, F sf * Is a special solution of a differential equation.
From the eigenvalues obtained from the eigen equation and a special solution of the differential equation, the general solution of the radial internal force can be obtained as:
Figure BDA0002964960800000072
in the formula, c 1 And c 2 Are real numbers.
(5) General solution for solving tangential internal force and bending moment
From formulas (8), (10) and (18):
Figure BDA0002964960800000073
Figure BDA0002964960800000074
(6) Solving radial deformation
For the small-curvature ring, the bending moment and the tangential internal force are respectively related to the radial deformation and the tangential deformation according to the knowledge of material mechanics as follows:
Figure BDA0002964960800000075
Figure BDA0002964960800000076
wherein v is the radial deformation of the ring, u is the tangential deformation of the ring, and E, I, A, and μ are the elastic modulus, moment of inertia, cross-sectional area, and Poisson's ratio, respectively, of the ring.
From equations (10), (11), (21) and (22), a binary high-order heterogeneous differential equation system can be obtained:
Figure BDA0002964960800000077
for a binary high order differential equation, an operator can be used to solve.
According to the definition of the operator method, equation (23) is simplified to obtain:
Figure BDA0002964960800000081
elimination of u can be according to equation (24):
Figure BDA0002964960800000082
the characteristic equation of the fifth-order inhomogeneous differential equation of radial deformation obtained by the formula (25) is as follows:
λ v 5 +2λ v 3 +λ v ＝0 (26)
solving to obtain λ v1 ＝0,λ v2,3 ＝±i,λ v4,5 (= ± i, i) is an imaginary unit.
To solve a general solution of a differential equation, let a special solution of the differential equation be:
v * ＝θ 2 (a 2 cosθ+b 2 sinθ) (27)
in the formula, a 2 And b 2 Is a real number, v * Is a special solution of a differential equation. .
By substituting equation (27) into equation (25), the following can be obtained:
Figure BDA0002964960800000083
a is obtained by undetermined coefficient method 2 And b 2 Respectively as follows:
Figure BDA0002964960800000084
b 2 ＝0 (30)
thus, the special solution of the differential equation is:
Figure BDA0002964960800000085
from the eigenvalues obtained from the eigen equation and a particular solution of the differential equation, the general solution for radial deformation can be found as:
Figure BDA0002964960800000086
in the formula, a v1 ～a v5 Are all real numbers.
Similarly, by eliminating v according to the formula (24), and then using the undetermined coefficient method, the general solution of the tangential deformation can be obtained as follows:
Figure BDA0002964960800000091
in the formula, a u1 ～a u5 Are all real numbers.
(7) Using boundary conditions to find coefficients of general solutions
The material mechanics knowledge shows that for the small-curvature ring, the corner on any section
Figure BDA0002964960800000092
Comprises the following steps:
Figure BDA0002964960800000093
taking a section at the position of the circular ring theta =0 for analysis, wherein the radial internal force is as follows:
Figure BDA0002964960800000094
the radial deformation of the ring at θ =0 is zero, i.e.:
v (θ＝0) ＝0 (36a)
the radial deformation of the ring is symmetrical about θ =0, i.e.:
v (θ) ＝v (2π-θ) (36a)
the tangential distortion of the ring at θ =0 and θ = pi is zero, i.e.:
u (θ＝0) ＝0 (37a)
u (θ＝π) ＝0 (37b)
the tangential deformation of the ring is centrosymmetric about θ =0, i.e.:
u (θ) ＝-u (2π-θ) (38)
the corners of the ring at θ =0 and θ = pi are both zero, i.e.:
Figure BDA0002964960800000095
Figure BDA0002964960800000096
(8) Solving for
The general solutions of radial deformation (formula (32)) and tangential deformation (formula (33)) are brought into formulas (21) and (22), and are respectively compared with formulas (19) and (20) to obtain two groups of relational expressions, and then the coefficients are obtained by combining formulas (35) to (39):
Figure BDA0002964960800000101
therefore, under a single radial load based on the assumption of extension, the radial internal force, tangential internal force, bending moment, radial deformation and tangential deformation of the ring are respectively:
Figure BDA0002964960800000102
Figure BDA0002964960800000103
Figure BDA0002964960800000104
Figure BDA0002964960800000105
Figure BDA0002964960800000111
from the relationship between internal force and stress in material mechanics and the equations (41) to (42), the tangential stress and the radial stress of the ring are respectively:
Figure BDA0002964960800000112
Figure BDA0002964960800000113
(9) Superposition of stresses
Taking stress as an example, as shown in fig. 2, the circular ring is at θ = θ k Is subjected to a radial load F ef The action of (1) distributing a virtual force balanced with the load on the circumference of the ring, wherein the radial load and the virtual force are equal in magnitude and rotate counterclockwise by theta in the direction of fig. 1a k And (4) an angle.
Fig. 3a is a graph of the distribution of tangential stress, radial deformation and tangential deformation of a ring under a single radial load, the magnitudes of which are shown in fig. 3 b-3 f. All parameters taken are taken from tables 1-1 below. FIG. 3g is a rotation of θ from FIG. 3a k The resulting profile of each statics parameter after the corner.
TABLE 1-1 permanent magnet rotor basic parameters
Figure BDA0002964960800000114
Because the static parameters such as stress have vectorial properties and can not be directly rotated, added and subtracted, and the like, in order to facilitate subsequent operations, a distribution function needs to be decomposed into a Fourier series form. The fourier series is a linear combination of expanding some periodic function that satisfies the dirichh condition into a trigonometric function. Since the period of the tangential stress and the radial stress of the ring under a single radial load is 2 pi, the fourier series expansion of the equations (46), (47), (43), (44) and (45) can be set as follows:
Figure BDA0002964960800000121
Figure BDA0002964960800000122
Figure BDA0002964960800000123
Figure BDA0002964960800000124
Figure BDA0002964960800000125
in the formula (I), the compound is shown in the specification,
Figure BDA0002964960800000126
and
Figure BDA0002964960800000127
respectively as follows:
Figure BDA0002964960800000128
Figure BDA0002964960800000129
Figure BDA00029649608000001210
Figure BDA00029649608000001211
Figure BDA00029649608000001212
Figure BDA0002964960800000131
Figure BDA0002964960800000132
Figure BDA0002964960800000133
Figure BDA0002964960800000134
Figure BDA0002964960800000135
Figure BDA0002964960800000136
Figure BDA0002964960800000137
Figure BDA0002964960800000138
Figure BDA0002964960800000141
Figure BDA0002964960800000142
this is obtained by simplifying formulae (53) to (55) and substituting formula (48):
Figure BDA0002964960800000143
this is obtained by simplifying formulae (56) to (58) and substituting formula (49):
Figure BDA0002964960800000144
this is obtained by simplifying formulae (59) to (61) and substituting the formula (50):
Figure BDA0002964960800000145
this is obtained by simplifying formulae (62) to (64) and substituting them into formula (51):
Figure BDA0002964960800000151
this is obtained by simplifying formulae (65) to (67) and substituting them into formula (52):
Figure BDA0002964960800000152
radial load rotation θ k After the angle, the distribution of the individual statics parameters of the ring should also be rotated by the same angle, i.e.:
Figure BDA0002964960800000153
Figure BDA0002964960800000154
Figure BDA0002964960800000155
Figure BDA0002964960800000156
Figure BDA0002964960800000161
Figure BDA0002964960800000162
in the formula, theta k Is the angle of rotation from the first to the kth concentration force, F tfθN 、F sfrN 、M bm 、v N 、u N Respectively, the first concentrated force rotation theta k Then obtaining tangential internal force, radial internal force, bending moment, radial deformation and tangential deformation.
As shown in fig. 4, one revolution of the ring is subjected to N (N = N) 1* N 2 ) The function of N radial loads, the N radial loads are arranged in a grouping topology, and the radial loads are divided into N 1 Group, as in figure G i1 (i 1 ＝1,2,...N 1 ) Shown as N in each group 2 As shown in figure L i1,j1 (i 1 ＝1,2,...N 1 ,j 1 ＝1,2,...N 2 ) As shown. Psi i1 Described is the ith 1 The position angle of the 1 st radial load in the group, wherein,
Figure BDA0002964960800000163
α i1,j1 described is the ith 1 J in group 1 Radial load and ith 1 Angle between 1 st radial load in the group, wherein ij ＝(j-1)α i2 . By using the superposition method, when the ring is subjected to N radial concentration forces, the first radial concentration force is set to act at a position of theta =0, and each statics parameter of the ring is respectively as follows:
Figure BDA0002964960800000164
Figure BDA0002964960800000165
Figure BDA0002964960800000166
Figure BDA0002964960800000171
Figure BDA0002964960800000172
Figure BDA0002964960800000173
Figure BDA0002964960800000174
in summary, the embodiment of the present invention provides a method for stacking stresses of a packet topology radial loaded ring. The method establishes a statics model on the micro-segment of the ring by applying the Dalabel principle, obtains a distribution function of each statics parameter of the ring under the action of a single radial load with an extension hypothesis by adopting a cross section method, obtains a function of each statics parameter of a grouped topological radial loaded ring based on the extension hypothesis by utilizing an overlap method, obviously improves the accuracy, universality and authenticity of the statics analysis of the ring, and is better close to the engineering practice.
In the embodiment of the present invention, except for the specific description of the model of each device, the model of other devices is not limited, as long as the device can perform the above functions.
The present invention is not limited to the above-described embodiments. The foregoing description of the specific embodiments is intended to describe and illustrate the technical solutions of the present invention, and the above specific embodiments are merely illustrative and not restrictive. Those skilled in the art can make many changes and modifications to the invention without departing from the spirit and scope of the invention as defined in the appended claims.

Claims (3)

1. A static analysis method for a radial loaded ring of an extended hypothesis packet topology is characterized by comprising the following steps:
firstly, intercepting a micro-segment on a circular ring, and establishing a static model of the circular ring under the action of a single radial load by adopting a Dalnbell principle and a cross-section method;
secondly, establishing a distribution function of each static parameter of the ring under the action of a single radial load with an extension hypothesis through the static model;
the distribution function of each static parameter of the ring under the action of a single radial load based on the extension hypothesis is specifically as follows:
Figure FDA0003889377550000011
Figure FDA0003889377550000012
Figure FDA0003889377550000013
Figure FDA0003889377550000014
Figure FDA0003889377550000015
wherein, theta is the angle of a certain mass point on the circular ring, and R is the circular ringRadius of neutral circle, F ef For radial loading, M bm Is a bending moment, F tfθ As tangential stress, F sfr Radial stress, v radial deformation, u tangential deformation, A cross-sectional area of the ring, and A = bh; h is the radial thickness, b is the axial thickness, E is the Young's modulus, I is the moment of inertia, and μ is the Poisson's ratio;
and thirdly, obtaining a function of each static parameter of the grouped topological radial loaded circular ring based on the extension hypothesis by using an overlap method.
2. The method for statically analyzing the radially loaded ring with the extended hypothesis packet topology according to claim 1, wherein the static model of the ring under the action of the single radial load is specifically:
Figure FDA0003889377550000021
Figure FDA0003889377550000022
Figure FDA0003889377550000023
where θ is the angle of a certain point on the ring, R is the neutral radius of the ring, and F ef For radial loading, F ff For virtual loads, F sf As a radially internal force, F tf As a tangential internal force, M bm Is a bending moment.
3. The method for analyzing statics of the packet topology radial loaded ring with the extended hypothesis according to claim 1, wherein the function of each static parameter of the packet topology radial loaded ring with the extended hypothesis is specifically:
Figure FDA0003889377550000024
Figure FDA0003889377550000025
Figure FDA0003889377550000026
Figure FDA0003889377550000027
Figure FDA0003889377550000028
where θ is the angle of a certain mass point on the ring, R is the neutral radius of the ring, and F ef For radial load, A = bh is the cross-sectional area of the ring, E is Young's modulus, I is the moment of inertia, μ is Poisson's ratio, N 1 Number of groups of radial loads, i 1 ＝1,2,...N 1 ，N 2 The number of radial loads in each group, N is the total number of radial loads, N = N 1 N 2 ，α i,j Is the position angle of the jth radial load in the ith group, F tfθN For tangential stress of the ring under N radial loads, F sfrN For radial stress of the ring under N radial loads, M bmN Bending moment of the ring under N radial loads, v N For radial deformation of the ring under N radial loads, u N Is the tangential deformation of the ring under N radial loads.
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