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

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

A Static Analysis Method for Radially Loaded Circular Ring with Extended Assumption Grouping Topology

technical field

The invention relates to the field of mechanical stress distribution and elastic deformation of materials, in particular to a method for superimposing stress and deformation of a grouped topology radially loaded circular ring with an extension assumption.

Background technique

The ring structure is widely used in the fields of machinery, hydraulics, chemical industry, water conservancy, transportation and aerospace, such as gear rings, automobile hubs, impellers of centrifugal pumps, valve plates of plunger pumps, stators and rotors of motors, gyroscopes and aeroengines and other structures. Their working states can be divided into stationary, fixed-axis rotation and multi-axis rotation. In engineering practice, static and dynamic characteristics are the main concerns, especially in the case of high load, high speed and high precision requirements. When the annular structure bears unbalanced radial force and tangential force, the stress distribution state will change, which will reduce the structural precision, make the vibration more severe, and even shorten the service life. Therefore, in order to make the structure have the best performance, it is necessary to first study the stress distribution of the structure under the action of radial force and tangential force, so that the structure can obtain the best strength, stiffness and stability.

Literature (Sakamoto S, Hirata T, Kobayashi T, et al. Vibration analysis considering higher harmonics of electromagnetic forces for rotating electric machines. J IEEE T Magn, 1999, 35 (3): 1662-1665) studied the radial magnetic force produced by permanent magnets. Effect of tension on stress distribution. The results show that the radial magnetic pull is the main reason for the unbalanced stress distribution of the annular stator, and it is also the main reason for the stator to vibrate and radiate noise.

The literature (Barber J R, Force and displacement influence functions for the circular ring. J Journal of Strain Analysis, 1978, 13(2): 77-81) studies the problem of solving the internal force of a statically indeterminate ring under the assumption of no extension. A general method is established, so that various complex situations under the condition of no extension can be obtained through the distribution functions given in the literature.

But it is worth mentioning that, at present, many scholars are conducting research on rings based on the assumption of no extension, that is, assuming that the length of the neutral circle of the loaded ring structure remains constant. But in fact, the structure subjected to external loads will inevitably deform, especially in static research, the assumption of no extension cannot meet the actual situation.

Contents of the invention

The purpose of the present invention is to overcome the deficiencies in the prior art, and to provide a static analysis method for radially loaded circular rings with extended hypothetical grouping topology. The present invention aims at problems such as the static analysis of radially loaded circular rings, based on With the assumption of extension, a static model is established on the micro-segment of the ring, and the superposition method is used to calculate the stress distribution of the radially loaded ring based on the group topology with the assumption of extension, so that the obtained results are closer to the actual engineering.

The purpose of the present invention is achieved through the following technical solutions:

A static analysis method for radially loaded rings with extended assumptions grouping topology, comprising the following steps:

Step 1, intercepting micro-segments on the ring, adopting D'Alembert's principle, and establishing a static model of the ring under the action of a single radial load through the section method;

Step 2, establishing the distribution function of each static parameter of the ring under the action of a single radial load based on the assumption of extension through the static model;

Step 3, using the superposition method to obtain the functions of the static parameters of the grouped topology radially loaded ring based on the extension assumption.

Further, the static model of the ring under the action of the single radial load is specifically:

θ is the angle of a particle on the ring, R is the radius of the neutral circle of the ring, F _ef is the radial load, F _ff is the virtual load, F _sf is the radial internal force, F _tf is the tangential internal force, M _bm is the bending force 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:

In the formula, F _tfθ is the tangential stress, F _sfr is the radial stress, v is the radial deformation, u is the tangential deformation, A(A=bh) is the cross-sectional area of the ring, h is the radial thickness, b is Axial thickness, E is Young's modulus, I is moment of inertia, μ is Poisson's ratio.

Further, the function of each static parameter of the group topology radially loaded ring based on the extension hypothesis is specifically:

In the formula, N ₁ is the number of groups of radial loads, N ₂ is the 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 number of radial loads The position angle of the jth radial load in group i, F _tfθN is the tangential stress of the ring when it is subjected to N radial loads, F _sfrN is the radial stress of the ring when it is subjected to N radial loads, and M _bm is Bending moment of the ring when subjected to N radial loads, v _N is the radial deformation of the ring when subjected to N radial loads, u _N is the tangential deformation of the ring when subjected to N radial loads.

Compared with the prior art, the beneficial effects brought by the technical solution of the present invention are:

1. The present invention adopts d'Alembert's principle to set up a statics model on the micro-segment of the annulus by the section method based on the assumption of extension, and then utilizes the operator method to solve each of the annulus under the action of a single radial load based on the assumption of extension. The distribution function of the static parameters not only obtains the distribution function, but also simplifies the process of solving the differential equation by using the operator method;

2. The present invention utilizes the principle of superposition and adopts the superposition method to solve the function of each static parameter based on the grouping topology radially loaded ring with extension hypothesis, and provides a method for solving each static parameter of the loaded ring under complex situations New ideas and methods for reference;

3. The present invention has the characteristics of universality, convenience, precision and novelty. According to this method, the relationship between radial load and various static parameters can be studied, for example, the effect of radial load on the tangential stress of the ring, the relationship between radial load and radial deformation of the ring, etc., can also be Solve the functions of various static parameters under the action of various complex radial loads, such as radial stress, tangential deformation, etc. With these basic parameters and the relationship between parameters, it can be used for subsequent research on statics and dynamics. convenient.

Description of drawings

Fig. 1a and Fig. 1b are the distribution schematic diagrams of force on the whole ring and micro-segment under the action of single radial load provided by the present invention;

Fig. 2 is a schematic diagram of the distribution of the force on the whole ring under the action of a single radial load based on the extension assumption of the rotation θ _k angle provided by the present invention;

Figure 3a is a schematic diagram of the tangential and radial stress and deformation distribution of the ring under the action of a single radial load based on the assumption of extension provided by the present invention;

Figures 3b to 3f are schematic diagrams of the tangential and radial stress and deformation of the ring under the action of a single radial load based on the assumption of extension provided by the present invention;

Figure 3g is a schematic diagram of the tangential and radial stress and deformation distribution of the ring under the action of a single radial concentrated load based on the assumption of extension after the rotation θ _k angle provided by the present invention;

Fig. 4 is a schematic diagram of the force distribution on the radially loaded ring of the group topology based on the assumption of extension provided by the present invention.

detailed description

The present invention will be described in further detail below in conjunction with the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described here are only used to explain the present invention, not to limit the present invention.

The embodiment of the present invention provides a group topology radially loaded ring stress superposition method. This method uses d'Alembert's principle to establish a static model on the micro-segment of the ring, uses the section method to obtain the distribution function of each static parameter of the ring under the action of a single radial load with the assumption of extension, and then uses the superposition method, The function of each static parameter of the grouped topology radially loaded ring based on the extension assumption is obtained, which significantly improves the accuracy, universality and authenticity of the static analysis of the ring, and is closer to engineering practice. This method can also be used to solve various static parameters of typical periodic structures such as gears, ring gears, stators and rotors of rotating electrical machines, and ring components in precision instruments.

The circular ring is subjected to the radial load of the group topology; the basic feature of the superposition method of the distribution function of the static parameters is: the parameter solution of the circular ring is realized by the superposition method, and the specific steps are as follows:

(1) Using the cross-section method, the static model of the ring under the action of a single radial concentrated force is established on the micro-segment of the ring according to the principle of force and moment balance:

In the formula, θ is the angle of a particle on the ring, R is the radius of the neutral circle of the ring, F _ef is the radial concentrated force, F _ff is the virtual support force, F _sf is the radial internal force, F _tf is the tangential internal force , M _bm is the bending moment.

Figure 1a and Figure 1b are the distribution diagrams of the force on the ring and the micro-segment under the action of a single radial concentrated force. As shown in Figure 1a, the radius of the neutral circle 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 radial concentrated force F _ef at θ=0, and the direction is leftward. Around the ring, uniform virtual supports are distributed, and the virtual supports generate virtual force F _ff , and the direction is rightward. The ring is balanced by a concentrated force and a uniform virtual force. In order to study the stress distribution of the ring under the action of a single radial concentrated force, the micro-segment of dθ is intercepted at the ring θ(θ∈(0,2π)), and the force analysis of the micro-segment is carried out by using the section method, as shown in Figure 1b As shown, O and O' are the geometric center of the ring and the midpoint of the micro-segment, respectively, and F _sf , F _tf and M _bm are the radial internal force, tangential internal force and bending moment, respectively.

(2) Since the micro-segment of the ring is studied, dθ is a micro-quantity, and the idea of limit is used, therefore, the trigonometric function containing a micro-quantity can be simplified as:

(3) Substituting formulas (4)-(7) into formulas (1)-(3) and simplifying can get:

dM _bm = RF _sf dθ (10)

(4) Find the general solution of the radial internal force

The relationship between radial internal force F _sf and radial concentrated force F _ef can be obtained from formulas (8) and (9):

Equation (11) is the second-order inhomogeneous differential equation of radial internal force, and the characteristic equation can be obtained from Equation (11):

λ _sf ² +1＝0 (12)

In the formula, λ _sf is the eigenvalue of the characteristic equation, and the solution is λ _sf1,2 =±i, where i is the imaginary unit.

In order to solve the general solution of the differential equation, let a special 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), it can be simplified to get:

According to the undetermined coefficient method, a ₁ and b ₁ can be obtained as:

b ₁ =0 (16)

Therefore, the particular solution of the differential equation is:

In the formula, F _sf ^* is the special solution of the differential equation.

From the eigenvalue obtained from the characteristic equation and a special solution of the differential equation, the general solution of the radial internal force can be obtained as:

In the formula, c ₁ and c ₂ are real numbers.

(5) Find the general solution of tangential internal force and bending moment

From formulas (8), (10) and (18), we can get:

(6) Find the general solution of radial deformation

For small curvature rings, it can be known from the knowledge of material mechanics that the bending moment, tangential internal force and radial deformation, tangential deformation have the following relationship:

In the formula, v is the radial deformation of the ring, u is the tangential deformation of the ring, E, I, A and μ are the elastic modulus, moment of inertia, cross-sectional area and Poisson's ratio of the ring, respectively.

From equations (10), (11), (21) and (22), we can obtain binary high-order non-homogeneous differential equations:

For binary high-order differential equations, operators can be used to solve them.

According to the definition of the operator method, formula (23) is simplified to get:

Eliminate u according to formula (24):

From formula (25), the characteristic equation of the fifth-order non-homogeneous differential equation of radial deformation can be obtained as:

λ _v ⁵ +2λ _v ³ +λ _v ＝0 (26)

The solution is λ _v1 =0, λ _v2,3 =±i, λ _v4,5 =±i, i is the imaginary unit.

In order to solve the general solution of the 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 are real numbers, and v ^* is a special solution of the differential equation. .

Substituting formula (27) into formula (25), it can be simplified to get:

According to the undetermined coefficient method, a ₂ and b ₂ can be obtained as:

b ₂ =0 (30)

Therefore, the particular solution of the differential equation is:

From the eigenvalue obtained from the characteristic equation and a special solution of the differential equation, the general solution of the radial deformation can be obtained as:

In the formula, a _v1 ~a _v5 are all real numbers.

Similarly, according to formula (24) to eliminate v, and then use the undetermined coefficient method, the general solution of tangential deformation can be obtained as follows:

In the formula, a _u1 ~a _u5 are all real numbers.

(7) Use the boundary conditions to find the coefficients of each general solution

为：According to the knowledge of material mechanics, for a ring with small curvature, the rotation angle on any section

for:

Taking the section analysis of the ring at θ=0, the radial internal force is:

The radial deformation of the ring at θ=0 is zero, that is:

v _{(θ = 0)} = 0 (36a)

The radial deformation of the ring is symmetrical about θ=0, namely:

v _(θ) = v _(2π-θ) (36a)

The tangential deformation of the ring at θ=0 and θ=π is zero, that is:

u _(θ=0) =0 (37a)

u _(θ=π) =0 (37b)

The tangential deformation of the ring is symmetrical about the center of θ=0, that is:

u _(θ) = -u _(2π-θ) (38)

The rotation angles of the ring at θ=0 and θ=π are both zero, that is:

(8) Solving

Bring the general solution of radial deformation (Equation (32)) and the general solution of tangential deformation (Equation (33)) into Equation (21) and Equation (22), and compare them with Equation (19) and Equation (20), respectively, Obtain two sets of relational expressions, and then combine the vertical equations (35) to (39) to obtain each coefficient:

Therefore, when subjected to 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:

From the relationship between internal force and stress in material mechanics and formulas (41)-(42), it can be known that the tangential stress and radial stress of the ring are respectively:

(9) Superposition of stress

Taking stress as an example, as shown in Figure 2, the ring is subjected to a radial load F _ef at θ=θ _k , and a virtual force that is balanced with the load is distributed around the ring. Compared with Figure 1a, The magnitude of the radial load and the virtual force are respectively equal, and the direction is rotated counterclockwise by θ _k angle respectively.

Fig. 3a is a distribution diagram of tangential stress, radial stress, radial deformation and tangential deformation of the ring under a single radial load, and its magnitude is shown in Fig. 3b to Fig. 3f. All parameters are taken from Table 1-1 below. Fig. 3g is a distribution diagram of various static parameters obtained after rotating Fig. 3a by an angle θ _k .

Table 1-1 Basic parameters of permanent magnet rotor

Due to the vector nature of stress and other static parameters, operations such as rotation, addition and subtraction cannot be performed directly. Therefore, for the convenience of subsequent operations, the distribution function needs to be decomposed into the form of Fourier series. The Fourier series is a linear combination that expands a periodic function that satisfies the Dilich condition into a trigonometric function. Since under the action of a single radial load, the period of the tangential stress and radial stress of the ring is 2π, formula (46), formula (47), formula (43), formula (44) and formula (45) can be set )’s Fourier series expansion is:

和

分别为：In the formula,

and

They are:

By simplifying formulas (53)-(55) and substituting them into formula (48), we can get:

By simplifying equations (56)-(58) and substituting them into equation (49), we can get:

By simplifying formulas (59)-(61) and substituting them into formula (50), we can get:

By simplifying formulas (62)-(64) and substituting them into formula (51), we can get:

By simplifying formulas (65)-(67) and substituting them into formula (52), we can get:

After the radial load is rotated by θ _k angle, the distribution of each static parameter of the ring should also be rotated by the same angle, namely:

In the formula, θ _k is the rotation angle when the first concentrated force rotates to the kth concentrated force, and F _tfθN , F _sfrN , M _bm , v _N , u _N are obtained after the first concentrated force rotates θ _k Tangential internal force, radial internal force, bending moment, radial deformation, tangential deformation.

α_i1,j1描述的是第i₁组中第j₁个径向载荷与第i₁组中第1个径向载荷间的夹角，其中，α_ij＝(j-1)α_i2。利用叠加法，当圆环受到N个径向集中力作用时，设第一个径向集中力作用在θ＝0处，圆环的各个静力学参数分别为：As shown in Figure 4, the circle of the ring is affected by N (N=N _1* N ₂ ) radial loads, and the N radial loads are arranged in group topology, and the radial loads are divided into N ₁ groups , as shown in figure G _i1 (i ₁ =1,2,...N ₁ ), N ₂ for each group, as shown in figure L _i1,j1 (i ₁ =1,2,...N ₁ ,j ₁ =1,2,...N ₂ ). ψ _i1 describes the position angle of the first radial load in group i ₁ , where,

α _i1,j1 describes the angle between the j _1st radial load in the i _1st group and the 1st radial load in the i _1st group, where α _ij =(j-1)α _i2 . Using the superposition method, when the ring is subjected to N radial concentrated forces, the first radial concentrated force is assumed to act at θ=0, and the static parameters of the ring are respectively:

To sum up, the embodiment of the present invention provides a method for stress superposition of grouped topology radially loaded circular rings. This method uses d'Alembert's principle to establish a static model on the micro-segment of the ring, uses the section method to obtain the distribution function of each static parameter of the ring under the action of a single radial load with the assumption of extension, and then uses the superposition method, The function of each static parameter of the grouped topology radially loaded ring based on the extension assumption is obtained, which significantly improves the accuracy, universality and authenticity of the static analysis of the ring, and is closer to engineering practice.

In the embodiments of the present invention, unless otherwise specified, the models of the devices are not limited, as long as they can complete the above functions.

The present invention is not limited to the embodiments described above. The above description of the specific embodiments is intended to describe and illustrate the technical solution of the present invention, and the above specific embodiments are only illustrative and not restrictive. Without departing from the gist of the present invention and the scope of protection of the claims, those skilled in the art can also make many specific changes under the inspiration of the present invention, and these all belong to the protection scope of the present invention.

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:

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:

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:

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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