CN103218483B - A kind of strength calculation method that is threaded based on beam-spring model - Google Patents

Abstract

The invention discloses a kind of strength calculation method that is threaded based on beam-spring system model, first the method utilizes the stress balance be threaded, deformation state, boundary condition and distortion continuously/compatibility conditions set up the beam-spring simplified model that is reasonably threaded, again by solving this beam-spring system model, obtain the stressed of beam-spring model, deformation, stressed finally by this beam-spring system model, deformation reverts in original thread connection status, thus set up and solve based on beam-spring model each tooth distribution of force for the method for the Strength co-mputation that is threaded of being threaded.The present invention not only enormously simplify original solid model, and sets up rational and exercisable mechanical model, makes the strength condition by model analysis inspection threaded connector but not has become possibility by structural inspection; Compared with modeling method in the past, this mechanical model is more close to solid model, and simulation, computational accuracy are more increased, and apply more extensive.

Description

A kind of strength calculation method that is threaded based on beam-spring model

Technical field

The present invention is applicable to the analysis of each tooth distribution of force of threaded connector, in order to check the intensity that is threaded whether up to standard, and can be used for instructing by changing profile of tooth, material parameter optimizes threaded connector distribution of force, thus improves the load-bearing capacity of web member.

Background technology

Threaded connector has a wide range of applications at each industrial sector, has data to show, and only the U.S. will produce 1,000 hundred million threaded connectors for 1 year.Threaded connector in use also exists two subject matters, is not also well solved at present, namely bolt fatigure failure and from loosen problem.

For different loadings, the failure mode of bolt makes a big difference.Under static load, the damage of tension bolt mostly is plastic yield and the fracture of threaded portion; The fatigue break of tightening latch root portions mostly is under variable load.According to statistics, in ruinate bolt part, the ratio of fatigure failure is up to 85%, and its harmfulness is far away higher than other failure mode.The main cause of bolt fatigure failure is caused to be that the carrying of each tooth distributes inequality.In addition, there is tired source in bolt self, has direct relation with its production technology, is also the major reason causing bolt fatigure failure.Available data shows, and load assignment inequality is the deciding factor that body of bolt destroys.Large quantity research shows that the first tooth assume responsibility for the load more than 1/3, and first three tooth approximately assume responsibility for 70% of whole load.

Under Axial Loads, each tooth be threaded is stressed very uneven, seriously constrains the load-bearing capacity of web member entirety.For improving each tooth distribution of force of bolt, Chinese scholars is done a lot of work, and mainly concentrates on two aspects: be that the stress reducing tooth root portion is concentrated on the one hand, main method is radius of corner at the bottom of increase tooth, increases stress-relief grooves, reduces the depth of thread; Be make bolt each tooth load be tending towards even on the other hand, mainly change the structural parameters of each tooth of screw thread, as variable pitch screw, variable-tooth type screw thread, become profile nut etc.These researchs improve stressed unbalanced of each tooth of bolt to a certain extent, but also do not have a kind of simple and clear effective theoretical model at present, stressed in order to each tooth of analysis thread, for Tooth Form Optimizition provides theoretical direction and reference.

Summary of the invention

The technical problem to be solved in the present invention is to provide a kind of strength calculation method that is threaded based on beam-spring model, the method is threaded in the process of each tooth distribution of force solving, the part that will be threaded is reduced to beam-spring model, for the ruggedness test be threaded provides a kind of method concisely and comparatively accurately newly, solved by the mechanics of beam-spring model, obtain the distribution that each tooth is stressed, thus determine the strength condition that is threaded.In addition, also by adjustment profile of tooth and material parameter, the optimal design be threaded is carried out.

For solving the problems of the technologies described above, the technical solution used in the present invention is: set up beam-spring model, the force and deformation situation that simulation actual thread connects, thus by solving beam-spring model, obtains the force and deformation situation of actual connection.Technological approaches comprises the following steps: first utilize mechanical balance, distortion, boundary condition and the compatibility conditions be threaded to set up rational beam-spring model, again by solving this beam-spring model, obtain the force and deformation situation of beam-spring model, finally the force and deformation situation of this beam-spring model is reverted in original thread connection status, thus set up and solve based on beam-spring model each tooth distribution of force for the rule of the ruggedness test that is threaded of being threaded.

The invention has the advantages that:

(1) simplify original solid model, set up simple and exercisable mechanical model, make the strength condition by local pressure analytical control threaded connector but not check by experiment to become possibility;

(2) compared with modeling method in the past, this mechanical model is more close to solid model, and simulation, computational accuracy are more increased;

(3) compared with modeling method in the past; This model by changing the material, the tooth profile parameter that are threaded, can carry out being threaded optimal design.

Accompanying drawing explanation

Fig. 1 is stressed, the boundary condition schematic diagram of beam-spring model of the present invention;

Fig. 2 is the beam-spring model partial balancing schematic diagram of bolt and nut in the present invention.

Embodiment

Below in conjunction with drawings and Examples, the be threaded strength calculation method of one provided by the invention based on beam-spring model is described in detail.The concrete steps of the inventive method are as follows:

The first step, the foundation of the beam-spring method be threaded.

From accuracy and the practicality consideration of engineer applied, the present invention proposes the beam-spring method that analysis thread connects each tooth load-bearing analysis, as shown in Figure of description 1, ignore the impact of screw thread minor spiral lift angle, suppose to be threaded is axisymmetric problem, the whole part that is threaded is divided into several fan-shaped parts along radius, the spiral shell tooth of screw thread (comprising screw bolt and nut) is simplified to fanning beam, i.e. variable-section variable width beam (hereinafter referred to as uprising the beam that broadens): for bolt, can regard one end as from the axis of screw rod to tooth ends be without corner support, the other end is Model Beam freely, wherein, spiral shell teeth portion is divided into variable-section variable jowar (hereinafter referred to as uprising the beam that broadens), axis is that normal height broadens beam to tooth root portion, same, for nut, can regard one end as without corner support, other end Model Beam freely from nut outer surfaces to tooth ends, wherein, spiral shell teeth portion is divided into and uprises the beam that broadens, and nut outer surfaces is that normal height broadens beam to tooth root portion.It should be noted that, so-called " without corner support " i.e. beam-ends face without corner, but allows (i.e. screw axis direction) vertically mobile.Connection transition portion concentric circles between bolt spiral shell tooth is divided into little fan-shaped and N-1 four limit fan-shaped, and accordingly, nut is divided into M four limits fan-shaped; For bolt, these fan-shaped outers are respectively r/N, 2r/N apart from the distance of axis of symmetry ... r is R+c/M, R+2c/M for nut ... R+c, wherein, r is the internal diameter of bolt, and R is the external diameter of bolt, and c is the length of the normal high widened sections part of nut Model Beam.Each fan-shaped and fan-shaped model spring being reduced to elastic modulus gradual change in four limits.Thus, total N/M root model spring between adjacent two Model Beam of bolt/nut.For simplicity, get

Second step, determines the force and deformation situation of beam-spring method.

Described beam-spring method is stressed, deformation analysis following (referring to Figure of description 1):

(1) suppose that screw core divides by the homogeneous state of stress (having stress to concentrate at root portions), accordingly, the normal height of Model Beam of screw rod first tooth broadens partial cross sectional by uniform external applied load;

(2) the normal height of the nut Model Beam partial cross sectional that broadens suffers restraints due to fixation, can be simplified to fixed support (also can regard elastic restraint as);

(3) same, can suppose by the homogeneous state of stress corresponding to the normal height of the nut Model Beam partial cross sectional that broadens;

(4) connect each model spring of the transition portion of adjacent two spiral shell teeth, homogeneous deformation (stretch or compression) can be regarded as, be out of shape and determined by the broaden relative displacement of beam of adjacent two normal height up and down;

Consider that the stress in tooth root portion is concentrated, also mock bomb spring force can be regarded as function; In addition, mock bomb spring force can be regarded as discrete, also can to regard continuous distribution as.

Therefore the axial displacement of Model Beam comprises two parts: a part is the global displacement produced due to the carrying of spiral shell tooth, and namely because the last tooth of carrying is subjected to displacement, corresponding rigid body displacement occurs each tooth below therefrom; Another part is the amount of deflection (displacement) of the Model Beam formed due to tooth ends carrying.

(5) the between cog contact area of screw bolt and nut can think that Model Beam uprises the part that broadens, and due to the similarity of bolt and nut model structural form, can suppose that contact load is to contact region central point, and parabolically distribute.Therefore, contact load also can be simplified to concentrated force and act on contact area central point; Because thread taper is 1:16, so can think that contact force direction is axis.

Contact load also can be considered to be reduced to the symmetrical of other form, as triangulated linear distribution etc.

3rd step, the mechanics of beam-spring method solves.

1, the Model Beam that is threaded and model spring parameter calculate:

As shown in Figure of description 1, from top to bottom, 1., 3., 5., 7., 9. Model Beam (left side) sequence of bolt is, 2., 4., 6., 8., 10. Model Beam (the right) sequence of nut is.The internal diameter of bolt is the half of r(Minor diameter of screw), external diameter is the half of R(bolt large diameter), it is R-r that the Model Beam of bolt uprises the part (i.e. the spiral shell toothed portion of bolt) that broadens long; Correspond, the Model Beam total length of nut is C, and the broaden length of part of the normal height of Model Beam is c, and uprising the part (i.e. the spiral shell toothed portion of nut) that broadens long is C-c.The Model Beam of nut to uprise the partial-length that broadens the same with the length uprising the part that broadens of the Model Beam of bolt, i.e. R-r=C-c.If radial coordinate is x, the normal height of the Model Beam of the bolt part height that broadens is h ₀, wide b (x), wherein (0, r), uprise the part height that broadens is h (x) to x ∈, and wide is b (x), wherein x ∈ (r, R).

Easily obtain, the Model Beam of screw bolt and nut (comprise normal height broaden beam and uprise the beam that broadens) be widely x ∈ (0, R), the θ fan-shaped central angle corresponding to got Model Beam.In order to more realistic stress deformation, the whole fan-shaped quantity being threaded segmentation got greatly, therefore θ is very little, then for bolt, its model value is wide to be expressed as far as possible:

$b\left(x\right)=2x\sin \left(\frac{\mathrm{\θ}}{2}\right)\≈\mathrm{x\θ}$

For nut, its model value is wide to be expressed as:

$b\left(x\right)=2(R+c-x)\sin \left(\frac{\mathrm{\θ}}{2}\right)\≈(R+c-x)\mathrm{\θ}$

If the elastic modulus of bolt and nut material is E, the equivalent initial length of all model springs is that l(can be considered that in adjacent two teeth, distance between centers of tracks deducts monodentate tooth root thickness), be that the spacing of two adjacent teeth center lines cuts the normal height of Model Beam and to broaden the thickness (also can think the thickness in tooth root portion) of part; For convenience's sake, model spring numbering is as follows: in Model Beam downside is numbered spring i1 from the model spring of meshing point farthest, draws near and is followed successively by i2, i3 ... iN(nut is followed successively by i (N+1) by meshing point to outward flange, i (N+2) ... i (N+M)).The rigidity of Definition Model spring i1 is K ₁, then wherein S ₁for the area of first model spring (namely first little fan-shaped),

The rigidity easily being obtained spring in by each fan-shaped area is:

$K_n=\frac{{\mathrm{ES}}_n}{l}=\frac{[n^2-{(n-1)}^2]{\mathrm{ES}}_1}{l}=\frac{(2n-1){\mathrm{ES}}_1}{l}$

Wherein in=i1, i2, i3 ..., iN, Sn are the area that the n-th model spring is corresponding.So, model spring global stiffness in parallel between adjacent two Model Beam, for bolt, for:

$K_0=\underset{n=1}{\overset{N}{\mathrm{\Σ}}}{K}_n=\frac{N^2{\mathrm{ES}}_1}{l}=\frac{{\mathrm{E\θr}}^2}{2l}$

In like manner, nut is:

$K_0=\frac{{\mathrm{E\θr}}^2}{2l}\frac{(M-1)(M+1+2N)}{N^2}\≈\frac{\mathrm{E\θc}(c+r)}{2l}$

2, the foundation of model mechanics equation

As shown in invention book accompanying drawing 1, Model Beam 1. upside is subject to equally distributed known external applied load, and load collection degree is q, and by the overall stress balance of model spring, known in Model Beam, 2. upside is subject to intensity is q ', and meets q=q '.By Model Beam 1. downside model spring stress be not set to F11 by bolt axis to root portions, F12 ..., F1n, n=1,2 ..., N.By that analogy, Model Beam (i=1,2 ..., 8) and the model spring stress of downside Wei Fi1, Fi2 to root portions by bolt axis ..., Fin.For bolt, n=1 ..., N; For nut, n=N+1 ..., N+M.

The contact area be threaded is x ∈ (r, R), and 1. Model Beam with Model Beam contact load intensity is 2.:

$F_{1-2}=A_{1-2}\{{(x-r-\frac{R-r}{2})}^2-{\left(\frac{R-r}{2}\right)}^2\}=A_{1-2}(x-r)(x-R)---\left(1\right)$

Wherein, A _1-2for contact load distribution amplitude.Therewith roughly the same, load diatibution amplitude is respectively A to the distribution of other contact load _3-4, A _5-6, A _7-8, A _9-10.

Section the model spring under any one group of bolt and nut Model Beam contacted with each other, it is made a concerted effort as shown in Figure 2, according to the whole machine balancing of model structure, and considers that each group of camber of spring is evenly equal, then has (for screw bolt and nut):

$F_i=\underset{n=1}{\overset{N}{\mathrm{\Σ}}}{F}_{\mathrm{in}}$

(2)

$=F_{i+1}=\underset{m=N+1}{\overset{N+M}{\mathrm{\Σ}}}{F}_{i+1,m}$

i=1，3，5，7。

Generally speaking, bolt is in tension state, there is larger tension concentrated area in tooth root portion, is easy to generate crackle, causes screw rod to rupture along this cross section; On the contrary, nut is in pressured state, although also there is compressive stress concentrated area in tooth root portion, is not easy to generate crackle and then damage inefficacy.As can be seen here, the screw rod that can suitably select external diameter larger, to increase screw core area, reduces its stress; Meanwhile, the nut that can suitably select external diameter less, to reduce nut outer ring area, increases nut stress, while guarantee fastener strength, also makes securing member be optimized, weight reduction.

Model Beam 1. stress balance equation is:

$\frac{1}{2}{\mathrm{ar}}^2-F_1-\frac{A_{1-2}}{6}{(R-r)}^3=0---\left(3a\right)$

In like manner, Model Beam 3. balance equation be:

$F_1-F_3-\frac{A_{3-4}}{6}{(R-r)}^3=0---\left(3b\right)$

By that analogy, Model Beam balance equation is 5.:

$F_3-F_5-\frac{A_{5-6}}{6}{(R-r)}^3=0---\left(3c\right)$

Model Beam balance equation is 7.:

$F_5-F_7-\frac{A_{7-8}}{6}{(R-r)}^3=0---\left(3d\right)$

Model Beam 9. downside is free, and balance equation is:

$F_7-\frac{A_{9-10}}{6}{(R-r)}^3=0---\left(3e\right)$

The balance of each Model Beam of nut, is weighed by whole machine balancing and the screw model Liangping that contacts with it, obtains nature and meets.Here no longer repeat.

As previously mentioned, under external force and Model Beam interphase interaction, there is rigid body displacement and flexural deformation in each Model Beam of bolt and nut.Because bolt core stress distribution is even, can think that the normal height of the Model Beam part that broadens only has rigid body displacement (without bending); The Relative Deflection being deformed into up and down two Model Beam adjacent with model spring of model spring; Model Beam uprise broaden part normal height broaden part rigid body movement basis on also occur bending and deformation, suppose that the distributed contact power of contact region is equivalent to the concentrated force with distributed contact power equivalence, act on center, contact region, i.e. point (R+r)/2 place (that is point (C+c)/2 place of nut) of bolt.

Due to constraint, Model Beam 2. without rigid body displacement, i.e. rigid body displacement w _2,0=0, in addition, Model Beam (i=1,3,4,5 ..., 9,10) rigid body displacement be w _{i, 0}, the Relative Deflection that corresponding each Model Beam produces because of moment of flexure is , the Relative Deflection of equivalent contact force application point is , then mock bomb spring force is:

For bolt:

F _ij＝K _j(w _i,0-w _i+2,0)j＝1,2,3...Ni＝1,3,5,7(4a)

F _i＝K ₀(w _i,0-w _i+2,0)i＝1,3,5,7(4b)

W _i+2,0be the i-th+2 Model Beam rigid body displacements.

For nut:

F _ij＝K _j(w _i,0-w _i+2,0)j＝N+1,N+2,N+3...N+Mi＝2,4,6,8(4c)

F _i＝K ₀(w _i,0-w _i+2,0)i＝2,4,6,8(4d)

At (bolt/nut) model herein and in discussing, all using restrained end (i.e. bolt axis or nut outer rim) as the starting point of Model Beam length direction coordinate x.Spiral shell tooth depth (i.e. model beam length) is very little relative to screw inner diameter (or nut periphery is wide), and transverse tooth thickness (i.e. model deck-molding) is relatively comparatively large, if spiral shell tooth is considered as beam, not only has larger detrusion, and inherently beyond the category of beam.Therefore, Model Beam in this paper, from screw axis or nut outer rim to increment, meets the physical connotation of beam substantially, thus adopts beam model to carry out analyzing also comparatively rigorous.

Spring force also can adopt continuous function to represent.Make F _i(x)=K (x) (w _{i, 0}-w _i+2,0), wherein, for bolt, nut is meanwhile, the moment of flexure of Model Beam is collectively expressed as:

To Model Beam (i=1,3,5,7,9), in the normal high section of broadening, moment of flexure can be expressed as:

$M_{0i}=\underset{x}{\overset{r}{\∫}}\frac{\mathrm{E\θx}}{l}(w_{i+\mathrm{4,0}}+w_{i,0}-2w_{i+\mathrm{2,0}})(r-x)\mathrm{dx}+\frac{A_{i-(i+1)}}{12}{(R-r)}^3(\frac{R+r}{2}-x)+{\mathrm{\γ}}_i$

$=\frac{\mathrm{E\θ}}{6l}(w_{i+\mathrm{4,0}}+w_{i,0}-{2w}_{i+\mathrm{2,0}})(r^3-{3x}^{2}r+{2x}^3)+\frac{A_{i-(i+1)}}{12}{(R-r)}^3(\frac{R+r}{2}-x)+{\mathrm{\γ}}_i$

$=\frac{{(R-r)}^3{A}_{i-(i+1)}}{12}[\frac{3R+7r}{6}+\frac{{4x}^3-{6\mathrm{rx}}^2-{3r}^{2}x}{{3r}^2}]+{\mathrm{\γ}}_i$

Obtained by equilibrium equation:

$F_i-F_{i+2}-\frac{A_{i+2-i+3}}{6}{(R-r)}^3=0$

Above formula is converted into moment of flexure form:

$\underset{0}{\overset{r}{\∫}}[F_i\left(x\right)\mathrm{dx}-F_{i+2}\left(x\right)\mathrm{dx}]-\frac{A_{i+2-i+3}}{6}{(R+r)}^3\frac{R+r}{2}=0$

Simultaneous solution can obtain undetermined constant γ _i

${\mathrm{\γ}}_i=-\frac{{(R-r)}^3{A}_{i-(i+1)}}{12}\frac{3R+7r}{6}$

To Model Beam (i=1,3,5,7,9), in the normal high section of broadening, moment of flexure can be expressed as

$M_{0i}=\frac{{(R-r)}^3{A}_{i-(i+1)}}{12}\frac{{4x}^3-{6\mathrm{rx}}^2-{3r}^{2}x}{{3r}^2}---\left(5a\right)$

Meanwhile, to Model Beam (i=2,4,6,8,10), in the normal high section of broadening, moment of flexure can be expressed as:

$M_{0i}=\underset{x}{\overset{c}{\∫}}\frac{\mathrm{E\θ}(R+c-x)}{l}(w_{i+\mathrm{4,0}}+w_{i,0}-{2w}_{i+\mathrm{2,0}})(c-x)\mathrm{dx}-\frac{A_{(i-1)-i}}{12}{(R-r)}^3(\frac{C+c}{2}-x)+{\mathrm{\γ}}_i$

$=\frac{\mathrm{E\θ}}{6l}(w_{i+\mathrm{4,0}}+w_{i,0}-{2w}_{i+\mathrm{2,0}})({2c}^3+{3\mathrm{Rc}}^2-{2x}^3+\frac{R+2c}{2}{x}^2-(R+c)\mathrm{cx})-\frac{A_{(i-1)-i}}{12}{(R-r)}^3(\frac{C+c}{2}-x)+{\mathrm{\γ}}_i$

$=-\frac{{(R-r)}^3{A}_{(i-1)-i}}{12}\left[\frac{{5c}^3+{9\mathrm{Rc}}^2-3\mathrm{CRc}-{3\mathrm{Cc}}^2-{8x}^3+(2R+4c)x^2+2(R+c)\mathrm{cx}}{6c(c+r)}\right]+{\mathrm{\γ}}_i$

In like manner, can obtain,

${\mathrm{\γ}}_i=\frac{{(R-r)}^3{A}_{(i-1)-i}}{12}({5c}^3+{9\mathrm{Rc}}^2-3\mathrm{CRc}-{3\mathrm{Cc}}^2)$

So,

$M_{0i}=-\frac{{(R-r)}^3{A}_{(i-1)-i}}{12}\frac{{-8x}^3+(2R+4c)x^2+2(R+c)\mathrm{cx}}{6c(c+r)}---\left(5b\right)$

Model Beam the moment of flexure that (i=1,3,5,7,9) uprise widened sections section (r<x< (R+r)/2) is:

${\hat{M}}_i=\frac{A_{i-\&lang;i+1\&rang;}}{12}{(R-r)}^3(\frac{R+r}{2}-x)---\left(6a\right)$

Model Beam the moment of flexure that (i=2,4,6,8,10) uprise widened sections section (c<x< (C+c)/2) is:

${\hat{M}}_i=-\frac{A_{(i-1)-i}}{12}{(R-r)}^3(\frac{C+c}{2}-x)---\left(6b\right)$

For uprising widened sections section, if the angle of wedge of Model Beam is β, then this segment model beam section height can be expressed as:

i=1，3，5，7，9h(x)＝h ₀-2(x-)rtanβ(7a)

i=2，4，6，8，10h(x)＝h ₀-2(x-c)tanβ(7b)

Therefore Model Beam the bending equations of (i=1,3,5,7,9) can be expressed as:

$\frac{d^2{\hat{w}}_i}{{\mathrm{dx}}^2}=\left\{\begin{array}{cc}\frac{M_{0i}}{\mathrm{EI}\left(x\right)}=\frac{{(R-r)}^3{A}_{i-(i+1)}}{{\mathrm{Eh}}_0^3\mathrm{x\&theta;}}\frac{{4x}^3-{6\mathrm{rx}}^2-{3r}^{2}x}{{3r}^2}=\frac{{(R-r)}^3{A}_{i-(i+1)}}{{\mathrm{Eh}}_0^3\mathrm{\&theta;}}\frac{{4x}^2-6\mathrm{rx}-{3r}^2}{{3r}^2}& (0<x<r)\\ \frac{{\hat{M}}_i}{\mathrm{EI}\left(x\right)}=\frac{12}{{\mathrm{Eh}}^3\left(x\right)\mathrm{x\&theta;}}\left[\frac{A_{i-\&lang;i+1\&rang;}}{12}{(R-r)}^3(\frac{R+r}{2}-x)\right]& (r<x<\frac{R+r}{2})\end{array}\right.---\left(8a\right)$

Model Beam simultaneously the bending equations of (i=2,4,6,8,10) can be expressed as:

$\frac{d^2{\hat{w}}_i}{{\mathrm{dx}}^2}=\left\{\begin{array}{cc}\frac{M_{0i}}{\mathrm{EI}\left(x\right)}=-\frac{{(R-r)}^3{A}_{i-(i+1)}}{{\mathrm{Eh}}_0^3\mathrm{\&theta;}(R+c-x)}\frac{{-8x}^3+(2R+4c)x^2+2(R+c)\mathrm{cx}}{6c(c+r)}& \\ & (0<x<c)\\ \frac{{\hat{M}}_i}{\mathrm{EI}\left(x\right)}=-\frac{12}{{\mathrm{Eh}}^3\left(x\right)\mathrm{\&theta;}(R+c-x)}\left[\frac{A_{\&lang;i-1\&rang;-i}}{12}{(R-r)}^3(\frac{C+c}{2}-x)\right]& \\ & (c<x<\frac{C+c}{2})\end{array}\right.---\left(8b\right)$

Boundary condition is:

Clamped end (axis of screw rod or the outer rim of nut):

${\hat{w}}_i{|}_{x=0}=0,$ $\frac{d{\hat{w}}_i}{\mathrm{dx}}{|}_{x=0}=0---\left(9a\right)$

Tooth root portion and normal high widened sections section and the interface uprising widened sections section:

$\hat{w}{|}_{{x=r}^{-}}=\hat{w}{|}_{{x=r}^{+}},$ $\frac{d\hat{w}}{\mathrm{dx}}{|}_{{x=r}^{-}}=\frac{d\hat{w}}{\mathrm{dx}}{|}_{{x=r}^{+}}$ (bolt)

$\hat{w}{|}_{{x=c}^{-}}=\hat{w}{|}_{{x=c}^{+}},$ $\frac{d\hat{w}}{\mathrm{dx}}{|}_{{x=c}^{-}}=\frac{d\hat{w}}{\mathrm{dx}}{|}_{{x=c}^{+}}$ (nut) (9b)

Because the normal high section of broadening of bolt (nut) and the interface coordinate uprising the section of broadening are x=r(x=c), and the bending equations of the right and left is different herein.Therefore for the purpose of difference, with represent the numerical value obtained by the normal high section of broadening bending equations, with represent by the result uprising the section of broadening and obtain.

Twice integration type (8a), (8a) and consider boundary condition formula (9a), (9b) can obtain:

I=1,3,5,7, when 9 (bolt):

${\hat{w}}_i=\left\{\begin{array}{cc}\frac{A_{i-\&lang;i+1\&rang;}}{{9r}^2{{\mathrm{Eh}}_0}^3\mathrm{\&theta;}}{(R-r)}^3(x^3-{3\mathrm{rx}}^2-{9r}^{2}x)x& \\ & (0<x<r)\\ \frac{A_{i-\&lang;i+1\&rang;}{(R-r)}^3}{4\mathrm{E\&theta;}{(h_0+2r\tan \mathrm{\&beta;})}^3}\left(2(R+r)x\ln \frac{2x\tan \mathrm{\&beta;}}{h_0+2(r-x)\tan \mathrm{\&beta;}}\right)+\frac{{(h_0+2r\tan \mathrm{\&beta;})}^2(h_0+(R+r)\tan \mathrm{\&beta;})}{2(h_0+2(r-x)\tan \mathrm{\&beta;}){\tan }^2\mathrm{\&beta;}}& \\ +{\mathrm{\&Pi;}}_{1}x+{\mathrm{\&Xi;}}_1& (r<x<\frac{R+r}{2})\end{array}\right.---\left(10a\right)$

Wherein:

${\mathrm{\&Pi;}}_1=-\frac{A_{i-\&lang;i+1\&rang;}{(R-r)}^3}{E{h_0}^2\mathrm{\&theta;}}[\frac{14r}{{9h}_0}+\frac{(h_0+2r\tan \mathrm{\&beta;})(2R\tan \mathrm{\&beta;}-h_0)h_0+\left[{{2h}_0}^2(R+r)\tan \mathrm{\&beta;}\right]\ln \left(\frac{2r\tan \mathrm{\&beta;}}{h_0}\right)}{4{(h_0+2r\tan \mathrm{\&beta;})}^3\tan \mathrm{\&beta;}}]$

${\mathrm{\&Xi;}}_1=\frac{A_{i-\&lang;i+1\&rang;}}{\mathrm{E\&theta;}}{(R-r)}^3[\frac{{11r}^2}{{{9h}_0}^3}-\frac{2(R+r)r\ln \frac{2r\tan \mathrm{\&beta;}}{h_0}+\frac{{(h_0+2r\tan \mathrm{\&beta;})}^2(h_0+(R-r)\tan \mathrm{\&beta;})}{2h_0{\tan }^2\mathrm{\&beta;}}}{4{(h_0+2r\tan \mathrm{\&beta;})}^3}]$

(10b)

$+\frac{A_{i-\&lang;i+1\&rang;}{(R-r)}^3}{{{\mathrm{Eh}}_0}^2\mathrm{\&theta;}}[\frac{14r}{{9h}_0}+\frac{(h_0+2r\tan \mathrm{\&beta;})(2R\tan \mathrm{\&beta;}-h_0)h_0+\left[{{2h}_0}^2(R+r)\tan \mathrm{\&beta;}\right]\ln \left(\frac{2r\tan \mathrm{\&beta;}}{h_0}\right)}{4{(h_0+2r\tan \mathrm{\&beta;})}^3\tan \mathrm{\&beta;}}]r$

I=2,4,6,8, when 10 (nut):

${\hat{w}}_i=\left\{\begin{array}{c}-\frac{A_{(i-1)-i}{(R-r)}^3}{6c(c+r){{\mathrm{Eh}}_0}^3\mathrm{\&theta;}}\left[\begin{array}{c}\frac{x}{3}(-{6c}^3-18R^3+{9R}^{2}x+{3\mathrm{Rx}}^2+{2x}^3+{3c}^2(-10R+x)+2c(-{21R}^2+6\mathrm{Rx}+x^2))\\ -2{(c+R)}^2(c+3R)(c+R-x)\ln (R+c-x)\end{array}\right]\\ (0<x<C)\\ -\frac{A_{<i-1>-i}{(R-r)}^3}{2\mathrm{E\&theta;}}\&times;\\ \left\{\begin{array}{c}\frac{(2r+c-C)[\left(2(R+c-x)\tan \mathrm{\&beta;}\right)\ln (h_0+2(c-x)\tan \mathrm{\&beta;})-(R+c-x)\ln (R+c-x)]}{2{(h_0-2R\tan \mathrm{\&beta;})}^3\tan \mathrm{\&beta;}}\\ +\frac{h_0+(c-C)\tan \mathrm{\&beta;}}{2{(h_0-2R\tan \mathrm{\&beta;})(h_0+2(c-x)\tan \mathrm{\&beta;})\tan }^2\mathrm{\&beta;}}\end{array}\right\}\\ +{\mathrm{\&Pi;}}_{2}x+{\mathrm{\&Xi;}}_2\\ (C<x<C+\frac{R-r}{2})\end{array}\right.$

(10c)

Wherein:

${\mathrm{\&Pi;}}_2=\frac{A_{(i-1)-i}{(R-r)}^3}{\mathrm{E\&theta;}}\&times;$

$\left[\begin{array}{c}\frac{\ln \left(\frac{R}{h_0}\right)(2R+c-C)}{2{(2R\tan \mathrm{\&beta;}+h_0)}^3}+\frac{(2R\tan \mathrm{\&beta;}-h_0)[(C-c)\tan \mathrm{\&beta;}-h_0]+2(2R+c-C)h_0\tan \mathrm{\&beta;}}{4{(2R\tan \mathrm{\&beta;}-h_0)}^2{h_0}^2\tan \mathrm{\&beta;}}\\ -\frac{{20c}^3+{9\mathrm{Rc}}^2+{18R}^{2}c-6{(c+R)}^2(C+3R)\ln R}{18c(c+r){h_0}^3}\end{array}\right]$

${\mathrm{\&Xi;}}_2=-\frac{A_{(i-1)-i}{(R-r)}^3}{\mathrm{E\&theta;}}\left\{\begin{array}{c}\left[\begin{array}{c}\frac{c}{3}(c^3-{15c}^{2}R-33c{R}^2-18R^3)\\ -2{(c+R)}^2(c+3R)R\ln R\end{array}\right]\frac{1}{6c(c+r){h_0}^3}\\ +\frac{(2R+c-C)[\left(2R\tan \mathrm{\&beta;}\right)\ln h_0-R\ln R]}{4{(h_0-2R\tan \mathrm{\&beta;})}^3\tan \mathrm{\&beta;}}+\frac{h_0+(c-C)\tan \mathrm{\&beta;}}{8(h_0+2R\tan \mathrm{\&beta;})h_0{\tan }^2\mathrm{\&beta;}}\\ +\left[\begin{array}{c}\frac{\ln \left(\frac{R}{h_0}\right)(2R+c-C)}{2{(2R\tan \mathrm{\&beta;}-h_0)}^3}+\frac{(2r\tan \mathrm{\&beta;}-h_0)[(C-c)\tan \mathrm{\&beta;}-h_0]+2(2R+c-C)h_0\tan \mathrm{\&beta;}}{4{(2R\tan \mathrm{\&beta;}-h_0)}^2{h_0}^2\tan \mathrm{\&beta;}}\\ -\frac{{20c}^3+{9\mathrm{Rc}}^2+1{8R}^{2}c-6{(c+R)}^2(c+3R)\ln R}{18c(c+r){h_0}^3}\end{array}\right]c\end{array}\right\}$

(10d)

Because screw thread pair is in loading process, each tooth of bolt remains with each tooth of corresponding nut and contacts, and supposes the relative sliding in contactless, can obtain deformation compatibility condition as follows:

$w_{i,0}-{\hat{w}}_i{|}_{x=\frac{R+r}{2}}=w_{i+\mathrm{1,0}}+{\hat{w}}_{i+1}{|}_{x=\frac{C+c}{2}},i=\mathrm{1,3,5,7,9}---\left(11\right)$

3, the solving of beam-spring method

Nine separate balance equations can be obtained altogether by the equilibrium equation of equilibrium equation (3a) ~ (3e) and nut corresponding with it.Add formula (4a) ~ (4d) eight physical equation, ten bending equations (10a) ~ (10d), five Coordinate deformation equation (11), totally 32 equations; Wherein containing F _i(i=1,2 ..., 8), A _{<i-(i+1) >}(i=1,3,5,7,9), and w _i, 0(i=1,3,4 ..., 10) and (i=1,2 ..., 10) totally 32 unknown numbers.By solving the system of equations of above-mentioned 32 equations composition, can ask and obtain each tooth carrying distribution.

4th step, the reduction of beam-spring model and the ruggedness test that is threaded, the optimization of profile of tooth material parameter.

By the model mechanics equation solution in the 3rd step, wherein A _{<i-(i+1) >}(i=1,3,5,7,9) namely represent and are threaded the 1st, 2,3,4, the stressed size of 5 teeth.According to engineering experience, the carrying of the first tooth often accounts for total outer year 35% ~ 50%, much larger than the stand under load of other each teeth.Therefore, we can check the intensity be threaded whether to meet standard according to the number percent of the first tooth carrying.Thus reach the inspection object of the intensity that is threaded.Simultaneously, this method can also change connecting thread profile of tooth, material parameter (comprises the diameter of bolt and nut, nut outer ring thickness, and the elasticity modulus of materials of bolt and nut, pitch, root radius, half of thread angle), calculate each tooth distribution of force that is threaded under different parameters, thus reach the object of material, Tooth Form Optimizition.

Embodiment

Elastic modulus of drawing materials is E=2.1 × 10 ⁵n/m ²(plain carbon steel), bolt large diameter and path are respectively R=5.000mm, r=4.026mm, nut large footpath path is respectively C=3.97mm, c=3mm, screw thread teeth portion thickness h=1.125, between cog coupling part thickness l=0.375, half of thread angle π/6(standard M10 is threaded), calculate each tooth carrying ratio by MATLAB.Result of calculation is as follows:

Table 1 comparison of computational results

Respectively by changing the elasticity modulus of materials be threaded, tooth proportions parameter, half of thread angle, the distribution of force situation carrying out double thread connection is optimized design.

First, change elasticity modulus of materials, obtain distribution situation (E1 is bolt elastic modulus, and E2 is nut elastic modulus) as shown in table 2:

According to table 2, by changing nut material, reducing the elastic modulus of nut, the carrying to 28% of the first tooth can be reduced, make distribution of force more even.

Table 2 elasticity modulus of materials distribution situation

Secondly, change tooth profile parameter, only discuss with regard to the change of pitch (that is screw thread teeth portion thickness, between cog coupling part thickness) here.

Table 3 pitch affects distribution situation

According to table 3, by changing the pitch (screw thread teeth portion thickness, between cog coupling part thickness) be threaded, reducing root of the tooth radius of corner, the carrying to 43.6% of the first tooth can be reduced.DeGrain.

In addition, change pitch, obtain the variable effect that distribution situation is as shown in table 4.

Table 4 pitch affects distribution situation

According to table 4, by reducing the pitch be threaded, the carrying to 27.4% of the first tooth can be reduced.

Finally, change the half of thread angle that is threaded, obtain distribution situation as follows:

Table 5 half of thread angle affects distribution situation

According to table 5, by increasing the half of thread angle be threaded, can reduce the carrying to 43% of the first tooth, effect is also not obvious.

To sum up, by reducing elasticity modulus of materials, can reduce root of the tooth radius of circle, reducing pitch, it is stressed that the modes such as increase half of thread angle reduce the first tooth, each tooth carried more even, thus improve the overall load-bearing capacity that is threaded.

Claims (1)

1. the strength calculation method that is threaded based on beam-spring model, the technical scheme adopted comprises the following steps: first set up beam-spring system model according to the stress balance be threaded, deformation state, boundary condition and continuously/compatibility conditions, stressed, deformation that simulation actual thread connects, and then solve beam-spring system model and obtain concrete stressed, deformation, thus set up and to be threaded each tooth distribution of force for the method for the ruggedness test that is threaded, concrete steps are as follows based on beam-spring system model solution:

The first step: the foundation of the beam-spring method be threaded;

It is axisymmetric for supposing to be threaded, the whole part that is threaded is divided into several fan-shaped parts along radius, fan-shaped part from the axis of screw rod to its tooth ends is simplified to fanning beam, screw axis one end is without corner support, the other end freely, wherein, spiral shell teeth portion is divided into and uprises the beam that broadens, and screw axis is that normal height broadens beam to tooth root portion; Equally, for nut, the fan-shaped part from nut outer surfaces to tooth ends is simplified to fanning beam, and nut outer surfaces end is without corner support, and freely, spiral shell teeth portion is divided into and uprises the beam that broadens the other end, and nut outer surfaces is that normal height broadens beam to tooth root portion; Described threaded joints divides and comprises screw bolt and nut; So-called without corner support and beam-ends face without corner, but allow to be that screw axis direction is moved vertically;

Above-mentioned fan-shaped connection transition portion between bolt Model Beam, with concentric circles by radial decile, be divided into one little fan-shaped and N-1 four limits are fan-shaped, described little fan-shaped and outer that four limits are fan-shaped is respectively r/N, 2r/N apart from the distance of axis of symmetry ... r, wherein, r is Minor diameter of screw; Equally, connection transition portion between nut Model Beam, with concentric circles by radial decile, be divided into M four limits fan-shaped, the distance of the outer distance axis of symmetry that described four limits are fan-shaped is respectively R+ (c/M), R+ (2c/M),, R+c, wherein, R is bolt external diameter, and R+c is nut external diameter; Each block of above-mentioned screw bolt and nut is reduced to model spring, and its elastic modulus is determined by elasticity modulus of materials, transition portion height and cross-sectional area; So far, total N root model spring between two adjacent bolt Model Beam, total M root model spring between two adjacent nut Model Beam, and have c/M=r/N, wherein, N is the quantity of bolt segmentation, M is the quantity of nut segmentation, and c is the length of nut outer surfaces to tooth root portion; Described little fan-shaped refer near a piece of axis fan-shaped;

Second step: stressed, the deformation of determining beam-spring system model;

(1) screw core divides, and the normal height of the Model Beam corresponding to screw rod broadens part, supposes that its xsect is by the homogeneous state of stress;

(2) the normal height of Model Beam of nut first tooth broadens part due to fixation, and its simplified cross sectional becomes by fixed constraint;

(3) same, the normal height of nut Model Beam is broadened part, supposes that its xsect is by the homogeneous state of stress;

(4) at bolted on part, each model spring of the transition portion between adjacent two spiral shell teeth, its distortion is determined by the broaden relative displacement of beam of adjacent two normal height up and down, and supposes that each model camber of spring is even;

(5) the between cog contact area of screw bolt and nut thinks that Model Beam uprises the part that broadens, and due to the similarity of bolt and nut model structural form, supposes that contact load distribution is to contact region central point, and parabolically distributes; Therefore, contact load is also simplified to concentrated force and acts on contact area central point; Because thread taper is 1:16, so think that contact force direction is axis; 3rd step: the mechanics of beam-spring system model solves;

(1) Model Beam that is threaded and model spring parameter calculate:

From top to bottom, 1., 3., 5., 7., 9. the Model Beam sequence of bolt is, 2., 4., 6., 8., 10. the Model Beam sequence of nut is; The internal diameter of bolt is r, and external diameter is R, and the Model Beam total length of bolt equals the external diameter R of bolt, and the broaden length of part of normal height is r, and uprising the part that broadens long is R-r; Correspond, the Model Beam total length of nut is C, and the broaden length of part of normal height is c, and uprising the part that broadens long is C-c; If the Model Beam of nut uprises broaden partial-length and the Model Beam of bolt, to uprise the partial-length that broadens identical, i.e. R-r=C-c;

If radial coordinate is that (0, r), the normal height of Model Beam of the bolt part deck-molding that broadens is h to x ∈ ₀, wide is b (x), wherein x ∈ (0, r); Uprising the part height that broadens is h (x), and wide is b (x), wherein x ∈ (r, R); From setting up of model in the first step, bolt model deck-siding b (x)=2xsin (θ/2), wherein, x ∈ (0, R), the θ fan-shaped central angle corresponding to Model Beam; Equally, the relative dimensions of the Model Beam of nut and the relative dimensions of above-mentioned bolt Model Beam similar, no longer repeat;

It should be noted that, in order to more meet the virtual condition of the part stress deformation that is threaded, splitting the whole fan-shaped quantity be threaded should take large values as far as possible, and therefore θ is very little, then have for bolt:

$b\left(x\right)=2xsin\left(\frac{\mathrm{\&theta;}}{2}\right)\&ap;x\mathrm{\&theta;}$

Nut is had:

$b\left(x\right)=2(R+c-x)sin\left(\frac{\mathrm{\&theta;}}{2}\right)\&ap;(R+c-x)\mathrm{\&theta;},$ Wherein x remembers from nut outer surfaces

If the elastic modulus of bolt and nut material is E, the equivalent initial length of all model springs is l, and each model spring numbering is as follows: in bolt Model Beam downside is numbered i1 from the model spring of meshing point farthest, draws near and is followed successively by i2, i3 ..., iN; Nut is followed successively by i (N+1) by meshing point to outward flange, i (N+2) ..., i (N+M), the rigidity of Definition Model spring i1 is in like manner, the rigidity of all the other each model springs is determined by respective sector area:

$K_{iN}=\frac{{\mathrm{ES}}_{iN}}{l}=\frac{E\&lsqb;{\mathrm{iN}}^2-{(iN-1)}^2\&rsqb;S_{i1}}{l}=(2iN-1)K_{i1}$

Wherein iN=2,3 ..., N, S _iNit is the area of i-th N number of model spring;

So model spring global stiffness in parallel between adjacent two Model Beam, for bolt be:

$K_{LS}=\underset{iN=1}{\overset{N}{\&Sigma;}}{K}_{iN}=N^2\frac{{\mathrm{ES}}_{i1}}{l}=\frac{E}{l}\&CenterDot;\frac{r^2}{2}\mathrm{\&theta;}$

In like manner, for nut be:

$K_{LM}=\frac{{\mathrm{E\&theta;r}}^2}{2l}\frac{(M-1)(M+1+2N)}{N^2}\&ap;\frac{E\mathrm{\&theta;}c(c+r)}{2l}$

(2) foundation of model mechanics equation:

Model Beam 1. upside is subject to equally distributed known external applied load, and load collection degree is q; From the overall stress balance of model, be 2. subject to uniform constraining force in upside in Model Beam, intensity is q', and meets q=q'; By Model Beam 1. downside model spring stress be not set to F by bolt axis to root portions ₁₁, F ₁₂..., F _1n, n=1,2 ..., N; By that analogy, Model Beam the model spring stress of downside Wei F to root portions by bolt axis _i1, F _i2..., F _in, i=1,2 ..., 8, for bolt, n=1 ..., N, for nut, n=N+1 ..., N+M;

The contact area be threaded is x ∈ (r, R), and 1. Model Beam with Model Beam contact load intensity is 2.:

$F_{1-2}=A_{1-2}\{{(x-r-\frac{R-r}{2})}^2-{\left(\frac{R-r}{2}\right)}^2\}=A_{1-2}(x-r)(x-R)---\left(1\right)$

Wherein, A _1-2for contact load distribution amplitude; Therewith roughly the same, load diatibution amplitude is respectively A to the distribution of other contact load _3-4, A _5-6, A _7-8, A _9-10;

Section the model spring under any one group of bolt and nut Model Beam contacted with each other, according to the whole machine balancing of model structure, and consider for each group of spring, in group, the distortion of each spring is evenly equal, then have for screw bolt and nut:

$\begin{array}{c}{F}_i=\underset{n=1}{\overset{N}{\&Sigma;}}{F}_{in}\\ =F_{i+1}=\underset{m=N+1}{\overset{N+M}{\&Sigma;}}{F}_{i+1,m}\end{array}---\left(2\right)$

i＝1，3，5，7；

Model Beam 1. stress balance equation is:

$\frac{1}{2}{\mathrm{ar}}^2-F_1-\frac{A_{1-2}}{6}{(R-r)}^3=0---\left(3a\right)$

In like manner, Model Beam 3. balance equation be:

$F_1-F_3-\frac{A_{3-4}}{6}{(R-r)}^3=0---\left(3b\right)$

By that analogy, Model Beam balance equation is 5.:

$F_3-F_5-\frac{A_{5-6}}{6}{(R-r)}^3=0---\left(3c\right)$

Model Beam balance equation is 7.:

$F_5-F_7-\frac{A_{7-8}}{6}{(R-r)}^3=0---\left(3d\right)$

Model Beam 9. downside is free, and balance equation is:

$F_7-\frac{A_{9-10}}{6}{(R-r)}^3=0---\left(3e\right)$

The balance of each Model Beam of nut, is weighed by whole machine balancing and the screw model Liangping that contacts with it, obtains nature and meets;

Due to constraint, Model Beam 2. without rigid body displacement, i.e. rigid body displacement w _2,0=0, in addition, Model Beam rigid body displacement be w _{i, 0}, i=1,3,4,5 ..., 9,10; The Relative Deflection that corresponding each Model Beam produces because of moment of flexure is the Relative Deflection of equivalence contact force application point is then mock bomb spring force is:

Bolt:

F _ij＝K _j(w _i,0-w _i+2,0)j＝1,2,3...Ni＝1,3,5,7(4a)

F _i＝K ₀(w _i,0-w _i+2,0)i＝1,3,5,7(4b)

W _i+2,0be the i-th+2 Model Beam rigid body displacements;

Nut:

F _ij＝K _j(w _i,0-w _i+2,0)j＝N+1,N+2,N+3...N+Mi＝2,4,6,8(4c)

F _i＝K ₀(w _i,0-w _i+2,0)i＝2,4,6,8(4d)

By spring force by discrete form serialization, order: F _i(x)=K (x) (w _{i, 0}-w _i+2,0), wherein, for bolt, nut is meanwhile, the moment of flexure of Model Beam is unitized, obtains Model Beam in normal high widened sections section, i=1,3,5,7,9, moment of flexure is expressed as:

$M_{0i}=\frac{{(R-r)}^3{A}_{i-(i+1)}}{12}\frac{4x^3-6{\mathrm{rx}}^2-3r^{2}x}{3r^2}---\left(5a\right)$

Simultaneously:

To Model Beam i=2,4,6,8,10, in normal high widened sections section, moment of flexure is expressed as:

$M_{0i}=-\frac{{(R-r)}^3{A}_{(i-1)-i}}{12}\frac{-8x^3+(2R+4c)x^2+2(R+c)cx}{6c(c+r)}---\left(5b\right)$

Work as i=1,3,5,7,9, and time r<x< (R+r)/2, Model Beam the moment of flexure uprising widened sections section is:

${\hat{M}}_i=\frac{A_{i-<i+1>}}{12}{(R-r)}^3(\frac{R+r}{2}-x)---\left(6a\right)$

Work as i=2,4,6,8,10, and time c<x< (C+c)/2, Model Beam the moment of flexure uprising widened sections section is:

${\hat{M}}_i=-\frac{A_{(i-1)-i}}{12}{(R-r)}^3(\frac{C+c}{2}-x)---\left(6b\right)$

For uprising widened sections section, if the angle of wedge of Model Beam is β, then this segment model beam section is high is expressed as:

i＝1，3，5，7，9h(x)＝h ₀-2(x-r)tanβ(7a)

i＝2，4，6，8，10h(x)＝h ₀-2(x-c)tanβ(7b)

Therefore i=1,3,5, the Model Beam of 7,9 bending equations be expressed as:

$\frac{d^2{\hat{w}}_i}{{\mathrm{dx}}^2}=\{\begin{array}{cc}\frac{M_{0i}}{EI\left(x\right)}=\frac{{(R-r)}^3{A}_{i-(i+1)}}{{\mathrm{Eh}}_0^{3}x\mathrm{\&theta;}}\frac{4x^3-6{\mathrm{rx}}^2-3r^{2}x}{3r^2}=\frac{{(R-r)}^3{A}_{i-(i+1)}}{{\mathrm{Eh}}_0^3\mathrm{\&theta;}}\frac{4x^2-6rx-3r^2}{3r^2}& (0<x<r)\\ \frac{{\hat{M}}_i}{EI\left(x\right)}=\frac{12}{{\mathrm{Eh}}^3\left(x\right)x\mathrm{\&theta;}}\&lsqb;\frac{A_{i-<i+1>}}{12}{(R-r)}^3(\frac{R+r}{2}-x)\&rsqb;& (r<x<\frac{R+r}{2})\end{array}---\left(8a\right)$

I=2 simultaneously, 4,6, the Model Beam of 8,10 bending equations be expressed as:

$\frac{d^2{\hat{w}}_i}{{\mathrm{dx}}^2}=\left\{\begin{array}{c}\frac{M_{0i}}{EI\left(x\right)}=-\frac{{(R-r)}^3{A}_{i-(i+1)}}{{\mathrm{Eh}}_0^3\mathrm{\&theta;}(R+c-x)}\frac{-8x^3+(2R+4c)x^2+2(R+c)cx}{6c(c+r)}\\ (0<x<c)\\ \frac{{\hat{M}}_i}{EI\left(x\right)}=-\frac{12}{{\mathrm{Eh}}^3\left(x\right)\mathrm{\&theta;}(R+c-x)}\&lsqb;\frac{A_{<i-1>-i}}{12}{(R-r)}^3(\frac{C+c}{2}-x)\&rsqb;\\ (c<x<\frac{C+c}{2})\end{array}\right.---\left(8b\right)$

Boundary condition is:

Clamped end:

${\hat{w}}_i{|}_{x=0}=0,\frac{d{\hat{w}}_i}{dx}{|}_{x=0}=0---\left(9a\right)$

Tooth root portion:

$\hat{w}{|}_{x=r^{-}}=\hat{w}{|}_{x=r^{+}},\frac{d\hat{w}}{dx}{|}_{x=r^{-}}=\frac{d\hat{w}}{dx}{|}_{x=r^{+}}$ (bolt)

$\hat{w}{|}_{x=c^{-}}=\hat{w}{|}_{x=c^{+}},\frac{d\hat{w}}{dx}{|}_{x=c^{-}}=\frac{d\hat{w}}{dx}{|}_{x=c^{+}}$ (nut) (9b)

Because the normal height of the screw bolt and nut section of broadening and the interface coordinate uprising the section of broadening are x=r and x=c, and the bending equations of the right and left is different herein, therefore for the purpose of distinguishing, with represent the numerical value obtained by the normal high section of broadening bending equations, with represent by the result uprising the section of broadening and obtain;

Twice integration type (8a), (8a) consider boundary condition formula (9a), (9b):

I=1,3,5,7, when 9, bolt:

${\hat{w}}_i=\left\{\begin{array}{c}\frac{A_{i-<i+1>}}{9r^2{{\mathrm{Eh}}_0}^3\mathrm{\&theta;}}{(R-r)}^3(x^3-3{\mathrm{rx}}^2-9r^{2}x)x\\ (0<x<r)\\ \frac{A_{i-<i+1>}{(R-r)}^3}{4E\mathrm{\&theta;}(h_0+2r\tan \mathrm{\&beta;})}=\left(2\right(R+r)x\ln \frac{2x\tan \mathrm{\&beta;}}{h_0+2(r-x)\tan \mathrm{\&beta;}}+\frac{{(h_0+2r\tan \mathrm{\&beta;})}^2(h_0+(R-r)\tan \mathrm{\&beta;})}{2(h_0+2(r-x)\tan \mathrm{\&beta;}){\tan }^2\mathrm{\&beta;}})\\ \begin{array}{cc}+{\mathrm{\&Pi;}}_{1}x+{\mathrm{\&Xi;}}_1& (c<x<\frac{C+c}{2})\end{array}\end{array}\right.---\left(10a\right)$

Wherein:

${\mathrm{\&Pi;}}_1=-\frac{A_{i-<i+1>}{(R-r)}^3}{{{\mathrm{Eh}}_0}^2\mathrm{\&theta;}}\&lsqb;\frac{14r}{9h_0}+\frac{(h_0+2rtan\mathrm{\&beta;})(2Rtan\mathrm{\&beta;}-h_0)h_0+\&lsqb;2{h_0}^2(R+r)tan\mathrm{\&beta;}\&rsqb;ln\left(\frac{2rtan\mathrm{\&beta;}}{h_0}\right)}{4{(h_0+2rtan\mathrm{\&beta;})}^{3}tan\mathrm{\&beta;}}\&rsqb;$

$\begin{array}{c}{\mathrm{\&Xi;}}_1=\frac{A_{i-<i+1>}}{E\mathrm{\&theta;}}{(R-r)}^3\&lsqb;\frac{11r^2}{9{h_0}^3}-\frac{2(R+r)rln\frac{2rtan\mathrm{\&beta;}}{h_0}+\frac{{(h_0+2rtan\mathrm{\&beta;})}^2(h_0+(R-r)tan\mathrm{\&beta;})}{2h_0{\tan }^2\mathrm{\&beta;}}}{4{(h_0+2rtan\mathrm{\&beta;})}^3}\\ +\frac{A_{i-<i+1)}{(R-r)}^3}{{{\mathrm{Eh}}_0}^2\mathrm{\&theta;}}\&lsqb;\frac{14r}{9h_0}+\frac{(h_0+2rtan\mathrm{\&beta;})(2R\tan \mathrm{\&beta;}-h_0)h_0+\&lsqb;2{h_0}^2(R+r)tan\mathrm{\&beta;}\&rsqb;\ln \left(\frac{2rtan\mathrm{\&beta;}}{h_0}\right)}{4{(h_0+2rtan\mathrm{\&beta;})}^{3}tan\mathrm{\&beta;}}\&rsqb;r\end{array}---\left(10b\right)$

I=2,4,6,8, when 10, nut:

${\hat{w}}_i=\left\{\begin{array}{c}-\frac{A_{(i-1)-i}{(R-r)}^3}{6c(c+r){{\mathrm{Eh}}_0}^3\mathrm{\&theta;}}\left[\begin{array}{c}\frac{x}{3}(-6c^3-18R^3+9R^{2}x+3{\mathrm{Rx}}^2+2x^3+3c^2(-10R+x)+2c(-21R^2+6Rx+x^2))\\ -2{(c+R)}^2(c+3R)(c+R-x)\ln (R+c-x)\end{array}\right]{(R-r)}^3(x^3-3{\mathrm{rx}}^2-9r^{2}x)x\\ (0<x<r)\\ -\frac{A_{<i-1>-i}{(R-r)}^3}{2E\mathrm{\&theta;}}\&times;\\ \left\{\begin{array}{c}\frac{(2R+c-C)\&lsqb;\left(2\right(R+c-x\left)\tan \mathrm{\&beta;}\right)\ln (h_0+2(c-x\left)\tan \mathrm{\&beta;}\right)-(R+c-x)\ln (R+c-x)\&rsqb;}{2{(h_0-2R\tan \mathrm{\&beta;})}^3\tan \mathrm{\&beta;}}\\ +\frac{h_0+(c-C)\tan \mathrm{\&beta;}}{4(h_0-2R\tan \mathrm{\&beta;})(h_0+2(c-x)\tan \mathrm{\&beta;}){\tan }^2\mathrm{\&beta;}}\end{array}\right\}\\ +{\mathrm{\&Pi;}}_{2}x+{\mathrm{\&Xi;}}_2\\ (C<x<C+\frac{R-r}{2})\end{array}\right.---\left(10c\right)$

Wherein:

$\begin{array}{c}{\mathrm{\&Pi;}}_2=\frac{A_{(i-1)-i}{(R-r)}^3}{E\mathrm{\&theta;}}\&times;\\ \left[\begin{array}{c}\frac{\ln \left(\frac{R}{h_0}\right)(2R+c-C)}{2{(2R\tan \mathrm{\&beta;}-h_0)}^3}+\frac{(2R\tan \mathrm{\&beta;}-h_0)\&lsqb;(C-c)\tan \mathrm{\&beta;}-h_0\&rsqb;+2(2R+c-C)h_0\tan \mathrm{\&beta;}}{4{(2R\tan \mathrm{\&beta;}-h_0)}^2{h_0}^2\tan \mathrm{\&beta;}}\\ -\frac{20c^3+9{\mathrm{Rc}}^2+18R^{2}c-6{(c+R)}^2(c+3R)\ln R}{18c(c+r){h_0}^3}\end{array}\right]\end{array}$

${\mathrm{\&Xi;}}_2=-\frac{A_{(i-1)-i}{(R-r)}^3}{E\mathrm{\&theta;}}\left\{\begin{array}{c}\left[\begin{array}{c}\frac{c}{3}(c^3-15c^{2}R-33{\mathrm{cR}}^2-18R^3)\\ -2{(c+R)}^2(c+3R)R\ln R\end{array}\right]\frac{1}{6c(c+r){h_0}^3}\\ +\frac{(2R+c-C)\&lsqb;\left(2R\tan \mathrm{\&beta;}\right)\ln h_0-R\ln R\&rsqb;}{4{(h_0-2R\tan \mathrm{\&beta;})}^3\tan \mathrm{\&beta;}}+\frac{h_0+(c-C)\tan \mathrm{\&beta;}}{8(h_0-2R\tan \mathrm{\&beta;})h_0{\tan }^2\mathrm{\&beta;}}\\ +\left[\begin{array}{c}\frac{\ln \left(\frac{R}{h_0}\right)(2R+c-C)}{2{(2R\tan \mathrm{\&beta;}-h_0)}^3}+\frac{(2R\tan \mathrm{\&beta;}-h_0)\&lsqb;(C-c)\tan \mathrm{\&beta;}-h_0\&rsqb;+2(2R+c-C)h_0\tan \mathrm{\&beta;}}{4{(2R\tan \mathrm{\&beta;}-h_0)}^2{h_0}^2\tan \mathrm{\&beta;}}\\ -\frac{20c^3+9{\mathrm{Rc}}^2+18R^{2}c-6{(c+R)}^2(c+3R)\ln R}{18c(c+r){h_0}^3}\end{array}\right]c\end{array}\right\}---\left(10d\right)$

Because screw thread pair is in loading process, each thread of bolt remains with each thread of corresponding nut and contacts, and supposes the relative sliding in two contactless of screw thread threads contacted, and obtains deformation compatibility condition as follows:

$w_{i,0}-{\hat{w}}_i{|}_{x=\frac{R+r}{2}}=w_{i+1,0}+{\hat{w}}_{i+1}{|}_{x=\frac{C+c}{2}},i=1,3,5,7,9---\left(11\right);$

(3), the solving of beam-spring method;

Nine separate balance equations are obtained altogether by the equilibrium equation of balance equation (3a) ~ (3e) and nut corresponding with it, add formula (4a) ~ (4d) eight physical equation, ten bending equations (10a) ~ (10d), five Coordinate deformation equation (11), totally 32 equations; Wherein containing F _i, i=1,2 ..., 8; A _{<i-(i+1) >}, i=1,3,5,7,9; And w _i, 0, i=1,3,4 ..., 10; i=1,2 ..., 10; Totally 32 unknown numbers; By solving the system of equations of above-mentioned 32 equations composition, namely asking and obtaining each tooth carrying distribution;

4th step, the reduction of beam-spring model and the ruggedness test that is threaded:

By the model mechanics equation solution in the 3rd step, wherein A _{<i-(i+1) >}, wherein i=1,3,5,7,9, namely representative be threaded the 1st, 2,3,4, the stressed size of 5 teeth;

Number percent according to the first tooth carrying checks the intensity be threaded whether to meet standard, thus reaches the inspection object of the intensity that is threaded.