CN104699895B - A kind of method for calculating filmatic bearing bushing creep stress - Google Patents

A kind of method for calculating filmatic bearing bushing creep stress Download PDF

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CN104699895B
CN104699895B CN201510049757.4A CN201510049757A CN104699895B CN 104699895 B CN104699895 B CN 104699895B CN 201510049757 A CN201510049757 A CN 201510049757A CN 104699895 B CN104699895 B CN 104699895B
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msup
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王建梅
孟凡宁
梁宜楠
苗克军
张笑天
张亚南
李璞
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Taiyuan University of Science and Technology
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Abstract

A kind of method for calculating filmatic bearing bushing creep stress, belong to filmatic bearing design field, based on multidimensional steady state creep basic theory, with reference to thick cyclinder plane stress analytic method, derive creep stress calculation formula of the filmatic bearing bushing in the case where bearing oil film pressure, according to this computational methods, only it is to be understood that the external diameter of filmatic bearing bushing and internal diameter and suffered non-uniform load, you can try to achieve the creep stress value of arbitrfary point on filmatic bearing bushing section.The method have the advantages that the actual loading situation of filmatic bearing bushing has been taken into full account, with higher accuracy, simplicity and practicality.

Description

A kind of method for calculating filmatic bearing bushing creep stress
Technical field：
The invention belongs to filmatic bearing design field, and in particular to a kind of calculating filmatic bearing bushing creep stress Method.
Technical background：
Filmatic bearing is widely used in the key equipments such as steel, mine, metallurgy, electric power, space flight, aviation, with friction system Number is small, the advantages of low, rigidity is high is lost.Its most weak part is exactly bushing babbit, and babbit creep is its failure A kind of form.In the course of the work, babbit bears oil film pressure to filmatic bearing for a long time, is contacted with oil film so that Pasteur closes When golden temperature reaches 65 DEG C or so, microcreep can be produced, the greasy property and life-span to filmatic bearing produce what be can not ignore Negative effect.For the theoretical calculation of creep stress, conventional method is confined to the situation that part bears uniform load, but oil film Bearing insert inwall bears the effect of non-homogeneous oil film pressure, using conventional method calculate its creep stress obviously can not meet will Ask.
The content of the invention
It is an object of the invention to provide a kind of method for calculating filmatic bearing bushing creep stress.Based on steady under multaxial stress The basic theory of state creep, with reference to the computational methods of thick cyclinder plane problem, derives filmatic bearing bushing creep stress Computational methods.
What the present invention was realized in：As shown in figure 1, selection bushing inner boundary axially bears cutting for maximum oil film pressure Face, its external boundary is fixed by bearing block 3, and inner boundary supporting region 4 bears the effect of Radial Rotation Error oil film pressure 5.It is compacted in stable state In the change stage, influence of the temperature to creep is not considered, interception section is in plane stress state.
The invention is characterised in that calculation procedure is as follows：
1st, creep stress component is calculated：
Radial Rotation Error oil film pressure is suffered by bushing：
In formula, θ is any point radial pressure and horizontal line angle, a in inner boundary supporting region 4mFor coefficient correlation, m and n are Integer and m≤n.
According to geometry and load characteristic, choosing stress function is：
In formula, Am, Bm, CmAnd DmFor unknown parameter
A≤r≤b, a are bushing inside radius, and b is bushing outer radius
The creep stress component of filmatic bearing bushing is obtained by the derivative of stress function：
In formula, σr、σθ、τrθRespectively radial stress, circumferential stress and shear stress, its concrete form is：
2nd, creep strain component is calculated：
Multidimensional steady state creep stress analysis：Equivalent stress and equivalent creep are respectively defined as to the stress point under multaxial stress Amount and creep strain component, based on Norton steady state creep formula, obtaining steady state creep plane stress strain stress relation is：
In formula, εr、εθ、γrθRespectively radial strain, circumferential strain and shear strain
For equivalent creep stress
For equivalent creep strain
Creep stress weight expression (4) is substituted into stress-strain relation (5), creep strain weight expression is obtained：
3rd, creeping displacement component is calculated：
Geometric equation is：
Creep strain weight expression (6) is substituted into geometric equation (7), and to radial strain and the expression of circumferential strain Formula is integrated, and obtaining creeping displacement weight expression is：
In formula, urFor radial displacement
vθTo be circumferentially displaced
f1(θ)、f2(r) with ∫ f1(θ) d θ are integral constant
4th, the boundary condition of filmatic bearing bushing computation model：
5th, the unknown parameter of creep stress component is determined：
In the shear strain expression formula that formula (8) is substituted into geometric equation (7), according to boundary condition (9), creep stress is obtained The equation group of component unknown parameter：
Wherein, the unknown parameter of creep stress component is：
By the unknown parameter back substitution of creep stress component in plane into creep stress weight expression and according to section Stress state, obtains the creep stress component of filmatic bearing bushing：
In formula, σzFor axial stress.
It can be seen from above-mentioned calculation procedure, only it is to be understood that the external diameter b and internal diameter a of filmatic bearing bushing with it is suffered non-homogeneous Load, you can the creep stress component value of arbitrfary point on filmatic bearing bushing section is tried to achieve, using fourth strength theory Obtain equivalent creep stress：
Advantage of the present invention and good effect are the actual loading situations for taking into full account filmatic bearing bushing, are filmatic bearing lining The structure design of set provides more accurately design method.
Brief description of the drawings：
Fig. 1 filmatic bearing bushing creep stress computation model diagrams
Gained bushing creep stress distribution diagram in Fig. 2 examples
Bushing creep stress distribution diagram obtained by FInite Element in Fig. 3 examples
Contrast diagram of the invention with FEM calculation acquired results during r=110mm in Fig. 4 examples
Contrast diagram of the invention with FEM calculation acquired results during θ=90 ° in Fig. 5 examples
In figure：1-bushing inner boundary, 2-bushing external boundary, 3-bearing block, 4-inner boundary supporting region, 5-oil film pressure Power, 6-bushing.
Specific implementation method：
Example of the present invention is：The internal diameter of certain filmatic bearing bushing is a=110mm, and external diameter is b= 128mm, the oil film pressure for selecting bushing stress maximum cross-section under certain operating mode is：
In formula, Poil(θ) unit is MPa, 30 °≤θ≤150 °
bmFor oil film pressure parameter, wherein b1=-0.1503, b2=-0.07025, b3=-0.0364, b4=-0.01809, b5=-0.007537, b6=-0.002128
The creep stress calculation formula provided according to the present invention, calculation procedure is as follows：
1st, creep stress component is calculated：
According to geometry and load characteristic, choosing stress function is：
Substitution formula (3), the creep stress weight expression obtained under this example is：
2nd, creep strain component is calculated：
Creep stress component is substituted into formula (5), creep strain component is obtained：
3rd, creeping displacement component is calculated：
Creep strain component is substituted into formula (8), creeping displacement component is obtained：
4 list this boundary condition：
ur(r,θ+2π)-ur(r, θ)=0, vθ(r,θ+2π)-vθ(r, θ)=0
5th, the unknown parameter of creep stress component is determined：
By boundary condition, the equation group of creep stress component unknown parameter is obtained：
The unknown parameter for obtaining creep stress component is：
By a=110mm, b=128mm, b1=-0.1503, b2=-0.07025, b3=-0.0364, b4=-0.01809, b5=-0.007537, b6=-0.002128 substitutes into the unknown parameter of above-mentioned creep stress component, then substitutes into formula (12), obtains This bearing is under this operating mode, and the creep stress component of stress maximum cross-section is：
σz=0
In formula, Mrn, Nrn, KrnAnd HrnIt is the parameter of Radial creep stress, Mθn, Nθn, Kθn, and HθnIt is circumferential creep stress Parameter, n be 1~6 positive integer, above-mentioned value be following table shown in.
n 1 2 3 4 5 6
Mrn 0.066 -9.520×1010 -1.120×1015 -1.069×1019 -7.778×1022 -3.579×1026
Nrn -5.397× 106 1.430×107 1.369×1011 1.174×1015 7.997×1018 3.527×1022
Krn 1469.840 3.404×10-6 1.549×10-10 5.965×10-15 1.771×10-19 3.335×10-24
Hrn — -1.017×10-10 -6.382×10-15 -2.798×10-19 -8.897×10-24 -1.748×10-28
Mθn -0.066 9.520×1010 1.120×1015 1.069×1019 7.778×1022 3.579×1026
Nθn 5.397×106 -4.768×106 -6.845×1010 -7.041×1014 -5.332×1018 -2.519×1022
Kθn 6.911×10-6 -3.404×10-6 1.549×10-10 -5.965×10-15 -1.771×10-19 -3.335×10-24
Hθn — 3.052×10-10 1.276×10-14 4.663×10-19 1.335×10-23 2.448×10-28
According to above-mentioned result of calculation, the equivalent creep stress distribution map of bushing can be obtained, as shown in Figure 2.Utilize ANSYS Finite element software sets up two dimensional model, using implicit creep algorithm, and selecting unit solution ability is strong and supports implicit creep algorithm PLANE183 units, solve the equivalent creep stress cloud atlas of bushing supporting region as shown in figure 3, selection bushing r=110mm and θ= 90 ° of corresponding two positions, the application method and equivalent creep stress Comparative result such as Fig. 4 and Fig. 5 institutes obtained by finite element method Show.
Contrasted from Fig. 2 and Fig. 3, the application method is identical with the creep stress distribution situation of finite element method.By Fig. 4 Understood with Fig. 5, the application method is coincide with finite element method acquired results, and worst error occurs in r=110mm, θ=90 ° Position, the creep stress that the application method is tried to achieve in this position is 0.09088MPa, and the creep stress that FInite Element is tried to achieve is 0.09443MPa, its relative error is only 3.9%, shows that the creep stress that the application method is calculated is accurate, disclosure satisfy that work The actual demand of journey.
By above-mentioned calculating, as long as the application method determines the external diameter of filmatic bearing bushing, internal diameter and suffered oil film The situation of pressure, you can try to achieve the creep stress of the section every bit.The application method has taken into full account filmatic bearing bushing institute The non-uniform load situation received, is set out with loading, and calculating obtains its creep stress, only considers to bear compared to conventional method Even load, the present invention more meets reality.Meanwhile, the application method calculates simple, saves the time.

Claims (1)

1. a kind of method for calculating filmatic bearing bushing creep stress, it is characterised in that calculation procedure is as follows：
(1) creep stress component, is calculated：
Radial Rotation Error oil film pressure is suffered by bushing：
<mrow> <mi>P</mi> <mrow> <mo>(</mo> <mi>&amp;theta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> <mi>n</mi> </munderover> <msub> <mi>a</mi> <mi>m</mi> </msub> <mi>cos</mi> <mi>m</mi> <mi>&amp;theta;</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
In formula, θ is the interior any point radial pressure of inner boundary supporting region (4) and horizontal line angle, amFor coefficient correlation, m and n are whole Number and m≤n；
According to geometry and load characteristic, choosing stress function is：
<mrow> <msub> <mi>&amp;phi;</mi> <mi>m</mi> </msub> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> <mi>n</mi> </munderover> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>m</mi> </msub> <msup> <mi>r</mi> <mrow> <mi>m</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>B</mi> <mi>m</mi> </msub> <msup> <mi>r</mi> <mi>m</mi> </msup> <mo>+</mo> <msub> <mi>C</mi> <mi>m</mi> </msub> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mi>m</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>D</mi> <mi>m</mi> </msub> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mi>m</mi> </mrow> </msup> <mo>)</mo> </mrow> <mi>cos</mi> <mi>m</mi> <mi>&amp;theta;</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
In formula, Am, Bm, CmAnd DmFor unknown parameter
A≤r≤b, a are bushing inside radius, and b is bushing outer radius
The creep stress component of filmatic bearing bushing is obtained by the derivative of stress function：
<mrow> <msub> <mi>&amp;sigma;</mi> <mi>r</mi> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;phi;</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <mi>r</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <msup> <mi>r</mi> <mn>2</mn> </msup> </mfrac> <mfrac> <mrow> <msup> <mo>&amp;part;</mo> <mn>2</mn> </msup> <msub> <mi>&amp;phi;</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msup> <mi>&amp;theta;</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>,</mo> <msub> <mi>&amp;sigma;</mi> <mi>&amp;theta;</mi> </msub> <mo>=</mo> <mfrac> <mrow> <msup> <mo>&amp;part;</mo> <mn>2</mn> </msup> <msub> <mi>&amp;phi;</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msup> <mi>r</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>,</mo> <msub> <mi>&amp;tau;</mi> <mrow> <mi>r</mi> <mi>&amp;theta;</mi> </mrow> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mo>&amp;part;</mo> <mrow> <mo>&amp;part;</mo> <mi>r</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;phi;</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <mi>&amp;theta;</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
In formula, σr、σθ、τrθRespectively radial stress, circumferential stress and shear stress, its concrete form is：
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;sigma;</mi> <mi>r</mi> </msub> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> <mi>n</mi> </munderover> <mo>&amp;lsqb;</mo> <mo>-</mo> <msub> <mi>A</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>-</mo> <mi>m</mi> <mo>-</mo> <mn>2</mn> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mi>m</mi> </msup> <mo>-</mo> <msub> <mi>B</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>-</mo> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mn>2</mn> <mo>+</mo> <mi>m</mi> </mrow> </msup> <mo>-</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>C</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>m</mi> <mo>-</mo> <mn>2</mn> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mi>m</mi> </mrow> </msup> <mo>-</mo> <msub> <mi>D</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mi>m</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mo>&amp;rsqb;</mo> <mi>cos</mi> <mi>m</mi> <mi>&amp;theta;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;sigma;</mi> <mi>&amp;theta;</mi> </msub> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> <mi>n</mi> </munderover> <mo>&amp;lsqb;</mo> <msub> <mi>A</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>3</mn> <mi>m</mi> <mo>+</mo> <mn>2</mn> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mi>m</mi> </msup> <mo>+</mo> <msub> <mi>B</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>-</mo> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mn>2</mn> <mo>+</mo> <mi>m</mi> </mrow> </msup> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>C</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>-</mo> <mn>3</mn> <mi>m</mi> <mo>+</mo> <mn>2</mn> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mi>m</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>D</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mi>m</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mo>&amp;rsqb;</mo> <mi>cos</mi> <mi>m</mi> <mi>&amp;theta;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;tau;</mi> <mrow> <mi>r</mi> <mi>&amp;theta;</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> <mi>n</mi> </munderover> <mo>&amp;lsqb;</mo> <msub> <mi>A</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mi>m</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>B</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>-</mo> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mn>2</mn> <mo>+</mo> <mi>m</mi> </mrow> </msup> <mo>-</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>C</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>-</mo> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mi>m</mi> </mrow> </msup> <mo>-</mo> <msub> <mi>D</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mi>m</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mo>&amp;rsqb;</mo> <mi>sin</mi> <mi>m</mi> <mi>&amp;theta;</mi> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
(2) creep strain component, is calculated：
Multidimensional steady state creep stress analysis：By equivalent stress and equivalent creep be respectively defined as the components of stress under multaxial stress and Creep strain component, based on Norton steady state creep formula, obtaining steady state creep plane stress strain stress relation is：
<mrow> <msub> <mi>&amp;epsiv;</mi> <mi>r</mi> </msub> <mo>=</mo> <mfrac> <mover> <mi>&amp;epsiv;</mi> <mo>&amp;OverBar;</mo> </mover> <mover> <mi>&amp;sigma;</mi> <mo>&amp;OverBar;</mo> </mover> </mfrac> <mrow> <mo>(</mo> <msub> <mi>&amp;sigma;</mi> <mi>r</mi> </msub> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mi>&amp;sigma;</mi> <mi>&amp;theta;</mi> </msub> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>&amp;epsiv;</mi> <mi>&amp;theta;</mi> </msub> <mo>=</mo> <mfrac> <mover> <mi>&amp;epsiv;</mi> <mo>&amp;OverBar;</mo> </mover> <mover> <mi>&amp;sigma;</mi> <mo>&amp;OverBar;</mo> </mover> </mfrac> <mrow> <mo>(</mo> <msub> <mi>&amp;sigma;</mi> <mi>&amp;theta;</mi> </msub> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mi>&amp;sigma;</mi> <mi>r</mi> </msub> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>&amp;gamma;</mi> <mrow> <mi>r</mi> <mi>&amp;theta;</mi> </mrow> </msub> <mo>=</mo> <mn>3</mn> <mfrac> <mover> <mi>&amp;epsiv;</mi> <mo>&amp;OverBar;</mo> </mover> <mover> <mi>&amp;sigma;</mi> <mo>&amp;OverBar;</mo> </mover> </mfrac> <msub> <mi>&amp;tau;</mi> <mrow> <mi>r</mi> <mi>&amp;theta;</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
In formula, εr、εθ、γrθRespectively radial strain, circumferential strain and shear strain
For equivalent creep stress
For equivalent creep strain
Creep stress weight expression (4) is substituted into stress-strain relation (5), creep strain weight expression is obtained：
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;epsiv;</mi> <mi>r</mi> </msub> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> <mi>n</mi> </munderover> <mo>&amp;lsqb;</mo> <mfrac> <mover> <mi>&amp;epsiv;</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mn>2</mn> <mover> <mi>&amp;sigma;</mi> <mo>&amp;OverBar;</mo> </mover> </mrow> </mfrac> <mrow> <mo>(</mo> <mo>-</mo> <msub> <mi>A</mi> <mi>m</mi> </msub> <mo>(</mo> <mn>3</mn> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>m</mi> <mo>-</mo> <mn>2</mn> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mi>m</mi> </msup> <mo>-</mo> <msub> <mi>B</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mn>3</mn> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>-</mo> <mn>3</mn> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mn>2</mn> <mo>+</mo> <mi>m</mi> </mrow> </msup> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>C</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mo>-</mo> <mn>3</mn> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>m</mi> <mo>+</mo> <mn>2</mn> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mi>m</mi> </mrow> </msup> <mo>-</mo> <msub> <mi>D</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mn>3</mn> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>3</mn> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mn>2</mn> <mo>-</mo> <mi>m</mi> </mrow> </msup> <mo>)</mo> <mo>&amp;rsqb;</mo> <mi>cos</mi> <mi>m</mi> <mi>&amp;theta;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;epsiv;</mi> <mi>&amp;theta;</mi> </msub> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> <mi>n</mi> </munderover> <mo>&amp;lsqb;</mo> <mfrac> <mover> <mi>&amp;epsiv;</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mn>2</mn> <mover> <mi>&amp;sigma;</mi> <mo>&amp;OverBar;</mo> </mover> </mrow> </mfrac> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>m</mi> </msub> <mo>(</mo> <mn>3</mn> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>5</mn> <mi>m</mi> <mo>+</mo> <mn>2</mn> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mi>m</mi> </msup> <mo>+</mo> <msub> <mi>B</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mn>3</mn> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>-</mo> <mn>3</mn> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mn>2</mn> <mo>+</mo> <mi>m</mi> </mrow> </msup> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>C</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mn>3</mn> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>-</mo> <mn>5</mn> <mi>m</mi> <mo>+</mo> <mn>2</mn> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mi>m</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>D</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mn>3</mn> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>3</mn> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mn>2</mn> <mo>-</mo> <mi>m</mi> </mrow> </msup> <mo>)</mo> <mo>&amp;rsqb;</mo> <mi>cos</mi> <mi>m</mi> <mi>&amp;theta;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;gamma;</mi> <mrow> <mi>r</mi> <mi>&amp;theta;</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> <mi>n</mi> </munderover> <mo>&amp;lsqb;</mo> <mfrac> <mrow> <mn>3</mn> <mover> <mi>&amp;epsiv;</mi> <mo>&amp;OverBar;</mo> </mover> </mrow> <mover> <mi>&amp;sigma;</mi> <mo>&amp;OverBar;</mo> </mover> </mfrac> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>m</mi> </msub> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mi>m</mi> </msup> <mo>+</mo> <msub> <mi>B</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>-</mo> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mn>2</mn> <mo>+</mo> <mi>m</mi> </mrow> </msup> <mo>-</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>C</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>-</mo> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mi>m</mi> </mrow> </msup> <mo>-</mo> <msub> <mi>D</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mi>m</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mo>)</mo> <mo>&amp;rsqb;</mo> <mi>sin</mi> <mi>m</mi> <mi>&amp;theta;</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
(3) creeping displacement component, is calculated：
Geometric equation is：
<mrow> <msub> <mi>&amp;epsiv;</mi> <mi>r</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>u</mi> <mi>r</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <mi>r</mi> </mrow> </mfrac> <mo>,</mo> <msub> <mi>&amp;epsiv;</mi> <mi>&amp;theta;</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>v</mi> <mi>&amp;theta;</mi> </msub> </mrow> <mrow> <mi>r</mi> <mo>&amp;part;</mo> <mi>&amp;theta;</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <msub> <mi>u</mi> <mi>r</mi> </msub> <mi>r</mi> </mfrac> <mo>,</mo> <msub> <mi>&amp;gamma;</mi> <mrow> <mi>r</mi> <mi>&amp;theta;</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>v</mi> <mi>&amp;theta;</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <mi>r</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>u</mi> <mi>r</mi> </msub> </mrow> <mrow> <mi>r</mi> <mo>&amp;part;</mo> <mi>&amp;theta;</mi> </mrow> </mfrac> <mo>-</mo> <mfrac> <msub> <mi>v</mi> <mi>&amp;theta;</mi> </msub> <mi>r</mi> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
Creep strain weight expression (6) is substituted into geometric equation (7), and the expression formula of radial strain and circumferential strain is accumulated Point, obtaining creeping displacement weight expression is：
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>u</mi> <mi>r</mi> </msub> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> <mi>n</mi> </munderover> <mo>&amp;lsqb;</mo> <mfrac> <mi>&amp;epsiv;</mi> <mrow> <mn>2</mn> <mover> <mi>&amp;sigma;</mi> <mo>&amp;OverBar;</mo> </mover> </mrow> </mfrac> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>m</mi> </msub> <mo>(</mo> <mn>2</mn> <mo>-</mo> <mn>3</mn> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mn>1</mn> <mo>+</mo> <mi>m</mi> </mrow> </msup> <mo>-</mo> <msub> <mi>B</mi> <mi>m</mi> </msub> <mn>3</mn> <msup> <mi>mr</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo>+</mo> <mi>m</mi> </mrow> </msup> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>C</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mn>1</mn> <mo>-</mo> <mi>m</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>D</mi> <mi>m</mi> </msub> <mn>3</mn> <msup> <mi>mr</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo>-</mo> <mi>m</mi> </mrow> </msup> <mo>)</mo> <mo>&amp;rsqb;</mo> <mi>cos</mi> <mi>m</mi> <mi>&amp;theta;</mi> <mo>+</mo> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>&amp;theta;</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>v</mi> <mi>&amp;theta;</mi> </msub> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> <mi>n</mi> </munderover> <mo>&amp;lsqb;</mo> <mfrac> <mover> <mi>&amp;epsiv;</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mn>2</mn> <mover> <mi>&amp;sigma;</mi> <mo>&amp;OverBar;</mo> </mover> </mrow> </mfrac> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>m</mi> </msub> <mo>(</mo> <mn>8</mn> <mo>+</mo> <mn>3</mn> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mn>1</mn> <mo>+</mo> <mi>m</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>B</mi> <mi>m</mi> </msub> <mn>3</mn> <msup> <mi>mr</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo>+</mo> <mi>m</mi> </mrow> </msup> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>C</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mo>-</mo> <mn>8</mn> <mo>+</mo> <mn>3</mn> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mn>1</mn> <mo>-</mo> <mi>m</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>D</mi> <mi>m</mi> </msub> <mn>3</mn> <msup> <mi>mr</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo>-</mo> <mi>m</mi> </mrow> </msup> <mo>)</mo> <mo>&amp;rsqb;</mo> <mi>sin</mi> <mi>m</mi> <mi>&amp;theta;</mi> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>&amp;Integral;</mo> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>&amp;theta;</mi> <mo>)</mo> </mrow> <mi>d</mi> <mi>&amp;theta;</mi> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>
In formula, urFor radial displacement
vθTo be circumferentially displaced
f1(θ)、f2(r) with ∫ f1(θ) d θ are integral constant
(4), the boundary condition of filmatic bearing bushing computation model：
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mrow> <mo>(</mo> <msub> <mi>&amp;sigma;</mi> <mi>r</mi> </msub> <mo>)</mo> </mrow> <mrow> <mi>r</mi> <mo>=</mo> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mo>-</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> <mi>n</mi> </munderover> <msub> <mi>a</mi> <mi>m</mi> </msub> <mi>cos</mi> <mi>m</mi> <mi>&amp;theta;</mi> <mo>,</mo> <msub> <mrow> <mo>(</mo> <msub> <mi>&amp;tau;</mi> <mrow> <mi>r</mi> <mi>&amp;theta;</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow> <mi>r</mi> <mo>=</mo> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mrow> <mo>(</mo> <msub> <mi>u</mi> <mi>r</mi> </msub> <mo>)</mo> </mrow> <mrow> <mi>r</mi> <mo>=</mo> <mi>b</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mrow> <mo>(</mo> <msub> <mi>v</mi> <mi>&amp;theta;</mi> </msub> <mo>)</mo> </mrow> <mrow> <mi>r</mi> <mo>=</mo> <mi>b</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>u</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <mi>r</mi> <mo>,</mo> <mi>&amp;theta;</mi> <mo>+</mo> <mn>2</mn> <mi>&amp;pi;</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>u</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <mi>r</mi> <mo>,</mo> <mi>&amp;theta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>v</mi> <mi>&amp;theta;</mi> </msub> <mrow> <mo>(</mo> <mi>r</mi> <mo>,</mo> <mi>&amp;theta;</mi> <mo>+</mo> <mn>2</mn> <mi>&amp;pi;</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>v</mi> <mi>&amp;theta;</mi> </msub> <mrow> <mo>(</mo> <mi>r</mi> <mo>,</mo> <mi>&amp;theta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>
(5) unknown parameter of creep stress component, is determined：
In the shear strain expression formula that formula (8) is substituted into geometric equation (7), according to boundary condition (9), creep stress component is obtained The equation group of unknown parameter：
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>A</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>-</mo> <mi>m</mi> <mo>-</mo> <mn>2</mn> <mo>)</mo> </mrow> <msup> <mi>a</mi> <mi>m</mi> </msup> <mo>+</mo> <msub> <mi>B</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>-</mo> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>a</mi> <mrow> <mo>-</mo> <mn>2</mn> <mo>+</mo> <mi>m</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>C</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>m</mi> <mo>-</mo> <mn>2</mn> <mo>)</mo> </mrow> <msup> <mi>a</mi> <mrow> <mo>-</mo> <mi>m</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>D</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>a</mi> <mrow> <mo>-</mo> <mi>m</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>a</mi> <mi>m</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>A</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>a</mi> <mi>m</mi> </msup> <mo>+</mo> <msub> <mi>B</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>-</mo> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>a</mi> <mrow> <mo>-</mo> <mn>2</mn> <mo>+</mo> <mi>m</mi> </mrow> </msup> <mo>-</mo> <msub> <mi>C</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>-</mo> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>a</mi> <mrow> <mo>-</mo> <mi>m</mi> </mrow> </msup> <mo>-</mo> <msub> <mi>D</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>a</mi> <mrow> <mo>-</mo> <mi>m</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>A</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mn>2</mn> <mo>-</mo> <mn>3</mn> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>b</mi> <mrow> <mn>1</mn> <mo>+</mo> <mi>m</mi> </mrow> </msup> <mo>-</mo> <msub> <mi>B</mi> <mi>m</mi> </msub> <mn>3</mn> <msup> <mi>mb</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo>+</mo> <mi>m</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>C</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>b</mi> <mrow> <mn>1</mn> <mo>-</mo> <mi>m</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>D</mi> <mi>m</mi> </msub> <mn>3</mn> <msup> <mi>mb</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo>-</mo> <mi>m</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>A</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mn>8</mn> <mo>+</mo> <mn>3</mn> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>b</mi> <mrow> <mn>1</mn> <mo>+</mo> <mi>m</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>B</mi> <mi>m</mi> </msub> <mn>3</mn> <msup> <mi>mb</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo>+</mo> <mi>m</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>C</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mo>-</mo> <mn>8</mn> <mo>+</mo> <mn>3</mn> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>b</mi> <mrow> <mn>1</mn> <mo>-</mo> <mi>m</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>D</mi> <mi>m</mi> </msub> <mn>3</mn> <msup> <mi>mb</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo>-</mo> <mi>m</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>
Wherein, the unknown parameter of creep stress component is：
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>A</mi> <mi>m</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <msup> <mi>a</mi> <mrow> <mn>2</mn> <mo>+</mo> <mi>m</mi> </mrow> </msup> <mo>&amp;lsqb;</mo> <mo>-</mo> <mn>5</mn> <msup> <mi>a</mi> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </msup> <msup> <mi>b</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>3</mn> <msup> <mi>a</mi> <mn>2</mn> </msup> <msup> <mi>b</mi> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </msup> <mi>m</mi> <mo>-</mo> <mn>3</mn> <msup> <mi>b</mi> <mrow> <mn>2</mn> <mi>m</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mi>m</mi> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>a</mi> <mi>m</mi> </msub> </mrow> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mi>m</mi> <mo>)</mo> </mrow> <mo>&amp;lsqb;</mo> <mn>15</mn> <msup> <mi>a</mi> <mrow> <mn>2</mn> <mo>+</mo> <mn>4</mn> <mi>m</mi> </mrow> </msup> <msup> <mi>b</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>15</mn> <msup> <mi>a</mi> <mn>2</mn> </msup> <msup> <mi>b</mi> <mrow> <mn>2</mn> <mo>+</mo> <mn>4</mn> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>a</mi> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </msup> <msup> <mi>b</mi> <mrow> <mn>2</mn> <mi>m</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mn>9</mn> <msup> <mi>a</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>8</mn> <msup> <mi>b</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mo>+</mo> <mn>9</mn> <msup> <mi>a</mi> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </msup> <msup> <mi>b</mi> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </msup> <msup> <mrow> <mo>(</mo> <msup> <mi>a</mi> <mn>2</mn> </msup> <mo>-</mo> <msup> <mi>b</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>&amp;rsqb;</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>B</mi> <mi>m</mi> </msub> <mo>=</mo> <mfrac> <mrow> <msup> <mi>a</mi> <mrow> <mn>2</mn> <mo>+</mo> <mi>m</mi> </mrow> </msup> <msup> <mi>b</mi> <mn>2</mn> </msup> <mo>&amp;lsqb;</mo> <mn>15</mn> <msup> <mi>a</mi> <mrow> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mi>m</mi> </mrow> </msup> <mi>m</mi> <mo>-</mo> <mn>9</mn> <msup> <mi>a</mi> <mn>2</mn> </msup> <msup> <mi>b</mi> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </msup> <mrow> <mo>(</mo> <mo>-</mo> <mi>m</mi> <mo>+</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>b</mi> <mrow> <mn>2</mn> <mi>m</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mn>16</mn> <mo>+</mo> <mn>9</mn> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>a</mi> <mi>m</mi> </msub> </mrow> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mo>-</mo> <mi>m</mi> <mo>+</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mo>&amp;lsqb;</mo> <mn>15</mn> <msup> <mi>a</mi> <mrow> <mn>2</mn> <mo>+</mo> <mn>4</mn> <mi>m</mi> </mrow> </msup> <msup> <mi>b</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>15</mn> <msup> <mi>a</mi> <mn>2</mn> </msup> <msup> <mi>b</mi> <mrow> <mn>2</mn> <mo>+</mo> <mn>4</mn> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>a</mi> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </msup> <msup> <mi>b</mi> <mrow> <mn>2</mn> <mi>m</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mn>9</mn> <msup> <mi>a</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>8</mn> <msup> <mi>b</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mo>+</mo> <mn>9</mn> <msup> <mi>a</mi> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </msup> <msup> <mi>b</mi> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </msup> <msup> <mrow> <mo>(</mo> <msup> <mi>a</mi> <mn>2</mn> </msup> <mo>-</mo> <msup> <mi>b</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>&amp;rsqb;</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mfenced>
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>C</mi> <mi>m</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <msup> <mi>a</mi> <mrow> <mn>2</mn> <mo>+</mo> <mi>m</mi> </mrow> </msup> <msup> <mi>b</mi> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </msup> <mo>&amp;lsqb;</mo> <mn>5</mn> <msup> <mi>b</mi> <mrow> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mi>a</mi> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </msup> <msup> <mi>b</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>m</mi> <mo>)</mo> </mrow> <mo>+</mo> <mn>3</mn> <msup> <mi>a</mi> <mrow> <mn>2</mn> <mi>m</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mi>m</mi> <mo>&amp;rsqb;</mo> <msub> <mi>a</mi> <mi>m</mi> </msub> </mrow> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mo>-</mo> <mn>1</mn> <mo>+</mo> <mi>m</mi> <mo>)</mo> </mrow> <mo>&amp;lsqb;</mo> <mn>15</mn> <msup> <mi>a</mi> <mrow> <mn>2</mn> <mo>+</mo> <mn>4</mn> <mi>m</mi> </mrow> </msup> <msup> <mi>b</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>15</mn> <msup> <mi>a</mi> <mn>2</mn> </msup> <msup> <mi>b</mi> <mrow> <mn>2</mn> <mo>+</mo> <mn>4</mn> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>a</mi> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </msup> <msup> <mi>b</mi> <mrow> <mn>2</mn> <mi>m</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mn>9</mn> <msup> <mi>a</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>8</mn> <msup> <mi>b</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mo>+</mo> <mn>9</mn> <msup> <mi>a</mi> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </msup> <msup> <mi>b</mi> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </msup> <msup> <mrow> <mo>(</mo> <msup> <mi>a</mi> <mn>2</mn> </msup> <mo>-</mo> <msup> <mi>b</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>&amp;rsqb;</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>D</mi> <mi>m</mi> </msub> <mo>=</mo> <mfrac> <mrow> <msup> <mi>a</mi> <mrow> <mn>2</mn> <mo>+</mo> <mi>m</mi> </mrow> </msup> <msup> <mi>b</mi> <mrow> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mi>m</mi> </mrow> </msup> <mo>&amp;lsqb;</mo> <mo>-</mo> <mn>15</mn> <msup> <mi>a</mi> <mn>2</mn> </msup> <msup> <mi>b</mi> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </msup> <mi>m</mi> <mo>-</mo> <mn>9</mn> <msup> <mi>a</mi> <mrow> <mn>2</mn> <mi>m</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mi>m</mi> <mo>+</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>a</mi> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </msup> <msup> <mi>b</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mn>16</mn> <mo>+</mo> <mn>9</mn> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>a</mi> <mi>m</mi> </msub> </mrow> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mi>m</mi> <mo>+</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mo>&amp;lsqb;</mo> <mn>15</mn> <msup> <mi>a</mi> <mrow> <mn>2</mn> <mo>+</mo> <mn>4</mn> <mi>m</mi> </mrow> </msup> <msup> <mi>b</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>15</mn> <msup> <mi>a</mi> <mn>2</mn> </msup> <msup> <mi>b</mi> <mrow> <mn>2</mn> <mo>+</mo> <mn>4</mn> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>a</mi> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </msup> <msup> <mi>b</mi> <mrow> <mn>2</mn> <mi>m</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mn>9</mn> <msup> <mi>a</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>8</mn> <msup> <mi>b</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mo>+</mo> <mn>9</mn> <msup> <mi>a</mi> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </msup> <msup> <mi>b</mi> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </msup> <msup> <mrow> <mo>(</mo> <msup> <mi>a</mi> <mn>2</mn> </msup> <mo>-</mo> <msup> <mi>b</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>&amp;rsqb;</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>
By the unknown parameter back substitution of creep stress component in plane stress into creep stress weight expression and according to section State, obtains the creep stress component of filmatic bearing bushing：
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;sigma;</mi> <mi>r</mi> </msub> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> <mi>n</mi> </munderover> <mo>&amp;lsqb;</mo> <mo>-</mo> <msub> <mi>A</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>-</mo> <mi>m</mi> <mo>-</mo> <mn>2</mn> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mi>m</mi> </msup> <mo>-</mo> <msub> <mi>B</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>-</mo> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mn>2</mn> <mo>+</mo> <mi>m</mi> </mrow> </msup> <mo>-</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>C</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>m</mi> <mo>-</mo> <mn>2</mn> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mi>m</mi> </mrow> </msup> <mo>-</mo> <msub> <mi>D</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mi>m</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mo>&amp;rsqb;</mo> <mi>cos</mi> <mi>m</mi> <mi>&amp;theta;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;sigma;</mi> <mi>&amp;theta;</mi> </msub> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> <mi>n</mi> </munderover> <mo>&amp;lsqb;</mo> <msub> <mi>A</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>3</mn> <mi>m</mi> <mo>+</mo> <mn>2</mn> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mi>m</mi> </msup> <mo>+</mo> <msub> <mi>B</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>-</mo> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mn>2</mn> <mo>+</mo> <mi>m</mi> </mrow> </msup> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>C</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>-</mo> <mn>3</mn> <mi>m</mi> <mo>+</mo> <mn>2</mn> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mi>m</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>D</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>m</mi> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mi>m</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mo>&amp;rsqb;</mo> <mi>cos</mi> <mi>m</mi> <mi>&amp;theta;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;sigma;</mi> <mi>z</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>
In formula, σzFor axial stress.
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