CN101832881A - Method for detecting dynamic characteristics of fixing combination portion of machine tool - Google Patents

Abstract

The invention discloses a method for detecting the dynamic characteristics of a fixing combination portion of a machine tool. In the method, the fixing combination portion is taken as a virtual material with equal section, and the virtual material and the parts at two sides of the fixing combination portion are in rigid connection. Through detecting the parameters of elastic modulus, the Poisson ratio, the density, the yield strength, the hardness and the geometric size of parts constituting the combination portion part, the elastic modulus, the Poisson ratio and the thickness of the virtual material are obtained, the virtual material and the two parts at two sides of the fixing combination portion are in rigid connection; the three calculated parameters of the elastic modulus, the Poisson ratio and the density of the virtual material are input to a finite element software, and some dynamic characteristics (such as, mode of vibration, natural frequency, displacement, and the like) of the complex parts of the combination portion are detected. The relative error of the theory and the natural frequency of the first 6-step of an experiment is (-10 to 10) %, while the maximum relative error of the traditional influential Yoshimura model is about 4 times that of the virtual material mode. When damage experiments are not directly allowed for some precise digital control machine tools, and the experimental cost can be greatly reduced by using the predictor method of the theory.

Description

A kind of detection method of dynamic characteristics of fixing combination portion of machine tool

Technical field

The invention belongs to the technical field that the machine dynamic characteristics parameter is analyzed, relate to a kind of dynamic perfromance detection method of fixing combination portion of machine tool.

Background technology

Complete machine structure is made up of the many parts that link together by many joint portions.If consider mechanical characteristic (as pretightning force), physical attribute (as size, surface working mode, material behavior etc.), be difficult to set up the dynamic (dynamical) complex relationship in joint portion comprehensively.Traditional experimental modal analysis system has been widely used on the Modal Parameter Identification of numerical control equipment structure, belong to known in this area, for example see that Chinese patent application number is: 200610171541.6, denomination of invention is: the numerical control equipment processing dynamics is learned the characteristic test analytic system, the software platform of a whole set of test analysis of analyzing based on input/output signal.Chinese patent application number is: 87107568.7, and denomination of invention is: model analysis computer optimization algorithm provides hypothesis to be input as the modal parameter extracting method of white noise.More than these systems or algorithm all be by testing equipment test and excitation response signal, or the hypothesis situation that is input as white noise gets off to carry out.And the method for carrying out the detection of the virtual material build-in attribute of fixing combination portion of machine tool parametrization belongs to innovation in this area.

Summary of the invention:

Purpose of the present invention is intended to overcome the deficiency of prior art, and a kind of detection method of dynamic characteristics of fixing combination portion of machine tool is provided, and the error that this method is brought is less.

The detection method of a kind of dynamic characteristics of fixing combination portion of machine tool provided by the invention is characterized in that, this method comprises the steps:

The 1st step was considered as a kind of prismatic virtual material respectively with each fixing combination portion on the lathe, obtained elastic modulus, Poisson ratio and the density of each virtual material;

Be respectively first part and second part if constitute two parts of fixing combination portion, then the computation process of elastic modulus, Poisson ratio and the density of the pairing virtual material of this fixing combination portion is:

Step (1.1): the elastic modulus E and the shear modulus G that obtain virtual material _x:

Utilize formula 1., 2. formula calculate the elastic modulus of virtual material:

The elastic modulus E function representation of virtual material is:

$E(A_r>A_{\mathrm{rc}})=\frac{2\mathrm{<figure-callout\; id="3"\; label="D"\; filenames="HDA0000021381540000022.png,HDA0000021381540000071.png"\; state="\{\{state\}\}">D</figure-callout>}}{3{\mathrm{\&pi;}}^2}{E}^{\&prime;}{G}^{1-D}{a}_L^{0.5D}(a_c^{-0.5}-a_L^{-0.5})$ ①

The normal load P function representation of virtual material is:

$P(A_r>A_{\mathrm{rc}})=\left\{\begin{array}{cc}\frac{4\sqrt{\mathrm{\&pi;}}D}{3(3-2D)}{E}^{\&prime;}{G}^{D-1}{a}_L^{0.5D}(a_L^{1.5-D}-a_c^{1.5-D})+\frac{D}{2-D}{\mathrm{K\&sigma;}}_y{a}_L^{0.5D}-a_c^{1-0.5D}& \mathrm{forD}\&NotEqual;1.5\\ \sqrt{\mathrm{\&pi;G}}{E}^{\&prime;}{a}_L^{0.75}\ln \frac{a_L}{a_c}+3K{\mathrm{\&sigma;}}_y{a}_L^{0.75}{a}_c^{0.25}& \mathrm{forD}=1.5\end{array}\right.$ ②

Wherein, A _rRepresent first, second part real contact area, A _ReRepresent the actual critical contact area of first, second part, D represents the surface profile fractal dimension of first part or second part, the contacted equivalent elastic modulus of first, second part of E ' expression, G represents the fractal roughness parameter of first part or second part, a _LThe area of representing the little contact point of maximum flexibility on first, second part surface of contact, a _eThe critical area of representing first, second part surface of contact division bullet, plastic region; K represents related coefficient, σ _yRepresent the low YIELD STRENGTH of hardness in first part and second part;

The shear modulus G of virtual material _xFor

$G_x(A_r>A_{\mathrm{rc}})=\frac{16}{\mathrm{\&pi;}}\sqrt[3]{1-\frac{\mathrm{\&beta;}}{f}}{G}^{\&prime;}[{\left(\frac{a_L}{a_c}\right)}^{0.5D}-1]$ ③

Wherein, β represents the ratio of circumferential load and normal load, and f represents friction factor, the equivalent shear modulus of G ' expression two contact rough surfaces

Step (1.2): according to formula 4., calculate the Poisson ratio υ of virtual material:

$\mathrm{\&upsi;}=\frac{(1+{\mathrm{\&mu;}}^{\&prime;})E^{*}}{G_x^{*}}-1$ ④

Wherein, E ^*The elastic modulus of representing corresponding nondimensional virtual material, 5. the employing formula is calculated:

$E^{*}(A_r^{*}>A_{\mathrm{rc}}^{*})=\frac{2\mathrm{<figure-callout\; id="3"\; label="D"\; filenames="HDA0000021381540000022.png,HDA0000021381540000071.png"\; state="\{\{state\}\}">D</figure-callout>}}{3{\mathrm{\&pi;}}^2}{G}^{*1-D}{\left(\frac{2-D}{D}{A}_r^{*}\right)}^{0.5D}[a_c^{*-0.5}-{\left(\frac{2-D}{D}{A}_r^{*}\right)}^{-0.5}]$ ⑤

G _x ^*The shear modulus of the virtual material of expression corresponding dimensionless, 6. the employing formula is calculated:

$G_x^{*}(A_r^{*}>A_{\mathrm{rc}}^{*})=\frac{16}{\mathrm{\&pi;}}\sqrt[3]{1-\frac{\mathrm{\&beta;}}{f}}[{\left(\frac{2-D}{D}\frac{A_r^{*}}{a_c^{*}}\right)}^{0.5D}-1]$ ⑥

Wherein, A _r ^*Expression dimensionless real contact area, A _Rc ^*The actual critical contact area of expression dimensionless, a _c ^*The critical area that the expression dimensionless is divided bullet, plastic region;

Step (1.3): the density that 7. calculates virtual material according to formula.

The density p of virtual material is

$\mathrm{\&rho;}=\frac{{\mathrm{\&rho;}}_1{\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="HDA0000021381540000071.png"\; state="\{\{state\}\}">h</figure-callout>}}_1+{\mathrm{\&rho;}}_2{h}_2}{{\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="HDA0000021381540000071.png"\; state="\{\{state\}\}">h</figure-callout>}}_1+{\mathrm{<figure-callout\; id="2"\; label="h"\; filenames="HDA0000021381540000022.png,HDA0000021381540000071.png"\; state="\{\{state\}\}">h</figure-callout>}}_2}$ ⑦

ρ ₁, ρ ₂The density of representing first, second part respectively, h ₁, h ₂The normal direction height of representing first, second part respectively;

Elastic modulus, Poisson ratio and the density of the 2nd step with each virtual material is input to respectively in the finite element software, obtains the finite element model of the pairing fixing combination portion of each virtual material;

The 3rd step was utilized the finite element model of each fixing combination portion, calculated the lathe complete machine dynamic perfromance that comprises the influence of lathe joint portion according to rigidly connected method between each part.

Promptly, reach the purpose that fixing combination portion of machine tool detects by elastic modulus, Poisson ratio, density, yield strength, hardness and the physical dimension parameter of experimental measurement machine tool component.Implement the method for the parametrization detection of a kind of fixing combination portion of machine tool provided by the invention, its beneficial effect is: the present invention has broken through the constraint of existing traditional joint portion spring-damper theoretical model, joint portion experimental formula, proposed a cover detection method of joint portion parameter, particularly had breakthrough aspect the Bolt Connection fixing combination portion parameter detecting at fixing combination portion; In addition, to some precise numerical control machines, when not allowing directly it to be carried out damaging breaking test, use this method, experimental cost can reduce greatly.And the error that this detection method is brought is less

Description of drawings

Fig. 1 is the synoptic diagram of three class types of attachment of fixing combination portion of machine tool, wherein, is that crossbeam-column connects (a), (b) is that crossbeam-guide rail connects, and (c) is that slide seat body-slide block connects;

Fig. 2 is the synoptic diagram that contains the complex component of a joint portion;

Fig. 3 is the simple part synoptic diagram that contains a kind of virtual material;

Fig. 4 is the Elastic Contact synoptic diagram of the little contact point of two spheroids, wherein, is that the contact of the little contact point of two spheroids does not have distortion (a), (b) is that the little contact point contact of two spheroids has distortion, (c) is the little contact point of equivalent sphere and the contacting of rigid plane;

Fig. 5 is tangential little cunning of the little contact point of two spheroids and the synoptic diagram that adheres to contact;

Fig. 6 is a dumbbell shaped test test specimen synoptic diagram;

Fig. 7 is the synoptic diagram of kinetic test system.

Embodiment

Joint portion of the present invention detection method can be predicted the dynamic perfromance of complete machine structure, the general type of attachment of fixing combination portion of machine tool as shown in Figure 1, wherein L is the length of outer joint portion, b is the wide of outer joint portion, L ₁Be the length of interior joint portion, b ₁Wide for interior joint portion.

Dynamic characteristics of fixing combination portion of machine tool is meant and is connected a kind of response of part under external excitation, fixing combination portion refers to the bolt joint portion more, be connected part and be generally 2, the dynamic perfromance of fixing combination portion of machine tool comprises rigidity, damping of the vibration shape of complete machine, natural frequency, joint portion etc.

The parts that contain a fixing combination portion have relative displacement between the part 1,2 as shown in Figure 2.Now fixing combination portion 3 is regarded as a kind of prismatic virtual material 4, by detecting the attribute (being elastic modulus, Poisson ratio, density, yield strength, hardness and physical dimension parameter) of part 1,2, calculate elastic modulus, Poisson ratio, density, the thickness of this virtual material.The part 1,2 of supposing virtual material and fixing combination portion both sides is rigidly connected, and can be a part with the complex component equivalence of Fig. 2 by increasing an element (virtual material 4) like this, the fixing combination portion problem can be oversimplified.The simple part that contains a kind of virtual material as shown in Figure 3.Among Fig. 3, h represents the thickness of virtual material.

Described above is the general case of a fixing combination portion.In the reality, a lathe has a plurality of fixing combination portions certainly, because utilize the characteristic of classic method single part to be easy to solve, actual fixing combination portion of machine tool has several fixing combination portions, corresponding just have several virtual materials, and the parameter of grasping these several virtual materials just can detect the dynamic perfromance of complete machine.

The joint portion detection method comprises following 5 steps:

Step (1): elastic modulus and the shear modulus of obtaining virtual material

Method by experiment only need detect the attribute of constitutional detail 1,2, by simultaneous formula (1) and formula (2), can calculate the elastic modulus of virtual material,

The elastic modulus E function representation of virtual material:

$E(A_r>A_{\mathrm{rc}})=\frac{2\mathrm{<figure-callout\; id="3"\; label="D"\; filenames="HDA0000021381540000022.png,HDA0000021381540000071.png"\; state="\{\{state\}\}">D</figure-callout>}}{3{\mathrm{\&pi;}}^2}{E}^{\&prime;}{G}^{1-D}{a}_L^{0.5D}(a_c^{-0.5}-a_L^{-0.5})---\left(1\right)$

The normal load P function representation of virtual material:

$P(A_r>A_{\mathrm{rc}})=\left\{\begin{array}{cc}\frac{4\sqrt{\mathrm{\&pi;}}D}{3(3-2D)}{E}^{\&prime;}{G}^{D-1}{a}_L^{0.5D}(a_L^{1.5-D}-a_c^{1.5-D})+\frac{D}{2-D}{\mathrm{K\&sigma;}}_y{a}_L^{0.5D}-a_c^{1-0.5D}& \mathrm{forD}\&NotEqual;1.5\\ \sqrt{\mathrm{\&pi;G}}{E}^{\&prime;}{a}_L^{0.75}\ln \frac{a_L}{a_c}+3K{\mathrm{\&sigma;}}_y{a}_L^{0.75}{a}_c^{0.25}& \mathrm{forD}=1.5\end{array}\right.---\left(2\right)$

The elastic modulus of corresponding nondimensional virtual material and nondimensional joint portion normal load are respectively

$E^{*}(A_r^{*}>A_{\mathrm{rc}}^{*})=\frac{2\mathrm{<figure-callout\; id="3"\; label="D"\; filenames="HDA0000021381540000022.png,HDA0000021381540000071.png"\; state="\{\{state\}\}">D</figure-callout>}}{3{\mathrm{\&pi;}}^2}{G}^{*1-D}{\left(\frac{2-D}{D}{A}_r^{*}\right)}^{0.5D}[a_c^{*-0.5}-{\left(\frac{2-D}{D}{A}_r^{*}\right)}^{-0.5}]---\left(3\right)$

$p_a^{*}(A_r^{*}>A_{\mathrm{rc}}^{*})=\left\{\begin{array}{cc}\frac{4\sqrt{\mathrm{\&pi;}}D}{3(3-2D)}{G}^{*D-1}{\left(\frac{2-D}{D}{A}_r^{*}\right)}^{0.5D}[{\left(\frac{2-D}{D}{A}_r^{*}\right)}^{1.5-D}-a_c^{*1.5-D}]+\mathrm{K\&phi;}{A}_r^{*0.5D}{\left(\frac{D}{2-D}{a}_c^{*}\right)}^{1-0.5D}& \mathrm{forD}\&NotEqual;1.5\\ \sqrt{{\mathrm{\&pi;G}}^{*}}{\left(\frac{A_r^{*}}{3}\right)}^{0.75}\ln \frac{A_r^{*}}{3a_c^{*}}+\mathrm{K\&phi;}{A}_r^{*0.75}{\left({3a}_c^{*}\right)}^{0.25}& \mathrm{forD}=1.5\end{array}\right.---\left(4\right)$

In the formula, the surface profile fractal dimension of D---part 1 or part 2

E '-part 1 and part 2 contacted equivalent elastic modulus, unit is Pa

The fractal roughness parameter of G---part 1 or part 2, unit is m

a _L---the area of the little contact point of maximum flexibility on part 1 and part 2 surface of contact, unit is m ²

A _r---part 1 and part 2 real contact areas, unit is m ²

a _e---part 1 and part 2 surface of contact are divided the critical area of bullet, plastic region, and unit is m ²

K---related coefficient, H/ σ _y

Than the hardness of soft material, unit is Pa in H---part 1 and the part 2

σ _y---than the yield strength of soft material, unit is Pa in part 1 and the part 2

φ---material behavior, σ _y/ E '

The elastic modulus of E---virtual material, unit are Pa

The normal load of P---part 1 and part 2 joint portions, unit is N

A _Re---part 1 and part 2 actual critical contacts area, unit is m ²

E ^*---the dimensionless elastic modulus of virtual material, E/E '

A _r ^*---dimensionless real contact area, A _r/ A _a

A _a---the apparent contact area of part 1 and part 2, unit is m ²

A _Rc ^*---the actual critical contact area of dimensionless, A _Rc/ A _a

G ^*---the fractal roughness parameter of dimensionless,

a _c ^*---dimensionless is divided the critical area of bullet, plastic region, a _c/ A _a

p _a ^*---the apparent compressive stress of dimensionless of virtual material

p _a---the apparent compressive stress of virtual material, unit is Pa

The tangential contact that is subjected to normal direction, the little contact point of circumferential load two spheroids is keeping as shown in Figure 4

Under the constant situation of power, apply the x axle effect that is parallel to again

Power;

It is a suffered circumferential load of the little contact point of spheroid.Less than threshold friction

Circumferential load

Effect be at a part of interface S ₁=(ρ, θ) | c≤ρ≤r, 0≤θ≤2 π } go up the little relative motion of generation, be referred to as little cunning, wherein f is a constant coefficient of kinetic friction, its value is determined by the physical condition of material and surface of contact; C is little inside radius that touches annulus that slips; R is little external radius of touching annulus of slipping; Polar radius

Polar angle θ=arctan (y/x).The remainder S of interface ₂=(ρ, θ) | 0≤ρ＜c, 0≤θ≤2 π } do not have relative motion and only deform, the surface in this zone is called adhesion or adhesion, and wherein c is also referred to as and adheres to the contact radius of a circle.Attachment zone S ₂With little skating area S ₁Form whole surface in contact S={ (ρ, θ) | 0≤ρ≤r, 0≤θ≤2 π }.

Simultaneous formula (2) and formula (5) can calculate the shear modulus G of virtual material _xFor

$G_x(A_r>A_{\mathrm{rc}})=\frac{16}{\mathrm{\&pi;}}\sqrt[3]{1-\frac{\mathrm{\&beta;}}{f}}{G}^{\&prime;}[{\left(\frac{a_L}{a_c}\right)}^{0.5D}-1]---\left(5\right)$

The shear modulus of the virtual material of corresponding dimensionless is

$G_x^{*}(A_r^{*}>A_{\mathrm{rc}}^{*})=\frac{16}{\mathrm{\&pi;}}\sqrt[3]{1-\frac{\mathrm{\&beta;}}{f}}[{\left(\frac{2-D}{D}\frac{A_r^{*}}{a_c^{*}}\right)}^{0.5D}-1]---\left(6\right)$

In the formula:

G _x---the shear modulus of virtual material

The ratio of β---circumferential load and normal load, Q _x/ P

Q _x---the circumferential load of virtual material

The normal load of P---virtual material

G _x ^*---the dimensionless shear modulus of virtual material, G _x/ G '

Step (2):, calculate the Poisson ratio of virtual material according to formula (3), (6) and (7).

The Poisson ratio of virtual material is

$\mathrm{\&upsi;}=\frac{(1+{\mathrm{\&mu;}}^{\&prime;})E^{*}}{G_x^{*}}-1---\left(7\right)$

In the formula:

The equivalent Poisson ratio of μ '---part 1 and part 2 surface in contacts

Step (3): the density that calculates virtual material according to formula (8).

The density of virtual material is

$\mathrm{\&rho;}=\frac{{\mathrm{\&rho;}}_1{\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="HDA0000021381540000071.png"\; state="\{\{state\}\}">h</figure-callout>}}_1+{\mathrm{\&rho;}}_2{h}_2}{{\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="HDA0000021381540000071.png"\; state="\{\{state\}\}">h</figure-callout>}}_1+h_2}---\left(8\right)$

In the formula:

ρ ₁, ρ ₂---the density of part 1,2

h ₁, h ₂---the normal direction height of part 1,2

Step (4): the virtual material Poisson ratio that calculated of the virtual elasticity modulus of materials that is calculated according to step (1), step (3), density 3 parameters altogether at first, then these 3 parameters are input in the finite element software (as MSC.Nastran), just can obtain the finite element model of the pairing fixing combination portion of this virtual material.

Step (5): at a plurality of joint portions of complete machine lathe, set up the finite element model of each joint portion according to step (1)～(4), thereby in finite element software (as MSC.Nastran), can realize comprising lathe complete machine dynamic perfromance (as the vibration shape, natural frequency, the displacement etc.) calculating of lathe joint portion influence according to rigidly connected method between existing each part.

Described finite element software can also adopt softwares such as ANSYS, ADINA and ABAQUS except that MSC.Nastran.

2 instance analysis:

2.1 the design of test specimen

In order fully to reflect the characteristic of joint portion and structure to be simplified as far as possible, designed the structure of fixing combination portion as shown in Figure 6, two hexagon socket head cap screws connect part 1,2, and fixing combination portion shown in Figure 6 is also referred to as the bolt joint portion, and this joint portion is the general type of lathe joint portion.

2.2 dynamic experiment device

The kinetic test block diagram as shown in Figure 7, this equipment comprises Belgian LMS Test.Lab 9B vibration-testing and analytic system, specifically comprise input equipment (as the power hammer), output device (as the computing machine of display result) etc., mark 5 among Fig. 7 is a bolt, 6 is bolt, and 7 are the power hammer, and 8 is LMS Test.Lab9B analysis software, 9 is vibration-testing and analytic system, and 10 is sensor.In the process of mode test, the hammer of exerting oneself is the 086C04 piezoelectric type impulsive force hammer of U.S. PCB company, and the impact head of impulsive force hammer is a nylon material, and the testing acceleration sensor is the 356A15 piezoelectric acceleration transducer.

2.3 the validation verification of the virtual material parameter of fixing combination portion of machine tool

Finish-milling processing, acetone clean fixing combination portion of machine tool two surface in contacts, and the physical parameter of part shown in Figure 5 sees Table 1.

The physical parameter of table 1 part 1 and part 2 surface in contacts

Parameter	HT250 surface	1	HT250 surface 2
				??E/GPa	??116	??116
??μ	??0.27	??0.27
			??ρ/(kg/m 3)	??7340	??7340
??R a/μm	??3.2	??3.2
			??σ y/MPa	??240	??240
??H/MPa	??700	??700

According to table 1 data, all to bear under 3 kinds of screw-down torque 30Nm, 60Nm, the 90Nm operating mode at each bolt, the parameter of the virtual material in joint portion sees Table 2, and the parameter of Yoshimura model sees Table 3.

The parameter of virtual material under table 23 kind of the operating mode

Screw-down torque/the Nm of single bolt	??E/GPa	??υ	??ρ/(kg/m 3)
				??30	??0.631	??0.22	??7340

Screw-down torque/the Nm of single bolt	??E/GPa	??υ	??ρ/(kg/m 3)
				??60	??0.736	??0.24	??7340
??90	??1.38	??0.27	??7340

The parameter of Yoshimura model under 3 kinds of operating modes of table 3

Moment/Nm	??k x/(N/m)	??k y/(N/m)	??k z/(N/m)	??c x/(N·s/m)	??c y/(N·s/m)	??c z/(N·s/m)
							??30	??8.13e7	??8.13e7	??1.36e10	??6.48e5	??6.48e5	??2.14e8
??60	??1.08e8	??1.08e8	??2.14e11	??2.19e6	??2.19e6	??3.57e9
							??90	??5.43e9	??5.43e9	??7.62e12	??3.59e7	??3.59e7	??6.41e10

Bear respectively under 30Nm, 60Nm, the 90Nm operating mode at each bolt, various Model Identification natural frequencys and experimental identification natural frequency relatively see Table 4.The relative error of Yoshimura model is far longer than the relative error of virtual material model, and the relative error of virtual material model is between (9～9) %.

Various model natural frequencys of table 4 (Hz) and the comparison of testing natural frequency

The above only is a better embodiment of the present invention, so all equivalences of doing according to the described structure of patent claim of the present invention, feature and principle change or modify, is included in the patent claim of the present invention.

Claims (1)

1. the detection method of a dynamic characteristics of fixing combination portion of machine tool is characterized in that, this method comprises the steps:

The 1st step was considered as a kind of prismatic virtual material respectively with each fixing combination portion on the lathe, obtained elastic modulus, Poisson ratio and the density of each virtual material;

Be respectively first part and second part if constitute two parts of fixing combination portion, then the computation process of elastic modulus, Poisson ratio and the density of the pairing virtual material of this fixing combination portion is:

Step (1.1): the elastic modulus E and the shear modulus G that obtain virtual material _x:

Utilize formula 1., 2. formula calculate the elastic modulus of virtual material:

The elastic modulus E function representation of virtual material is:

$E(A_r>A_{\mathrm{rc}})=\frac{2D}{3{\mathrm{\&pi;}}^2}{E}^{\&prime;}{G}^{1-D}{a}_L^{0.5D}(a_c^{-0.5}-a_L^{-0.5})$ ①

The normal load P function representation of virtual material is:

$P(A_r>A_{\mathrm{rc}})=\left\{\begin{array}{c}\frac{4\sqrt{\mathrm{\&pi;}}D}{3(3-2D)}{E}^{\&prime;}{G}^{D-1}{a}_L^{0.5D}(a_L^{1.5-D}-a_c^{1.5-D})+\frac{D}{2-D}K{\mathrm{\&sigma;}}_y{a}_L^{0.5D}{a}_c^{1-1.5D}\mathrm{forD}\&NotEqual;1.5\\ \sqrt{\mathrm{\&pi;G}}{E}^{\&prime;}{a}_L^{0.75}\ln \frac{a_L}{a_c}+3K{\mathrm{\&sigma;}}_y{a}_L^{0.75}{a}_c^{0.25}\mathrm{forD}=1.5\end{array}\right.$ ②

Wherein, A _rRepresent first, second part real contact area, A _ReRepresent the actual critical contact area of first, second part, D represents the surface profile fractal dimension of first part or second part, the contacted equivalent elastic modulus of first, second part of E ' expression, G represents the fractal roughness parameter of first part or second part, a _LThe area of representing the little contact point of maximum flexibility on first, second part surface of contact, a _eThe critical area of representing first, second part surface of contact division bullet, plastic region; K represents related coefficient, σ _yRepresent the low YIELD STRENGTH of hardness in first part and second part;

The shear modulus G of virtual material _xFor

$G_x(A_r>A_{\mathrm{rc}})=\frac{16}{\mathrm{\&pi;}}{\sqrt[3]{1-\frac{\mathrm{\&beta;}}{f}}G}^{\&prime;}[{\left(\frac{a_L}{a_c}\right)}^{0.5D}-1]$ ③

Wherein, β represents the ratio of circumferential load and normal load, and f represents friction factor, the equivalent shear modulus of G ' expression two contact rough surfaces

Step (1.2): according to formula 4., calculate the Poisson ratio υ of virtual material:

$\mathrm{\&upsi;}=\frac{(1+{\mathrm{\&mu;}}^{\&prime;})E^{*}}{G_x^{*}}-1$ ④

Wherein, E ^*The elastic modulus of representing corresponding nondimensional virtual material, 5. the employing formula is calculated:

$E*(A_r^{*}>A_{\mathrm{rc}}^{*})=\frac{2D}{3{\mathrm{\&pi;}}^2}{{G*}^{1-D}\left(\frac{2-D}{D}{A}_r^{*}\right)}^{0.5D}[a_c^{*-0.5}-{\left(\frac{2-D}{D}{A}_r^{*}\right)}^{-0.5}]$ ⑤

G _x ^*The shear modulus of the virtual material of expression corresponding dimensionless, 6. the employing formula is calculated:

$G_x^{*}(A_r^{*}>A_{\mathrm{rc}}^{*})=\frac{16}{\mathrm{\&pi;}}\sqrt[3]{1-\frac{\mathrm{\&beta;}}{f}}[{\left(\frac{2-D}{D}\frac{A_r^{*}}{a_c^{*}}\right)}^{0.5D}-1]$ ⑥

Wherein, A _r ^*Expression dimensionless real contact area, A _Re ^*The actual critical contact area of expression dimensionless, a _c ^*The critical area that the expression dimensionless is divided bullet, plastic region;

Step (1.3): the density that 7. calculates virtual material according to formula.

The density p of virtual material is

$\mathrm{\&rho;}=\frac{{\mathrm{\&rho;}}_1{h}_1+{\mathrm{\&rho;}}_2{h}_2}{h_1+h_2}$ ⑦

ρ ₁, ρ ₂The density of representing first, second part respectively, h ₁, h ₂The normal direction height of representing first, second part respectively;

Elastic modulus, Poisson ratio and the density of the 2nd step with each virtual material is input to respectively in the finite element software, obtains the finite element model of the pairing fixing combination portion of each virtual material;

The 3rd step was utilized the finite element model of each fixing combination portion, calculated the lathe complete machine dynamic perfromance that comprises the influence of lathe joint portion according to rigidly connected method between each part.

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