Junction surface contact stiffness calculation method considering interaction of micro-convex body and matrix

Abstract

The invention discloses a joint surface contact stiffness calculation method considering interaction of a micro-bulge and a matrix, which is characterized in that additional deformation generated by the matrix is calculated on the basis of an elastic theory and micro-bulge stress analysis, the actual normal deformation of the micro-bulge is obtained, and then the critical sectional areas of plastic deformation, elastic plastic deformation and elastic deformation of the micro-bulge are calculated. The interaction is considered on the basis of previous research, the real contact area and the contact load expression of the joint surface are obtained through integration, and the maximum contact sectional area is obtained according to the actual load condition. The contact stiffness of the joint surface is obtained by integrating the stiffness of the micro-convex bodies with elastic deformation and elastic plasticity, and the accuracy of a theoretical model is verified by comparing modal simulation with an experimental result. Compared with other technologies, the method can more accurately predict the theoretical stiffness value of the joint surface.

Description

Technical Field

[0001] The present invention belongs to the field of dynamic characteristics of machine tool mechanics, and relates to a method for calculating the contact stiffness of a bonding surface taking into account the interaction between a micro-convex body and a substrate. The three stages of elasticity, elastoplasticity and complete plastic deformation of the material are considered simultaneously, and a bonding surface contact stiffness model is derived based on fractal theory and elasticity theory. Background Art

[0002] Complex mechanical systems are composed of many parts, and the contact surface between the parts is called the joint surface. Mechanical joints have both stiffness and damping characteristics, and have an important impact on the dynamic performance of the mechanical system. Machine tools, as industrial mother machines, are typical representatives of complex mechanical systems. Studies have shown that the contact stiffness of the joints in machine tools accounts for about 40% to 60% of the total stiffness of the machine tools, and the contact damping of the joints accounts for more than 90% of the total damping of the machine tools, which shows that the stability of the machine tool structure is significantly related to the contact stiffness of the joint surface.

[0003] At present, most prediction models for the contact stiffness of the joint surface do not consider the interaction of micro-convex bodies. Due to the effect of the normal load, the two contact surfaces will gradually approach each other, the ratio of the real contact area to the nominal contact area between the joint surfaces will increase, and the interaction effect between the rough surfaces will be more obvious. The calculation results of the existing prediction models for the contact stiffness of the joint surface are not accurate enough. The present invention, starting from a microscopic perspective, based on fractal theory and elasticity theory, obtains an improved calculation method for the contact stiffness of the joint surface that takes into account the interaction between micro-convex bodies and substrates, which can more accurately predict the dynamic characteristics of machine tools. Summary of the invention

[0004] The technical focus of the present invention is to obtain the local normal deformation of the substrate, and to propose a more accurate method for calculating the contact stiffness of the bonding surface, so as to solve the problem that the existing method based on fractal theory has insufficient accuracy in predicting the contact stiffness.

[0005] The technical solution adopted by the present invention is a method for calculating the contact stiffness of the bonding surface taking into account the interaction between the micro-convex body and the substrate. The implementation process of the method is as follows:

[0006] Step 1: Obtain the height data of the bonding surface profile through a three-dimensional profiler, and use the roughness length method to calculate the equivalent fractal parameters D and G of the bonding surface, where 2 <D<3;

[0007] Step 2: Construct the additional deformation equation of the matrix based on the elastic theory, and establish the deformation geometric parameters of the single micro-convex body through micro-stress analysis.

[0008] According to the stress condition of a single micro-convex body and combined with the elastic theory, the local normal deformation equation of the matrix is ​​constructed as follows:

[0009]

[0010] Where m is the area factor, for the Hertzian pressure distribution f is the pressure on the micro-convex body; A is the local deformation area of ​​the substrate; E is the equivalent Young's modulus of the contact material, which is defined by the relationship between the elastic modulus and Poisson's ratio of the two contact materials,

[0011] According to the definition of the WM function, the micro-convex body is in the shape of a cosine wave. Without considering the deformation of the matrix, the normal deformation of the micro-convex body is:

[0012]

[0013] where r a is the cross-sectional radius of the micro-convex body intercepted by the ideal rigid plane; γ is the contour frequency density, which is generally taken as 1.5; the actual deformation of the micro-convex body δ = δ a -δ b ;

[0014] The equivalent curvature radius R of the microconvex body satisfies the formula R 2 =(R-δ a ) 2 +r a 2 , since R>>r a , which can be simplified to r a 2 =2Rδ a Therefore, the equivalent curvature radius R of the micro-convex body can be expressed as:

[0015]

[0016] When a single micro-convex body transitions from elastic deformation to elastic-plastic deformation, its critical deformation is δ ec , the critical deformation cross-sectional area is a' ec :

[0017]

[0018] When a single micro-convex body transitions from elastic-plastic deformation to complete plastic deformation, its critical deformation is δ pc , the critical deformation cross-sectional area is a' pc :

[0019]

[0020] Where H is the hardness of the softer material; K is the hardness coefficient, which is related to the Poisson's ratio of the softer material, K=0.464+0.41v.

[0021] Step 3: Considering the three stages of elastic, elastoplastic and completely plastic deformation of the micro-convex body, the mathematical expressions of the real contact area and contact load of the bonding surface are established.

[0022] (1) Elastic deformation stage (a' ec <a'<a' l )

[0023] When the cross-sectional area a' obtained by the micro-convex body being cut by an ideal rigid plane satisfies a' ec <a'<a' l , a' l is the maximum micro-contact cross-sectional area. According to Hertz theory, the elastic contact area of ​​the micro-convex body is a e and elastic contact load f e satisfy:

[0024]

[0025] Where λ is the interaction factor between the micro-asperity and the matrix, λ = δ / δ a , which indicates the ratio of the normal deformation of the micro-convex body to the total normal deformation of the bonding surface. In the elastic deformation stage, λ is a constant.

[0026] The modified rough surface micro-convexity size distribution function n(a') is used to integrate equations (8) and (9) to obtain the actual contact area A of the bonding surface in the elastic deformation stage: re and contact load F e :

[0027]

[0028] (2) Elastic-plastic deformation stage (a' pc <a'<a' ec )

[0029] According to the study by Liou et al., the contact area a of a single micro-convex body is ep and contact load f ep The relationship can be expressed as δ / δ ec Function:

[0030]

[0031] When the micro-convex body interacts with the substrate at this stage, the load acting on the micro-convex body acts on the substrate through the micro-convex body. The influence area is a'. According to equations (1) and (13), the relationship between the additional deformation of the substrate and the deformation of the micro-convex body can be expressed as:

[0032]

[0033] Among them, 3 / (400K)<δec / δ<1. When δ ec When / δ=1, λ is the interaction factor between the micro-asperity and the substrate in the elastic deformation stage.

[0034] The fractal model expressions of the true contact area and contact load of a single asperity are as follows:

[0035]

[0036] where λ x It can be written as The form of , i corresponds to area, load, normal stiffness and tangential stiffness respectively. Parameter m i , b i , n i Related to material properties and fractal parameters of the bonding surface.

[0037] Similarly, the actual contact area A of the bonding surface in the elastic-plastic deformation stage can be obtained rep and contact load F ep :

[0038]

[0039] in

[0040] (3) Complete plastic deformation stage (0 <a'<a' pc )

[0041] Similarly, the actual contact area A in the complete plastic deformation stage can be obtained: rp and contact load F p for:

[0042]

[0043] From the three deformation stages of the material, the total real contact area A of the bonding surface can be obtained: r And the contact load F:

[0044]

[0045] Step 4: Contact Stiffness Modeling

[0046] When a single micro-convex body undergoes elastic deformation, its ability to resist normal deformation is the elastic contact stiffness k ne

[0047]

[0048] Therefore, the normal elastic contact stiffness K of the joint surface is ne for

[0049]

[0050] When a single micro-convex body undergoes elastic-plastic deformation, its ability to resist normal deformation is the elastic-plastic contact stiffness k nep

[0051]

[0052] Normal elastic-plastic contact stiffness k of the joint surface nep for

[0053]

[0054] in

[0055] In summary, the normal contact stiffness K of the joint surface n for

[0056]

[0057] The tangential stiffness of a single asperity can be expressed as

[0058]

[0059] Where r is the true contact radius between the micro-convex body and the rigid plane, and v is the equivalent Poisson's ratio. represents the equivalent shear modulus of the contact material,

[0060] Similarly, the total tangential stiffness of the joint surface in the elastic deformation stage and the elastic-plastic deformation stage is

[0061]

[0062] in

[0063] Therefore, the total tangential contact stiffness K of the joint surface is t for

[0064]

[0065] Compared with the prior art, the present invention takes into account the interaction between the micro-protrusions and the substrate, and further refines the normal deformation of the micro-protrusions by introducing the additional deformation of the substrate under the normal load. At the same time, the modified rough surface micro-protrusion size distribution function is adopted to optimize the contact stiffness fractal model of the fixed joint, providing theoretical support for predicting the contact stiffness value of the joint surface, and solving the problem of insufficient accuracy of the calculation results of the existing prediction model. BRIEF DESCRIPTION OF THE DRAWINGS

[0066] Figure 1 It is the equivalent contact diagram of a single asperity.

[0067] Figure 2 This is the macro-micro contact diagram of the bolted joint;

[0068] Figure 3 This is the surface profile measurement result diagram;

[0069] Figure 4 This is the roughness length method diagram of the bonding surface. DETAILED DESCRIPTION

[0070] The present invention implements a method for calculating the contact stiffness of a bonding surface taking into account the interaction between the micro-convex body and the substrate. Figure 1-4 The present invention is described with detailed examples, but is not limited to the implementation scope of the present invention.

[0071] Step 1: Obtain the surface characteristic parameters of the joint surface

[0072] like Figure 2 As shown in the figure, the surface of the joint surface is in the form of peaks and valleys, and the two connecting parts are actually in contact through the micro-convex bodies on the upper and lower surfaces. The surface morphology of the specimen is scanned by ST400 to obtain the surface profile data points of the joint surface. One of the data points is as follows Figure 3 shown.

[0073] The roughness length method is used to calculate the fractal dimension D and fractal roughness G of the surface. The results are as follows: Figure 4 In this embodiment, the equivalent fractal parameters of the bonding surface are D=2.5038, G=4.0254×10 -9 .

[0074] Step 2: Asperity deformation parameter modeling

[0075] The contact between two asperities is simplified to the contact between a rigid plane and a single asperity (e.g. Figure 1 ). By analyzing the force on the micro-convex body, the deformation geometric parameters of the single micro-convex body are obtained based on fractal theory and elasticity theory.

[0076] The deformation equation of the micro-convex body considering the deformation of the base

[0077]

[0078] The equivalent radius of curvature R of the convex body

[0079]

[0080] When the micro-convex body transitions from elastic deformation to elastic-plastic deformation, its critical deformation δ ec and critical deformation cross-sectional area a' ec They are

[0081]

[0082] When the convex body transitions from elastic-plastic deformation to complete plastic deformation, its critical deformation is δ pc and critical deformation cross-sectional area a' pc They are

[0083]

[0084] In the above equation, m is the area factor. f is the pressure on the micro-convex body; A is the local deformation area of ​​the substrate, that is, Figure 1 The red circular area shown; E is the equivalent Young's modulus of the contact material, which is defined by the relationship between the elastic modulus and Poisson's ratio of the two contact materials, H is the hardness of the softer material; K is the material hardness coefficient, K=0.464+0.41v. In this embodiment, the contact material is 45# steel.

[0085] Step 3: Joint surface contact modeling

[0086] (1) Elastic deformation stage (a' ec <a'<a'l)

[0087] When the cross-sectional area a' obtained by the micro-convex body being cut by an ideal rigid plane satisfies a' ec <a'<a' l According to Hertz theory, the elastic contact area a of the microconvex body is e and elastic contact load f e for

[0088]

[0089] Where λ is the interaction factor between the micro-asperity and the matrix, λ = δ / δ a , which indicates the ratio of the normal deformation of the micro-convex body to the total normal deformation of the bonding surface. In the elastic deformation stage, λ is a constant.

[0090] According to the modified rough surface micro-convex body size distribution function n(a'), equations (7) and (8) are integrated to obtain the real contact area A of the bonding surface in the elastic deformation stage: re and contact load F e :

[0091]

[0092] (2) Elastic-plastic deformation stage (a' pc <a'<a' ec )

[0093] When the cross-sectional area a' of a single micro-protrusion satisfies a' pc <a'<a' ec , elastic-plastic contact area a ep and contact load f ep for

[0094]

[0095] The relationship between the additional deformation of the substrate and the deformation of the asperity can be expressed as

[0096]

[0097] Among them, 3 / (400K)<δ ec / δ<1. When δ ec When / δ=1, λ is the interaction factor between the micro-asperity and the substrate in the elastic deformation stage.

[0098] The fractal model expressions of the true contact area and contact load of a single asperity are as follows:

[0099]

[0100] where λ x It can be written as The form of , i corresponds to area, load, normal stiffness and tangential stiffness respectively. Parameter m i , b i , n i Related to material properties and fractal parameters of the bonding surface.

[0101] Therefore, the actual contact area A of the bonding surface during the elastic-plastic deformation stage is rep and contact load F ep for

[0102]

[0103]

[0104] in

[0105] (3) Complete plastic deformation stage (0 <a'<a' pc )

[0106] Similarly, the actual contact area A in the complete plastic deformation stage can be obtained: rp and contact load F p for:

[0107]

[0108] From the three deformation stages of the material, the total real contact area A of the bonding surface can be obtained: r And the contact load F:

[0109]

[0110] Step 4: Contact Stiffness Modeling

[0111] When a single micro-convex body undergoes elastic deformation, its ability to resist normal deformation is the elastic contact stiffness k ne

[0112]

[0113] Therefore, the normal elastic contact stiffness K of the joint surface is ne for

[0114]

[0115] When a single micro-convex body undergoes elastic-plastic deformation, its ability to resist normal deformation is the elastic-plastic contact stiffness k nep

[0116]

[0117] The total normal contact stiffness K of all asperities at this stage nep for

[0118]

[0119] in

[0120] In summary, the normal contact stiffness K of the joint surface n for

[0121]

[0122] Similarly, the tangential contact stiffness of the joint surface in the elastic deformation stage and the elastic-plastic deformation stage is

[0123]

[0124] in

[0125] Therefore, the total tangential contact stiffness K of the joint surface is t for

[0126]

[0127] Step 5: Construction of stiffness physical model and experimental verification

[0128] The present invention calculates the maximum cross-sectional area a' by substituting the normal load on the joint surface into formula (20): l , and the total stiffness value of the contact area is calculated by equations (25) and (28). The stress value of each node in the contact area of ​​the specimen is calculated using Workbench, and the actual stiffness value of each node is calculated based on the node stress. The stiffness of the joint has an important influence on the dynamic characteristics of the overall structure. The node-node stiffness assignment of the two contact surfaces is established through the Matrix27 stiffness damping unit in MechanicalAPDL. Finally, the modal analysis of the specimen is performed to obtain the dynamic characteristic parameters, and the results are compared and analyzed with the hammer test results to verify the accuracy of the present invention.

Claims

1. A method for calculating contact stiffness of a bonding surface considering the interaction between a micro-convex body and a substrate, characterized in that: The specific steps include: Step 1: Obtain the height data of the bonding surface profile through a three-dimensional profiler, and calculate the equivalent fractal parameters D and G of the bonding surface using the roughness length method, where 2 <D<3; Step 2: construct the additional deformation equation of the matrix based on the elastic theory, and establish the deformation geometric parameters of the single micro-convex body through micro-stress analysis; Step 3, considering the three stages of elastic, elastoplastic and complete plastic deformation of the micro-convex body, establish the mathematical expression of the real contact area and contact load of the bonding surface; Step 4: Establish the contact stiffness model of the joint surface according to the above steps.

2. A method for calculating contact stiffness of a bonding surface taking into account the interaction between a micro-convex body and a substrate as claimed in claim 1, characterized in that: The step 2 comprises: According to the stress of a single micro-convex body and combined with the elastic theory, the micro-convex body deformation equation δ considering the deformation of the matrix is ​​obtained. where r a is the radius of the cross section of the micro-convex body intercepted by the ideal rigid plane; γ is the profile frequency density; m is the area factor, for the Hertzian pressure distribution f is the pressure on the micro-convex body; A is the local deformation area of ​​the substrate; E is the equivalent Young's modulus of the contact material, which is defined by the relationship between the elastic modulus and Poisson's ratio of the two contact materials, According to the geometric relationship of the micro-convex body, the equivalent radius of curvature R of the micro-convex body is calculated: When the micro-convex body transitions from elastic deformation to elastic-plastic deformation, its critical deformation δ ec and critical deformation cross-sectional area a' ec They are: When the convex body transitions from elastic-plastic deformation to complete plastic deformation, its critical deformation is δ pc and critical deformation cross-sectional area a' pc They are: Where H is the hardness of the softer material; K is the hardness coefficient, which is related to the Poisson's ratio of the softer material, K = 0.464 + 0.41ν.

3. The method for calculating contact stiffness of a bonding surface taking into account the interaction between a micro-convex body and a substrate as claimed in claim 1, characterized in that: The step 3 comprises: When the micro-convex body interacts with the substrate, the load acting on the micro-convex body affects the substrate through the micro-convex body. The area a' is the relationship between the additional deformation of the substrate and the deformation of the micro-convex body. Among them, 3 / (400K)<δ ec / δ<1; when δ ec When / δ=1, λ is the interaction factor between the micro-convex body and the substrate in the elastic deformation stage; λ x Rewrite as The form of i corresponds to area, load, normal stiffness and tangential stiffness respectively; parameter m i , b i , n i Related to material properties and fractal parameters of the bonding surface; i = 1, 2, 3, 4; (1) Elastic deformation stage; when the cross-sectional area a' obtained by cutting the micro-convex body by the ideal rigid plane satisfies a' ec <a'<a' l , a' l is the maximum micro-contact cross-sectional area. According to Hertz theory, the elastic contact area of ​​the micro-convex body is a e and elastic contact load f e , and the modified rough surface micro-convex body size distribution function n(a') is integrated to obtain the real contact area A of the bonding surface in the elastic deformation stage re and contact load F e : (2) Elastic-plastic deformation stage; when the cross-sectional area a' obtained by cutting the micro-convex body by the ideal rigid plane satisfies a' pc <a'<a' ec , the modified rough surface micro-convex body size distribution function n(a') is integrated to obtain the real contact area A of the bonding surface in the elastic-plastic deformation stage rep and contact load F ep : (3) Complete plastic deformation stage: when the cross-sectional area a' obtained by cutting the micro-convex body by the ideal rigid plane satisfies 0 <a'<a' pc , the actual contact area A in the deformation stage is obtained rp and contact load F p for: From the three deformation stages of the material, the total real contact area A of the bonding surface is obtained r And the contact load F:

4. The method for calculating contact stiffness of a bonding surface taking into account the interaction between a micro-convex body and a substrate as claimed in claim 1, characterized in that: The step 4 comprises: (1) Modeling of normal contact stiffness of the joint surface When a single micro-convex body undergoes elastic deformation, its ability to resist normal deformation is the elastic contact stiffness k ne , the normal elastic contact stiffness K of the joint surface ne for When a single micro-convex body undergoes elastic-plastic deformation, its ability to resist normal deformation is the elastic-plastic contact stiffness k nep , the normal elastic-plastic contact stiffness K of the joint surface nep for Therefore, the normal contact stiffness K of the joint surface is n for (2) Modeling of tangential contact stiffness of the joint surface Similar to the modeling of normal contact stiffness, the total tangential stiffness of the joint surface in the elastic deformation stage and the elastic-plastic deformation stage is Therefore, the total tangential contact stiffness K of the joint surface is t for 。

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