CN110940587A - Lathe bed-foundation joint surface contact rigidity calculation method based on multi-scale theory - Google Patents

Lathe bed-foundation joint surface contact rigidity calculation method based on multi-scale theory Download PDF

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CN110940587A
CN110940587A CN201911302168.7A CN201911302168A CN110940587A CN 110940587 A CN110940587 A CN 110940587A CN 201911302168 A CN201911302168 A CN 201911302168A CN 110940587 A CN110940587 A CN 110940587A
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contact
critical
deformation
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赵永胜
牛娜娜
刘志峰
左伟亮
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Beijing University of Technology
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Abstract

The invention discloses a method for calculating the contact rigidity of a bed body-foundation junction surface based on a multi-scale theory, which aims at concrete materials of a foundation, obtains stress and strain data of the materials by an experimental method, and fits the data to the existing curve equation to find a relatively accurate concrete material stress-strain curve equation. Based on a multi-scale theory, real contact areas and critical deformation parameters under different scale sequences are obtained, and then a contact layer is divided into a crushing stage, a plastic deformation stage and an elastic deformation stage. And obtaining the contact stiffness of a single microprotrusion on the elastic deformation sequence by using a Hertz contact theory, and finally obtaining the contact stiffness of the whole joint surface through an iterative relationship of real contact areas among all frequency levels.

Description

Lathe bed-foundation joint surface contact rigidity calculation method based on multi-scale theory
Technical Field
The invention belongs to the field of joint surface dynamics, and relates to a method for calculating the contact rigidity of a bed-foundation joint surface based on a multi-scale theory.
Background
The dynamic characteristics of the machine tool-foundation junction surface have very important influence on the machining precision and the service life of the machine tool. The foundation is usually made of concrete material and is connected with the machine tool body through anchor bolts and adjusting sizing blocks. Because the machine tool has the characteristics of heavy weight and large load, the machine tool can deform due to insufficient rigidity in the machining process, and the machining precision is influenced. The joint surface between the lathe bed and the foundation is used as an important connecting unit, the dynamic characteristic of the joint surface is mainly researched by identifying parameters of the joint part through experimental tests, so that a large number of experiments are needed, the nonlinear relation between materials, surface roughness, pretightening force, processing modes and the like and the rigidity damping of the joint part is difficult to represent, and the crushing deformation of concrete materials is not considered in the existing analytical method. Therefore, considering the crushing deformation of the concrete material, it is necessary to accurately model the contact rigidity of the bed body-foundation junction surface.
Disclosure of Invention
The method aims at the concrete material of the foundation, obtains the stress and strain data of the material through an experimental method, and fits the data to the existing curve equation to find a relatively accurate concrete material stress-strain curve equation. Based on a multi-scale theory, real contact areas and critical deformation parameters under different scale sequences are obtained, and then a contact layer is divided into a crushing stage, a plastic deformation stage and an elastic deformation stage. And obtaining the contact stiffness of the microprotrusions on the given frequency level in the elastic deformation stage based on the Hertz contact theory, and finally obtaining the contact stiffness of the whole joint surface through the iterative relationship of the real contact areas among the frequency levels.
The invention is realized by adopting the following technical means:
s1, obtaining the critical contact area from elastic deformation to plastic deformation through a stress-strain curve equation obtained by a concrete material uniaxial experiment
Figure BDA0002322113650000011
And plastic deformation to critical contact area for fracture
Figure BDA0002322113650000012
And S2, obtaining the real contact area and the contact load of the single micro-convex body with the given frequency level under the conditions of an elastic deformation stage, a plastic deformation stage and a crushing stage according to a multi-scale theory, a Hertz contact theory and the actual total external load. And dividing the scale sequence into elastic deformation, plastic deformation and crushing according to the critical contact parameters.
S3, obtaining the contact stiffness of a single micro-convex body on a given frequency level through the relation between the contact force and the contact deformation amount in the elastic deformation stage, and obtaining the contact stiffness on the given frequency level through the distribution density of the micro-convex bodies.
S4, calculating the contact stiffness of all elastic deformation sequences through the real contact area iterative relationship among the frequency levels, wherein the total normal stiffness is regarded as the result of the series connection of the contact stiffness of all the elastic deformation sequences.
The invention is characterized in that the crushing stage of the foundation concrete material is considered, and the hypothesis that the load borne by the crushed microprotrusions will be evenly dispersed to the next layer of microprotrusions is provided. The idea of layering the contact surface by combining a multi-scale theory is combined, and different sequences of the surface at the joint are divided into three contact states of elastic deformation, plastic deformation and crushing according to the critical parameter change under different sequences. The invention provides a more accurate characterization method for the contact characteristic of the combined surface of the lathe bed and the foundation. And theoretical reference is provided for improving the integral rigidity and the machining precision of the machine tool. The following figures illustrate the invention more clearly.
Drawings
FIG. 1 is a graph of uniaxial compressive stress-strain of a concrete material.
FIG. 2 is a schematic diagram of a multi-scale contact surface model.
Fig. 3 is a flow chart of the implementation of the present invention.
Detailed Description
The invention discloses a method for calculating the contact rigidity of a bed-foundation junction surface based on a multi-scale theory, which is specifically described by combining the following drawings:
step (1) setting critical contact parameters of a single frequency-level microprotrusion body;
according to the stress-strain curve equation-Popovics formula of the concrete material uniaxial experiment, the concrete mainly undergoes three steps of elastic deformation, plastic deformation and crushing deformation from initial loading to failure under the condition of uniaxial compressionAnd (4) carrying out each stage. As shown in FIG. 1, the concrete microprotrusions exhibit elastic deformation at the initial stage of deformation, the stress-strain curve approaches a linear relationship at this stage, the end point A of the stage is called the proportional limit, and the corresponding critical stress σ iscThe peak point C stress value, i.e., the compressive strength, is σ0The limit of the normal concrete proportion is sigmac＝(0.3～0.5)σ0C60 concrete value sigmac＝0.408σ0(ii) a Beyond the limit of proportionality, the microprotrusions begin to transform into plastic deformation when the fracture critical stress σ is exceededu＝(0.88～0.97)σ0When the micro-convex body is subjected to yield and crushing deformation, the value of the concrete with the C60 label is sigmau＝0.902σ0The stress-strain curve drops sharply because the yielding and broken microprotrusions cannot continue to bear the load.
Figure BDA0002322113650000021
In the formula (I), the compound is shown in the specification,
Figure BDA0002322113650000031
n＝5.7×10-3σ0+1
α＝n+β
β -Curve camber adjustment constant;
σ0concrete compressive strength, see table 1;
ε0-peak strain;
the relationship between the contact area and the contact force of a single microprotrusion obtained from the hertzian contact theory is:
ae＝πRω
Figure BDA0002322113650000032
the critical real contact area and contact force thus obtained are:
Figure BDA0002322113650000033
Figure BDA0002322113650000034
Figure BDA0002322113650000035
critical parameters for plastic deformation and fracture:
the critical strain value when the breaking occurs can be known from the stress-strain curve equation of the concrete material
Figure BDA0002322113650000036
The critical deformation of a single microprotrusion body at a given frequency level can be determined
Figure BDA0002322113650000037
The critical contact area of a single microprotrusion body at a given frequency level when the microprotrusion body is broken can be obtained:
Figure BDA0002322113650000038
Figure BDA0002322113650000039
Figure BDA00023221136500000310
step (2) giving the real contact area of a single microprotrusion body of a frequency level
Elastic phase
According to the literature, "Surface separation and contact resistance coherent elastic-planar multi-scale rod Surface contact",
Figure BDA00023221136500000311
the method comprises the following steps:
Figure BDA0002322113650000041
Figure BDA0002322113650000042
approach to
Figure BDA0002322113650000043
The method comprises the following steps:
Figure BDA0002322113650000044
wherein:
Figure BDA0002322113650000045
Figure BDA0002322113650000046
-average pressure at full contact of the surfaces of a given frequency level;
the equivalent modulus of elasticity of the E' -contact surface,
Figure BDA0002322113650000047
E1、E2、v1、v2respectively the elastic modulus and Poisson's ratio of the two surfaces in contact;
according to experimental and computational data, the document "a multi-scale model for contact between roughsources" fits a coupling equation to obtain an approximate solution of the real contact area of a single microprotrusion of a given frequency level:
Figure BDA0002322113650000048
plastic phase
According to classical hertzian contact theory, the contact area and contact force in the plastic deformation phase are, respectively:
ap＝2πRω
fp＝Hap＝2HπRω
substituting the elastic-plastic deformation critical contact force into the actual contact area of a single microprotrusion with a given frequency level in the deformation stage of the plastic stage:
Figure BDA0002322113650000049
Figure BDA00023221136500000410
Figure BDA00023221136500000411
wherein:
Δ β — elasto-plastic critical amplitude for a given frequency level;
step (3) contact stiffness in the elastic phase;
the influence of different scale ordinal numbers n on the critical contact state is researched in a scale-related fractal rough surface elastoplastic contact mechanical model, and the conclusion is obtained: 6 consecutive scale ordinals are decisive for the contact parameters and can fully represent the current contact state. And because the microprotrusions have no bearing capacity in the crushing stage and have no contact rigidity in the plastic deformation stage, the calculation of the contact rigidity only considers the elastic deformation sequence.
Hertzian contact mechanics in the elastic phase:
Figure BDA0002322113650000051
Figure BDA0002322113650000052
the following can be obtained:
Figure BDA0002322113650000053
the contact stiffness of a single microprotrusion at a given frequency level is:
Figure BDA0002322113650000054
(Kn)i＝Ni(kn)i＝ηiAi-1(kn)i
Figure BDA0002322113650000055
step (4) an iterative method of contact stiffness under different sequences;
an iterative relationship between a given frequency level and a true contact area below the given frequency level can be obtained according to the Amulti-scale model for contact between the surface surfaces:
Figure BDA0002322113650000056
the contact stiffness at various frequency levels below the critical frequency level is considered as a series spring model, the total normal stiffness:
Figure BDA0002322113650000057

Claims (5)

1. a method for calculating the contact rigidity of a bed-foundation junction surface based on a multi-scale theory is characterized by comprising the following steps: the method comprises the following steps of,
s1, obtaining the critical contact area from elastic deformation to plastic deformation through a stress-strain curve equation obtained by a concrete material uniaxial experiment
Figure FDA0002322113640000011
And plastic deformation to critical contact area for fracture
Figure FDA0002322113640000012
S2, obtaining the real contact area and the contact load of a single micro-convex body of a given frequency level under the conditions of an elastic deformation stage, a plastic deformation stage and a crushing stage according to a multi-scale theory, a Hertz contact theory and actual total external load; dividing the scale sequence into three types of elastic deformation, plastic deformation and crushing according to critical contact parameters;
s3, obtaining the contact stiffness of a single micro-convex body on a given frequency level through the relation between the contact force and the contact deformation amount in the elastic deformation stage, and obtaining the contact stiffness on the given frequency level through the distribution density of the micro-convex bodies;
s4, calculating the contact stiffness of all elastic deformation sequences through the real contact area iterative relationship among the frequency levels, wherein the total normal stiffness is regarded as the result of the series connection of the contact stiffness of all the elastic deformation sequences.
2. The method for calculating the contact rigidity of the combined surface of the lathe bed and the foundation based on the multi-scale theory as claimed in claim 1, wherein: the critical contact parameters for a single microprotrusion at a given frequency level are implemented as follows,
according to a uniaxial experimental stress-strain curve equation-Popovics formula of a concrete material, when the concrete is under uniaxial compression, the concrete mainly undergoes three stages of elastic deformation, plastic deformation and crushing deformation from initial loading to failure; the concrete microprotrusions exhibit elastic deformation at the initial stage of deformation, the stress-strain curve approaches the linear relationship at this stage, the end point A of the stage is called the proportional limit, and the corresponding critical stress sigmacThe peak point C stress value, i.e., the compressive strength, is σ0The limit of the ordinary concrete proportion is sigmac＝(0.3～0.5)σ0C60 concrete value sigmac＝0.408σ0(ii) a Beyond the limit of proportionality, the microprotrusions begin to transform into plastic deformation when the fracture critical stress σ is exceededu＝(0.88～0.97)σ0When the micro-convex body is subjected to yield and crushing deformation, the value of the concrete with the C60 label is sigmau＝0.902σ0Because the yielding and broken micro-convex bodies cannot continuously bear the load, the stress-strain curve is sharply reduced;
Figure FDA0002322113640000013
in the formula (I), the compound is shown in the specification,
Figure FDA0002322113640000014
n＝5.7×10-3σ0+1
α＝n+β
β -curve camber adjustment constant, sigma0-concrete compressive strength; epsilon0-peak strain;
the relationship between the contact area and the contact force of a single microprotrusion obtained from the hertzian contact theory is:
ae＝πRω
Figure FDA0002322113640000021
the critical real contact area and contact force thus obtained are:
Figure FDA0002322113640000022
Figure FDA0002322113640000023
Figure FDA0002322113640000024
critical parameters for plastic deformation and fracture:
the critical strain value when the breaking occurs can be known from the stress-strain curve equation of the concrete material
Figure FDA0002322113640000025
The critical deformation of a single microprotrusion body at a given frequency level can be determined
Figure FDA0002322113640000026
The critical contact area of a single microprotrusion body at a given frequency level when the microprotrusion body is broken can be obtained:
Figure FDA0002322113640000027
Figure FDA0002322113640000028
Figure FDA0002322113640000029
3. the method for calculating the contact rigidity of the combined surface of the lathe bed and the foundation based on the multi-scale theory as claimed in claim 1, wherein: the step of implementing the real contact area of a single microprotrusion at a given frequency level is as follows in the elastic phase,
Figure FDA00023221136400000210
the method comprises the following steps:
Figure FDA00023221136400000211
Figure FDA00023221136400000212
approach to
Figure FDA00023221136400000213
The method comprises the following steps:
Figure FDA00023221136400000214
wherein:
Figure FDA00023221136400000215
Figure FDA0002322113640000031
-average pressure at full contact of the surfaces of a given frequency level;
the equivalent modulus of elasticity of the E' -contact surface,
Figure FDA0002322113640000032
E1、E2、v1、v2respectively the elastic modulus and Poisson's ratio of the two surfaces in contact;
fitting a coupling equation to obtain an approximate solution of the real contact area of a single microprotrusion body of a given frequency order according to experimental and computational data:
Figure FDA0002322113640000033
plastic phase
According to classical hertzian contact theory, the contact area and contact force in the plastic deformation phase are, respectively:
ap＝2πRω
fp＝Hap＝2HπRω
substituting the elastic-plastic deformation critical contact force into the actual contact area of a single microprotrusion with a given frequency level in the deformation stage of the plastic stage:
Figure FDA0002322113640000034
Figure FDA0002322113640000035
Figure FDA0002322113640000036
wherein:
Δ β — the elasto-plastic critical amplitude for a given frequency level.
4. The method for calculating the contact rigidity of the combined surface of the lathe bed and the foundation based on the multi-scale theory as claimed in claim 1, wherein:
hertzian contact mechanics in the elastic phase:
Figure FDA0002322113640000037
Figure FDA0002322113640000038
obtaining:
Figure FDA0002322113640000041
the contact stiffness of a single microprotrusion at a given frequency level is:
Figure FDA0002322113640000042
(Kn)i＝Ni(kn)i＝ηiAi-1(kn)i
Figure FDA0002322113640000043
5. the method for calculating the contact rigidity of the combined surface of the lathe bed and the foundation based on the multi-scale theory as claimed in claim 1, wherein: iterative methods of contact stiffness under different sequences;
iterative relationship of a given frequency level and one true contact area below the given frequency level:
Figure FDA0002322113640000044
the contact stiffness at various frequency levels below the critical frequency level is considered as a series spring model, the total normal stiffness:
Figure FDA0002322113640000045
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Publication number Priority date Publication date Assignee Title
CN105004662A (en) * 2015-07-08 2015-10-28 中国科学院力学研究所 Method for testing contact rigidity of rock discontinuity structural plane, and apparatus thereof
CN106768741A (en) * 2016-12-07 2017-05-31 北京工业大学 A kind of machine tool basic joint surface contact stiffness computational methods for considering that concrete micro-bulge is broken
WO2018086160A1 (en) * 2016-11-09 2018-05-17 北京工业大学 Rough surface-based three-dimensional contact stiffness calculation method for spur gear
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Publication number Priority date Publication date Assignee Title
CN105004662A (en) * 2015-07-08 2015-10-28 中国科学院力学研究所 Method for testing contact rigidity of rock discontinuity structural plane, and apparatus thereof
WO2018086160A1 (en) * 2016-11-09 2018-05-17 北京工业大学 Rough surface-based three-dimensional contact stiffness calculation method for spur gear
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CN108052756A (en) * 2017-12-22 2018-05-18 太原科技大学 A kind of method that faying face exposure parameter is determined based on FFT
CN109446655A (en) * 2018-10-30 2019-03-08 东北大学 A kind of double three-dimensional fractal prediction techniques for dividing shape faying face normal contact stiffness
CN109960883A (en) * 2019-03-28 2019-07-02 北京工业大学 A kind of calculation method of joint surface contact stiffness containing inclination angle based on fractal theory

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