CN110378018B - Method for calculating steady-state bearing capacity of hydrodynamic and hydrostatic ball bearing - Google Patents
Method for calculating steady-state bearing capacity of hydrodynamic and hydrostatic ball bearing Download PDFInfo
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Abstract
The method for calculating the steady-state bearing capacity of the liquid ball bearing comprises the following steps: 1, establishing an oil film pressure calculation model; 2, establishing a differential expression of a liquid lubricating oil film pressure calculation model in a spherical coordinate system; 3, calculating the oil film thickness of the bearing; 4, setting an initial pressure value and calculating oil film pressure; 5, decomposing the oil film pressure along the lower three directions of the rectangular coordinate system; 6, integrating the pressure in the circumferential direction and the radial direction of the oil film to obtain the bearing capacity in the circumferential x, y and z directions respectively; 7, dimensionless bearing capacity formula to obtain dimensionless bearing capacity; and 8, integrating the dimensionless bearing capacity formula to obtain the bearing capacity of the bearing. The method for calculating the steady bearing capacity of the liquid ball bearing is based on the three-dimensional shafting, so that the obtained steady bearing capacity result of the liquid ball bearing can better reflect the actual situation and can better provide theoretical guidance for the structural design of the liquid ball bearing.
Description
Technical Field
The invention belongs to the field of machinery, and particularly relates to a method for calculating the steady-state bearing capacity of a hydrodynamic and hydrostatic ball bearing.
Background
Ultra-high precision machine tools and materials, also known as "industrial masters", precision ultra-precision machining techniques and equipment are one of the key technologies in international competition for sophisticated products and defense industries. The main key of the ultra-precise equipment is that a rotating part, namely a machine tool headstock with an ultra-precise main shaft rotating shaft system, can only measure and process ultra-precise parts. The ultra-precise spindle is required to achieve high rigidity, extremely high rotation precision, stable rotation and no vibration, and the key point is the precise bearing used.
The liquid ball bearing is an important component of ultra-precise and heavy machine tools, can bear axial and radial loads simultaneously, and is convenient for centering. The bearings have the performance advantages of high rotation precision, high dynamic stiffness, high vibration damping and long service life. The fluid ball bearing can reduce bearing wear at low speeds of conventional hybrid bearings.
The spherical liquid bearing consists of a rotor (convex ball) and a stator (concave ball), and a certain number of oil holes are formed in the stator. On the one hand, oil with certain pressure is output through the oil pump, flows into a bearing gap through the restrictor to form a static pressure oil film and a static pressure bearing capacity. On the other hand, by high-speed operation between the bearing surfaces, wedge oil is formed between the rotor and the stator, generating a dynamic pressure oil film and forming a dynamic pressure bearing capacity.
The liquid bearing has uneven oil film thickness distribution due to gravity and rotation centrifugal force in the rotation process of the supporting main shaft, so that the uneven oil film pressure distribution is caused, and the bearing capacity of the bearing is greatly influenced. A suitable computational model needs to be built for the design, production and manufacture. At present, scholars mainly concentrate on a gas ball bearing in academic journal research, and the compressibility of gas ensures that the gas ball bearing has lower rigidity and poorer bearing capacity, and meanwhile, design strategies and technical schemes considered by the gas bearing and the liquid bearing have larger differences, so that the oil film bearing capacity of the liquid ball bearing is greatly different from that of the gas ball bearing.
The journal paper "analysis of steady bearing capacity of hemispherical gas bearing" also samples and establishes a gas lubrication Reynolds equation to obtain the gas film pressure, when the gas lubrication equation is established, the gas density changes along with the change of pressure and temperature due to the compressibility of the gas, and the density of the liquid is approximately constant, i.e. the density of the liquid hardly changes along with the change of temperature and pressure, so the liquid lubrication Reynolds equation and the gas lubrication Reynolds equation have the difference; in calculating the bearing capacity of the ball bearing, the bearing capacity of the liquid ball bearing is much larger than that of the gas ball bearing, and the calculation method also has small differences due to the difference of three-dimensional models of the ball bearing. The above paper only discusses the radial bearing capacity and does not calculate the axial bearing capacity.
Disclosure of Invention
In order to solve the problem of the oil film bearing capacity of the liquid ball bearing, the three direction bearing capacities of the liquid ball bearing are calculated under a given certain pressure and spindle rotating speed on the basis of establishing a three-dimensional geometric model of the ball bearing and establishing a liquid lubrication Reynolds equation of the ball bearing.
The invention provides a method for calculating the steady-state bearing capacity of a hydrostatic ball bearing, which has the characteristics that the method comprises the following steps:
step 1, establishing a lubrication equation of a hydrodynamic and hydrostatic ball bearing, wherein the expression is as follows:
wherein:h represents the dimensionless oil film thickness; />Representing dimensionless oil film pressure; w represents the rotational speed; η represents the liquid viscosity coefficient; p (P) a Represents atmospheric pressure; h is a 0 Indicating oil film gap; r represents the radius of the bearing;
step 2, establishing a differential expression of a liquid lubricating oil film pressure calculation model in a spherical coordinate system:
atmospheric boundary conditions:
liquid of a certain pressure is injected through the orifice, and static pressure boundary conditions are as follows:
wherein n is the number of orifices, the number is selected to be even,
pressure continuous conditions:
correcting the pressure of each node by using a relaxation method, wherein the expression of the relaxation method is as follows:
wherein: omega is a relaxation factor and is generally 0-2; k is an iteration coefficient.Represents the oil film pressure value k+1 times,represents the value of the oil film pressure k times,/>Representing the value of the dimensionless oil film pressure of k+1 times;
step 3, calculating the oil film thickness of the bearing according to the bearing parameters and boundary conditions;
step 4, setting an initial pressure value, and calculating oil film pressure through the liquid lubricating oil film pressure calculation model;
step 5, the oil film pressure in the step 4 is expressed along the rectangular coordinate system of three x, y and zDecomposing in each direction to obtain P x ，P y P z ，
Step 6, adopting a Simpson integration method to integrate the pressure in the circumferential direction and the radial direction of the oil film to respectively obtain the bearing capacity in the circumferential x direction, the bearing capacity in the circumferential y direction and the bearing capacity in the radial z direction,
wherein: r represents the radius of the oil film,indicating the radial initial angle of the bearing +.>Representing the radial integral wrap angle of the ball bearing;
step 7, dimensionless treatment of formula (4) to obtain
Wherein: w (W) x ，W y W is provided z The dimensionless bearing capacity along the x, y and z directions are respectively shown;
and 8, integrating the formula (5) by adopting the simpson to obtain the bearing capacity of the bearing as follows:
F x ＝p a ·R 2 W x (6)
calculating the bearing capacity of the other two directions, wherein the radial bearing capacity expressions are respectively as follows:
the above method is dimensionless to obtain:
wherein: w represents dimensionless radial bearing capacity.
The method for calculating the steady-state bearing capacity of the hydrostatic ball bearing provided by the invention can also have the following characteristics: in step 3, the bearing oil film pressure distribution is calculated according to the bearing parameters and the boundary conditions.
Effects and effects of the invention
The bearing capacity of the liquid ball bearing is an important index of the static characteristic of the bearing, and is mainly influenced by the oil film pressure and the bearing stress area. The method for calculating the steady-state bearing capacity of the hydrostatic-dynamic-hydrostatic ball bearing provides a method for calculating the steady-state bearing capacity of the hydrostatic-dynamic-hydrostatic ball bearing under a spherical coordinate system, and the method comprises the steps of calculating the bearing capacity of the hydrostatic-dynamic-hydrostatic ball bearing after the pressure distribution of the ball bearing is obtained; and integrating the pressure in the circumferential direction and the radial direction of the oil film by adopting a Simpson integration method to respectively obtain the bearing capacity in the circumferential x direction, the bearing capacity in the circumferential y direction and the bearing capacity in the radial z direction.
The steady-state bearing capacity result of the liquid ball bearing, which can be obtained by the calculation method, is based on a three-dimensional shafting. Therefore, the result can better reflect the real situation and provide theoretical guidance for the structural design of the liquid ball bearing.
The calculation method fills the gap of calculating the steady-state bearing capacity of the liquid ball bearing.
Drawings
FIG. 1 is a schematic illustration of a fluid ball bearing geometry model in an embodiment of the invention;
FIG. 2 is a schematic diagram of solving a domain meshing in an embodiment of the invention;
FIG. 3 is a schematic illustration of the thickness of oil film and pressure distribution in an embodiment of the present invention; and
FIG. 4 is a schematic diagram of the equilibrium position of a rotor in an embodiment of the invention.
Detailed Description
In order to make the technical means, creation characteristics, achievement purposes and effects achieved by the present invention easy to understand, the following embodiments specifically describe a method for calculating the steady-state bearing capacity of the hydrostatic ball bearing according to the present invention with reference to the accompanying drawings.
Examples
The dynamic-static pressure liquid ball bearing outputs oil with certain pressure through an oil pump, flows into a bearing gap through a restrictor to form a static pressure oil film and a static pressure bearing capacity, and on the other hand, forms wedge-shaped oil between a rotor and a stator through high-speed operation between bearing surfaces to generate a dynamic pressure oil film and a dynamic pressure bearing capacity. By analyzing the oil film pressure in the oil cavity, the bearing capacity of the oil film can be calculated.
The object under investigation in this embodiment is a liquid ball bearing, first a three-dimensional model of the ball bearing is built. As shown in fig. 1, an orifice 4 is provided in a stator 1, a rotor 2 is provided in the stator 1, a main shaft 3 is provided in the rotor 2 and drives the rotor 2 to rotate, and an oil supply system 5 supplies oil to the stator 1 through the orifice 4. Wherein: p (P) a Represents atmospheric pressure, d 0 Represents the diameter of the throttle hole, R represents the radius of the convex sphere, O represents the sphere center, R represents the radius of the concave sphere, and P s Representing a small Kong Gongyou pressure, the pressure,indicating the oil supply tangential angle->Indicating wrap angle, h 0 Represents the average oil film gap, d 1 Representing the spindle diameter.
1. Establishing a steady-state dimensionless liquid lubrication oil film pressure calculation model of the hydrodynamic and hydrostatic ball bearing according to the geometric model of the hydrodynamic and hydrostatic ball bearing and the liquid lubrication principle in FIG. 1:
wherein:h represents the dimensionless oil film thickness; />Representing dimensionless oil film pressure; w represents the rotational speed; η represents the liquid viscosity coefficient; p (P) a Represents atmospheric pressure; h is a 0 Indicating oil film gap; r represents the bearing radius R, < >>And +.>Respectively representing three coordinate directions in the spherical coordinate system.
2. The oil film pressure calculation model of the ball bearing is a two-dimensional partial differential equation, a finite difference method is generally adopted to solve the partial differential equation, a central differential method is respectively used for discretizing the pressure P and the oil film thickness H in the above formula on an interlaced grid under a generalized coordinate system, a finite differential expression under a steady-state condition is deduced, and as shown in fig. 2, the oil film pressure distribution is obtained by combining the oil film pressure calculation model to obtain an atmospheric boundary condition, a pressure continuous condition and a static pressure condition.
The differential expression of the liquid lubricating oil film pressure calculation model in the spherical coordinate system is as follows:
atmospheric boundary conditions:
liquid of a certain pressure is injected through the orifice, and static pressure boundary conditions are as follows:
where n is the number of orifices and is an even number, typically 4,6, or 8.
Pressure continuous conditions:
correcting the pressure of each node by using a relaxation method, wherein the expression of the relaxation method is as follows:
wherein: omega is a relaxation factor and is generally 0-2; k is an iteration coefficient.Represents the oil film pressure value k+1 times,represents the value of the oil film pressure k times,/>Represents the value of the dimensionless oil film pressure in the k+1 times.
The calculated parameters for the bearings are shown in table 1.
Table 1 bearing parameter table
From the bearing parameters and boundary conditions, the oil film thickness (left hand graph) and oil film pressure (right hand graph) distributions of the bearing as shown in fig. 3 can be calculated.
3. Setting an initial pressure value, obtaining oil film pressure distribution by solving an oil film pressure calculation model, enabling the oil film pressure direction to be always perpendicular to a spherical surface, and decomposing the oil film pressure along the x, y and z directions in a rectangular coordinate system to obtain P x ，P y P z Specifically, as shown in fig. 4, the oil film pressure in each direction is multiplied by the unit area acted by the pressure, and the pressure in the circumferential direction and the radial direction of the oil film is integrated by the simpson integration method, so as to obtain the bearing capacity in the circumferential x direction, the bearing capacity in the circumferential y direction and the bearing capacity in the radial z direction.
In fig. 4: e, e x Indicating eccentricity with respect to x-direction, e y Indicating eccentricity in the y-direction, e z Represents eccentricity with respect to the z direction, ω represents angular velocity, t represents time, θ represents circumferential angle,represents the radial direction included angle, F x ,F y And F z Respectively, the bearing forces in the x, y and z directions, and P represents the oil film pressure.
The pressure direction of the liquid oil film is always perpendicular to the tangential direction of the spherical surface of the bearing, and the pressure is decomposed along the directions of a rectangular coordinate system x, y and z to obtain:
wherein: r represents the radius of the oil film,indicating the radial initial angle of the bearing +.>Representing the radial integral wrap angle of the ball bearing.
The dimensionless calculation of the bearing capacity formula can be obtained:
wherein: w (W) x ，W y W is provided z The dimensionless bearing forces in the x, y and z directions are shown, respectively.
The simpson is adopted to integrate the above formula, and the bearing capacity of the bearing obtained by numerical solution is as follows:
F x ＝p a ·R 2 W x (6)
the bearing capacities in the other two directions can also be calculated, wherein the radial bearing capacity expressions are respectively:
the above method is dimensionless to obtain:
wherein: w represents dimensionless radial bearing capacity
In summary, firstly, the ball bearing parameters are determined, the oil film thickness of each node is calculated, then, the oil film pressure of each node is calculated by setting an initial pressure value, the pressure value of each node is continuously updated, and finally, the bearing capacity in the x, y and z directions, the axial bearing capacity and the radial bearing capacity are calculated respectively.
Effects and effects of the examples
The bearing capacity of the liquid ball bearing is an important index of the static characteristic of the bearing, and is mainly influenced by the oil film pressure and the bearing stress area. The method for calculating the steady-state bearing capacity of the hydrostatic ball bearing in the embodiment provides a method for calculating the steady-state bearing capacity of the hydrostatic ball bearing in a spherical coordinate system, and the method comprises the steps of calculating the bearing capacity of the hydrostatic ball bearing after the pressure distribution of the hydrostatic ball bearing is obtained; and integrating the pressure in the circumferential direction and the radial direction of the oil film by adopting a Simpson integration method to respectively obtain the bearing capacity in the circumferential x direction, the bearing capacity in the circumferential y direction and the bearing capacity in the radial z direction.
The calculation method of the embodiment can obtain the steady-state bearing capacity result of the liquid ball bearing based on a three-dimensional shafting. Therefore, the result can better reflect the real situation and provide theoretical guidance for the structural design of the liquid ball bearing.
The calculation method fills the gap of calculating the steady-state bearing capacity of the liquid ball bearing.
The above embodiments are preferred examples of the present invention, and are not intended to limit the scope of the present invention.
Claims (1)
1. The method for calculating the steady-state bearing capacity of the hydrodynamic and hydrostatic ball bearing is characterized by comprising the following steps of:
step 1, firstly, a three-dimensional model of a hydrostatic-dynamic ball bearing is built, an orifice is arranged in a stator, a rotor is arranged in the stator, a main shaft is arranged in the rotor and drives the rotor to rotate, the hydrostatic-dynamic ball bearing outputs oil with certain pressure through an oil pump, the oil flows into a bearing gap through a restrictor to form a hydrostatic oil film and a hydrostatic bearing capacity, on the other hand, a wedge-shaped oil groove is formed between the rotor and the stator through high-speed operation between bearing surfaces to generate a dynamic pressure oil film and form the dynamic pressure bearing capacity, and then a lubrication equation of the hydrostatic-dynamic ball bearing is built, wherein the expression is as follows:
wherein:h represents the dimensionless oil film thickness; />Representing dimensionless oil film pressure; w represents the rotational speed; η represents the liquid viscosity coefficient; p (P) a Represents atmospheric pressure; h is a 0 Indicating oil film gap; r represents the radius of the bearing;
step 2, establishing a differential expression of a liquid lubricating oil film pressure calculation model in a spherical coordinate system:
atmospheric boundary conditions:
liquid of a certain pressure is injected through the orifice, and static pressure boundary conditions are as follows:
wherein n is the number of orifices, the number is selected to be even,
pressure continuous conditions:
correcting the pressure of each node by using a relaxation method, wherein the expression of the relaxation method is as follows:
wherein: omega is a relaxation factor, and the value is 0-2; k is an iteration coefficient;represents the oil film pressure value k+1 times, +.>Represents the value of the oil film pressure k times,/>Representing the value of the dimensionless oil film pressure of k+1 times;
step 3, calculating the oil film thickness of the bearing and the oil film pressure distribution of the bearing according to the bearing parameters and boundary conditions;
step 4, setting an initial pressure value, and calculating oil film pressure through the liquid lubricating oil film pressure calculation model;
step 5, decomposing the oil film pressure in the step 4 along the x, y and z directions under a rectangular coordinate system to obtain P x ，P y P z ，
Step 6, adopting a Simpson integration method to integrate the pressure in the circumferential direction and the radial direction of the oil film to respectively obtain the bearing capacity in the circumferential x direction, the bearing capacity in the circumferential y direction and the bearing capacity in the radial z direction,
wherein: r represents the radius of the oil film,indicating the radial initial angle of the bearing +.>Representing the radial integral wrap angle of the ball bearing;
step 7, dimensionless treatment of formula (4) to obtain
Wherein: w (W) x ，W y W is provided z The dimensionless bearing capacity along the x, y and z directions are respectively shown;
and 8, integrating the formula (5) by adopting the simpson to obtain the bearing capacity of the bearing as follows:
F x ＝p a ·R 2 W x (6)
calculating the bearing capacity of the other two directions, wherein the radial bearing capacity expressions are respectively as follows:
the above method is dimensionless to obtain:
wherein: w represents dimensionless radial bearing capacity.
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