CN106354974B - A Calculation Method of Equivalent Stiffness and Equivalent Damping of Rolling Bearings - Google Patents

Abstract

The invention discloses a method for calculating equivalent stiffness and equivalent damping of a rolling bearing, comprising the following steps: 1) based on the theory of elastic hydrodynamic lubrication, refine the contact area between the rolling element, the oil film and the raceway; 2) Through the bilinear approximation function of the pressure distribution and the difference method, the elastic deformation stiffness, oil film pressure and oil film thickness of the contact in each area after refinement are obtained, and the linear disturbance equation is used to solve the oil film dynamic stiffness in each area after refinement and oil film damping; then calculate the contact stiffness and contact damping of the rolling element in contact with the raceway; 3) On the basis of the contact stiffness and contact damping of the rolling element in contact with the raceway calculated in step 2), calculate the contact stiffness of the rolling bearing. Equivalent stiffness and equivalent damping. The invention has the advantages of high accuracy of calculation results, can provide technical support for the design of the bearing rotor system, and reduces the failure rate of the designed bearing rotor system.

Description

A Calculation Method of Equivalent Stiffness and Equivalent Damping of Rolling Bearings

technical field

The invention relates to a method for calculating equivalent stiffness and equivalent damping of a rolling bearing.

Background technique

Bearing rotor system design requires consideration of bearing stiffness and damping. When analyzing the dynamic characteristics of rolling bearings such as critical speed, the dynamic characteristic parameters of rolling bearings must be given: equivalent stiffness and equivalent damping. The inaccuracy of equivalent stiffness and equivalent damping directly brings great errors to the analysis of the dynamic characteristics of the rolling bearing rotor system, resulting in large errors in the design of the bearing rotor system. fatal accident.

Now, in the dynamic analysis of rolling bearings, the internal contact elastic stiffness of the rolling bearing is often regarded as the overall stiffness of the rolling bearing without considering the influence of the oil film, and the substantial influence of damping on the system under steady state is ignored, and the acquisition of damping and stiffness mainly depends on experience. value, the obtained equivalent stiffness and equivalent damping are less accurate. Fang Bing et al. used experiments to measure the equivalent damping and equivalent stiffness of the bearing when analyzing the bearing characteristics. He Zhixian et al. considered stiffness and ignored the influence of damping in the dynamic analysis of bearings. Gupta's model simplifies the effects of damping. Hagiu proposes a dynamic theoretical analysis model that emphasizes that the high-speed rolling contact dynamics depend on the elastic stiffness of the mechanical Hertz contact and the lubricant stiffness and damping at the entrance of the contact zone. Liu Xiuhai simulates the viscous damping of the roller by using the damping of the rolling element when it translates in the fluid, but ignores the damping squeezing characteristic. Elsermans and Walford believe that the radial and axial stiffness and damping experimental results of ball bearings are larger than expected values. This result can be explained by the analysis of outer ring-rolling element-inner ring-shaft, but the theoretical calculation is not perfect. Harsha and Kankar proposed a nonlinear model of ball bearing based on Hertz elastic deformation, and introduced empirical damping to analyze its vibration transmission. Chen Bin et al. performed theoretical calculations on oil film damping, but lacked strong experimental verification.

SUMMARY OF THE INVENTION

In order to solve the above technical problems, the present invention provides a method for calculating the equivalent stiffness and equivalent damping of a rolling bearing with high calculation results of the equivalent stiffness and equivalent damping of the rolling bearing, which provides technical support for the design of the bearing rotor system and reduces the design cost. The failure rate of the bearing rotor system.

The technical solution of the present invention to solve the above technical problems is: a method for calculating the equivalent stiffness and equivalent damping of a rolling bearing, comprising the following steps: 1) Based on the theory of elastic hydrodynamic lubrication, the rolling element, the oil film and the raceway are divided into three parts: 2) Through the bilinear approximation function of the pressure distribution and the difference method, the elastic deformation stiffness, oil film pressure and oil film thickness of the contact in each area after the refinement are obtained, and the local oil film is solved by using the linear disturbance equation. Dynamic stiffness and oil film damping; 3) According to the dynamic load distribution of the rolling bearing, combined with the elastic deformation stiffness, oil film stiffness and oil film damping calculated in step 2), calculate the equivalent stiffness and equivalent damping of the rolling bearing.

In the above calculation method of equivalent stiffness and equivalent damping of rolling bearing, when the contact between the rolling elements, oil film and raceway is refined in step 1), the relative sliding between the rolling elements and the inner and outer raceways is not considered, and the rolling elements are The contact between the oil film and the raceway is divided into three areas: the oil film inlet area, the elastic contact area and the oil film outlet area according to the elastic deformation of the contact.

In the above calculation method of equivalent stiffness and equivalent damping of rolling bearing, the solution method of oil film dynamic stiffness and oil film damping in each region after refinement in step 2) is: simultaneous elastic deformation equation and Reynolds equation, and use compound direct iteration Using the method, the pressure and thickness of the oil film under static conditions are obtained, and then the disturbance equations are established by using the disturbance equation and the micro-momentum and first-order micro-momentum are solved to obtain the linear approximate oil film stiffness value and oil film damping value.

In the above method for calculating the equivalent stiffness and equivalent damping of the rolling bearing, when solving the elastic deformation in step 2), a bilinear function is used on the four-node rectangular element to approximate the pressure distribution to solve the elastic deformation.

In the above calculation method of equivalent stiffness and equivalent damping of rolling bearing, when solving the disturbance equation system, the oil film thickness and oil film pressure are expanded by Taylor series, and the origin of coordinates is taken as the static equilibrium position.

In the above calculation method of equivalent stiffness and equivalent damping of rolling bearings, in step 2), the oil film stiffness and oil film damping in the oil film outlet area are ignored, and the oil film stiffness and oil film damping in the oil film inlet area and the rolling elements and raceways in the elastic contact area are ignored. The elastic contact stiffness, oil film stiffness and oil film damping are calculated separately, and then the contact stiffness and contact damping between the rolling element and the raceway are calculated on this basis.

The above calculation methods for equivalent stiffness and equivalent damping of rolling bearings assume the following conditions when calculating the oil film damping and oil film stiffness in the oil film inlet area:

Since the shape of the contact surface between the rolling element and the raceway in the rolling bearing is a long and narrow ellipse, the short semi-axis of the rolling direction is much smaller than the long semi-axis, so the shape of the contact surface can be approximated to the rectangular contact surface with the same ellipse long and short diameters. and ignore oil spills on the edges of the contact area;

The gap between the rolling element and the raceway is a parabola;

The thickness of the oil film in the bearing is greater than the surface roughness of the rolling elements, inner and outer rings;

The value of the inertial force is less than the viscous force;

Ignore the effect of gravity;

Considering the applicability of the Reynolds equation, the viscosity is considered constant;

Neglect the back pressure due to cavitation during contact.

In the above calculation method of equivalent stiffness and equivalent damping of rolling bearing, when calculating the radial equivalent stiffness and equivalent damping of bearing, consider the existence of radial clearance between rolling element and raceway in rolling bearing and under the radial load of bearing. Relative displacement occurs between the inner and outer rings of the bearing.

Compared with the prior art, the present invention has the following beneficial effects:

(1) The present invention refines the contact between the rolling element, the oil film and the raceway, and divides it into three areas according to the elastic deformation of the contact, the oil film inlet area, the elastic contact area, and the oil film outlet area, and considers the rolling bearing in the There is radial clearance between the rolling element and the raceway. Under the radial load of the bearing, the relative displacement of the inner and outer rings of the bearing is more in line with the actual situation, and the calculated result is more accurate; it provides a useful tool for the design of the bearing rotor system. Technical support reduces the failure rate of the designed bearing rotor system.

(2) On the basis of the simultaneous elastic deformation equation and the Reynolds equation, the present invention uses the compound direct iteration method to obtain the pressure and thickness values under the static state of the oil film, and uses the perturbation equation to solve the micro-momentum and first-order micro-motion of the equation system Finally, the linear approximate oil film stiffness value and oil film damping value are obtained, which has the advantages of fast calculation speed and high calculation accuracy.

(3) The calculation of the present invention does not require various tests on the rolling bearing to obtain relevant parameters, but can be directly applied, which has the advantages of simplicity and practicality.

Description of drawings

Figure 1 is a simplified elastic fluid lubrication contact model of the present invention.

Figure 2 shows the geometry of the oil film of the rolling bearing of the present invention.

Figure 3 shows the load distribution and deformation of the rolling bearing of the present invention.

FIG. 4 shows the equivalent stiffness and equivalent damping of the contact of a single rolling element of the rolling bearing of the present invention.

Fig. 5 is the position angle between the rolling element and the radial direction of the rolling bearing of the present invention The oil film pressure value between the outer ring and the rolling element.

Fig. 6 is the position angle between the rolling element and the radial direction of the rolling bearing of the present invention The oil film pressure value of the inner ring in contact with the rolling element.

Fig. 7 is the position angle between the rolling element and the radial direction of the rolling bearing of the present invention When the outer ring is in contact with the rolling element, the oil film pressure value.

Fig. 8 is the position angle between the rolling element and the radial direction of the rolling bearing of the present invention The oil film pressure value of the inner ring in contact with the rolling element.

FIG. 9 is the first four order natural frequency measurement results of the shaft-bearing-base system used for experimental verification of the present invention.

FIG. 10 is a simplified analysis result of the shaft-bearing-base system for experimental verification based on Dyrobes rotor-bearing of the present invention.

Detailed ways

The present invention will be further described below in conjunction with the accompanying drawings.

The present invention comprises the following steps:

The first step: establish the EHL contact model between the rolling element and the raceway.

Without considering the relative sliding between the rolling element and the inner and outer raceways, the contact area between the rolling element, the oil film and the raceway (as shown in Figure 1) is divided into three areas according to the elastic deformation of the contact: the oil film inlet area A. Elastic contact area B and oil film outlet area C. k _ef and c _ef are the oil film stiffness and oil film damping in the oil film inlet area A. k _c , k _f , and c _f are the elastic contact stiffness, oil film stiffness and oil film damping of the rolling element and the raceway in the elastic contact zone B, respectively. Since the oil film in the oil film outlet area C begins to stretch, its influence is weakened, and the influence of the overall contact damping and stiffness is very small. The oil film stiffness and oil film damping of the oil film outlet area C are not considered here. In this way, the main stiffness and damping are five parameters of k _ef , c _ef , k _c , k _f , and c _f . In order to calculate the equivalent stiffness and equivalent damping of the rolling bearing, these five parameters should be analyzed and calculated first.

(1) Elastic deformation contact stiffness k _c

The Hertz contact theory is derived from the static contact conditions of a complete elastic body, and is usually used as the basis for the calculation of the elastic deformation and stress field of the contact pair of anisotropic surfaces. When the rolling body is in contact with the channel, the width of its contact surface Much smaller than the radius of curvature of the contact surface, since the contact is regarded as a point contact, the contact surface can be considered as an ellipse.

The long axis a and short axis b of the contact ellipse and the contact deformation δ are obtained by calculation as follows:

K(e)=1.5277+0.6023ln(R _y /R _x ), E(e)=1.0003+0.5968(R _x /R _y )

k=1.0339(R _y /R _x ) ^0.6360 ,

The sign of curvature specifies that the two convex surfaces in contact are positive and the concave surfaces are negative. E(e), k(e) are the first and second complete elliptic integral functions, respectively, a is the long-axis radius of the ellipse contact surface, b is the short-axis radius of the ellipse contact surface, and Q is the mutual pressure of the contact surface , δ is the maximum elastic change, ∑ρ is the curvature sum, ν is the Poisson's ratio, E is the elastic modulus, and e is the ellipse parameter.

The Hertz theory elastic deformation of a single rolling element can be obtained by using the formula (3), and the formula (3) can be simplified as:

δ=GQ ^2/3 (4)

in The Hertz contact stiffness of a single rolling element to the inner or outer ring is:

It can be seen from this that its stiffness is not a constant, it will vary with displacement (or load).

(2) Oil film stiffness k _f and oil film damping c _f in elastic contact area

As shown in Fig. 2, the point contact problem between two elastic objects can be regarded as the contact between an elastic ellipsoid with equivalent principal curvature radii R _z , R _y and equivalent elastic modulus E' and a rigid plane. There is a lubricating oil film between the two surfaces, and the actual thickness of the oil film at the contact center point o is h _c . Under the action of the oil film pressure, the elastic deformation of the contact surface is δ(x, y). The expression of the oil film thickness can be written as:

h _c =h ₀ +δ(0,0), h ₀ is the thickness of the oil film at the center of the rigid body.

Based on the parameter calculation of the oil film by the Reynolds equation under isothermal conditions, the general form of the Reynolds equation under isothermal conditions (assuming that u ₂ and v ₂ do not vary with x and y) is as follows:

The equations (6) and (7) are solved simultaneously, and the oil film stiffness and oil film damping in the contact area are obtained based on the relationship between the oil film thickness and the oil film force. Some scholars directly derive the stiffness value from the formula of the minimum oil film thickness at point contact derived by Hamrock and Dowson, which has two problems: (1) Oil film stiffness refers to the stiffness value under micro-deformation, (2) because of the influence of elastic deformation The variation of the lower local oil film thickness is inconsistent, the oil film stiffness cannot be directly obtained by derivation, and the oil film damping value cannot be directly obtained. Therefore, on the basis of the simultaneous elastic deformation equation and Reynolds equation, the combined direct iteration method is used to obtain the pressure and thickness of the oil film under static conditions, and the micro-momentum and first-order micro-momentum of the equation system are solved by using the perturbation equation. A linear approximation of the oil film stiffness and oil film damping values is obtained.

The elastic deformation is solved by using a bilinear function to approximate the pressure distribution on a four-node rectangular element. Under the action of the oil film pressure, the sum of the normal displacements of the two contact surfaces is:

Ω is the solution area, e is the unit area, p _ij is the central pressure value of the unit area e, and λ(x, y) is the coefficient value.

Using the dimensionless form, equations (6) and (7) are simplified to the point-contact lubrication dimensionless Reynolds equation:

Oil film thickness equation:

Film Thickness Parameters Load parameters Speed parameter The material parameter G=αE ¹ α is the coefficient of viscosity and pressure, is the maximum Hertz pressure, where w is the load value.

is the dimensionless Roelands viscosity-pressure relation.

is the dimensionless density equation.

Combining Equation (9) and Equation (10), and using the three-point central difference format to replace the partial derivative, the following difference equations can be obtained after finishing:

The static oil film thickness and oil film pressure are obtained. On the basis of the perturbation equation, the oil film thickness and oil film pressure are expanded by Taylor series, and the origin of the coordinates is taken as the static equilibrium position. When the oil film pressure fretting near the static value, the oil film pressure and The oil film thickness can be expressed by the following linear relationship:

h=h ₀ +Δh (13)

where: p ₀ is the static equilibrium oil film pressure, h ₀ is the static equilibrium oil film thickness, k is the approximate oil film stiffness, c is the oil film approximate damping, Δh, Both are disturbance parameters, and their magnitudes are small. Then the simplified dimensionless form of the Reynolds equation under isothermal conditions is:

h ₀ and the time variable Regardless, the dimensionless formulas of formula (12) and formula (13) are substituted into formula (14), and Δh, The quadratic and above terms are omitted, and the terms of the same degree are normalized to obtain the following three equations:

Using the direct compound iteration method to calculate equations (16) and (17), the oil film stiffness k _f and oil film damping _cf are obtained.

(3) Oil film stiffness k _ef and oil film damping c _ef in the oil film inlet area

Simplify the analysis of oil film damping and oil film stiffness in the inlet region, assuming the following conditions:

① Since the shape of the contact surface between the rolling element and the raceway in the rolling bearing is a narrow and long ellipse, the short semi-axis b of the rolling direction is much smaller than the long semi-axis a, so the shape of the contact surface can be approximated to a rectangle with the same long and short diameters of the ellipse contact surface, and ignore oil spills on the edges of the contact area.

②The gap between the rolling element and the raceway is a parabola And x≥b, where R is the comprehensive radius of curvature in the y direction.

③ The thickness of the oil film in the bearing is greater than the surface roughness of the rolling elements and the inner and outer rings.

④The value of inertial force is smaller than the viscous force.

⑤ Ignore the effect of gravity.

⑥ Considering the applicability of Reynolds equation, the viscosity can be considered to be constant.

⑦ Ignore the back pressure generated by cavitation during the contact process.

Based on the above assumptions, Reynolds equation (7) is simplified to:

where u _s =u ₁ +u ₂ , and u _z is the normal extrusion speed. Both η and ρ are fixed values. Integrating x in equation (18), taking into account the sommerfeld condition and the semi-sommerfeld condition, we get:

η is oil film viscosity, u rolling element peripheral speed, L contact ellipse area long axis, v oil film entry speed. This gives the oil film damping in the inlet zone:

Through numerical simulation, it is found that the deformation of the oil film inlet area (including the elastic deformation of the contact surface and the thickness change of the oil film) is very small compared with the deformation of the elastic contact area. Therefore, under the condition of slight load changes, the stiffness of the oil film inlet area k _ef can be ignored. Bring the calculated c _ef , c _f , k _c , and k _f into equations (19) and (20) to finally obtain the contact stiffness k and contact damping c of the rolling element in contact with the raceway:

c = c _ef + c _f (22)

Step 2: Calculate the equivalent damping and equivalent stiffness of the rolling bearing.

By calculating the contact stiffness k and contact damping c of the rolling element in contact with the raceway, the radial equivalent stiffness k _rc and equivalent damping c _re of the bearing are calculated. Considering the existence of radial clearance between the rolling element and the raceway in the rolling bearing, under the radial load F _r of the bearing, the relative displacement generated by the inner and outer rings of the bearing is δ _r , as shown in Figure 3.

Under equilibrium conditions, the radial load on the inner ring must be equal to the sum of the vertical components of the rolling element load:

here is the angle between the single rolling element and the radial position, Note that after the integral conversion is continuous, and α is the correction coefficient.

The load distribution between the rolling element 2 and the raceway can be obtained by combining the equations (24) and (25), so as to obtain the contact stiffness and damping of a single rolling element 2 with the inner ring 1 and the outer ring 3, as shown in Figure 4 The contact stiffness and contact damping of a single rolling element 2 in contact with the inner ring 1 and the outer ring 3 are obtained:

The contact stiffness and contact damping of each rolling element 2 in contact with the inner ring 1 and the outer ring 3 are combined, and the radial equivalent stiffness k _rc and equivalent damping c _re of the rolling bearing are obtained:

Likewise, here is the angle between the position of a single rolling element and the radial direction.

Experimental verification

Using the B&K test system, on the basis of the modal of the bearing component, the hammer method is used to measure the rigid body natural frequency of the system, and the equivalent damping of the bearing is obtained by using the formula.

Considering that the bearing itself is light in weight and small in size, it belongs to the modal test of the component, so it is not conducive to the arrangement of the sensor, and the calculation of the transfer function is difficult. The inner ring is fastened by locking screws, so that the shaft-bearing-base becomes a system.

Using the single-point vibration pick-up method, the sensor is arranged on a section of the shaft, and the white dots marked on the shaft and the bearing seat are the hammering points. The measured transfer function is used to obtain the rigid body natural frequency of the bearing system components through modal analysis. The test method is carried out after the bearing is continuously operated at 600rpm for 10 minutes. The natural frequency obtained by measuring the component system is shown in Figure 9.

From Fig. 9, we can know: the natural frequency value ω _n and damping ratio ζ of the first few orders of the component mode, here we take the natural frequency and damping ratio of the second order mode (the rigid body mode of the component) and substitute it into c _b =2ζmω _n calculated in , the calculated damping is 483.2, which is 0.17% off from our theoretically calculated value of cre = _484.0255 .

In order to verify the stiffness, the first-order critical speed of 57.97Hz under the double bearing-single shaft is softly simulated through the Dyrobes rotor-bearing analysis, as shown in Figure 10, and a 0.82kg mass block is added in the middle.

Here, the shaft is equivalent, and the equivalent mass m=1.64kg, the equivalent bearing stiffness ks= ₂₁₉₄₅₀ , and the equivalent damping _ck =121.3705 are obtained respectively.

Combining equations (30), (31) and (32), the radial dynamic stiffness of the bearing is obtained as k _b =5.4743×10 ⁷ , which is 3.06% different from our theoretical calculated value k _rc =5.3067×10 ⁷ .

Claims (5)

1. A method for calculating equivalent stiffness and equivalent damping of a rolling bearing, comprising the following steps: 1) Based on the theory of elastic hydrodynamic lubrication, the contact area between rolling elements, oil film and raceway is refined; 2) Through the pressure distribution bilinear approximation function and the difference method, the elastic deformation stiffness, oil film pressure and oil film thickness of the contact in each area after refinement are obtained, and the linear disturbance equation is used to solve the oil film dynamic stiffness and oil film in each area after refinement. Then calculate the contact stiffness and contact damping between the rolling element and the raceway; 3) On the basis of the contact stiffness and contact damping between the rolling element and the raceway calculated in step 2), calculate the equivalent of the rolling bearing stiffness and equivalent damping; The solution method of oil film dynamic stiffness and oil film damping in each region after refinement in step 2) is: Simultaneous elastic deformation equation and Reynolds equation, and use compound direct iteration method to find the pressure and thickness value of oil film under static state, and then use The perturbation equations are set up, and the micro-momentum and first-order micro-momentum are solved to obtain the linear approximate oil film stiffness and oil film damping value; When solving the perturbation equations, the oil film thickness and oil film pressure are expanded by Taylor series, and the coordinate origin is taken as the static equilibrium position; When solving the elastic deformation in step 2), a bilinear function is used on the four-node rectangular element to approximate the pressure distribution to solve the elastic deformation. 2. The method for calculating the equivalent stiffness and equivalent damping of a rolling bearing according to claim 1, in step 1), when the contact between the rolling elements, the oil film and the raceway is refined, the rolling elements and the inner and outer raceways are not considered For relative sliding, the contact between the rolling element, oil film and raceway is divided into three areas: the oil film inlet area, the elastic contact area and the oil film outlet area according to the elastic deformation of the contact. 3. The method for calculating the equivalent stiffness and equivalent damping of a rolling bearing according to claim 2, in step 2), the oil film stiffness and oil film damping in the oil film outlet area are ignored, and the oil film stiffness and oil film damping and elastic contact in the oil film inlet area are ignored. The elastic contact stiffness, oil film stiffness and oil film damping of the rolling element and the raceway are calculated separately, and then the contact stiffness and contact damping of the rolling element and the raceway are calculated on this basis. 4. The method for calculating the equivalent stiffness and equivalent damping of a rolling bearing according to claim 3, when calculating the oil film damping and oil film stiffness in the oil film inlet area, the following conditions are assumed: Since the shape of the contact surface between the rolling element and the raceway in the rolling bearing is a long and narrow ellipse, the short semi-axis of the rolling direction is much smaller than the long semi-axis, so the shape of the contact surface can be approximated to the rectangular contact surface with the same ellipse long and short diameters. and ignore oil spills on the edges of the contact area; The gap between the rolling element and the raceway is a parabola; The thickness of the oil film in the bearing is greater than the surface roughness of the rolling elements, inner and outer rings; The value of the inertial force is less than the viscous force; Ignore the effect of gravity; Considering the applicability of the Reynolds equation, the viscosity is considered constant; Neglect the back pressure due to cavitation during contact. 5. The method for calculating the equivalent stiffness and equivalent damping of a rolling bearing according to claim 1, when calculating the radial equivalent stiffness and equivalent damping of the bearing, consider the existence of radial clearance between the rolling element and the raceway in the rolling bearing and Relative displacement occurs between the inner and outer rings of the bearing under the radial load of the bearing.