CN106354974B - A kind of calculation method of rolling bearing equivalent stiffness and equivalent damping - Google Patents
A kind of calculation method of rolling bearing equivalent stiffness and equivalent damping Download PDFInfo
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Abstract
The invention discloses the calculation method of a kind of rolling bearing equivalent stiffness and equivalent damping, include the following steps: 1) to refine the contact area between rolling element, oil film, raceway three based on elastohydrodynamic lubrication theory；2) bilinearity approximating function and calculus of finite differences are distributed by pressure, find out elastic deformation rigidity, oil film pressure and the oil film thickness of the contact after refining in each region, oil film dynamic stiffness and oil-film damping after utilizing linear perturbation equation solution to refine in each region；And then calculate contact stiffness and contact damping that rolling element is in contact with raceway；3) on the basis of the contact stiffness and contact damping that are in contact in the rolling element that step 2 is calculated with raceway, the equivalent stiffness and equivalent damping of rolling bearing are calculated.The present invention has the advantages that calculated result accuracy is high, can provide technical support for the design of bearing rotor system, reduce the failure rate of the bearing rotor system of design.
Description
Technical field
The present invention relates to the calculation methods of a kind of equivalent stiffness of rolling bearing and equivalent damping.
Background technique
Bearing rotor system designs the rigidity and damping for needing to consider bearing.It is dynamic carrying out rolling bearing such as critical speed etc.
The dynamic characteristic parameter of rolling bearing: equivalent stiffness and equivalent damping must be provided when mechanical characteristic analysis.Equivalent stiffness with etc.
The inaccurate dynamical property analysis directly to rolling bearing rotor-support-foundation system of effect damping brings very big error, and bearing is caused to turn
It is big that subsystem sets timing error, will cause critical speed when serious and rated speed is close, cause the major break down of fatal crass.
Now, when carrying out dynamic analysis to rolling bearing, often using rolling bearing interior contact elastic stiffness as rolling
Bearing overall stiffness has ignored damping to the materially affect under systematic steady state, for damping without considering the influence of oil film
Acquisition with rigidity relies primarily on empirical value, and obtained equivalent stiffness is low with the accuracy of equivalent damping.Square soldier etc. is to bearing spy
Property the equivalent damping and equivalent stiffness of experiment measurement bearing are utilized when being analyzed.When He Zhixian etc. analyzes bearing system dynamics
It considers rigidity and ignores the influence of damping.The influence of damping is simplified processing by the model of Gupta.Hagiu proposes one and moves
State theoretical analysis model highlights elastic stiffness and touching area's entrance that high speed scroll contact dynamic depends on mechanism Hertz contact
The lubricant rigidity and damping at place.Suffered damping is when Liu Xiuhai is translated in a fluid using rolling element to simulate suffered by roller
Viscous damping, but have ignored damping compression properties.Elsermans and Walford think the rigidity and resistance of ball bearing radially, axially
Value is big than expected for Buddhist nun's experimental measurements, this result can be explained by outer ring-rolling element-inner ring-axis analysis,
But theoretical calculation is incomplete.Harsha and Kankar proposes the ball bearing nonlinear model based on Hertz flexible deformation, and draws
Experience damping is entered to analyze its vibration transmitting.Chen Bin etc. has carried out theoretical calculation to oil-film damping, but lacks strong experiment
Verifying.
Summary of the invention
In order to solve the above technical problem, the present invention provides a kind of rolling bearing equivalent stiffness and equivalent damping calculation result
Rolling bearing equivalent stiffness and equivalent damping calculation method with high accuracy, it provides technology branch for the design of bearing rotor system
It holds, reduces the failure rate of the bearing rotor system of design.
The technical solution that the present invention solves above-mentioned technical problem is: a kind of equivalent stiffness and equivalent damping of rolling bearing
1) calculation method includes the following steps: based on elastohydrodynamic lubrication theory, will be between rolling element, oil film, raceway three
Contact area is refined；2) bilinearity approximating function and calculus of finite differences are distributed by pressure, found out after refining in each region
Elastic deformation rigidity, oil film pressure and the oil film thickness of contact, using linear perturbation equation solution part oil film dynamic stiffness with
Oil-film damping；3) it is distributed according to rolling bearing dynamic load, and the elastic deformation rigidity that is calculated in conjunction with step 2), oil film are rigid
Degree and oil-film damping, calculate the equivalent stiffness and equivalent damping of rolling bearing.
In the calculation method of above-mentioned rolling bearing equivalent stiffness and equivalent damping, rolling element, oil film, raceway in step 1)
Between three contact refinement when, do not consider rolling element it is opposite with inner and outer ring raceway slide, by rolling element, oil film, raceway three it
Between contact, according to contact resilient deformation be divided into three oil film inlet region, flexible contact areas and oil film outlet area areas.
In the calculation method of above-mentioned rolling bearing equivalent stiffness and equivalent damping, in step 2) after refinement in each region
Oil film dynamic stiffness and oil-film damping method for solving are as follows: simultaneous elastic deformation equation and Reynolds equation, and with compound direct
Iterative method finds out pressure and thickness value under oil film static state, then establishes perturbation equation group using perturbation equation and solves fine motion
Amount and single order fine motion amount, acquire linear approximation oil film rigidity value and oil-film damping value.
When solving flexible deformation in the calculation method of above-mentioned rolling bearing equivalent stiffness and equivalent damping, in step 2),
Pressure distributed problem solving flexible deformation is approached using bilinear function on the four nodes element of rectangle.
In the calculation method of above-mentioned rolling bearing equivalent stiffness and equivalent damping, when perturbation equation group solves, utilize
Oil film thickness and oil film pressure are unfolded Taylor series, and taking coordinate origin is equipoise.
In the calculation method of above-mentioned rolling bearing equivalent stiffness and equivalent damping, in step 2), ignore oil film outlet area
Oil film rigidity and oil-film damping, to the oil film rigidity of oil film inlet region and oil-film damping and flexible contact areas rolling element and raceway
Elastic contact stiffness, oil film rigidity and oil-film damping calculate separately, then calculate rolling element and raceway phase on this basis
The contact stiffness and contact damping of contact.
The calculation method of above-mentioned rolling bearing equivalent stiffness and equivalent damping calculates oil film inlet region oil-film damping and oil
When film rigidity, it is assumed that the following conditions:
Since the shape in rolling element and raceway contact face in rolling bearing is long and narrow ellipse, rotating direction it is short by half
Axis is small more than major semiaxis, so contact surface shape can be similar to the rectangular contact face of same major diameter and minor axis, and ignores
Oil leak on contact area edge；
Gap between rolling element and raceway is parabola；
Oil film thickness is greater than the surface roughness of rolling element, Internal and external cycle in bearing；
The value of inertia force is less than viscous force；
Ignore gravity influence；
In view of Reynolds equation applicability, it is believed that viscosity is constant；
Ignore the counter-pressure that cavitation generates in contact process.
In the calculation method of above-mentioned rolling bearing equivalent stiffness and equivalent damping, calculate bearing radial direction equivalent stiffness with etc.
When effect damping, there are radial internal clearance and in bearing radial load lower bearing between rolling element and raceway in consideration rolling bearing
Relative displacement is generated between outer snare.
Compared with prior art, the invention has the following advantages:
(1) present invention has refined rolling element, oil film, the contact between raceway three, is divided into three according to contact resilient deformation
A area, oil film inlet region, flexible contact areas and oil film outlet area, and consider in rolling bearing and deposited between rolling element and raceway
In radial internal clearance, under bearing radial load, the relative displacement that the inside and outside snare of bearing generates is more in line with practical feelings in this way
Condition, calculated result are more accurate；Technical support is provided for the design of bearing rotor system, the bearing for reducing design turns
The failure rate of subsystem.
(2) present invention is on the basis of simultaneous elastic deformation equation and Reynolds equation, with compound Direct Iterative Method,
The pressure and thickness value under oil film static state are found out, using the fine motion amount and single order fine motion amount of perturbation equation solve system of equation, finally
Linear approximation oil film rigidity value and oil-film damping value are acquired, has many advantages, such as that calculating speed is fast, computational accuracy is high.
(3) calculating of the invention does not need to carry out various tests to rolling bearing, obtains relevant parameter, but directly can be with
Using having the advantages that simple and practical.
Detailed description of the invention
Fig. 1 is the elastohydrodynamic behaviour contact model of simplification of the invention.
Fig. 2 is the oil film geometry of rolling bearing of the invention.
Fig. 3 is the distribution of rolling bearing load and deformation of the invention.
Fig. 4 is the equivalent stiffness and equivalent damping that the single rolling element of rolling bearing of the invention contacts.
Fig. 5 is the rolling element and radial position angle of rolling bearing of the inventionWhen outer ring and rolling element between
Oil film pressure value.
Fig. 6 is the rolling element and radial position angle of rolling bearing of the inventionWhen inner ring contact with rolling element
Oil film pressure value.
Fig. 7 is the rolling element and radial position angle of rolling bearing of the inventionWhen outer ring contacted with rolling element
Oil film pressure value.
Fig. 8 is the rolling element and radial position angle of rolling bearing of the inventionWhen inner ring contacted with rolling element
Oil film pressure value.
Fig. 9 is the preceding quadravalence intrinsic frequency measurement result of axis-bearing-base systems for experimental verification of the invention.
Figure 10 is that axis-bearing-base systems for experimental verification of the invention are simplified based on Dyrobes rotor-bearing
Analyze result.
Specific embodiment
The present invention will be further described below with reference to the drawings.
The present invention includes the following steps:
Step 1: establishing the EHL contact model of rolling element and raceway.
In the case where not considering opposite with the inner and outer ring raceway sliding of rolling element, rolling element, oil film, connecing between raceway three
It touches region (as shown in Figure 1), be divided into three areas according to contact resilient deformation: oil film inlet region A, flexible contact areas B and oil film go out
Mouth region C.kef、cefFor the oil film rigidity and oil-film damping of oil film inlet region A.kc、kf、cfRespectively flexible contact areas B rolling element
With Elastic contact stiffness, oil film rigidity and the oil-film damping of raceway.Since the oil film of its oil film outlet area C starts to stretch its shadow
Ring power reduction, it is very small compared to whole contact damping and stiffness effect, do not consider here the oil film rigidity of oil film outlet area C with
Oil-film damping.Rigidity main in this way and damping are kef、cef、kc、kf、cfThis five parameters, in order to equivalent just to rolling bearing
Degree is calculated with equivalent damping, and analysis meter is answered to calculate this five parameters first.
(1) resiliency deforming contact's rigidity kc
Hertz contact theory is drawn according to the Static Contact condition of perfectly elastic body, is usually used to as different
To the flexible deformation of curved face contact pair and the foundation of stress field calculation, when rolling element and channel contact, the width of contact surface
The far smaller than radius of curvature of contact surface is considered as point contact due to that will contact, then its contact surface is regarded as oval.
It is as follows by long axis a and short axle b and contact deformation δ that Contact Ellipse is calculated:
K (e)=1.5277+0.6023ln (Ry/Rx), E (e)=1.0003+0.5968 (Rx/Ry)
K=1.0339 (Ry/Rx)0.6360、
Two face nonreentrant surfaces of sign of curvature regulation contact are positive, and concave surface is negative.E (e), k (e) be respectively the first kind and
Complete elliptic integral of the second kind function, a are the major axis radius of elliptic contact surface, and b is the minor axis radius of elliptic contact surface, and Q is to connect
The mutual pressure of contacting surface, δ are the maximum variable quantity of elasticity, ∑ ρ be curvature and, ν is Poisson's ratio, and E is elasticity modulus, and e is oval ginseng
Number.
(3) formula of utilization can find out the Hertz theory flexible deformation of single rolling element, and (3) formula is simplified are as follows:
δ=GQ2/3 (4)
WhereinSingle rolling element and inner ring or the Hertz contact stiffness of outer ring are as follows:
There it can be seen that its rigidity is not a constant, it can change with the variation of displacement (or load).
(2) the oil film rigidity k of flexible contact areasfWith oil-film damping cf
As shown in Fig. 2, the Point contact between two elastomeric objects can be considered as with equivalent principal radius of curvature Rz, RyWith
Equivalent elastic modulus E ' contact of the ellipsoid of elasticity body with rigid plane.There are lubricating oil films between two surfaces, and in contact
The practical oil film thickness of heart point o is hc, under oil film pressure effect, the flexible deformation that contact surface generates is δ (x, y), and oil film is thick
The expression formula of degree can be write as:
hc=h0+ δ (0,0), h0For rigid body center oil film thickness.
It is calculated based on parameter of the Reynolds equation under isothermy to oil film, the Reynolds equation under isothermy
Common form is (assuming that u2And v2Do not change with x and y) as follows:
Joint type (6), formula (7) are solved, and on the basis of the correlation of oil film thickness and oil-film force, are connect
Touch the oil film rigidity and oil-film damping in area.Some scholars are in the Hamrock and Dowson point contact minimum oil film thickness derived
Derivation is carried out on formula directly to obtain rigidity value, there are problems that two in this way: 1. oil film rigidity refers to rigid under micro deformation
Angle value, 2. because the variation of local oil film thickness is inconsistent under the influence of elastic deformation, oil film rigidity is unable to immediate derivation and obtains
It takes, and oil-film damping value can not directly acquire.Therefore on the basis of simultaneous elastic deformation equation and Reynolds equation, fortune
With compound Direct Iterative Method, pressure and thickness value under oil film static state are found out, utilizes the fine motion amount of perturbation equation solve system of equation
With single order fine motion amount, linear approximation oil film rigidity value and oil-film damping value are finally acquired.
Pressure distributed problem solving flexible deformation is approached using bilinear function on the four nodes element of rectangle, is made in oil film pressure
Under, the Normal Displacement summation of two contact surfaces are as follows:
Ω is domain, and e is unit area, pijFor the center pressure value of unit area e, λ (x, y) is coefficient value.
It is that point contact lubricates dimensionless Reynolds equation by formula (6) and formula (7) abbreviation using Dimensionless Form:
Oil film thickness equation:
Film thickness parameterLoad parameterSpeed parameterMaterial parameter G=α E1α is
Viscous pressure coefficient,For maximum Hertz pressure, w is load value here.
For the viscous pressure relational expression of dimensionless Roelands.
For non-dimensional density equation.
Local derviation is substituted by formula (9) and formula (10) simultaneous, and using three dot center's difference schemes, it is collated to obtain as follows
DIFFERENCE EQUATIONS:
Acquire static oil film thickness and oil film pressure value, on the basis of perturbation equation, with Taylor series by oil film thickness with
Oil film pressure expansion, taking coordinate origin is equipoise, then oil film pressure is near quiescent value when fine motion, oil film pressure and oil
Film thickness can be indicated by following linear relationships formula:
H=h0+Δh (13)
In formula:p0For static balance oil film pressure, h0For static balance oil film thickness, k is that oil film is approximate
Rigidity, c are that oil film approximation damps, Δ h,It is disturbance parameter, magnitude very little.The then side Reynolds under isothermy
The simplified Dimensionless Form of journey are as follows:
h0With time variableIt is unrelated, by formula (12), formula (13) dimensionless formula substitute into formula (14) in, by Δ h,
Item more than quadratic power and quadratic power omits, and by homogeneous item normalizing, obtains following three equation:
Using direct combination iterative method calculating formula (16) and formula (17), oil film rigidity k is obtainedfWith oil-film damping cf。
(3) oil film inlet region oil film rigidity kefWith oil-film damping cef
Simplified analysis inlet region oil-film damping and oil film rigidity, it is assumed that the following conditions:
1. since the shape in rolling element and raceway contact face in rolling bearing is long and narrow ellipse, rotating direction it is short
Semiaxis b is small more than major semiaxis a, so contact surface shape can be similar to the rectangular contact face of same major diameter and minor axis, and
Ignore the oil leak on contact area edge.
2. the gap between rolling element and raceway is parabolaAnd x >=b, R is the direction y here
Composite curve radius.
3. oil film thickness is greater than the surface roughness of rolling element, Internal and external cycle in bearing.
4. the value of inertia force is less than viscous force.
5. ignoring gravity influence.
6. considering Reynolds equation applicability, it is believed that viscosity is constant.
7. ignoring the counter-pressure that cavitation generates in contact process.
According to above-mentioned it is assumed that Reynolds equation (7) are simplified are as follows:
Wherein us=u1+u2, uzFor normal direction extrusion speed.η and ρ is definite value.To the x integral in formula (18), and consider
Sommerfeld condition and half sommerfeld condition can obtain:
η is film viscosity, u rolling element peripheral speed, L Contact Ellipse region long axis, v oil film admission velocity.It can obtain in this way
The oil-film damping of inlet region:
By numerical simulation, deformation (the thickness change of flexible deformation and oil film including contact surface of oil film inlet region is found
Change) deformation of resilience in comparison contact zone be it is very small, therefore, in the case where load is slightly variable, the rigidity k of oil film inlet regionef
It can be ignored.The c that will be calculatedef、cf、kc、kfBringing formula (19), formula (20) into, finally acquisition rolling element connects with raceway
The contact stiffness k and contact damping c of touching:
C=cef+cf (22)
Step 2: calculating rolling bearing equivalent damping and equivalent stiffness.
The contact stiffness k and contact damping c that rolling element obtained by calculation is in contact with raceway calculate bearing radial direction etc.
Imitate rigidity krcWith equivalent damping cre.Consider that there are radial internal clearances between rolling element and raceway in rolling bearing, radially carry in bearing
Lotus FrUnder, the relative displacement that the inside and outside snare of bearing generates is δr, as shown in Figure 3.
In equilibrium conditions, the radial load of inner ring is necessarily equal to the sum of the vertical component of rolling element load:
HereFor single rolling element and radial position angle, After paying attention to Integral TransformationIt is continuously, α is correction factor.
Joint type (24), formula (25) formula can be obtained rolling element 2 and the load of raceway is distributed, so as to find out single rolling element
2 with inner ring 1, outer ring 3 ct clamping, be in contact as shown in figure 4, finding out single rolling element 2 with inner ring 1, outer ring 3
Contact stiffness and contact damping:
The contact stiffness and contact damping simultaneous that each rolling element 2 is in contact with inner ring 1, outer ring 3, acquire rolling bearing
Radial equivalent stiffness krcWith equivalent damping cre:
Similarly, hereFor single rolling element and radial position angle.
Experimental verification
It is measured on the basis of parts of bearings mode using hammering method using B&K test macro, obtains its system
Rigid body intrinsic frequency acquires bearing equivalent damping using formula.
In view of bearing itself light weight, small in size, belong to Appendage mode test, therefore be unfavorable for the arrangement of sensor,
And transmission function calculates the factors such as difficulty, and bearing outer ring is fixed on bearing spider, the axis of certain mass is mounted on bearing
Inner ring is simultaneously fastened by tightly locking screw, and such axis-bearing-pedestal becomes system.
Using single-point pick-up method, the white point marked in one section of placement sensor of axis, axis and bearing block is hammer point.
The transmission function that measurement is obtained obtains bearing arrangement component rigid body intrinsic frequency by model analysis.The test method is in axis
It holds and is carried out after ten minutes with 600rpm continuous work, the obtained intrinsic frequency of measuring part system is as shown in Figure 9.
From Fig. 9 it can be seen that: former rank natural frequency value ω of Appendage modenAnd dampingratioζ, is taken here
2 rank Modal frequencies and damping ratio (component rigid body mode) substitute into cb=2 ζ m ωnMiddle calculating, the damping being calculated are
483.2, the obtained value c of this and our theoretical calculationsre=484.0255 differences 0.17%.
To verify to rigidity, one under duplex bearing-single-turn axis is drawn up by Dyrobes rotor-bearing analysis soft mode
Rank critical speed 57.97Hz, as shown in Figure 10, centre joined the mass block of 0.82kg
Here shaft has been carried out it is equivalent, respectively obtain equivalent mass m=1.64kg, equivalent bearing rigidity ks=
219450, equivalent damping ck=121.3705.
Joint type (30), (31), (32), acquiring bearing radial direction dynamic rate is kb=5.4743 × 107, this is managed with us
By calculated value krc=5.3067 × 107Deviation be 3.06%.
Claims (5)
1. a kind of calculation method of rolling bearing equivalent stiffness and equivalent damping includes the following steps: 1) dynamic based on elastic fluid
Hydrodynamic lubrication is theoretical, and the contact area between rolling element, oil film, raceway three is refined；2) bilinearity is distributed by pressure
Approximating function and calculus of finite differences, elastic deformation rigidity, oil film pressure and the oil film for finding out the contact after refining in each region are thick
Degree utilizes the oil film dynamic stiffness and oil-film damping in each region after the refinement of linear perturbation equation solution；And then calculate rolling element with
The contact stiffness and contact damping that raceway is in contact；3) the rolling element contact with raceway is in contact being calculated in step 2 is rigid
On the basis of degree and contact damping, the equivalent stiffness and equivalent damping of rolling bearing are calculated；
Oil film dynamic stiffness and oil-film damping method for solving after being refined in step 2 in each region are as follows: simultaneous elastic deformation equation and
Reynolds equation, and compound Direct Iterative Method is used, the pressure and thickness value under oil film static state are found out, disturbance side is then utilized
Cheng Jianli perturbation equation group simultaneously solves fine motion amount and single order fine motion amount, acquires linear approximation oil film rigidity value and oil-film damping value；
When perturbation equation group solves, oil film thickness and oil film pressure are unfolded using Taylor series, it is quiet flat for taking coordinate origin
Weighing apparatus position；
When solving flexible deformation in step 2, pressure distributed problem solving bullet is approached using bilinear function on the four nodes element of rectangle
Property deformation.
2. the calculation method of rolling bearing equivalent stiffness and equivalent damping according to claim 1, rolling element in step 1),
When contact between oil film, raceway three refines, rolling element sliding opposite with inner and outer ring raceway is not considered, by rolling element, oil film, rolling
Contact between road three is divided into three oil film inlet region, flexible contact areas and oil film outlet area areas according to contact resilient deformation.
3. the calculation method of rolling bearing equivalent stiffness and equivalent damping according to claim 2, in step 2, ignore oil
The oil film rigidity and oil-film damping of film outlet area, oil film rigidity and oil-film damping and flexible contact areas to oil film inlet region roll
The Elastic contact stiffness of body and raceway, oil film rigidity and oil-film damping calculate separately, and then calculate rolling element on this basis
The contact stiffness and contact damping being in contact with raceway.
4. the calculation method of rolling bearing equivalent stiffness and equivalent damping according to claim 3 calculates oil film inlet region
When oil-film damping and oil film rigidity, it is assumed that the following conditions:
Since the shape of rolling element in rolling bearing and raceway contact face is long and narrow ellipse, the semi-minor axis of rotating direction is remote
It is smaller than major semiaxis, so contact surface shape can be similar to the rectangular contact face of same major diameter and minor axis, and ignore contact
Oil leak in edges of regions；
Gap between rolling element and raceway is parabola；
Oil film thickness is greater than the surface roughness of rolling element, Internal and external cycle in bearing；
The value of inertia force is less than viscous force；
Ignore gravity influence；
In view of Reynolds equation applicability, it is believed that viscosity is constant；
Ignore the counter-pressure that cavitation generates in contact process.
5. the calculation method of rolling bearing equivalent stiffness and equivalent damping according to claim 1 calculates bearing radial direction etc.
When imitating rigidity and equivalent damping, there are radial internal clearance and in bearing radial load between rolling element and raceway in consideration rolling bearing
Relative displacement is generated between the inside and outside snare of lower bearing.
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| CN110929419B (en) * | 2018-12-29 | 2021-08-13 | 山东大学 | Method for quickly predicting instability limit of steam turbine rotor system based on shroud zero damping |
| CN109946077B (en) * | 2019-02-15 | 2021-04-20 | 南昌航空大学 | Method for establishing fractional order damping rolling bearing fault dynamics gradual model |
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